All Questions
Tagged with co.combinatorics reference-request
335 questions with no upvoted or accepted answers
24
votes
0
answers
760
views
How much of the plane is 4-colorable?
In 1981, Falconer proved that the measurable chromatic number of the plane is at least 5. That is, there are no measurable sets $A_1,A_2,A_3,A_4\subseteq\mathbb{R}^2$, each avoiding unit distances, ...
- 7,633
18
votes
0
answers
579
views
What is the geometric intuition behind Wilf-Zeilberger theory?
This problem is somehow inspired by a bunch of impressive posts of combinatorial identities by T. Amdeberhan. Earlier this month I learnt from computer scientists that they have a generic algorithmic ...
- 8,071
15
votes
0
answers
398
views
References on Discrete field theory vs Discrete differential geometry vs Combinatorial topology
Let me ask several related questions on discretization of classical field theory:
In topological folklore, it is known that cochains are "discrete analogues" of differential forms, and coboundary ...
14
votes
0
answers
270
views
A symmetry of lattice paths
The number of $n$-step NSEW lattice paths from $(0,0)$ to $(a,b)$ that intersect the line $y=k$ precisely $t$ times is independent of $k$, for $0\leq k\leq b$, where we assume $b\geq0$ for simplicity.
...
- 2,896
14
votes
0
answers
618
views
Chasing a 1950s thesis from the University of Dhaka on block designs
On behalf of a friend I am searching for a thesis on block designs from the 1950s. The details are below.
Author: Qazi Motahar Husein (Sometimes Husain or Hussein).
Title of the Thesis: Symmetrical ...
- 6,162
14
votes
0
answers
522
views
Reconstruction conjecture and partial 2-trees
Reconstruction conjecture says that graphs (with at least three vertices) are determined uniquely by their vertex deleted subgraphs. This conjecture is five decades old.
Searching relevant literature,...
- 281
13
votes
0
answers
323
views
Reference request: exponential growth rates of subword-closed languages are integers
For a language $L$ over the finite alphabet $\Sigma$, let $L_n$ denote the set of words in $L$ of length $n$. The word $u$ is a subword of $w$ if $u$ can be obtained from $w$ by deleting letters (...
- 2,339
13
votes
0
answers
349
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Some $q-$analogues of $ \sum\limits_{j = - k}^k {{{( - 1)}^{ j}}}\binom{n}{k-j}\binom{n}{k+j}=\binom{n}{k}.$
Let ${\left( {a;q} \right)_n}=\prod\limits_{j = 0}^{n - 1} {(1-{q^j}a} )$ and
let $ {{n}\brack{k}}_q$ denote a $q-$binomial coefficient.
I am interested in $q-$analogues of the identity $ \sum\...
- 5,622
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
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
$$\sum_{n\geq0}a_c(n)q^n=\prod_{k\geq1}\...
- 43.2k
11
votes
0
answers
387
views
Inequality for symmetric polynomial functions of log concave variables
Let $(x_i)_{i \ge 1}$ be a log-concave (resp. log-convex) sequence of non-negative real variables. In other words, for $i \ge 2$, we have $x_i^2 \ge x_{i-1}x_{i+1}$ (resp. $x_i^2 \le x_{i-1}x_{i+1}$).
...
- 505
11
votes
0
answers
406
views
Relation between a continued fraction and partitions
I am interested in the continued fraction
$$\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 }}}}}}}}}.$$
OEIS A104977 states ...
- 5,622
11
votes
0
answers
228
views
Is there a term for this graph subset?
Suppose $G$ is a (finite) graph which is $k$-vertex colourable (i.e. $\chi(G)\leqslant k$). Suppose $S$ is a set of vertices of $G$ with the following property:
If $c:V(G)\rightarrow [k]$ is a vertex ...
- 211
11
votes
0
answers
870
views
Reference/quote request: "All of combinatorics is the representation theory of $S_n$"
I think I remember reading somewhere a glib (or is it deep?) quote, perhaps due to Rota?, which was something like
"All of combinatorics is essentially [or can be reduced to?] the representation ...
