Suggestions for good notation I occasionally come across a new piece of notation so good that it makes life easier by giving a better way to look at something. Some examples:


*

*Iverson introduced the notation [X] to mean 1 if X is true and 0 otherwise; so for example Σ1≤n<x [n prime] is the number of primes less than x, and the unmemorable and confusing Kronecker delta function δn becomes [n=0]. (A similar convention is used in the C programming language.) 

*The function taking x to x sin(x) can be denoted by x ↦ x sin(x). This has the same meaning as the lambda calculus notation λx.x sin(x) but seems  easier to understand and use, and is less confusing than the usual convention of just writing x sin(x), which is ambiguous: it could also stand for a number.

*I find calculations with Homs and ⊗ easier to follow if I write Hom(A,B) as A→B. Similarly writing BA for the set of functions from A to B is really confusing, and I find it much easier to write this set as A→B.

*Conway's notation for orbifolds almost trivializes the classification of wallpaper groups.
Has anyone come across any more similar examples of good notation that should be better known? (Excluding standard well known examples such as commutative diagrams, Hindu-Arabic numerals, etc.)
 A: This is probably the exact opposite of what the thread starter intended, but here's an instance where it might have been useful to have overloaded notation! I recently found that in propositional logic, $p \to q$ obeys much the same rules as exponentiation $q^p$: for instance, we have $(r^q)^p = r^{q \times p}$, and similarly, $p \to (q \to r)$ iff $p \land q \to r$. This is apparently due to the universal property for exponential objects, as applied to a Boolean algebra viewed as a poset category. I suppose it's also an instance of the Curry—Howard correspondence.
More generally, it seems like it isn't such a bad idea to conflate exponentiation and arrows - it looks nicer, to me at least, to write that a function of the type $A \to (B \to C)$ is naturally isomorphic to a function of the type $A \times B \to C$, than to write about $A \to C^B$. Even, as some have suggested, $A \to {}^B C$ or $C^B \leftarrow A$ would look nicer. On the other hand we'd lose the association with cardinal arithmetic if we do this...
A: If $\mathcal{C}$ is a category and $X,Y\in\mathrm{obj}(\mathcal{C})$, I like the notation $\mathcal{C}(X,Y)$ to denote $\mathrm{Hom}_{\mathcal{C}}(X,Y)$.
So, $\mathcal{C}(X,X)=\mathrm{End}_{\mathcal{C}}(X)$.
What do you think of the notation $\mathcal{C}(X):=\mathrm{Aut}_{\mathcal{C}}(X)$ ?
This would be consistent with the notation (or similar notations) $\mathsf{DIFF}(S^1)$ (resp. $\mathsf{TOP}(S^1)$ ) for diffeomorphisms (resp. homeomorphisms) of the circle, i.e. the $\mathrm{Aut}$ in the category $\mathsf{DIFF}$ of smooth manifolds (resp. $\mathsf{TOP}$ of topological manifolds), sometimes used in topology (see e.g. here and here. And (see e.g. here) $\mathsf{TOP}(n)=\mathrm{Aut}_{\mathsf{TOP}}(\mathbb{R}^n)$.
A: Forgive me for bumping such an old big-list question, but I can't resist. In linear algebra texts you will sometimes find the notation $[v]_B$ for the vector of coordinates of a vector $v \in V$ with respect to a basis $B$ of $V$, but you will not (to my knowledge) find corresponding notation for the matrix of a linear transformation $T : V \to W$ with respect to two bases $B, C$ of $V$ and $W$, except perhaps in the special case $V = W, B = C$, where I have seen $[T]_B$. The notation I use and advocate for this matrix is
$$_C[T]_B$$
and I taught this in a few linear algebra classes at UC Berkeley as a TA although I don't think it caught on. It has the pleasant property that its definition can be written
$$_C [T]_B [v]_B = [T(v)]_C.$$
This lets us give a transparent proof of the equally pleasant "functoriality" property that if $S : U \to V$ is another linear transformation and $A$ is a basis of $U$ then
$$_C[TS]_A = (_C[T]_B)(_B[S]_A)$$
("the $B$s just cancel"). Try even stating this result without notation for it! The notation makes it hard to misstate the basic facts about how such matrices change upon change of basis etc., because the correct statements "typecheck" properly and you won't be tempted to multiply matrices that haven't been expressed in compatible bases. Although things would be slightly better if either 1) the vector notation were written with the basis on the left or 2) matrices acted on the right and we switched the order of the two bases.
Importantly, the change-of-basis matrix between two bases $B, B'$ can just be written $_B[I]_{B'}$ and all of its properties become obvious (everything follows from "canceling" bases) and easy to remember, including e.g. that $_{B'}[I]_B$ is the inverse matrix and that if $T : V \to V$ is an endomorphism then
$$_{B'}[T]_{B'} = (_{B'} I_B) (_B[T]_B)(_B I_{B'}).$$
$\LaTeX$ balks a bit at the repeated subscripts, though.
(Incidentally, it's annoying that we use "vector" to denote both an "abstract vector" and a "concrete vector" while we have different terms for "linear transformation" and "matrix." Perhaps we should start using something like "array" or "list" or "tuple" to mean a concrete vector.)
A: Bourbaki dangerous bend symbol  to mark dangerous or difficult ideas.
A: I like $A \hookrightarrow B$ and $A \twoheadrightarrow B$ for "$A$ injects into $B$" and "$A$ surjects onto $B$" respectively.
A: If one needs to denote the fiber (not the stalk which is standardly denoted $\mathcal{F}_{x}$) of a sheaf $\mathcal{F}$ at the closed point $x$ of the $\Bbbk$-scheme $X$, one can write
$\mathcal{F}\mid_{x}$
After all, the fiber $\mathcal{F}\otimes_{\Bbbk}\;\kappa (x)$ is the restriction (pullback) of $\mathcal{F}$ to the point $x:\rm{Spec}\;\Bbbk\rightarrow X$.
The problem is that, when you identify vector bundles with locally free sheaves, the above notation clatches with the usual notation $E_x$ for the fiber of vector bundles. 
On the other hand almost always the context would be sufficient to clarify which of the two notations is being used.
A: *

*I also like the notation $x \prec y$ to denote majorization of a vector $x$ by a vector $y$; once defined, this notation relieves quite lot of burden.

*On a related note, I also prefer the notation $A \succeq 0$ to signify that $A$ is a positive semidefinite matrix (some prefer to use the perhaps "more natural" $A \ge 0$, but since I frequently deal with nonnegative matrices, the $\ge$ is out)
A: All of the notations created to simplify writing category theory. For instance, the idea of drawing a circular arrow inside of a diagram to indicate that that diagram is commutative.  As well as the idea of putting an angle in the top left or bottom right of a square diagram to indicate that it is a pushout or pullback.  And finally, the notation of augmenting any of these notations with $\simeq$ to indicate that the diagram is only "up to homotopy".
A: I find 
$$\lim_{x\nearrow0} f(x)$$
$$\lim_{x\searrow0}  f(x)$$
for the limit from below and from above much more intuitive that all other notation I've seen, including $\lim_{x\to 0^-}f(x)$ and $\lim_{x\to 0^+}f(x)$.
A: $D_j f$ to denote the partial derivative of a function between Euclidean spaces, w.r.t. the $j$'th coordinate. For some reason Jacobi's notation $\frac{\partial f}{\partial x_j}$ has become more popular. Jacobi's notation tends to cause much ambiguity and confusion, a point which is emphasized in the book "Multidimensional Real Analysis" by Duistermaat & Kolk. For instance (this example is taken from their book), let $e_1,e_2$ be the standard basis for $\mathbb{R}^2$ and define a new basis by $e'_1 = e_1 + e_2, e'_2 = e_2$. The passage from one basis to another is as follows: If $x_1 e_1 + x_2 e_2 = y_1 e'_1 + y_2 e'_2$ then $y_1 = x_1, y_2 = x_2 -x_1$. Now the meaning of $\frac{\partial y_2}{\partial y_1}$ is ambiguous: If one interprets $y_1$ and $y_2$ as independent coordinate functions, then $\frac{\partial y_2}{\partial y_1} = 0$. On the other hand, $\frac{\partial y_2}{\partial y_1} = \frac{\partial (x_2 -x_1)}{\partial x_1} = -1$, right? This was the source of much confusion for me when I was taught multivariate calculus and the notation $D_j f$ would have eliminated this confusion entirely.
A: The three-dot notation $f\mathrel{\scriptsize\vdots}A\to B$ to indicate that $f$ is a partial function from $A$ to $B$, meaning that $\text{dom}(f)\subseteq A$ rather than $\text{dom}(f)=A$. Partial functions are pervasive in logic, especially computability theory and set theory, and this notation is both compact and suggestive.
A: I really like $(-)^n$ instead of $(-1)^n$ for alternating signs in series etc. Its more aesthetic and slightly easier to write. I found it in a 1976 paper about multipolar expansions.
A: String diagram-notation
makes for example adjoint functors, monads, tensor categories,... much clearer.
A: To say that $u$ and $v$ are orthogonal you can spell out "The scalar product of $u$ and $v$ is equal to zero", i.e.:
$\langle u,v \rangle=0$
but you can also use the binary symbol $\perp$ to write the sentence "$u$ orthogonal to $v$" more directly, i.e. $u\perp v$.
Analogously, to say that sets $A$ and $B$ have empty intersection, of course you can spell out "$A$ intersection $B$ equals the empty set", i.e.:
$A \cap B = \emptyset$ 

