All Questions
Tagged with real-analysis co.combinatorics
32 questions with no upvoted or accepted answers
19
votes
0
answers
775
views
A Linear Order from AP Calculus
In teaching my calculus students about limits and function domination, we ran into the class of functions
$$\Theta=\{x^\alpha (\ln{x})^\beta\}_{(\alpha,\beta)\in\mathbb{R}^2}$$
Suppose we say that $...
- 433
10
votes
0
answers
287
views
Coefficients of polynomials vs trigonometric product
Let's consider the family of sequences of coefficients in the expansion
$$\prod_{i=0}^{n-1}(1+x^{3^i}+x^{3^{i+1}})=\sum_{k\geq0}a_n(k)\, x^k.$$
Remark. Evidently, the RHS is a finite sum.
Here is a ...
- 43.2k
9
votes
0
answers
180
views
Infinite series identities in search of a proof
This comes in relation to the Fishburn numbers.
I stumbled on the following relation for which I ask a proof if true.
Let $Q_i(z):=1-(1-t)^{i-1}(1-zt)$. Then
$$\sum_{n=0}^{\infty}\frac{(n+1)zt}{...
- 43.2k
7
votes
0
answers
203
views
Permutations which change the value of a convergent series
I'm interested in the following combinatorial problem: What is a necessary and sufficent condition on a permutation $\sigma : \mathbb{N} \rightarrow \mathbb{N}$, so that there exist a summable ...
- 71
5
votes
0
answers
190
views
Number of discrete Lipschitz functions with given Lipschitz constant
Fix $T, K, N \in \mathbb Z_+$. How many distinct Lipschitz functions $f: \{0, \dots, T\} \to \mathbb Z$ are there with Lipschitz constant $K$, and supremum norm at most $N$ satisfying $f(0) = 0$?
In ...
- 6,155
5
votes
0
answers
167
views
Bounding elementary symmetric polynomials away from zero
Let $2 \leq m \leq n$ be integers and let $\mathbf{x} \in \mathbb{R}^n$ (importantly, I am not assuming that the entries of $\mathbf{x}$ are non-negative). The elementary symmetric polynomials are ...
- 6,351
5
votes
0
answers
107
views
Generalized Puiseux series for diagonal reflections of the curves $y = \frac{x}{(1-ax)(1-bx)^m}$
Reflection of the curve $y = f_m(x) = \frac{x}{(1-ax)(1-bx)^m}$ through the diagonal line $y=x$ in the $xy$-plane can be regarded as local compositional inversion of the curve $y=f_m(x)$. ($x,y,a,b$ ...
- 10.5k
5
votes
0
answers
170
views
operation on Ord., Exp., Dri. generating functions
The ordinary, exponential and Dirichlet generating functions for a sequence $\{a_n\}_{n\geq0}$ are given (at least on the formal side), respectively, by
$$F(x)=\sum_{n\geq0}a_nx^n, \qquad E(x)=\sum_{n\...
- 43.2k
4
votes
0
answers
208
views
Extract this constant term
Given a Laurent polynomial $F$ in the variables $\mathbf{t}=(t_1,\dots,t_n)$, let $CT_{\vec{\mathbf{t}}}\,F$ denote its constant term.
For example, $CT_{t_1,t_2}((8t_1-\frac1{3t_1t_2})(5t_1t_2+t_2^2+\...
- 43.2k
4
votes
0
answers
141
views
Level sets of function of inner products of vectors on hypercube
Let $H = \{ 0, 1\}^d$ be the $d$-th Cartesian product of $\{0, 1\}$ in $\mathbb{R}^d$. Suppose $v_1, \ldots, v_k$ are $k$ vectors in $H$ in general position. We define function $F \colon H^{k}\...
- 1,127
4
votes
0
answers
188
views
Evaluate a multiple integral
I want to compute this integral and I would appreciate any help: $N\geq 1$ is fixed.
$$I_N=\int_{0\le r_n\le r_{n-1}\le\cdots\le r_1} e^{-(r_1^2+\cdots+r_n^2)} \prod_{i<j} \sinh(r_i-r_j) dr_1\...