- 3,230
10
votes
0
answers
287
views
Coefficients of polynomials vs trigonometric product
Let's consider the family of sequences of coefficients in the expansion
$$\prod_{i=0}^{n-1}(1+x^{3^i}+x^{3^{i+1}})=\sum_{k\geq0}a_n(k)\, x^k.$$
Remark. Evidently, the RHS is a finite sum.
Here is a ...
- 43.2k
10
votes
0
answers
349
views
A bijective proof for the odd companion to Shapiro's Catalan convolution
Shapiro's Catalan convolution is the following formula (where $C_n$ is the $n$th Catalan number):
$$
\sum_{k=0}^{n}{C_{2k}C_{2(n-k)}}=4^nC_n.
$$
In other words, letting $C(z)=\sum_{n=0}^{\infty}{C_nz^...
- 1,190
10
votes
0
answers
458
views
Proof of Dixon's identity only using Chu-Vandermonde
For any integers $a,b,c\ge 0$,
one has the well known identity or Dixon's Theorem:
$$
\sum_{k\in\mathbb{Z}} (-1)^k
\left(\begin{array}{c}a+b\\a+k\end{array}\right)
\left(\begin{array}{c}b+c\\b+k\end{...
10
votes
0
answers
191
views
Ideals in strong Bruhat order
Recall that a (lower) ideal in a poset $(P, \le)$ is a subset $I\subset P$ such that $x\in I \Rightarrow y\in I$ for all $y\le x$. In am interested in ideals in finite Coxeter groups $W$ equipped ...
- 31.2k
10
votes
0
answers
1k
views
Bound on the number of lattice points in d-dimensional ball
The following paper states that the number of lattice points in a $d$-dimensional ball of radius $R$ is $V_d R^d + O(R^\alpha)$ where $\alpha = d - 2$ and $V_d$ is the volume of the unit $d$-...
- 201
10
votes
0
answers
191
views
What is known about the number of permissible simplicial complexes given the number of k-cells?
Motivation: I am working on a problem that reduces to finding simplicial complexes given some data (details are unnecessary), but all I have managed to wrangle from my input is the number of cells of ...
- 1,488
9
votes
0
answers
358
views
Being even or odd in the product expansion $\prod(1+x^k+x^{k+1})$
Consider the generating function of "partitions with distinct parts"
$$\sum_nQ(n)x^n=\prod_k(1+x^k).$$
It's known that
$$\left[\prod_k(1+x^k)\right] \mod 2=\prod_m(1-x^m)=\sum_{j\in\mathbb{Z}...
- 43.2k
9
votes
0
answers
297
views
An abstract zero-sum problem
I would like to know whether the problem described below has appeared in the literature and/or whether similar questions have been studied. I would be very happy to find some references or, if none ...
- 1,169
9
votes
0
answers
1k
views
Balls and bins -- concentration bounds pertaining to the minimal load bin
Consider the standard balls and bins process, where $m$ balls are thrown uniformly at random into $n$ bins. Previous work has been done on estimating the value of the maximum load (i.e., the number of ...
- 201
9
votes
0
answers
213
views
A duality on partial permutations
A partial permutation matrix $\pi$ is one with at most one 1 in any row and column (the rest 0s). Given one, we can cross out to the East and South (but not Southeast) of each 1. Some boxes get ...
- 27.8k
9
votes
0
answers
241
views
An inequality for the ratio of standard Young tableau with {1,2,...,k} in the first row
For a partition $\lambda \vdash n$, define $\dim \lambda$ to be the number of standard Young tableaux of shape $\lambda$, and $\dim \lambda/(k)$ as the number of standard Young tableaux with $1,2,\...
- 181
9
votes
0
answers
205
views
Reference for sparseness of incomparability graphs implying sparseness of covering graphs
If a partial order on $n$ elements has $m$ incomparable pairs, then its covering graph (aka Hasse diagram aka transitive reduction, the graph of pairs of elements that are comparable but are not the ...
- 18.6k
9
votes
0
answers
778
views
Faa di Bruno and Free Probability?
It is possible to glean many combinatorial identities using Faa di Bruno’s formula for the coefficients of higher derivatives of a composite function. For many examples, see David Vella’s paper. The ...