But it would be nice if there was a binary symbol (like a barred $\cap$ symbol, not to be confused with the $\pitchfork$ symbol for transversality) to say directly "$A$ does not intersect $B$ (nontrivially)". 

I don't think this symbol already exists in LaTeX.
A: Whenever $X$ is a singleton set, denote by $!X$ the unique element in $X$, or put else (assuming some standard material set theory),
$$! \colon \mathrm{Singletons} → \mathrm{Set}$$
is the unique inverse for $\{\,\} \colon \mathrm{Set} → \mathrm{Singletons},~x ↦ \{x\}$. So $!\{x\} = x$ for all things $x$ and $X = \{!X\}$ for all singletons $X$.
This makes it possible to formulaically talk about “the unique element $x$ such that …” (which happens a lot) and it fits well with the notation “$∃!$” for saying “there exists some unique …”.
For example, one may define the minimal polynomial of an element $α$ in an algebraic field extension $E / F$ as
$$\operatorname{minpol}_F α = ~!\{f ∈ F[X];~\text{$f$ monic, irreducible with $f(α) = 0$}\}.$$
A: For a Lie group $G$ with Lie algebra $\mathfrak{g}$, and element $A\in\mathfrak{g}$, denote the left invariant vector field $\overrightarrow{A}(g)=L_{g*} A$, and the right invariant vector field $\overleftarrow{A}(g)=R_{g*}A$. On any matrix group, $\overrightarrow{A}(g)=gA$ and $\overleftarrow{A}(g)=Ag$, easy to remember. The notation uses fewer subscripts than the more common notation $X_A$, which is also not easy to adapt to right versus left invariant vector fields. When discussing group actions, we usually use left actions. The left action of the group on itself is generated by the right invariant vector fields. Some authors use $X_A$ for left invariant vector fields, but also for the generators of any action of the group, which can lead to sign mistakes when discussing the left action of the group on itself.
A: I like $f\colon\thinspace M\looparrowright N$ to denote an immersion of smooth manifolds.
A: I recommend the notation
$$
    a \equiv_n b
$$
in place of $a \equiv b \pmod{n}$.
It's much less verbose. The meaning is clear. And the $n$ is where it really belongs, next to the $\equiv$ it is describing.
We're stuck with $a \equiv b \pmod{n}$ as the standard notation (for now!), because that's what Gauss came up with. I've got nothing against Gauss for not using a subscript $\equiv_n$. It seems to me that Disquisitiones Arithmeticae doesn't have subscripts anywhere. Subscripts must have been outside the graphic design space or something. So I don't blame him for resorting to $a \equiv b \pmod{n}$. Gauss did a great thing by popularizing $n \mid a-b$ as an equivalence relation of $a$ and $b$. But if we were to invent the notation today, I dare say $a \equiv_n b$ would be the modern choice. (See this post on Math.SE, where Alexander Gruber suggests the same thing in a comment.)
Of course, if there's no ambiguity, you can still just write plain $a \equiv b$. I'm talking about the cases where you need to or want to indicate the modulus $n$. It may not seem like much, but "(mod n)" is surprisingly verbose to physically write. If you're hand-writing pages or blackboards full of congruences, chances are you've already succumbed to abbreviating "(mod n)" somehow. I've seen lots of different shorthand, based on dropping the parenthesis, or some or all of the text "mod" (which is itself an abbreviation of "modulo", or if you really go by Gauss's Latin, "secundum modulum" - be thankful you're not writing that):
\begin{align}
    a &\equiv b \quad \mathrm{mod}\ n \\
    a &\equiv b \quad (\mathrm{m}\ n) \\
    a &\equiv b \quad (n) \\
\end{align}
I've seen all of these used before, as well as $a \equiv_n b$. Certainly $a \equiv_n b$ is the cleanest notation.
As a free bonus, you get a cool-looking Fermat's theorem:
$$
    a^p\!\equiv_p\!a.
$$
A: The notation $M^{\oplus n}$ and $M^{\otimes n} $ to denote, respectively, nth direct sum and nth tensor product. 
The notation $X \mathbin{\pi} Y$ to denote product of objects in an abstract category, and the analogous with the "upside down $\pi$" for coproduct. I once have seen this being used by B.Keller in a talk.
It'd be nice to have a smaller $\Pi$ (resp. $\amalg$) symbol instead.
A: I am fond of subscripting asymptotic notation with the parameters that the implied constant is allowed to depend on (and on the asymptotic parameter, if needed).  e.g.

*

*$X = O_k(Y)$ (or $X \ll_k Y$, or $Y \gg_k X$) means that $|X| \leq C_k Y$ for some $C_k$ depending only on $k$.

*$X = o_{n \to \infty; k}(Y)$ means that $|X| \leq c_k(n) Y$ for some function $c_k(n)$ of both $k$ and $n$, which goes to zero as $n \to \infty$ for fixed $k$.

*(Rarer) $X = O_{n \to \infty; k}(Y)$ means that $|X| \leq C_k Y$ whenever $n \geq N_k$, for some $C_k$ and $N_k$ depending only on $k$.

Of course, if there is a parameter that influences all the constants (e.g. the ambient dimension) then it is better to explicitly state at the beginning that all constants will depend on this parameter so that one does not have to put in the explicit subscripts in all the time.
It can be instructive to rewrite some basic notions in analysis in this sort of notation, just to get a slightly different perspective.  For instance, if $f: {\bf R} \to {\bf R}$ is a function, then:

*

*$f$ is continuous iff one has $f(y) = f(x) + o_{y \to x; f,x}(1)$ for all $x,y \in {\bf R}$

*$f$ is uniformly continuous iff one has $f(y) = f(x) + o_{|y-x| \to 0; f}(1)$ for all $x,y \in {\bf R}$

*A sequence $F = (f_n)_{n \in {\bf N}}$ of functions is equicontinuous if one has $f_n(y) = f_n(x) + o_{y \to x; F,x}(1)$ for all $x,y \in {\bf R}$ and $n \in {\bf N}$ (note that the implied constant depends on the family $F$, but not on the specific function $f_n$ or on the index $n$)

*A sequence $F = (f_n)_{n \in {\bf N}}$ of functions is uniformly equicontinuous if one has $f_n(y) = f_n(x) + o_{|y-x| \to 0; F}(1)$ for all $x,y \in {\bf R}$ and $n \in {\bf N}$

*$f$ is differentiable iff one has $f(y) = f(x) + (y-x) f'(x) + o_{y \to x; f,x}(|y-x|)$ for all $x,y \in {\bf R}$;

*(similarly for uniformly differentiable, equidifferentiable, etc.)

