**2**

votes

**1**answer

60 views

### Coefficients for Powers of the Mittag-Leffler Function

Considering the one parameter Mittag-Leffler function,
$$E_{\alpha}(z)=\sum_{k=0}^\infty\frac{z^{k}}{\Gamma(\alpha k+1)}, \Re(\alpha)>0$$
Considering then the generating function for $E_\alpha(z^...

**1**

vote

**1**answer

91 views

### Counting tournaments with ties

An improper tournament, or tournament with ties, is a graph in which every pair of nodes is connected by a single uniquely directed edge or by a single undirected edge.
There are 1, 2, and 7 improper ...

**4**

votes

**0**answers

38 views

### Number of classes $\pmod p$ represented by $b_1s^{n-1} + \dots + b_n$ where $ord_p(s) = n$

Let $n \in \mathbb Z$ with $n \ge 3$ and let $p$ be a prime number such that $n|p-1$. Let $a_1,a_2,\dots,a_{2n-1} \in \mathbb Z/p\mathbb Z$. Suppose that the same class is represented by at most $n-1$ ...

**5**

votes

**2**answers

129 views

### Do character tables determine association schemes up to isomorphism?

I am interested in commutative, but not-necessarily-symmetric association schemes. I noticed that the conjugacy class scheme of the dihedral group of order 8 has the same character table as the one ...

**7**

votes

**3**answers

431 views

### Homotopy type of some lattices with top and bottom removed

The only reaction to this question on math.SE was 21 views in 4 days, so I decided to repost it here. I am not changing anything.
There was an interesting question on MO which OP removed by some ...

**8**

votes

**2**answers

176 views

### An identity related to partitions into $n$ parts and Schur polynomials

While working with Schur polynomials I found what seems like a nice identity, and I wonder if it has a simple proof.
Notation: Suppose $d,n\in\mathbb{N}$, and $\lambda =(\lambda_1,\dots,\lambda_n)$ ...

**3**

votes

**2**answers

143 views

### Combinatorial identity involving number of cycles (of any length) in a permutation

I am going through Phil Hanlon's paper and on page 127, right after the first paragraph, "It is well known that.."
which boils down to the following identity:
$$
\prod_{i=0}^{n-1}(\beta-i) = \sum_{\...

**6**

votes

**1**answer

247 views

### The current status of the conjecture on algebraic matroids

Can anyone point out some articles for the conjecture: the dual of an algebraic matroid is algebraic?
Thank you!

**0**

votes

**0**answers

93 views

### Sum of unit vectors always has a binary span after constrained permutations

Conjecture:
Let $e_1 = (1,0,\ldots,0), \ldots , e_{m_1+m_2} = (0,\ldots,0,1)$ be the unit vectors of the standard basis $E$ of $\mathbb{R}^{m_1+m_2}$.
An enumeration $ E \cup -E = \{f_1, \ldots, ...

**9**

votes

**0**answers

198 views

### A Combinatorial Game: the Snake and the Hunter

The Snake and the Hunter is a game for two players who play in two rounds interchanging the roles of snake and hunter. The game is played in a rectangular grid of points, say $6 \times 6$. In both ...

**-1**

votes

**1**answer

135 views

### number of partitions from 0 to n^2 [on hold]

You are given the numbers 0 to n^2. You must use n numbers with no number greater than n to form all the partitions of the numbers 0 to n^2. For example with n=4 you want to find the partitions of 7:...

**0**

votes

**1**answer

50 views

### Count Functional digraph [on hold]

Given a set of nodes, how can I count the number of different functional digraph containing a specific number of connected components? With a restriction that no node can have an edge point to itself. ...

