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bio website mit.edu/~darij/www
location Karlsruhe (home), Munich (until summer 2011), Cambridge/MA (2011-)
age 26
visits member for 5 years, 8 months
seen 1 hour ago
I'm just here for asking stupid questions.

2d
comment Important formulas in Combinatorics
See Proposition 1.3.3 in van Leeuwen's www-math.univ-poitiers.fr/~maavl/pdf/foata-fest.pdf for the simplest proof of $( * )$. The simplicitly of this proof suggests that the RSK algorithm is actually deeper than $( * )$.
Aug
27
awarded  Nice Answer
Aug
25
comment Dickson/determinant type polynomial (updated)
I still don't get it.
Aug
25
comment Dickson/determinant type polynomial (updated)
Not sure I get your argument.
Aug
25
revised Arithmetic product of symmetric functions: why is it integral?
wrong copypaste due to linux console bug
Aug
24
awarded  co.combinatorics
Aug
23
answered Dickson/determinant type polynomial (updated)
Aug
23
revised Important formulas in Combinatorics
edited body
Aug
23
comment Dickson/determinant type polynomial (updated)
If $\ell=1$, then the determinant is $0$ (at least for $k \geq 2$) since all of the $\alpha_j$ are $1$. But $P_{k,\ell} = x_1 x_2 \cdots x_k$. Where am I going wrong?
Aug
23
comment Dickson/determinant type polynomial (updated)
The $a_i$ are all supposed to be in $\left\{0,1\right\}$, right?
Aug
22
comment Schubert calculus and Pieri's formula
@JasonStarr: The word "Schubert" appears only once in Fulton&Harris, and that's in a reference. You might be talking of Fulton? (Or you mean the combinatorial Pieri rule? But that's in many places.)
Aug
22
comment Trace of a Product of Finitely Many Matrices with Cosine Entry
The product is such that the first matrix is the one for $j=0$, etc.?
Aug
22
comment Schubert calculus and Pieri's formula
I don't know the notations involved (what is a special Schubert cycle?), but I suspect that Fulton's Young tableaux (Chapter 10 particularly) is an introduction into this.
Aug
20
comment Important formulas in Combinatorics
@მამუკაჯიბლაძე: If this interpretation is what I think it is (with physical interpretations, I can never tell), then it is well-known under the names "edge sequences", "Maya diagrams" and others (the "abacus" usually stands for the edge sequence subdivided into length-$p$ blocks); see §2 in van Leeuwen's wwwmathlabo.univ-poitiers.fr/~maavl/pdf/edgeseqs.pdf .
Aug
20
comment Important formulas in Combinatorics
@GilKalai: Shouldn't your $L^-(G)$ rather be some kind of reduced incidence matrix? Otherwise it looks like you're getting the square of $\kappa(G)$.
Aug
20
comment Examples of unexpected mathematical images
Abelian sandpile model on a square grid, to be more precise. See other grids at math.cmu.edu/~wes/sandgallery.html .
Aug
20
comment Examples of unexpected mathematical images
This is neat, but probably needs a couple of qualifications: (1) The apparent randomness in the picture is due to the centers of the circles being chosen at random (there is no deterministic chaos here), and (2) the concentric circles are (as far as I can tell) just an artistic way to make the various regions easier to distinguish from each other.
Aug
19
comment Important formulas in Combinatorics
Ah, I forgot the Eulerian form -- that's nice (though probably a case of Möbius inversion?).
Aug
19
comment The Matrix-Tree Theorem without the matrix
There is a minor flaw in the argument: The directed graph in which $i$ points at $\alpha\left(i\right)$ will not always consist of a collection of cycles and a tree rooted at $n$. It will consist of a collection of cycles and some arcs from vertices outside this collection to vertices lying either outside or inside it. These latter arcs might not always form a tree rooted at $n$; they will form a forest whose roots are the vertices of the cycles and $n$. I think the proof is not hard to fix, though.
Aug
19
revised The Matrix-Tree Theorem without the matrix
trees don't have cycles