(These formulations are close to the nonstandard analysis formulations of these concepts, which uses similar but not quite identical asymptotic notation, but that is another story.)
A: $a \vee b$ and $a \wedge b$ to denote the maximum and minimum of the numbers $a$ and $b$.  (This seems to be well-known only among probabilists.)
A: I once came across the notation $\underline{n}$ for the set $\lbrace 1,2,\dots,n\rbrace$.  It came in very handy to write $i \in \underline{n}$ instead of $1\leq i \leq n$ or $i \in \lbrace 1,2,\dots,n\rbrace$.
A: The notation for transversality:
$M \pitchfork N$
A: Instead of $[X]$ one often sees $\mathbf 1_{X}$ (especially in probability work?). This is neat because it literally is 1 on $X$. Also it has the advantage over $[X]$ that you can write things like $(2+\mathbf 1_{X})^2$ for the function that is $9$ when $X$ occurs and $4$ otherwise; $(2+[X])^2$ would be less appealing here. On the other hand, if there is a lot of notation replacing "$X$" this is not so good:
$$\mathbf 1_{n_k\in \{n: n\text{ prime}\}}.$$
A: *

*For rising and falling factorials: $x^{\overline{n}}$ and $x^{\underline{n}}$ à la Knuth. Much better than the traditional way to write the Pochhammer symbol: $(x)_n := x^{\overline{n}}$. In a book I'm writing, I use the notation $x^{\uparrow n}$ and $x^{\downarrow n}$, which I find much less clumsy (consider $(2x+1)^{\overline{6k-2}}$ vs $(2x+1)^{\uparrow6k-2}$). Anyway, the utility in either of these notations is seen in the umbral calculus; it makes the connection to "ordinary" calculus much more apparent, such as with $$\Delta x^{\uparrow n} = n x^{\uparrow n-1}\qquad\text{compared to}\qquad D x^n = nx^{n-1}.$$

*The simple idea of omitting parentheses for function application: $f\,x$ as opposed to $f(x)$. I think this often makes some mathematics look cleaner, especially when the argument isn't especially complex. It also allows for some nice (= convenient) abuse of notation, such as in $$\left[ (-1)^{p - m - n} z \prod_{j = 1}^p \left( z D_z - a_j + 1 \right) - \prod_{j = 1}^q \left( z D_z - b_j \right) \right] G(z) = 0,$$ where $D_z:=d/dz$. Note this equation isn't a product (entirely); upon expansion, we'd have $D_z G(z)$ terms.

*Do fractions count? Imagine having to write $$\sqrt{(x^2 + 2x + 1)\div (5x^3 - 3x^2 + 2x - 7)}$$ instead of $$\sqrt{\frac{x^2 + 2x + 1}{5x^3 - 3x^2 + 2x - 7}}.$$

*Big-O notation. Though often abused, this is a much less clumsy way to express boundedness and asymptotics and errors and even lets you begin to do some algebra with them (provided you're careful). I don't think doing such is as obvious when you write it all out manually.

*$\square(x)$ for the square wave, $\triangle(x)$ for the triangle wave, $Ш(x)$ for the Dirac comb (seriously, see Appel's "Mathematics for Physics and Physicists"). These are more cute than explicitly useful.

*Notation used with musical isomorphisms as a way to do raising and lowering of indices. We have $X^\sharp$ which raises the index (in the context of Einstein summation) and $X^\flat$ which lowers the index. Here, $\flat$ and $\sharp$ are isomorphisms between tangent $TM$ and cotangent bundles $T^*M$: $\flat:TM\to T^*M$ and $\sharp:T^*M\to TM$.

*Using $\operatorname{cis}\theta = \cos\theta + \mathrm{i}\sin\theta$ (cosine i sine), which is nice for obvious reasons (yes, $\omega = e^{\mathrm{i}\theta}$ is nice too) and $\operatorname{cas}\theta = \cos\theta + \sin\theta$ (cosine and sine), which is used in e.g., the Hartley transform.

*Notations for hypergeometric functions $${}_pF_q \!\left( \left. \begin{matrix} a_1, \dots, a_p \\\\ b_1, \dots, b_q \end{matrix} \; \right| \, z \right) = {}_pF_q(\mathbf{a},\mathbf{b};z)$$ and Meijer-$G$ functions: $$G_{p,q}^{m,n} \!\left( \left. \begin{matrix} a_1, \dots, a_p \\\\ b_1, \dots, b_q \end{matrix} \; \right| \, z \right)=G_{p,q}^{m,n} \!\left( \left. \begin{matrix} \mathbf{a} \\\\ \mathbf{b} \end{matrix} \; \right| \, z \right)$$

*Notation for general continued fractions: $$\underset{j=1}{\overset{\infty}{\LARGE\mathrm K}}\frac{a_j}{b_j}=\cfrac{a_1}{b_1+\cfrac{a_2}{b_2+\cfrac{a_3}{b_3+\ddots}}}.$$ The $\mathrm{K}$ comes from German's "Kettenbruch", which is "continued fraction."
I think that's good for now. There are probably lots more. :)
To end, I'll say one notation I do not like: the use of fraktur. Most of the time it just looks ugly and no one can actually write fraktur letters.
A: (This would be a comment on notation for partial functions, but I don't have the reputation points, as I just joined MO.)  Though this is by no means standard, for personal use I've adopted the following system of arrow decorations that captures many standard types of binary relations.  For a relation f from A to B, use $\rightharpoonup$ to indicate $\forall x\in A ~\exists y \in B~~xfy$, $\rightharpoondown$ to indicate $\forall x\in A~\exists^{\leq 1} y\in B~~xfy$, $\leftharpoondown$ to indicate $\forall y\in B~\exists x\in A~~xfy$, and $\leftharpoonup$ to indicate $\forall y\in B~\exists^{\leq 1}x\in A~~xfy$.  So, $\rightarrow$ is for functions, $\leftrightarrow$ is for bijections, $\leftharpoonup\hspace{-1em}\to$ is for injections, $\leftharpoondown\hspace{-1em}\to$ is for surjections, $\rightharpoondown$ is for partial functions, $\rightharpoonup$ is for serial relations, and so on.
A: Has this already been mentioned?  If a group $G$ acts on a commutative group $A$ by homomorphisms, $G \to Aut(A)$, then use $a^g$ to denote the action. Especially if the group multiplication on $A$ is written multiplicatively, where we can say things like $(ab)^g = a^g b^g$. This can come up especially in Galois theory; I remember Lang using this notation in his Algebra to prove Hilbert's Theorem 90, and I thought it was very neat, and enhanced the readability of notation as well. 
A: Round brackets as Cartesian coordinates and square brackets as homogenous coordinates.
I picked this up idea from Needham's Visual Complex Analysis, and I'm not sure how commonly it's used elsewhere. In the book, the convention was to use round-bracketed matrices
$$
    \begin{pmatrix}
        a & b \\
        c & d \\
    \end{pmatrix}
$$
for representing linear transformations, and square-bracketed matrices
$$
    \begin{bmatrix}
        a & b \\
        c & d \\
    \end{bmatrix}
$$
for representing Möbius transformations. More generally, we can let round brackets be for all "ordinary", "Cartesian", "vector"-ish things, and square brackets be for all "projective" or "homogeneous" things. It's useful to be able to tell them apart quickly, as the algebra looks the same but the meaning is very different. 
Outside of complex analysis, there's an even plainer context where this convention would help:  computer graphics (a.k.a. the actual "real-life" application of projective geometry). In computer graphics, there are two common ways to think of a point, say, in 2-dimensional space. One way is a pair $(x,y)$, the way people normally think of coordinates. The other way is to add an extra coordinate: $x$ and $y$ along with an extra $1$. This is a practical representation, since it works naturally with affine and projective transformations, which we represent in homogeneous coordinates too. The use of both systems leads to a problem. Without any notational convention, is $(3,-4,1)$  supposed to mean a 2D point or a 3D point? Is
$$
    \begin{bmatrix}
        1 & 0 & 0 \\
        0 & 1 & 1 \\
        0 & 0 & 1 \\
    \end{bmatrix}
    \begin{bmatrix}
        -2 \\
        3 \\
        1 \\
    \end{bmatrix}
$$
supposed to be a translation applied to a 2D point, or a shear mapping applied to a 3D point? You need context and you can't tell at a glance. If we used round/square brackets with our designated interpretations, though, we'd know right away that $(3,-4,1)$ means a 3D point and the square-bracketed matrix above means a 2D translation.
Unfortunately, they don't actually make any notational distinction at all in computer graphics. They're just going to stay confused ;) - but you don't have to. You can use the round and square to distinguish ordinary and projective coordinates. It deserves to be more standard.
(There's another convention out there to make the distinction with commas and colons, i.e. $(a,b,c)$ is Cartesian and $(a \mathbin{:} b \mathbin{:} c)$ is homogeneous. It works well for some purposes, but not if you need the notation to generalize to matrices.)
A: Writing $\int_{x=0}^{2 \pi} \sin x dx$ rather than $\int_0^{2 \pi} \sin x dx$ can be very useful when there are integrals stacked several layers deep. EG
$$\int_{x=-\infty}^{\infty} \int_{y=-\infty}^{\infty} e^{-(x^2+y^2)/(2 \sigma)} dx dy = \int_{r=0}^{\infty} \int_{\theta=0}^{2 \pi} e^{-r^2/(2 \sigma)} r dr d\theta.$$
A: 
The abstract index notation.