- 41
3
votes
0
answers
315
views
When does the Taylor coefficient of $e^{\sin x}$ vanish?
If $f(x)=\frac{a_1}{1!}x+\frac{a_2}{2!}x^2+\frac{a_3}{3!}x^3+\frac{a_4}{4!}x^4+\cdots$ is an exponential generating function for $\{a_k\}_{k\geq1}$ then
$$e^{f(x)}=1+\frac{a_1}{1!}x+\frac{a_1^2+a_2}{2!...
- 43.2k
3
votes
0
answers
144
views
Noncrossing partitions in Hopf algebras/monoids via compositional inversion
Partition polynomials constructed from the face structures of the associahedra (OEIS A133437) and permutahedra (A133314) comprise the antipodes/compositional inverses in a Faa-di-Bruno-type Hopf ...
- 10.5k
3
votes
0
answers
181
views
Refined f- and h-partition polynomials of the associahedra
The f-polynomials, $F_n(x)$ (cf. OEIS A126216, A033282, and A086810), and the h-polynomials, $H_n(x)$ (cf. A001263, the Narayana polynomials), of the family of simple convex polytopes the associahedra ...
- 10.5k
3
votes
0
answers
155
views
asymptotics of the largest real root
Suppose you have a family of polynomials
$$P_n(x)=\sum_{k=0}^n(-1)^ka_k^{(n)}x^k$$
for $n=0,1,2,\dots$.
Further assumptions:
(1) the coefficients $a_k^{(n)}$ are recursively related to the $a_j^{(n-1)...
- 43.2k
2
votes
0
answers
231
views
Where does this trig. identity hold?
Fix an integer $n\geq1$.
QUESTION. Is it possible to find ALL pair of sequences of non-negative integers $(a_k,b_k)$, for $k=1,2,\dots,n$, such that
$$\sum_{k=1}^n \sin^{2a_k}\theta\cdot \cos^{2b_k}\...
- 43.2k
2
votes
0
answers
321
views
Distribution of $\frac{(\sin(n))^2}{2^n}$ in dyadic intervals?
Good morning all,
I was wondering what kind of methods could help in order to tackle the following problem :
Define the set $A = \left\{ \frac{(\sin(n))^2}{2^n}\right\}$ for $n$ integer. So A is a ...
- 125
2
votes
0
answers
58
views
Tail asymptotics of Durfee square identity
This post is related to the problem Asymptotics of a combinatorial series
According to the Durfee square identity:
$$\sum_{k \ge 0} \frac{q^{k^2}}{(q;q)_k^2} (q;q)_{\infty} = 1,$$
where $(q;q)_k$ is ...
- 51
2
votes
0
answers
99
views
Lower bound on iterated matrix application
Let $n \in \mathbb Z^2$ such that the non self-adjoint weighted Laplacian is
$$(\Delta u)(n)=u(n_1+1,n_2)-u(n_1-1,n_2) + i( u(n_1,n_2+1)- u(n_1,n_2-1))$$
the adjoint operator is then
$$(\Delta^* u)(n)=...
- 192
2
votes
0
answers
109
views
Average number of pieces of a random piecewise-linear function
Let $I$ be a (nonempty) compact interval in $\mathbb R$ and $a_1,b_1,\ldots,a_L,b_L \in \mathbb R$. Let $\varphi$ be a piecewise function with $T \ge 2$ pieces(for example $T=2$ for the choice $\...
- 6,853
2
votes
0
answers
251
views
Volume of bounded regions in hyperplane arrangements
I am given a hyperplane arrangement $\mathcal{H}_0$ in $\mathbb{R}^n$ and a function $\phi \colon \mathbb{R}^n \to \mathbb{Q}.$ I choose any enumeration on the set of primitive vectors (i.e. vectors ...
