All Questions
Tagged with combinatorics or co.combinatorics
3,202 questions with no upvoted or accepted answers
13
votes
0
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1k
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Generalization of Cauchy's identity
Let $ s_{\lambda} $ be the Schur function associated to the partition $ \lambda $.
Cauchy's identity (as in Macdonald) states that
$$
\sum_{\lambda} s_{\lambda}(X)s_{\lambda}(Y) = \prod_{i,j}(1-...
- 151
13
votes
0
answers
385
views
Are the zeros of Tutte polynomials dense in $\mathbb C^2$?
For the chromatic polynomials of graphs we have two nice theorems which describe the behavior of their zeros: Thomassen proved that the set of real zeros of all chromatic polynomials is the union of $\...
- 85.6k
13
votes
0
answers
751
views
$\epsilon$-nets with respect to the cut norm
The cut norm $||A||\_C$ of a real matrix $A = (a_{i,j}) \in \mathcal{R}^{n\times n}$ is the maximum over all $I \subseteq [n], J \subseteq [n]$ of the quantity $\left|\sum_{i \in I, j \in J}a_{i,j}\...
- 794
13
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0
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713
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Regular languages of matrices and their generating functions
My question is somewhat related to this question.
Let us fix natural numbers $k$ and $C$. Let $A$ be an automaton whose alphabet consists of $k\times k$ matrices with integer coefficients of ...
- 3,897
13
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0
answers
1k
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Finding a chromatic polynomial by polynomial fitting
I would like to find the chromatic polynomial χ for the n by m rook's graph Gn,m for as many values of n and m possible. The rooks graph is also (a) the line graph of the complete bipartite graph ...
- 4,229
12
votes
0
answers
530
views
Finding the diameter of an unknown tree: Is BFS optimal?
I'm interested on the following nice problem that is somewhat standard in CS, but I was surprised on the lack of references on the optimal algorithm to this problem.
Ana and Banana plays the ...
- 63
12
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0
answers
168
views
Can the optimal packing density in $\mathbb{Z}^d$ be irrational?
For a finite $S \subset \mathbb{Z}^d$, let $d_p(S)$ be its optimal packing density. That is, the maximal lower asymptotic density of $A+S$, where $A \subset \mathbb{Z}^d$ is such that $(a_1+S)\cap (...
- 1,209
12
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0
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629
views
$q$-analogue of the multinomial theorem?
The $q$-binomial theorem states that
$$
\prod_{k=0}^{n-1}(1+q^kt) = \sum_{k=0}^n q^{\binom k2}{n\brack k}_q t^k.
$$
This identity is a $q$-analogue of the binomial theorem
$$
(1+t)^n = \sum_{k=0}^n \...
- 5,654
12
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0
answers
267
views
Computing the number of ways to delete vertices sequentially without disconnecting a graph
Given a finite connected graph on $n$ vertices, we are trying to count the number of ways to label the vertices $1$ to $n$ so that deleting them sequentially in that order never disconnects the graph. ...
- 10.1k
12
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0
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229
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3-manifolds with stacked links
Stacked spheres
A triangulation of a 2-dimensional sphere is called a stacked sphere if it is obtained inductively from the boundary of a 3-simplex by deleting a 2-face (triangle) $T$ adding a new ...
- 24.7k
12
votes
0
answers
469
views
Around the Erdős-Ginzburg-Ziv theorem
(Here is a problem that emerged in a conversation with Fedor Petrov and should really be a sort of "joint posting" if this format were supported.)
For any positive integers $k_1\le k_2\le\...
- 23k
12
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0
answers
246
views
$\mathfrak{sl}_3$ webs without faces having a multiple of 4 sides
In settling the main conjecture of Cyclic action on Kreweras walks, see https://arxiv.org/abs/2005.14031, a rather interesting object popped up.
Recall from
Kuperberg, Greg, Spiders for
rank 2 Lie ...
- 5,822
12
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0
answers
321
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Combinatorial proof of invertibility of a symmetric matrix associated to the ring of matrices over a finite field
Let $F$ be a finite field of $q$ elements with characteristic $p$. Let $M_n(F)$ be the ring of $n\times n$ matrices over $F$. We define a $q^{n^2}\times q^{n^2}$ symmetric matrix $L$ over the ...