The original problem with the indices was that they were used to label coordinates, so mathematicians preferred more and more coordinate independent operators, while physicists continued to use indices. Then, Penrose realized that it has to be something beyond the indices that makes them useful - mainly the Einstein summation convention - and proposed the abstract index notation. This notation is almost identical in form with that of coordinate indices, but it is invariant, like the notation used by mathematicians, and maintains the simplifications due to the use of indices. The indices are not interpreted as labeling coordinates, but as representing the type of vectors and tensors and how they act on each other.
I think that there are advantages and disadvantages in both notations. Though, many tensor operations, especially contraction and type change, are easier to define and perform by using indices.
The following fields can benefit of this notation: Linear Algebra, Representation Theory, Group Theory, Differential Geometry.
This notation can naturally be related to Penrose's diagrammatic notation.
A: I like to interpret $f(x)$ as meaning $f\circ x$, otherwise known as the pullback $x^*f$. For instance $x$ could be the standard real valued coordinate on a line. This makes rigorous sense of the concept of a "variable" and hence also dependent and independent variables ($y=f(x)$). In the example of functions on a line, $f'=dy/dx$ is simply a ratio of 1-forms.
Such an interpretation also answers the common complaint that $f=f(x)$ confuses a function with its values. Instead it represents the very common shorthand of omitting pullbacks!
A: The notation $\perp$ to denote either orthogonality, or to indicate independent random variables, or perhaps even to indicate relatively prime numbers.
A: The lack of a nice obviously symmetric notation for $\binom{a+b}{b}$ has bothered me; Dijkstra suggested in EWD 782 the notation $P(a,b)$, generalizing it also to $P(a_1,\ldots,a_k)$ for $\binom{a_1+\ldots+a_k}{a_1,\ldots,a_k}$.  (Though I certainly disagree with him about $\binom{n}{k}$ being useless - you certainly do want to think about it that way a lot of the time.)  I haven't actually had any reason to use this since I saw it but I can certainly think of times I would have.
Also the double-parentheses multichoose notation $\left(\!\binom{n}{k}\!\right)$ is nice because it lets you say "...and this is n multichoose k (which is equal to this binomial coefficient)" instead of just jumping directly to a binomial coefficient whose relevance may not be immediately obvious.  But I suppose that's not really on the level of giving you a better way to look at things.
A: Using $(a, b, ... )$ is handy to denote a column vector, which is the transpose of the row vector $[a, b, ... ]$, especially in linear text. Correspondingly, all displayed matrices should be written with brackets, not parentheses. This notation agrees with the usual identification of coordinates with column vectors.
A: In the notation of Time scale calculus, the ordinary calculus derivative df/dt and the forward difference operator $\Delta f $ are both written as $f^\Delta$. Indefinite sums and indefinite integrals are both written as $\int{f(t)\Delta t}$ and called indefinite integrals. The context would say $\mathbb{T}=\mathbb{Z}, \mathbb{T}=\mathbb{R}$ or other $\mathbb{T}\subset\mathbb{R}$.
A: Multi-factorials are handy.  Sometimes results can be expressed compactly by introducing a double factorial or possibly higher factorial. For example
$$\int_0^{\pi/2} \sin^{2n+1} \theta \:\: d\theta = \frac{(2n)!! }{ (2n+1)!!}$$
A: I think that Inuit numerals are cool. (http://en.wikipedia.org/wiki/Inuit_numerals) They are useful for vigesimal type things. 
A: In type theory, the notation $(x : A) \to B (x)$ instead of $\Pi_{x : A} B (x)$ for the dependent product and $(x : A) \times B (x)$ instead of $\Sigma _{x : A} B (x)$ for the dependent sum. For non-type-theorists, a dependent product is a set-indexed product of a collection of sets, and a dependent sum is a set-indexed disjoint union of a collection of sets[1]. The former notation is used in the Agda, the latter is less widely spread but both are used for example in cubicaltt. Some advantages of this notation are:


*

*It is consistent with notation for non-dependent sums and products $A \times B$ and $A \to B$ (the latter is in fact an instance of the third example of good notation given in the original question, and is already widespread among type-theorists as a notation for the set of functions from $A$ to $B$)

*The fact that $A$ does not appear as a subscript certain complex types easier to follow, as in the equation
$$ (x : A) \times ((y : B (x)) \times C (x, y)) \cong (u : (x : A) \times B (x)) \times C (\pi_1 u, \pi_2 u) $$


*

*The fact that $A$ does not appear as a subscript makes it easier to represent this notation in plaintext, as is frequently done when programming with dependent types.


[1] Here and in the rest of my answer, I am ignoring the distinction between a set and a type.
A: When I teach topology, I write $A\stackrel{\textrm{o}}{\subset} X$ to mean that A is an open subset of the topological space $X$, and I write $A \sqsubset X$ to mean that $A$ is a closed subset of $X$. I think it would drive me crazy to write the words open and closed constantly.
A: A good notation and a bad notation (in my opinion).
Good: $p' = (1 - \frac1p)^{-1}$. It is commonly enough used in analysis (Holder inequality) that it is good to have a shorthand, and it makes clear that the conjugate exponents are dual pairs: $(p')' = p$. 
Bad: $p^* = \frac{np}{n-p}$ the Sobolev conjugate in Sobolev inequalities. It hides the dependence on the spatial dimension $n$, and overloads the $ * $ for something that does not have a duality: $(p^* )^* = \frac{(2p)^*}{2} \neq p$.   
A: I really like the arrow notation for limits: $$\frac{\sin(x)}{x} \xrightarrow{x\rightarrow 0} 1.$$ I've seen people use this on the blackboard, but I don't think I've seen it in print. The right-hand side of an arrow expression can be decorated with a "+" or "-": $$\frac{\sin(x)}{x} \xrightarrow{x\rightarrow 0^+} 1^-.$$ Arrow expressions can be treated as propositions (e. g., $x\rightarrow 0$ implies $\frac{\sin(x)}{x}\rightarrow 1$), but this is usually less succinct than the stacked arrows. However, it's easier to chain limits this way: 

If $f$ and $g$ are continuous [in the sense of elementary calculus], then so is $f\circ g$: if $a$ is fixed and $x\rightarrow a$, then $g(x)\rightarrow g(a)$ (since $g$ is continuous), so $f(g(x)) \rightarrow f(g(a))$ (since $f$ is continuous), QED.

This can be made rigorous, say, with nonstandard analysis, although there are probably more elementary ways. 
Sometimes, we need to use a limit as a subexpression in a formula, rather than just stating that the limit equals something. For this, I like the notation $f(x)|_{x\rightarrow a}$ in favor of $\lim_{x\rightarrow a}f(x)$. To me, it's an obvious and intuitive extension of the notation $A(x)|_{x=a}$, which is commonly used to denote the expression that results when $x$ is replaced by $a$ in the expression $A(x)$ (in which $x$ occurs free).
A: I found the notation $K_\bullet$ for a complex (in with objects an abelian category or as an objects of the derived category) is very helpful. Otherwise people have to write something like $\cdots \to K_{n}\to \cdots \to K_{2}\rightarrow K_{1} \to K_{0}$ which just contains exactly the same amount of information.
A: As a freshman, I "invented" the notation 
$H \lhd ! \; G$
to say that $H$ is a characteristic subgroup of $G$, i.e. a subgroup invariant under any automorphism of $G$ (whereas a normal subgroup $N\lhd G$ is only invariant under the inner automorphisms).
A: Since the standard notation for open interval $(a,b)$ can be confused with the coordinates, gcd, and other stuffs (open brackets have been used A LOT!), I've seem notations like 
$]a,b[$ 
occurred in the book "Elementary Classical Analysis" by Marsden,
and we can denote half-open half-closed interval like this:
$]a,b]$ or $[a,b[$.
A: To denote an action $\alpha: G\times X \rightarrow X$ of a group $G$ on a space $X$, there is the nice piece of notation:
$\alpha: G \curvearrowright X$
or simply
$G \curvearrowright X$  (the latter when the action is understood from the context).
E.g. you can say something like: $\rm{GL}(V) \curvearrowright V$ linearly. Or, to say that $W$ is an invariant subspace for $G \curvearrowright V$, you just write: $G \curvearrowright W$.
Another example: $\rm{Ad}:G \curvearrowright \mathfrak{g}$, and so on.
A: $$a^{\cdot \, n} = a\cdot a\cdots a$$
$$a^{\wedge \, n} = a\wedge a\wedge\dots\wedge a$$
$$a^{\,, \, n} = a,a,\dots,a$$
For example one could write
$$\langle(x+10y-z)^{\,, \, 2}\rangle= \langle(x+10y-z),(x+10y-z)\rangle.$$
or 
$$\sin^{\circ(-1)}x=\arcsin x$$
or
$$\sin^{\cdot(-1)}x=\frac1{\sin x}$$
A: One can decorate a subscript or superscript by additional symbols to indicate what the subscript or superscript is doing.  For instance, consider a truncation $f 1_{|f| \leq N}$ of a function to its values whose magnitude is at most $N$.  One could of course call such a function something like $f_N$, but why not call it $f_{\leq N}$ instead?  Then one can do things like "Decompose $f = f_{\leq N} + f_{>N}$, where $f_{\leq N} := f 1_{|f| \leq N}$ and $f_{>N} := f 1_{|f| > N}$."  Notation of this type is sometimes used in PDE, particularly with regard to Littlewood-Paley frequency projections.
Similarly, one could imagine the operation of shifting $f$ by $N$ to be denoted something like $f_{+N}$ rather than $f_N$, etc..
A: $G \circlearrowleft X$ (or $G \circlearrowright X$) to denote that $G$ acts on $X$.