**-2**

votes

**0**answers

23 views

### weighted restricted integer compositions and extended binomial coefficients [closed]

proof of
d_{S,f}(n,k) = \binom{k}{n}{(f(s)){s\in S}}

**2**

votes

**0**answers

47 views

### Average range of Motzkin path

Motzkin path are paths from (0,0) to (n,0) in $\mathbb{Z}^2$ such that we are allowed to move SE, E and NE.
More on this is here https://en.wikipedia.org/wiki/Motzkin_number
I would like to know if ...

**6**

votes

**1**answer

131 views

### q-analog of a combinatorial identity involving binomial coefficients

Using, e.g., properties of iterated finite differences it is easy to show that for any pair of integers $n$ and $m$ with $n>\!>m$ one has the identity
$$
\sum_{k=0}^m(-1)^{k-m} {n-k\choose m}{m\...

**3**

votes

**2**answers

183 views

### Examples of Sets with Positive Upper Density

While reading the statement of Roth's theorem I started asking myself what are examples of sets of positive upper density? It's not hard to come up with a few:
Flip a coin with probability $\mathbb{...

**7**

votes

**1**answer

132 views

### On a result of Frankl and Wilson

In the paper 'Intersection theorems with geometric consequences' (Combinatorica 1981) P. Frankl and R. M. Wilson consider families $\mathcal{F}$ of $k$-subsets of $\{1,\dots,n\}$ with the restriction ...

**1**

vote

**1**answer

132 views

### Rank 2 complex vector bundles over $S^2\times S^2$

How could people classify all rank $2$ complex vector bundles over $S^2\times S^2$ up to isomorphism?
Could you give a rank 2 complex vector bundle which cannot be split as a sum of two line bundles?

**4**

votes

**0**answers

95 views

### A color interpolation lemma

I need the following "color interpolation lemma". Actually I know a way to prove it, but I'm not very satisfied with that proof.
Lemma. Let $G=(V,E)$ be a (properly) colored graph with colors $1, \...

**5**

votes

**0**answers

97 views

### For which Ramsey type results density versions are wrong?

I look for examples of Ramsey-type statements, for which the density counterparts do not hold.
Example: usual Ramsey theorem. If all edges of a complete graph $K_n$ are colored in $c$ colors, there ...

**3**

votes

**0**answers

100 views

### A question about smooth convex lattice polygons

Let $P$ be a smooth convex lattice polygon in $\mathbb{R}^2$ (the lattice being $\mathbb{Z}^2$). Here smooth means that at any vertex of $P$, the two primitive integer vectors (i.e. vectors whose ...

**3**

votes

**0**answers

88 views

### The number of monotone Boolean functions

In the paper "The number of monotone Boolean functions" A. D. Korshunov calculates an asymptotic number of the number of monotone Boolean functions
(see https://en.wikipedia.org/wiki/Dedekind_number)...

**3**

votes

**0**answers

65 views

### Minimal algebraic degree of symmetric unit distance embedding of Heawood graph

I'm looking at embeddings of the Heawood graph in the plane as unit distance graph. Apparently the first such embedding was given by Gerbracht, 2009 and has algebraic (over the rationals) coordinates ...

**4**

votes

**0**answers

76 views

### Random polyominoes containing $2\times2$ squares

The construction quoted in the question "How to sample a uniform random polyomino?" claims to produce a "uniform random polyomino". But apart from the mentioned possibility of getting stuck, it also ...

**0**

votes

**1**answer

35 views

### What is the shatter coefficient / VC - dimension of some hypothesis set?

Let $H:=\{h:\mathbb{N}_0^n \rightarrow \{0,1\}| h(x_1,\cdots,x_n) = \mathbb{1}_0(\sum_{i\in I}{x_i}-\sum_{j \notin I}{x_j}) \text{ for some } I \subset \{1,\cdots,n\}\}$
where $\mathbb{1}$ is the ...

**0**

votes

**1**answer

63 views

### On some examples of critical families

I'm reading the book on Injective choice functions by Holz, Podewski and Steffens, and I find it to be at the same time well written and quite difficult. It has almost no examples - and in quite a few ...