- 38.6k
12
votes
0
answers
104
views
How many upper sets in this decomposition of finite posets
Let $X$ be a finite poset.
If
$$X = X_1 \cup X_2$$
where $X_1$ and $X_2$ are strict upper sets, then a lot of properties of $X$ can be inferred from the smaller posets $X_1, X_2$ and $X_1\cap X_2$ (...
- 221
12
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0
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191
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Non-Boolean Eulerian interval of finite groups
An Eulerian subgroup lattice is Boolean (see here), so it is natural to wonder whether it is also true for an interval of finite groups. The smallest non-Boolean Eulerian lattice is the following:
It ...
12
votes
0
answers
330
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The number of labeled pairs of edge disjoint trees and related questions
I wonder what is known on the following:
1) What is the number $T_k(n)$ of $k$-tuples of (pairwise) edge-disjoint trees $(T_1,T_2,\dots, T_k)$ with $n$ labelled vertices?
2) (harder, it seems) What ...
- 24.7k
12
votes
0
answers
643
views
Wilf's conjecture: complementary Bell numbers
The complementary Bell numbers or Uppuluri–Carpenter numbers, denoted $\tilde{B}_n$, can be delivered by
$$G(x):=\sum_{n\geq0}\tilde{B}_n\frac{x^n}{n!}=e^{1-e^x}.$$
Definition. Fix an integer $m\geq0$....
- 43.2k
12
votes
0
answers
513
views
Converse of Frobenius
Enumerate the elements of a finite group $G$ as follows: $g_1,g_2,\dots,g_n$. Introduce $n$ variables indexed by the elements of $G$: $x_{g_1},\dots,x_{g_n}$.
Consider the matrix $X_G$ with entries $...
- 43.2k
12
votes
0
answers
270
views
Number of updown sequences of $1,1,2,2,\cdots,n,n$
I would like to count the updown sequences of the set $\{1,1,2,2, \cdots, n,n \}.$
Sequence $a_1, a_2, a_3, \ldots$ is an updown sequence if the sequence satisfies the following: $ a_1 \lt a_2 \gt a_3 ...
- 121
12
votes
0
answers
570
views
Computer searches for the $g$-conjecture
McMullen's $g$-conjecture aims the classify possible $f$-vectors of simplicial $d$-spheres. The $g$-conjecture has been proven for polytopal spheres and for simplicial spheres of dimension $d < 5$. ...
- 7,875
12
votes
0
answers
219
views
Do the Laver tables converge to the Sierpinski triangle with a line segment sticking out in the hyperspace topology?
Let $(\{1,...,2^{n}\},*_{n})$ denote the $n$-th Laver table.
Let
$$C_{n}=\{(\frac{x}{2^{n}},\frac{x*_{n}y}{2^{n}})|x,y\in\{1,2,3,...,2^{n}\}\}$$
for all $n\in\mathbb{N}$.
Then since $C_{n}$ is a ...
12
votes
0
answers
731
views
Product of a Schubert polynomial and a double Schubert polynomial
Let $S_u(x)$ be a Schubert polynomial and let $S_v(x;y)$ be a double Schubert polynomial. Then their product can be expressed in terms of the double Schubert polynomials as
$$S_u(x)S_v(x;y)=\sum_w{c_{...
- 2,168
12
votes
0
answers
224
views
Is there a nice formula for the "non-crossing substitution" of linear combinatorial species?
Background
A linear species is a functor
$$F : \mathrm{Lin} \to \mathrm{FinSet},$$
where $\mathrm{Lin}$ is the category of totally ordered sets and bijections and $\mathrm{FinSet}$ is the category ...
- 221
12
votes
0
answers
359
views
Consequences of Zeeman's conjecture
Recall the Zeeman's conjecture: if $K$ is a contractible polyhedron of dimension 2, then $K\times I$ has a collapsible subdivision.
Zeeman showed that this implies the Poincaré conjecture in ...
