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Question 1

0votes

0answers

19views

Upper bound on the intersection of two radius-r Hamming balls in binary constant-weight codes when codewords are at Hamming distance 2

Let $\mathcal{B}(n,w)$ be the set of all binary vectors of length $n$ and constant weight $w$, i.e.,$\mathcal{B}(n,w) = \{ x \in \{0,1\}^n : \mathrm{wt}(x) = w \}$.The Hamming ball of radius $r$ (in ...

zchan

1

asked14 hours ago

Question 2

6votes

1answer

196views

Expected number of upsets in a knockout tournament

$2^n$ players $P_1, \dots, P_{2^n}$, ordered in decreasing order of skill are placed uniformly at random at the leaves of a binary tree of depth $n$.They play a knockout tournament according to the ...

Nate River

9,840

asked16 hours ago

Question 3

1vote

1answer

150views

Directly deducing a counting formula for plane partitions in a box from another

Please show (using arithmetic) that, for $r,s,t\in\mathbb N$,$$\prod_{i=1}^r\prod_{j=1}^s\prod_{k=1}^t\,\frac{i+j+k-1}{i+j+k-2}$$equals$$\prod_{i=1}^r\,\frac{\binom{s+t+i-1}{s}}{\binom{s+i-1}{s}}...

Tri

1,917

askedNov 26 at 3:31

Question 4

0votes

0answers

71views

Gibbs distribution via rigorous counting? [closed]

Consider the function $Dist: \mathbb{N} \times \mathbb{N} \to \mathbb{N}$ (natural numbers include $0$), by defining $Dist(E,N)$ to be the size of the set$$ \{ (s_1, s_2, \ldots, s_N ) \in \mathbb{...

Student

5,748

askedNov 25 at 23:39

Question 5

8votes

1answer

334views

Coefficient comparison between a double sum and a single sum

For a formal Laurent series $F(q)$, denote its coefficient of $q^j$ by $[q^j](F)$.QUESTION. For integers $r\geq1$, is this true?$$[q^{2r}]\sum_{n\geq1}\frac{q^n}{1-q^{2n}}\sum_{k=1}^n\frac{q^k}{1+q^...

T. Amdeberhan

43.9k

askedNov 25 at 17:27

Question 6

10votes

0answers

156views

Identities of Bernoulli numbers from rank 2 simple Lie algebras

Denote $B_n$ as Bernoulli numbers, it is known that the following three identities hold (for $n\in \mathbb{N}$):$$\tag{A}\label{504268_A}\frac{(2n)!}{(4n+1)!} \frac{-B_{6n+2}}{6n+2}= \sum_{k=0}^{2n} \...

pisco

1,013

askedNov 25 at 11:32

Question 7

2votes

0answers

146views

Simplicity for exterior powers of $\mathfrak{g}$-modules

Let $\mathfrak{g}$ be a complex semisimple Lie algebra. Denote by $V_{\pi_m}$ the $m$-th fundamental representation of $\mathfrak{g}$. When is it true that the $k$-th exterior power of $\Lambda^k(V_{\...

Mili Fishta

83

askedNov 23 at 11:23

Question 8

1vote

0answers

61views

What are some consequences of Graph eigenvectors delocalization?

Given a $d$-regular graph $G$ on $n$ vertices, suppose that every normalizedeigenvector $v$ of the adjacency matrix $A$ satisfies a strongdelocalization bound$$ \|v\|_\infty \;\ll\; \frac{\log n}{...

Alexander

387

askedNov 22 at 23:56

Question 9

0votes

0answers

288views

Apéry series for $\zeta(3)$

I am still now stumped on deriving the series equivalence$$\zeta(3)=\frac{5}{2}\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n^3\binom{2n}{n}}$$Like even I did not get the series for $\frac1{n^2}$ mentioned ...

vidyarthi

2,135

askedNov 22 at 18:14

Question 10

4votes

1answer

355views

Ramanujan's work on the central factorial numbers

BackgroundThe central factorial numbers are described on OEIS sequence A008955. Among the references, "Ramanujan's notebooks, part 1" (edited by Bruce Berndt) is listed. Upon checking this ...

Max Lonysa Muller

5,065

askedNov 21 at 13:21

Question 11

12votes

2answers

688views

Algorithm for selecting a fixed-point free permutation $\varphi:\{1,\ldots,n\}\to\{1,\ldots,n\}$

Motivation. In my younger son's class, everyone has to give a (small) Christmas present to one other student. Let $n\in\mathbb{N}$ be the number of students in the class. If you pick a permutation $\...

Dominic van der Zypen

54k

askedNov 19 at 13:25

Question 12

5votes

0answers

330views

Cases $n=7,8$ of a conjecture on semigroups

[Crossposted at math.stackexchange and AoPS].I would like to prove cases $n=7,8$ of this conjecture (general question asked here): given any commutative semigroup $S$ of order $n \ge 1$, there exist $...

Fabius Wiesner

1,166

askedNov 19 at 10:21

Question 13

4votes

1answer

103views

Ehrhart-Macdonald reciprocity for polytopal complex

Given a lattice polytopal complex $P$, one can still define the Ehrhart polynomial to be: scale up the polytopal complex and count the number of lattice points. When one plugs in a negative number ...

cacha

731

askedNov 19 at 2:11

Question 14

2votes

1answer

168views

Lower bounds for difference sets of finite integer sets

Let $S$ be a finite subset of integer. Let $\{p \leq X\}$ be the set of primes bounded by $X$. Is it true that the set $S-S$ has a subset $A$ of positive density such that$p \mid a$ for all prime $p \...

NumDio

317

askedNov 17 at 16:58

Question 15

7votes

1answer

343views

On the class forcing used in Jensen's proof of Con(CH+SH)

It seems that Jensen's proof of the consistency of CH + SH used class forcing, but the revelant properties are not clearly verified. I haven't learnt about class forcing, so I wonder whether it is ...

Ning Gan

73

askedNov 17 at 7:08

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Questions tagged [co.combinatorics]

Upper bound on the intersection of two radius-r Hamming balls in binary constant-weight codes when codewords are at Hamming distance 2

Expected number of upsets in a knockout tournament

Directly deducing a counting formula for plane partitions in a box from another

Gibbs distribution via rigorous counting? [closed]

Coefficient comparison between a double sum and a single sum

Identities of Bernoulli numbers from rank 2 simple Lie algebras

Simplicity for exterior powers of $\mathfrak{g}$-modules

What are some consequences of Graph eigenvectors delocalization?

Apéry series for $\zeta(3)$

Ramanujan's work on the central factorial numbers

Algorithm for selecting a fixed-point free permutation $\varphi:\{1,\ldots,n\}\to\{1,\ldots,n\}$

Cases $n=7,8$ of a conjecture on semigroups

Ehrhart-Macdonald reciprocity for polytopal complex

Lower bounds for difference sets of finite integer sets

On the class forcing used in Jensen's proof of Con(CH+SH)

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