Edit by A.H. :
Here are some latex definitions that produce the symbol that David Speyer describes in his comment:
\def\acts{
\hspace{.1cm}
{
\setlength{\unitlength}{.30mm}
\linethickness{.09mm}
\begin{picture}(8,8)(0,0)
    \qbezier(7,6)(4.5,8.3)(2,7)
    \qbezier(2,7)(-1.5,4)(2,1)
    \qbezier(2,1)(4.5,-.3)(7,2)
    \qbezier(7,6)(6.1,7.5)(6.8,9)
    \qbezier(7,6)(5,6.1)(4.2,4.4)
    \end{picture}
\hspace{.1cm}
}}

and
\def\acted{
\hspace{.1cm}
{
\setlength{\unitlength}{.30mm}
\linethickness{.09mm}
\begin{picture}(8,8)(0,0)
    \qbezier(1,6)(3.5,8.3)(6,7)
    \qbezier(6,7)(9.5,4)(6,1)
    \qbezier(6,1)(3.5,-.3)(1,2)
    \qbezier(1,6)(1.9,7.5)(1.2,9)
    \qbezier(1,6)(3,6.1)(3.8,4.4)
    \end{picture}
\hspace{.1cm}
}}

A: I know some people absolutely DESPISE using coordinates and components to do "tensor analysis", but sometimes there is no recourse, and then Einstein's summation convention is a big help.
A: As mentioned in a comment, $\lfloor x\rfloor$ is much better notation than $[x]$ for denoting the greatest-integer function. Most especially since it doesn't collide with the $10^6$ other things that $[]$ is used for, e.g. the $0,1$ function Richard Borcherds mentioned.
I very much like, though haven't had much use for, the notation $n{q\atop \cdot}$ for $|GL_n(q)/B|$, pronounced "$n$ $q$-torial". Famously, it extends to a polynomial function of $q$, and when $q=1$ we have $n{1\atop \cdot} = n!$
(Oops: I left out the $/B$ the first time, thanks Jim and David.)
A: I use the notation
$V \oplus^{\perp}W$
to denote orthogonal direct sum [Edit: direct sum of, say, subspaces of a given inner-product space]. 
Or
$(M,g) \times^{\perp} (N,g')$, or simply $M \times^{\perp} N$, to denote (orthogonal) cartesian product of Riemannian manifolds.
A: Cauchy-Binet as a generalized Pythagoras theorem.
Let $X$ be an $ n \times k$ matrix with $n \ge k$. For any $k$-index
$I=i_1...i_k, \; 1 \le i_1 < ... < i_k \le n$, there is some advantage
to denote by $X_I$, the determinant of the $k \times k$ submatrix of
$X$ with rows indexed by $I$. For any two such $X,Y$, we can state
the Cauchy-Binet formula as a pairing
$$ \det (X^TY)= \sum_{I} X_I Y_I $$ where the sum is over all $n
\choose k$ $k$-indices. This is a Pythagoras theorem for $X=Y$ since
it says that the the volume-squared of the parallelepiped spanned by
the $k$ columns of $X$ in $\mathbb{R}^n$ is the sum of squares of
the volume of the projections on the $n \choose k$ $k$-dimensional
coordinates.
For any $n \times m$ matrix $A$ with $m,n \ge k$ and $k$ indices
$I,J$, we also denote by $A_{IJ}$ the determinant of the $k \times
k$ submatrix of $A$ with rows indexed by $I$ and column indexed by
$J$. Then for $X(m \times k)$ and $Y(n \times k)$, we have by
Cauchy-Binet twice,
$$ \det(X^TAY)=\det(X^T(AY))=\sum_{I}X_I(AY)_I =\sum_I X_I \det(A^IY)=\sum_I X_I \sum_J A_{IJ} Y_J,$$
where $A^I$ is the $k \times n$ matrix given by the rows of $A$
indexed by $I$ and we note that $(AY)_I= \det(A^IY)$ and
$(A^I)^T_J=A_{IJ}$. This notation thus allows us to view
Cauchy-Binet (usually stated with $m=n,A=I$) as an extension of the
usual $x^TAy=\sum_{ij}A_{ij}x_iy_j$ for $k=1$.
A: Instead of writing 
$$|x-y|\le \varepsilon,$$
I used to write 
$$x\lessgtr y\pm \varepsilon.$$
You may read it as $x$ is more-or-less $y$ plus-minus $\varepsilon$.
One may also write something like 
$$x\lessgtr e^{\pm\varepsilon}\cdot y$$
which is much better than
$$|\ln(y/x)|\le\varepsilon$$
It is easier to read, 
especially if instead of $x$ and $y$  you have long expressions.
A: In computations in differential geometry, when we need to omit one entry of a list, I like to write $x^{\hat\imath}$ instead of $(x_1,\dots,\hat{x}_i,\dots,x_n)$, and similarly write $x_{\hat\imath}$ instead of $(x^1,\dots,\hat{x}^i,\dots,x^n)$. This makes the Einstein convention work, and simplifies the computations of relations between exterior and Lie derivatives of differential forms, for example. Similarly, $x^{\widehat{\imath\jmath}}$ to omit two entries. Careful that $x^{\hat\imath\hat\jmath}\ne x^{\widehat{\imath\jmath}}$
A: UMBRAL NOTATION (courtesy of Blissard)
$$(q.)^n = q_n$$
Concise, elegant, and suggestive, it often allows for both brevity and comprehensibility of presentation and short derivations of operational results.
Examples of umbral variables:
$q.^n = q_n = d$, a constant quantity, maybe a single variable or polynomial,
$q.^n = q_n = d^n $, a constant quantity raised to powers,
$q.^n = q_n = x^n$, a variable raised to a power,
$q.^n = b.^n = b_n$; an element of a sequence of numbers, such as the Bernoulli,
$q.^n = B.(x)^n = B_n(x)$, an element of a sequence of polynomials, .e.g., Bernoulli,
$q.^n = (.)!^n = (n)!$, an expression containing an integer parameter,
$q.^n = \binom{x}{.}^n = \binom{x}{n} $, ditto,
$q.^n = M.^n = M_n$, a sequence of matrices, differential ops, ... whatever.
The umbral maneuver in more complex expressions:

*

*reduce an expression that contains the umbral entity to an analytic series in monomial powers of $q.$ when possible


*then lower the positive integer exponent to a subscript


*when in doubt apply
$$e^{q.xD_y} \; f(y) \; |_{y=0} = f(q.x +y) \;|_{y=0} = f(q.x) \;$$


*sensible results can often be obtained for functions not analytic at the origin from

$$f(q.x) := e^{-(1-q.)x D_y} \; f(y) \; |_{y=x} = f(x-(1-q.)x) = f((1-(1-q.))x) $$
(See below on Newton interpolation. E.g., with $f(x)=x^s$, then $f(q.x) = (q.x)^s = q_s\; x^s \; := (1-(1-q.))^sx^s \; $ when convergent.)
Basic umbral operations:

*

*binomial convolution for commuting quantities:

$$c_n = c.^n = (a. + b.)^n = \sum_{k=0}^n \binom{n}{k} \; a.^k \; b.^{n-k} = \sum_{k=0}^n \binom{n}{k} \; a_k\; b_{n-k},$$


*formation of e.g.f.s and o.g.f.s:

$$E(t) = e^{q.t} = \sum_{n \geq 0} \; q.^n \; \frac{t^n}{n!} = \sum_{n \geq 0} \; q_n \; \frac{t^n}{n!}$$
and, with $q_0 =1$,
$$O(t) = \frac{1}{1-q.t} = \sum_{n \geq 0} (q.t)^n = \sum_{n \geq 0} q_n \; t^n = e^{\bar{q}.t}= \bar{O}(t)$$
with $\bar{q}_n = n! \; q_n$.
Then multiplication of formal series is
$$EA(t) \; EB(t) = e^{a.t} \; e^{b.t} =  e^{(a.+b.)t} = e^{c.t} = EC(t) \; .$$
We can even disregard convergence of both series if we are interested only in formal compositional or multiplicative inverses or the formal Laplace transform of an e.g.f. to an o.g.f. since the production of the $n$-th coefficient of the target series depends only on the coefficients of order $n$ or less of the source series for these transformations.
Brevity and intelligibility in umbral versus conventional presentations:
The Euler transformation for e.g.f.s in umbral notation

*

*$$ e^{a.x} =   e^x \; e^{-(1-a. )x}$$
versus conventional notation


*$$ \sum_{n \geq 0} a_n \; \frac{x^n}{n!} = e^x \; \sum_{n \geq 0} (-1)^n \; [\;  \sum_{k=0}^n \; (-1)^k \; \binom{n}{k} \; a_k \; ] \; \frac{x^n}{n!} \; ,$$
and derivation of the conventional formula via the umbral,


*$$ e^{a.x} = e^x \; e^{-x} e^{a.x} =  e^x \; e^{-(1-a. )x}$$
$$ = e^x \; \sum_{n \geq 0} (-1)^n \; (1-a.)^n \; \frac{x^n}{n!} = e^x \; \sum_{n \geq 0} \; (-1)^n \; [\;  \sum_{k=0}^n \; (-1)^k \; \binom{n}{k} a_k \; ] \; \frac{x^n}{n!} \;.$$
(23 keys tapped for eqn. 1 versus 102 keys for 2, not counting the appreciable difference in taps for blank spacing and latex spacing for readability. I could have saved a good fraction of a lifetime of math work by being able to use umbral notation in symbolic apps, study, and publications rather than sigma gymnastics.)
Natural generalizations via umbralization--monic to partition polynomials:
Stirling polynomials of the first kind, related to cyclic permutations, have the e.g.f.
$$e^{St1.(x)t}  = e^{x \ln(1+t)} = (1+t)^x\; ,$$
and Stirling polynomials of the second kind, related to partitions of sets,
$$e^{St2.(x)t}  = e^{x (e^t-1)} \, .$$
Flexibility and precision is afforded if some notation is used to indicate at what level the umbral exponent lowering is to occur. For example, with $\langle \cdots \rangle$ denoting evaluation of the expression between the angular brackets after reduction to a series in the umbral quantity and with $q_0=1$,
$$\langle \; e^{-\ln(1-q.t)} \;\rangle = \langle \frac{1}{1-q.t}\rangle =  \langle \sum_{n \geq 0} q.^n \; t^n \; \rangle = \sum_{n\geq 0} \langle q.^n \rangle \; t^n =\sum_{n \geq 0} q_n \; t^n $$
and
$$e^{-\langle \ln(1-q.t)\rangle} = \exp[\langle \sum_{n \geq 1} q.^n \frac{t^n}{n}\rangle] = \exp(\sum_{n \geq 1} q_n \frac{t^n}{n}) ,$$
which is the e.g.f. for the cycle index partition polynomials of the symmetric groups, a.k.a. the Stirling partition polynomials of the first kind, OEIS A036039.
The natural generalization of the Stirling polynomials of the second kind is via the umbralization, for $q_0 = 1$,
$$e^{\langle e^{q.t}-1 \rangle} =  \exp[ \; \sum_{n \ge 1} \; q_n \frac{t^n}{n!} \; ], $$
giving the complete Bell polynomials of the Faa di Bruno composition formula, a.k.a. the Stirling partition polynomials of the second kind, A036040.
Including an extra variable as in
$$e^{x\langle e^{q.t}-1 \rangle} = e^{Bell.(q_1,..,q.;x)t}$$
and
$$e^{-x\langle\ln(1-q.t)\rangle} = e^{Cyc.(q_1,..,q.;x)t}$$
allows better tracking of block sizes and combinatorics within each partition polynomial. (Note $q_n$ could be a commuting sequence of wedge products, curvature forms, or matrices, just as well as integers, giving basic formulas for the Chern and Pontryagin characteristic class polynomials.)
Now for a little umbral mojo,
$$ e^{St1.(St2.(x)) \; t} = e^{St2.(x) \; \ln(1+t)}=e^{x \; (\exp(\ln(1+t))-1)} = e^{xt} \; ,$$
so the two sets of binomial Sheffer polynomial sequences are an inverse pair under umbral composition; i.e.,
$$St1_n(St2.(x)) = x^n = St2_n(St1.(x)) \; .$$
This is reflected in their lower triangular coefficient matrices being a matrix inverse pair. The same umbral compositional inverse and matrix relationships hold for any Sheffer polynomial sequence. For a pair of inverse/reciprocal Appell Sheffer sequences, this is due to their moment e.g.f.s being multiplicative inverses rather than compositional inverse. The generic Sheffer sequence has a mix of these conditions, belonging to the semidirect product of the Appell and binomial Sheffer subgroups.
Actions of diff op often become more transparent with umbral calculus and/or underlying combinatorics are revealed:
With $a.$ treated as independent of $x$ and $D = d/dx$ for $f(x)= e^{b.x}$,
$$e^{a.D_x} \; f(x) = f(a.+x) = e^{b.(a.+x)}.$$
For the Bernoulli number and polynomials (or any Appell Sheffer sequence),
$B_n(x) = e^{b.D} \; x^n =  (b.+x)^n$,
$D \; B_n(x) = D \; (b.+x)^n = n \; (b.+x)^{n-1} = n \; B_{n-1}(x)$
$e^{a.D} \; B_n(x) = B(x + a.) = (x+b.+a.)^n = (B.(x)+a.)^n = (B.(a.)+x)^n \; .$
Adopt the additional notational convenience $(:AB:)^n := A^nB^n$. Alternatively, we could use $q.^n = q_n = A^nB^n$, but this would veil aspects/intuitions inherited from the properties of the derivative.
From our previous feat of umbral mojo,
$$St1_n(xD) = n! \; \binom{xD}{n} = ST1_n(ST2.(:xD:) = \; :xD:^n \; = x^nD^n,$$
consistent with the actions
$xD \; [x^n] = n x^n$,
$(1 + a.)^{xD} \; [x^n] = (1 + a.)^{n} \; x^n$,
and
$(1 + a.)^{xD} \; f(x) = \sum_{n \geq 0} a_n \; \binom{xD}{n} \; f(x) =  e^{a.:xD:} \; f(x) = f((1+a.)x).$
The operational definition of the Stirling polynomials of the second kind is
$$(xD)^n = (St2.(:xD:))^n =  St2_n(:xD:).$$
From this the e.g.f. given above for the polynomials can easily be derived, and, consequently,
$$e^{txD} \; f(x) = e^{t \; St2.(:xD:)} \; f(x) = e^{(e^t-1):xD:} \; f(x)$$
$$ =f((e^t-1)x + x) = f(e^t x),$$
a dilation, consistent with $e^{txD} \; x^n = e^{tn} \; x^n$.
The Lah polynomials $L_n(x)$ (the normalized, unsigned Laguerre polynomials of order -1, related to partitions of sets into ordered sets or lists) have several useful Stokes-Rodrigues diff op reps,
$$ x \; :Dx:^n  \; x^{-1} = x \; D^n x^n x^{-1} = x^{-n+1} (xDx)^n x^{-1} = x^{-n} (x^2D)^n =  L_n(:xD:) ,$$
and the e.g.f.
$$e^{L.(x)t} = e^{x \frac{t}{1-t}},$$
so
$$ e^{tx^2D} \; f(x) = e^{t \; :xL.(:xD:):} \; f(x) = e^{:\frac{tx^2}{1-tx}D:} \; f(x)$$
$$= f(\frac{tx^2}{1-tx} + x) = f(\frac{x}{1-tx}), $$
a special linear fractional, or Moebius, transformation.
(The inverse ops to the translation, $e^{tD}$; dilation, $e^{txD}$; and special linear fractional, $e^{tx^2D}$, transformations of SL2 are given by negating $t$, which is equivalent to additive ($t+(-t)=0$), multiplicative ($e^t \cdot 1/e^t =1$), and compositional inversion ($\frac{x}{1-tx}|_{x \to\frac{x}{1+tx}}=x$), respectively. SL2 becomes associated to the underlying combinatorics of the binomial Sheffer polynomials $St2_n(x)$ and $L_n(x)$ via the umbral symbolic calculus.)
Interpolation methods and results of integral transforms are naturally suggested and expressed in umbral notation:
Laplace transform of e.g.f. to o.g.f.s with $q_0=1$:
$$\int_{0}^\infty \; e^{q.xt} \; e^{-t} dt = \int_{0}^\infty \;  e^{-t(1-q.x)} dt = \frac{1}{1-q.x}$$
Mellin transform interpolation to continuous indices (related to Ramanujan's master formula, e.g., the Bernoulli polynomials are essentially discrete samples of the Hurwitz zeta function):
$$ q_{-s} = (q.)^{-s} := \int_{0}^\infty \; e^{-q.t} \; \frac{t^{s-1}}{(s-1)!} \; dt$$
Newton interpolation to continuous indices:
$$q_s = (q.)^s = (1-(1-q.))^s = \sum_{m=0}^\infty \; (-1)^m \; \binom{s}{m} \;\sum_{k=0}^m \; (-1)^k \; \binom{m}{k} \; q_k$$
$$= \int_{0}^\infty \; e^{-(1-(1-q.))t} \; \frac{t^{-s-1}}{(-s-1)!} \; dt= \int_{0}^\infty \; e^{-t} \; e^{(1-q.)t} \; \frac{t^{-s-1}}{(-s-1)!} \; dt$$
$$= \int_{0}^\infty \; e^{-q.t} \; \frac{t^{-s-1}}{(-s-1)!} \; dt$$
via analytic continuation. If $q_n = D_x^n$, the action of the integral on $H(x) \frac{x^\alpha}{\alpha!}$, where $H(x)$ is the Heaviside step function, gives a classic fractional calculus in terms of an analytically-continued (AC) Mellin convolution (essentially that for the AC Euler beta function integral) for the interpretation of $D_x^{s}$ for real or complex $s$.
See Boole, Blissard, Sylvester, Cayley, Appell, (Jensen?), Heaviside, Steffensen, Sheffer, Pincherle, Bell, Riordan, Rota, Taylor, Roman, Ray, and Lenart for contributions to and influences on the development of the umbral symbolic calculus.
A: For the value at $x$ of the conditional probability density function of a random variable (capital) $X,$ given the event that the value of a random variable (capital) $Y$ is $y,$ I prefer $f_{X\,\mid\,Y\,=\,y}(x)$ to the more frequently seen $f_{X\,\mid\,Y}(x\mid y).$ It emphasizes that we are concerned with a function of $x.$
A: To resolve the ambiguity when using multiple instances of $\pm$ within a single expression, I use  $\pm_n$ and $\mp_n$. $\pm_a$ must be equal to $\pm_b$ iff $a=b$.
A: $f_*$ and $f^*$ for direct and inverse image. We really should use this right from the beginning, for functions $f\colon X\to Y$, where $f_*\colon P(X)\to P(Y)$ ($P(X)$ being the power set) and $f^*\colon P(Y) \to P(X)$ instead of the awful notations $f(A)$ and $f^{-1}(B)$ for subsets $A$ of $X$ and $B$ of $Y$.
A: The ever-controversial reverse Polish notation for functions: $f(x) = xf$.  Thus in composition, the order makes sense: $(g \circ f)(x) = x f g$ (this point is moot for the fortunate Hebrew- and Arabic-speaking mathematicians).  I hate this notation in practice but I can't deny that it is objectively right and "just makes sense" in more or less the same way that the original post discusses writing $B^A = A \to B$.  Please no one vote this up.
A: 
Dirac's bra-ket notation