- 2,104
12
votes
0
answers
502
views
A $q$-analogue of Foulkes' character related to alternating permutations
My paper "Alternating permutations and symmetric functions" at
http://math.mit.edu/~rstan/papers/altenum.pdf enumerates certain
classes of alternating permutations, such as those whose inverse is
...
- 50.8k
12
votes
0
answers
450
views
Colouring a graph whose edge set is a special union of cliques
I am trying to show that a certain family of graphs can always be properly coloured with at most $6$ colours (where "properly coloured" means that each vertex gets a colour and no edge has both ends ...
- 12.7k
12
votes
0
answers
258
views
Proving that the set of $\lfloor n/3 \rfloor+1$ partial Latin squares given by Pebody is unavoidable?
Introduction
Cutler and Öhman (2006) attribute to Pebody (via personal communication) a construction of a set of $k:=\lfloor n/3 \rfloor+1$ partial Latin squares which are unavoidable (i.e., any ...
- 4,229
12
votes
0
answers
558
views
Possible orders of products of 2 involutions which interchange disjoint residue classes of the integers
Definition / Question
Definition: Let $r(m)$ denote the residue class $r+m\mathbb{Z}$, where
$0 \leq r < m$.
Given disjoint residue classes $r_1(m_1)$ and $r_2(m_2)$, let the class transposition
$...
- 19.6k
12
votes
1
answer
2k
views
Hobbled rook tour – Hamiltonian cycle on square grid
Consider a square grid of even side length ($2n \times 2n$). It is easy to see that there must exist a Hamiltonian cycle on the corresponding grid graph. Such a cycle is called balanced if the number ...
- 121
12
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0
answers
599
views
Once differentiable, piecewise degree three polynomials on triangulated planar domains
Here is an easily described, but very difficult, problem that I
(and a number of other people) really would like to see solved during
our life times. The basic problem is to compute the dimension of ...
- 121
12
votes
0
answers
910
views
Hamiltonian cycles and fundamental groups
I'm interested in the interplay between the Hamiltonian cycles of graphs and the compact surfaces they embed in. I was doing some reading on the Lovász conjecture for Cayley graphs, I started noticing ...
- 85.6k
12
votes
0
answers
871
views
Sums of Partitions and Stirling's formula
Stirling's formula $$N! \sim \sqrt{2 \pi}\ N^{N+ \frac{1}{2}} e^{-N}$$ follows easily from Laplace's method in light of the famous integral representation $$N! = \int_0^{\infty} e^{-z} z^N dz.$$ ...
- 952
12
votes
0
answers
456
views
Enumeration of Standard Young Tableau of bounded height
First for some notation
$$ l(\lambda) = \text{ number of parts in a partition } \lambda \vdash n$$
$$ f_{\lambda} = \text{number of standard Young tableau of shape } \lambda\vdash n$$
The number $f_{...
- 771
12
votes
0
answers
529
views
A commutative monoid associated with a finite abelian group
Let $M$ be a finite abelian group, and denote by $e_m$, for $m \in M$, the canonical basis of $\mathbb{Z}^M$. For $m, n \in M$ define elements $v_{m,n} \in \mathbb{Z}^M/\langle e_0\rangle$ as
$$
v_{m,...
- 181
12
votes
0
answers
349
views
Matroids with prescribed independent sets
Let $A$ be a finite set. Let $B$ be a family of subsets of $A$. We are interested in a matroid with a minimum rank such that every element of $B$ is independent. The answer is obvious - a uniform ...
- 1,791
11
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0
answers
492
views
Connection properties of a single stone on an infinite Hex board
This includes a series of questions.
One of the most typical examples is shown as the picture below.
An half-infinite Hex board with an one row of black stones. Black stones are separated by one ...
- 631
11
votes
0
answers
290
views
Color your partitions by parity
Let $a_c(n)$ be the number of ways to partition a positive integer $n$ where each even part comes in $c$ colors. Then, we can supply the generating function
$$\sum_{n\geq0}a_c(n)q^n=\prod_{k\geq1}\...
- 43.2k
11
votes
0
answers
488
views
Are there 100 points that are part of every half-density part of the plane?
Is there a configuration $P$ that consists of 100 points of the plane such that every $X\subset\mathbb R^2$ whose density is half contains an isometric copy of $P$?
