Questions tagged [combinatorial-identities]
The combinatorial-identities tag has no usage guidance.
164 questions
2
votes
2
answers
894
views
Proving generating functions equality
What do you use to prove the following equality (and possibly more general ones of the kind)?
\begin{align*}\sum_{r,s,t} \frac{q^{r^2+rs+s^2+st+t^2}}{(q)_r (q)_s (q)_t} z_1^{r+s} z_2^{s+t} = \sum_{a,...
- 22.5k
12
votes
3
answers
3k
views
Gauss sum (with sign) through algebra
Let $p$ be an odd prime, and $\zeta$ a primitive $p$-th root of unity over a field of characteristic $0$.
Let $G = \sum\limits_{j=0}^{p-1} \zeta^{j\left(j-1\right)/2}$ be the standard Gauss sum for $...
- 18.7k
5
votes
1
answer
585
views
Alternating sums of alternate Stirling numbers
Does anybody know of any identities or combinatorial interpretations for alternating sums of alternate Stirling numbers?
I am particularly interested in expressions of the form:
$$\pm\sum_{k}(-1)^k|...
- 85.6k
1
vote
2
answers
820
views
Converting a recursive definition to an explicit one
Is there an explicit form for $a_x$ (whole numbers x) given that $a_x = \displaystyle\sum_{i=1}^{x-1} \binom{x-1}{i} a_i$?
I've listed out the first few terms:
for $x=0,1,2,3,4,5,6, 7$
we have $a_x ...
CommunityBot
- 1
8
votes
1
answer
811
views
(0,1)-matrix congruence: is it known?
[[UPDATE: This work has now been published at SIAM J Discrete Math.: Formulae for the Alon–Tarsi Conjecture.]]
By equating two formulae (one congruence by Glynn (1) (which has just appeared) and one ...
- 4,229
5
votes
0
answers
345
views
When does a triangle of numbers have a zero row sum?
Suppose we have a triangle of numbers defined by the recurrence relation
$$\left| n \atop k \right| = f(n,k) \left| n-1 \atop k \right| +g(n,k) \left| n-1 \atop k-1 \right| + [n=k=0],$$
for some ...
CommunityBot
- 1
9
votes
2
answers
845
views
An identity involving an infinite integral with a sinh in the denominator
I recently encountered the rather appealing looking integral, which appears in the theory of random matrices :
$$\int_{-\infty}^{\infty}\prod_{j=1}^{p-1}(j^{2}+z^{2})\frac{zdz}{\mathrm{sinh}(2\pi z)} ...
- 79.9k
4
votes
0
answers
312
views
Generalization of Tamarkin’s ARO 1993, final round, problem 10/8: part II
Let us use the notations of my previous question about Tamarkin's problem.
Let $\ell\in\left\lbrace 0,1,...,p\right\rbrace$.
An element $f\in \mathbb Z^{\mathbb Z}$ is said to be $\ell$-average-...
CommunityBot
- 1
0
votes
1
answer
405
views
Finite sums with Binomial and Catalan inverses
In a recent failed-post about some partial sums with respect to the Central Binomial and Catalan number the formulas
$$\sum_{k=0}^n\frac{4^k}{B_k}=\frac{4^{n+1}(2n+1)}{3 B_{n+1}}+\frac{1}{3}$$
$$\sum_{...
- 345
32
votes
2
answers
2k
views
Generalization of Tamarkin's ARO 1993, final round, problem 10/8: still a conjecture?
This is from the category "problems I cannot believe that are still open". But then again, I don't know whether it is still open; it seems to have escaped the attention of most number theorists and ...
- 8,688
3
votes
2
answers
696
views
Open problems and known identities involving sums
As many people here, I know of a few identities involving expressions of the type $\sum_{i}\ f(i)$, with "arbitrarily complicated $f(\cdot)$", as well as closed formulas in some cases.
I also know ...
- 2,204
2
votes
1
answer
179
views
Inverse formula for counting marginals
I am interested in a formula which relating two functions over a multiset.
I have a multiset $X$ of sets where each element in $X$ is a set $x \subseteq \{1,2,\ldots,m\}$. Now I have two ``count'' ...
- 2,442
19
votes
2
answers
2k
views
What role does Cauchy's determinant identity play in combinatorics?
When studying representation theory, special functions or various other topics one is very likely to encounter the following identity at some point:
$$\det \left(\frac{1}{x _i+y _j}\right) _{1\le i,j \...
- 50.8k
18
votes
3
answers
2k
views
A binomial sum is divisible by p^2
This is a question I have since longer time, but I have absolutely no idea how to proceed on it.
Let $p>3$ be a prime. Prove that $\displaystyle\sum\limits_{k=1}^{p-1}\frac{1}{k}\binom{2k}{k}\...
CommunityBot
- 1