- 357
2
votes
0
answers
917
views
Guessing game with guess cost
This is a question about Problem 328 on the website Project Euler. A description of the problem is provided in the previous link. I was wondering if there has been any research done on this question. ...
- 4,952
1
vote
0
answers
175
views
Solution of recurrence relation with summation
I have the following recurrence relation:
$$b(n,k)=\sum _{\text{i}=0}^{2 n-1} \left(b(n-1,k-\text{i})+\frac{\text{i} (2 n-\text{i}) \binom{2 n-1}{\text{i}} \binom{(n-2)^2}{k-\text{i}}}{2 n-1} \right)$$...
- 47
1
vote
0
answers
162
views
Triangular and pentagonal numbers in $q$-series
Consider the following two infinite series
$$\sum_{n\geq0}a(n)q^n=\prod_{k\geq1}\frac1{(1-q^k)^2(1-q^{5k})^2} \,\,\,\, \text{and} \,\,\,
\sum_{n\geq0}b(n)q^n=\prod_{k\geq1}\frac1{(1-q^k)^2(1-q^{7k})^2}...
- 43.2k
1
vote
0
answers
91
views
limsup of sequence
Let $\mathbb{Z}_{\geq 0}[|t|]$ be the ring of power series with non-negative integer coefficients and consider the power series
$$P(t) = \sum_{i=0}^ \infty a_i t^i \in \mathbb{Z}_{\geq 0}[|t|]$$
$$P^2(...
- 81
1
vote
0
answers
134
views
Number of solutions to a diophantine equation
Given a positive integer $n$, consider the diophantine equation $4x^2+y^2+4x+y=2n$ with solutions in non-negative integers $x$ and $y$.
Define the proportion
$$\delta_n=\frac{\#\{(x,y)\in\mathbb{Z}^2_{...
- 43.2k
1
vote
0
answers
79
views
Bounds on the inverse multivariate beta function
Can i find some constants $\mathbf a$ and $\mathbf b$ in $\mathbb{R}_+^d$ such that for all $\mathbf{x} \in \mathbb{R}_{+}^{d}$, the inverse beta function :
$$IB(\mathbf x) = \frac{\Gamma\left(\lvert \...
- 686
1
vote
0
answers
75
views
Generalization of Lagrange-Burmann to system of self-consistency equations
In my research, I have come across a system of probability generating functions of the following form:
$$H_1(x) = x A(H_1(x))B(H_2(x)) \text{,}$$
$$H_2(x) = x C(H_1(x))D(H_2(x)) \text{,}$$
and I am ...
1
vote
0
answers
75
views
Characterization of certain families of functions
For $R$ equal $\mathbb{R}$ or $\mathbb{Z}$, let $D^+_R:=\{(x,y)\in R^2\colon x<y\}$. For each natural $n$, let $F_{n,R}$ denote the set of all Borel-measurable functions $f\colon D^+_R\to\mathbb{R}$...
- 128k
0
votes
0
answers
63
views
Arrangements of fixed $k$-polyplets in a $n\times n$ matrix
Recently, I asked a question about the number of arrangements of $k$ elements inside a $n\times n$ matrix with certain restrictions. The one I´m actually interested in for this question is in its 2. ...
- 47
0
votes
0
answers
161
views
Asymptotic analysis of a sum of complex summands using integral
I'm trying to find the exact asymptotics of a sum:
$$A = \sum^n_{i=0} \begin{pmatrix} 2n \\ i \end{pmatrix} x^{i} y^{2n-i} $$
as $n\rightarrow\infty$. Here $x,y$ are complex numbers, $|x|\leq1, |y|\...
- 93
0
votes
0
answers
145
views
A question about the duality principle
Suppose $X$ and $Y$ are finite sets and $K:X\times Y\to \mathbb R$ is some function. We get an integral transform from the space of real functions on $X$ to real functions on $Y$ given by $$\Phi_Kf(y)=...
- 133