I am deliberately being vague ...
- 18.7k
11
votes
0
answers
387
views
Inequality for symmetric polynomial functions of log concave variables
Let $(x_i)_{i \ge 1}$ be a log-concave (resp. log-convex) sequence of non-negative real variables. In other words, for $i \ge 2$, we have $x_i^2 \ge x_{i-1}x_{i+1}$ (resp. $x_i^2 \le x_{i-1}x_{i+1}$).
...
- 505
11
votes
0
answers
195
views
Number of triangle-free graphs with prescribed number of edges
This question is posted from StackExchange since it received no answer there.
Let $f(n, e)$ be the number of triangle-free graphs on $n$ vertices and $e$ edges. From empirical evidence, I am motivated ...
- 384
11
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0
answers
541
views
How to determine the sign for the sum over all simple paths in the graph
$\DeclareMathOperator\perm{perm}\DeclareMathOperator\len{len}$Let $A$ be the adjacency matrix of a tree $T$ for some ordering $v_1,...,v_n$ of the vertices, and let $D=xI-A$ its characteristic ...
- 1,362
11
votes
0
answers
285
views
Does every finite poset have a rigid endomorphism?
Crossposted on Mathematics.
In this post, an order-preserving self-map of a poset $X$ will be called an endomorphism of $X$, and such an endomorphism $f$ will be called rigid if the only automorphism ...
- 4,416
11
votes
0
answers
357
views
Recognizing reflection subgroups of Coxeter groups
Given a Coxeter system $(W,S)$ with reflections $T$, and any subset $A \subseteq T$, it is known that the reflection subgroup $W_A$ generated by $A$ has a canonical choice $S_A$ of generators so that $...
- 2,743
11
votes
0
answers
161
views
Bijections $\mathbb{Z}\times\mathbb{Z}\to\mathbb{Z}$ with vanishing local means
This is just a summer-time curiosity arisen after a recent question by Dominic van der Zypen.
For a finite subset $S$ of $\mathbb{Z}\times\mathbb{Z}$ and a function $f$ on $\mathbb{Z}\times\mathbb{...
- 60.5k
11
votes
0
answers
334
views
Conjectural nonvanishing of some combinatorial sums (6j symbols)
From various considerations and with the help of J. Van der Jeugt, I was led to conjecture the following property of a class of Wigner 6j-symbols:
for any integers $k,m$ with $m\ge k\ge 2$,
$$
\left\{...
11
votes
0
answers
186
views
Iterated automorphism groups of finite groups
Let $\mathcal{G}$ be the set of isomorphism classes of finite groups.
There is an operation $\mathrm{Aut} : \mathcal{G} \rightarrow \mathcal{G}$ which gives the automorphism group of a given group, ...
- 12.4k
11
votes
0
answers
406
views
Relation between a continued fraction and partitions
I am interested in the continued fraction
$$\sum\limits_k {{z^{{2^k} - 1}}} = \frac{1}{{1 - \frac{{{T_0}z}}{{1 - \frac{{{T_1}z}}{{1 - \frac{{{T_2}z}}{{1 -{ \ddots }}}}}}}}}.$$
OEIS A104977 states ...
- 5,612
11
votes
0
answers
253
views
Poset of nonvanishing minors of a matrix
This question was posed on MSE here three days ago, but hasn't gotten any answers or suggestions. I hope it's okay to ask it on MO, but if I should wait a little longer, please just let me know.
Say $...
- 6,237
11
votes
0
answers
450
views
A congruence involving roots of unity
Let $f(x) \in \mathbb{Z}[x]$ and suppose $f(\omega^j) \in \mathbb{Z}$ for all $j= 1, \dots, n$ where $\omega = e^{2 \pi i/n}$ is a primitive $n^{\text{th}}$ root of unity.
Computational evidence ...
- 305
11
votes
0
answers
228
views
Is there a term for this graph subset?
Suppose $G$ is a (finite) graph which is $k$-vertex colourable (i.e. $\chi(G)\leqslant k$). Suppose $S$ is a set of vertices of $G$ with the following property:
If $c:V(G)\rightarrow [k]$ is a vertex ...
- 211