This notation is very useful when applying the Hilbert spaces in Quantum Theory. It exploits some properties of duality, eigenvalues/eigenvectors, projectors and self-adjoint operators. In mathematics, perhaps it is difficult to adopt, because mathematicians are using notations that are more general, and cannot exploit these particularities. But if you know Hilbert spaces you can learn this notation in one minute, and then it makes visible many of these nice properties.
A: Here is a notation in algebraic geometry that in my opinion is very useful and self-explanatory but not used widely.
For a birational morphism $f:X\to Y$ there exists an open dense set $U\subseteq Y$ for which $f$ induces an isomorphism $f^{-1}U\to U$. For a closed subset $Z\subseteq Y$ such  that $Z\cap U\neq\emptyset$ the strict transform is defined as $$\overline{f^{-1}(Z\cap U)}\subseteq X,$$ i.e., the closure of the preimage of the part of $Z$ that lies on the part where the morphism is an isomorphism. This is a very important construction and there isn't a universally accepted notation for it. 
János Kollár invented the following notation for this: $$f^{-1}_*Z:= \overline{f^{-1}(Z\cap U)}\subseteq X$$ The genius of the notation is that anyone familiar with basic notation in algebraic geometry should understand what it is:
1) As $f$ is birational, $f^{-1}: Y\dashrightarrow X$ exists as a rational map.
2) For any map $g$, it is common to use $g_*$ to denote push-forward of cycles. 
The strict transform is really just the push-forward of cycles via the rational map $f^{-1}$.  
A: Diagrammatic notation for tensors (Penrose diagrams, birdtracks, etc.).  It makes many things like the invariance of tr(A B C) under cyclic permutation into empty statements.
A: I like $ A^{\text{H}} $ for the conjugate transpose of the matrix $ A $, ananlogously to how $ A^{\text{T}} $ and $ A^{\text{C}} $ means the transpose and the conjugate.  You call it the Hermitian of the matrix for short.  I learnt this notation from Rózsa Pál, but I can't tell who invented it.
A: For quite some time, Giovanni Sambin has been advocating the notation $A\between B$ for "overlapping" sets, that is, for inhabited intersections, where $\exists_{x} \ x \in A \cap B$. He makes his case in a constructivist frame, but I think that a lot of ink, chalk---and keyboard pushing...---would be spared if one wrote $A\between B$ instead of $A\cap B \neq \emptyset$, even in non-constructive mathematics.
A: The universal property of the univariate polynomial ring: For any commutative ring $A$, any commutative $A$-algebra $B$ and any $x\in B$, there exists one and only one $A$-algebra homomorphism from the polynomial ring $A\left[X\right]$ to $B$ which maps $X$ to $x$.
This is the so-called evaluation homomorphism at $x$. I denote this homomorphism by $\lim\limits_{X\to x}$. This has the advantage that we have $\lim\limits_{X\to 0}\dfrac{\left(X+1\right)^n-1}{X}=n$ and similar properties hold just as in classical analysis. The polynomial $\dfrac{\left(X+1\right)^n-1}{X}$ is well-defined (since $X$ is not a zero divisor in $A\left[X\right]$ and divides $\left(X+1\right)^n-1$), but if we would blindly replace $X$ by $0$ we would obtain a $\dfrac{0}{0}$ error.
A: Among recent introductions, I like the notation and names (introduced by Kenneth Iverson and popularized by Donald Knuth) for the ceiling function  $\lceil x\rceil$ and floor function $\lfloor x\rfloor$. Compare with the heavy "approximation by excess/defect"...  
A: A) Two notations I love are the rising factorial $x^\overline n$ and its falling factorial twin $x^\underline n$. They are used and advocated in the great book see http://en.wikipedia.org/wiki/Concrete_Mathematics . In passing this book uses great notations.
B) A general trick with binomials to reuse them with sets instead of numbers, here are some typical examples.  
1) $\binom S k $ to denote the set of all $k$-sets of the base set $S$ . 
2) $S^\underline 2$ to denote the pairs $(x,y)$ of $S$ where $x$ and $y$ are different.
3) $S^\underline k $ to denote  the $k$- uplets of $S$ (each uplet has $k$ different elements).
C) Another notation I find useful when listing some (big) families of examples in a combinatorial setting. Use as variables the very numerals $1$ $2$ .. themselves instead of $x_1$ , $x_2$ ... . For example ( very untelling because too small an example) : the intersection of $123$ and $34$ is $3$. 
D) I also often use {{ a,a,b,c}} for multiset. Any other standard or suggestion (or a way to avoid speaking about multiset) is welcome.
A: I like notation such as $2^X$ for the set of subsets of $X$ and ${X\choose k}$ for the set of $k$-element subsets. Also $[x^n]F(x)$ for the coefficient of $x^n$ in the power series $F(x)$, and multivariate notation like $x^\alpha$ for $x_1^{\alpha_1}\cdots x_n^{\alpha_n}$, where $x=(x_1,\dots,x_n)$ and $\alpha=(\alpha_1,\dots,\alpha_n)$. 
A: In commutative algebra with many variables, repeating lists of variables in polynomial arguments and various rings gets very tedious. I suggest using $X_{1..n}$ instead of $X_1,\ldots,X_n$.
Here's an excerpt from Bourbaki's Commutative Algebra, page 222:

For every formal power series $f\in A[[X_1,\ldots,X_n]]$,
$$f(X_1,\ldots,X_n)-f(Y_1,\ldots,Y_n)=\sum_{i=1}^n (X_i-Y_i)h_i(X_1,\ldots,X_n,Y_1,\ldots,Y_n)$$
where the $h_i$ belong to $A[[X_1,\ldots,X_n,Y_1,\ldots,Y_n]]$.

And here's how it looks with my suggested notation:

For every formal power series $f \in A[[X_{1..n}]]$,
$$ f(X_{1..n})-f(Y_{1..n})=\sum_{i=1}^n (x_i-Y_i)h_i(X_{1..n},Y_{1..n})$$
where the $h_i$ belong to $A[[X_{1..n},Y_{1..n}]]$.

I think this notation is maximally succinct, and helps a reader from getting lost in long lists of variables.
A: I would love for topologists to start differentiating between properties that hold “locally” and properties that hold “regionally”, that is: A local property holds on some neighbourhood basis of every point, a regional property holds on some neighbourhood of every point. For any space, having a property locally implies having it regionally.
This would resolve the permanent confusion of “which sort of local” is meant in a given context.
A: In algebra, it is very useful to write $J\cap A$ for the inverse image of an ideal $J$ in a ring $B$ under a homomorphism $f:A\to B$, rather than $f^{-1}(J)$. I normally also omit the morphism and write $IB$ for the ideal generated by the image in $B$ of an ideal $I$ in $A$, rather than $f(I)B$.
A: I recently saw the following notation in the context of divisors on algebraic varieties, and I liked it very much.  
Suppose that $D$ and $E$ are reduced divisors on a normal algebraic variety $X$.  One can use $D \vee E$ to denote the reduced divisor with support equal to $D + E$ and $D \wedge E$ to denote $(D + E) - (D \vee E)$.  I could imagine variants on this if $D$ and $E$ are non-reduced (involving taking max's, respecitvely mins, of the divisors component-wise).
EDIT:  I'm slightly curious as to why this was downvoted.  I guess it's too common to be interesting?
A: 
Has anyone come across any more
  similar examples of good notation that
  should be better known?

Some interesting glyphs:


*

*Combinatorial Principles in Set Theory:     


*

*♣ (Clubsuit)

*◊ (Diamondsuit)


*Bisimulation:


*

*Given two states p and q in S, p is bisimilar to q, written p ~ 
q, if there is a bisimulation R such
that (p,q) is in R.


*Boxplus operator in Coding Theory 


*

*⊞


A: I like the notation $A\mathrel{\in\in}\mathcal C$ for objects (rather than morphisms) in a category (not my own invention).
A: I find the standard notation $X \times_G Y := \frac{X \times Y}{G}$ for balanced products of $G$-spaces annoying, because it conflicts with the well-established notation for fibre products. But on the other hand writing out the full quotient $(X \times Y)/G$ is cumbersome, especially if you're dealing with associated bundles to principal bundles, or if you have spaces with compatible left and right actions of different groups, and you need to write something like $X \times_G Y \times_H Z$.
So I came up with that following notation which combines the product symbol "$\times$" and the quotient symbol "$\_$" into a single binary symbol:

The last one, without the group decorating it, is used in the same way as tensor products $\otimes$ when the ring involved is obvious (e.g. if you're working with associated bundles to a $G$-principal bundle $P$, and you're not planning on taking any reductions of the structure group). 
It's just a rotated semidirect product, so it's easy to implement in LaTeX, and looks similar enough to the usual notation that it's easy to introduce in a talk. Here's my LaTeX code:

A: The notation of $u \mathbin{[\wedge]} v$ for the product of Lie-algebra valued differential forms (I prefer this to another variant  $[u \wedge v]$).
Writing the tangent natural tranformation as $\tau: T \rightarrow \operatorname{Id}$ which expands to $\tau_M : TM \rightarrow M$.
I've seen some category theorists write $\operatorname{Hom}_A[a,b]$ for the homset for a category $A$, which gets a bit long-winded when $A$ is a category with a long name like $\mathrm{Ring}$ or $R\text-\mathrm{Mod}$. It seems much more natural to write $A[a,b]$.
Personally, I like $\bar{\mathbb{R}}$ to write the order completions of the reals. It's less clumsy than $\mathbb{R} \cup \infty$.
A: I like the notation $f:A\cong\subseteq B$ for "$f$ is an embedding of $A$ into $B$."  The idea is that the relation of embeddability is obtained by composing the relations "isomorphic to" and "substructure of."
A: $x\gtrless 0$ to denote that $x\in\mathbb{R}\setminus\{0\}$. Or, if $x\notin\mathbb{R}$ but rather $x\in\mathbb{Z}\setminus\{0\}$ for example, then we write $x\in\mathbb{Z}_{\gtrless 0}$. I prefer the $\gtrless$ notation because it is simpler and more understandable when first met with the eye, and I hope it is set in stone sooner or later.
I also came across the following notation: $$\sum_p'f(x).$$ The little dash $'$ on top of the sigma $\Sigma$ denotes that $p$ is prime. I like this, but I don't feel this is necessary because we can write $$\sum_{p \ \text{prime}}f(x)$$ but when looking at both conventions, the former looks a lot neater.
Also, we have notation $>$, $\gg$ and $\ggg$ with $<$, $\ll$, and $\lll$ but I was thinking, is there a notation to denote that a value $a$ is not much less than a value $b$? I came up with $$a\overline{<} b\tag*{$a$ is not much less than $b$}$$ $$a\overline{>}b\tag*{$a$ is not much greater than $b$}$$
I have also seen that the symbol for concatenation is sometimes $||$, but if I was to see $$A \ || \ B$$ then my first thought would be $A$ is parallel to $B$, but that is just me. With some research, I discovered that there is however some kind of official notation, such that $A^\frown B$ but I don't like it.
I also went extreme and invented this: $$\mathop{\LARGE\Omega}_{\substack{x=k \\ \\ R}}^{x_0} (x, y)$$ to define a set of coordinates; a relation. Here, $x\to x_0$ and $R$ simply denotes the rule, $y = \cdots$. This way, we can write stuff like $$\mathop{\LARGE\Omega}_{\substack{x=36 \\ \\ y = 2x+1}}^{144}(x, y)$$ And then if we don't have a rule, but something like $y\geq0$ for example, then we can write a double index to refer to what value $y$ tends towards, namely $y_0$.
Oh and I almost forgot, I thought that maybe we can symbolise contradiction? For instance, I want to prove that $\sqrt{2}$ is irrational. I would first suppose it is rational, then come to a conclusion that contradicts this statement. Since most of us use $\Box$ in notation of completing a proof, I decided that I could use $\Diamond$ in notation of a contradiction. I thought about it because it is like a titled box $-$ a bit like approaching the proof $\Box$ on a different angle $\Diamond$, if you get what I am saying.
I am not taking this too seriously, for I was just being creative, but I believe there's nothing wrong in trying new things out. Any thoughts?
A: Many years ago, I invented the notation $\mathcal{Lex}_{T}(X)$ as the function equal to $1$ if and only if the consideration of $X$ in the theory $T$ entails no contradiction and $0$ otherwise. Lex is both  "law" in Latin and a shortcut for "Logical existence", which in some sense is the only law of mathematics.