**10**

votes

**3**answers

292 views

### How to sample a uniform random polyomino?

A polyomino is formed by joining finitely many unit squares edge to edge. It may be regarded as a finite subset of the regular square tiling with a connected interior. In particular, for us, ...

**0**

votes

**0**answers

71 views

### Variation on stones in buckets

This is a spinoff, see Collecting stones in n buckets. Frankly speaking my only motivation is that I became curious: what happens if one redistributes the stones into the same buckets?
More ...

**0**

votes

**0**answers

44 views

### How many different solutions does this cube puzzle have?

I designed a 4x4x4 soma cube in AutoCad and then built it with wood cubes.
Now I want to know how many different solutions there are for it.
Similar to the Bedlam Cube, there are twelve pentacube and ...

**0**

votes

**1**answer

260 views

### How many subsets I of $\{1,\cdots,n\}$ exist?

How many subsets $I$ of $S:=\{1,\cdots,n\}$ exist such that
$\sum_{i \in I} x_i \neq \sum_{j \in S-I} x_j$
for all $0 < x_1 < \cdots < x_n$?
Let $b_n$ be this number. Then we are ...

**3**

votes

**0**answers

193 views

### On weight enumerators of codes

Are there $[n,k]_q$ constant rate $\frac kn$ and constant alphabet linear code families with automorphism group of size $\Omega((n-n^\beta)!)$ that have minimum distance $d=O(n^\alpha)$ and number of ...

**4**

votes

**0**answers

111 views

### Combinatorial fairness property in division of goods

Given $n$ agents, and $m$ items where $v_i(g) \geq 0$ is the value of item $g$ for agent $i$, does there always exist a partition $A_1, ..., A_n$ of the $m$ items into $n$ sets s.t. for all $i, j \in \...

**3**

votes

**1**answer

134 views

### A Combinatorial Identity Involving Characters of $S_n$ (Reference Request?)

It is a well-known exercise that $C_n = \chi_{(n,n)}(1)=\chi_{(n,n)}^{1^n}$ where $C_n$ is the $n$th Catalan number and $\chi_{(n,n)}^{1^n}$ is the character of the irrep $(n,n)$ on conjugacy class $1^...

**1**

vote

**0**answers

114 views

### Singular Locus of a Schubert variety

I am trying to compute the singular locus of the schubert variety $X_w$ in $G_{2,7}$ where $w=(4,7) \in I_{2,7}$. Following the notation in the book "The Grassmannian Variety: Geometric and ...

**6**

votes

**2**answers

232 views

### Seeking very regular $\mathbb Q$-acyclic complexes

This question was raised from a project with Nati Linial and Yuval Peled
We are seeking a $3$-dimensional simplicial complex $K$ on $12$ vertices with the following properties
a) $K$ has a complete $...

**0**

votes

**0**answers

47 views

### Solving Recursive Expression for Counting Unrooted Tree Topologies

The book Bayesian Evolutionary Analysis with BEAST by Alexei J. Drummond et al. (2015) states at section 2.2.1 that the number of rooted unlabelled (binary) tree topologies $a_n$ is given by the ...

**2**

votes

**0**answers

79 views

### Bounding Schur polynomials of a particular shape

Consider Schur polynomials $s_\lambda$ with $\lambda = (2m, m, m, \ldots, m, 0)$ and $\ell(\lambda) = n$ (that is, $\lambda$ has $n$ rows). Here $m \gg n$, which, for the sake of concreteness, let's ...

**2**

votes

**0**answers

57 views

### Induced matchings in a bipartite graph with every matching having the same number of edges

Suppose $n,k$ are positive integers such that $k\mid n$.
Consider a bipartite graph $H=(A,B,E)$ such that $|A|=|B|=n$ and the edge set $E$ consists of the union of $m(H)$ induced matchings with every ...

**0**

votes

**0**answers

118 views

### Counting integer solutions for a system of (in)equalities [migrated]

I wish to enumerate the number of solutions of the system of equations and inequalities for 3 non-negative integer unknowns $x,y,z \ge 0$: ($a$,$b$ specified)
\begin{align}
x+y+z&=a\\
x+y&>...

**1**

vote

**1**answer

206 views

### Approximation of sets

Is the following true? For every $\varepsilon>0$ there is a finite
subset $W$ of $\mathbb{N}\times \mathbb{N}\times \mathbb{N}$, such that
$$|p_1(W)\cap p_2(W)\cap \{p_1(x)+p_2(x):x\in W\}\cap \{...

**2**

votes

**2**answers

260 views

### Convergence issues with infinite product of formal series

Question first:
Show that if $s_1 < s_2 < \dots$ is an increasing sequence of positive integers and $P(x)$ is a nonzero polynomial then we cannot have
$$ P(x) \equiv \prod_{j=1}^\infty (1 -...

**24**

votes

**1**answer

526 views

### Inequality for hook numbers in Young diagrams

Consider a Young diagram $\lambda = (\lambda_1,\ldots,\lambda_\ell)$. For a square $(i,j) \in \lambda$, define hook numbers $h_{ij} = \lambda_i + \lambda_j' -i - j +1$ and complementary hook numbers $...

**-1**

votes

**1**answer

40 views

### Size of smallest set in critical covering of $\omega$

A covering of a non-empty set $X$ is a collection ${\cal U} \subseteq ({\cal P}(X)\setminus\{\emptyset\})$ such that $\bigcup {\cal U} = X$. If ${\cal U}$ is a covering of $X$ then a function $f:{\cal ...

**7**

votes

**1**answer

148 views

### Does an expander remain an expander after removing few vertices and edges?

Consider a sequence of expander graphs ($G_n$); say $G_n$ has $n$ vertices.
Remove $o(n)$ vertices (and the edges emanating from these vertices) and cut $o(n)$ edges. Call $G'_n$ the largest connected ...

**2**

votes

**0**answers

192 views

### Average minimum number of random k-sparse vectors in $\mathbb{F}_2^n$ to span a specific base vector?

A while back I posted a question in MO about the average minimum number of independent random k-sparse (having at most $k$ non-zero elements) vectors belonging to $\mathbb{F}_2^n$ to span the whole ...

**8**

votes

**1**answer

148 views

### How many edges can be added to two circles before the graph becomes Hamiltonian?

Start with two $n$-circles $(v_1\cdots v_n)$ and $(w_1\cdots w_n)$ of vertice sets $V$ and $W$, where $n\ge 5$. Add a number of vertex-disjoint edges between $V$ and $W$ (thus no chords) in a way ...

**4**

votes

**1**answer

383 views

### Three dimensional representations of Alternating group

The alternating group $A_5$ has $2$ irreducible representation of degree $3$. The characters for these representations have irrational values. I guess the ring of invariants of these representations ...

**1**

vote

**1**answer

184 views

### An extremal problem on matrices

Is it possible to determine (or give bounds for) the following extremal problem:
Let $k,m,r$ be positive integers such that $k,m \geq r$. What is the least number $n$ such that for any $r \times n$ ...

**3**

votes

**1**answer

83 views

### Probability of collision of some family of hash functions

Given $x$ and $y$ in $\mathbb{R}$, and let $\mathcal{H} = \{ h \mid \mathbb{R} \to \mathbb{N} \}$ be a family of hash functions where $ h(x) = \left\lfloor x + \sum^C_{i=1} U_i \right\rfloor$ for some ...

**0**

votes

**1**answer

84 views

### Critical coverings of $\omega$

A covering of a non-empty set $X$ is a collection ${\cal U} \subseteq ({\cal P}(X)\setminus\{\emptyset\})$ such that $\bigcup {\cal U} = X$. If ${\cal U}$ is a covering of $X$ then a function $f:{\cal ...