- 7,057
8
votes
0
answers
244
views
Strengthening of Frankl's union-closed sets conjecture: An algebraic approach
Let $\mathcal F$ be a union-closed family of subsets of $[n]=\{1,2,...n\}$ and $n$ real numbers $x_1,x_2,...,x_n\geq 1$.
Conjecture: There exists $k\in [n]$ such that:
$$\sum_{k\in A,A\in \mathcal F}\...
- 1,462
8
votes
0
answers
181
views
Self-avoiding walks on strips
A strip is a locally finite graph which admits a quasi-transitive (i.e. finitley many orbits on vertices) action of $\mathbb Z$. A self avoiding walk is a walk which visits no vertex more than once.
...
- 2,671
8
votes
0
answers
149
views
Edge-colorings of plane graphs: do you know references where the following questions are studied?
Let $G$ be a plane graph (or more generally, a graph embedded on a surface) with a proper edge-coloring of $G$ with $k$ colors $\{1,\ldots,k\}$. I am interested in studying the cyclic permutations of ...
- 537
8
votes
0
answers
2k
views
What is the best lower bound for the domination number in regular graphs of girth 5?
The following theorem is a classical result (see [Alon and Spencer, The probabilistic method, 2nd ed., Theorem 1.2.2]):
Theorem: Let $G$ be a graph on $n$ vertices with minimum degree $d$. Then $G$ ...
- 161
8
votes
0
answers
358
views
Coloring toroidal polyhedra with convex faces?
Consider a toroidal polyhedron, which is a topological torus, in which all faces are planar, two faces meet in at most an edge, and adjacent faces are not coplanar. The Szilassi polyhedron has 7 non-...
8
votes
0
answers
152
views
Disjoint Rooted Paths with Specified Patterns
Let $S:=$ { $s_i : i \in [k]$ } and $T:=$ { $t_i : i \in [k]$ } be disjoint subsets of vertices of a graph $G$. Furthermore, let $A$ be a subset of $S_k$ (the symmetric group on $[k]$). A set of ...
- 32.1k
7
votes
0
answers
218
views
Analytic continuation of Dixon's identity
Many well-known combinatorial identities has an analytic version. For example, the following identities
$$
2^n = \sum_{k=0}^n \binom{n}{k}
$$
$$
\binom{2n}{n} = \sum_{k=1}^n \binom{n}{k}^2
$$
can be ...
- 1,608
7
votes
0
answers
176
views
The quotient of a higher Specht polynomial over the corresponding regular Specht polynomial
I'll need some notation before I can phrase my question, so please bare with me for a little. I'll try to get there as fast as possible (it's also my first MO question...).
Let $\lambda$ be a ...
- 131
7
votes
0
answers
150
views
Question about function on permutations
The following question is motivated by my research.
Let's consider a permutation $\sigma$ of the set $\{1, \ldots, n\}$. We define an element $i \in \{1, \ldots, n\}$ to be locally minimal for $\sigma$...
- 4,736
7
votes
0
answers
98
views
Pattern avoidance and P-recursiveness
A sequence $\{a_n\}_{n \geq 0}$ is said to be P-recursive if there exist polynomials $p_0(n), p_1(n), \dots , p_k(n)$ such that
$$
\sum_{i=0}^k p_i(n) a_{n+i}=0
$$
for all $n \in \mathbb N$.
Let $ P$ ...
- 1,608
7
votes
0
answers
97
views
What is known about chromatic polynomial of hypergraph at $-1$
Let $H$ be a hypergraph and let $P_H$ denote its chromatic polynomial. I am interested in the best results interpreting $P_H(-1)$. I am interested both in the general case (which I think is hard) as ...
- 7,875
7
votes
0
answers
203
views
Primitive recursive bounds for the the Gallai-Witt theorem
Let me first recall some facts:
By the work of Gowers, the Van der Waerden numbers belong to class $\mathcal{E}^3$ of the Grzegorczyk hierarchy
By the work of Shelah, the Hales-Jewett numbers belong ...
- 32.1k
7
votes
0
answers
113
views
A question related to the union-closed sets conjecture
Let $f(n)$ denote the maximum possible cardinality of a collection $\mathcal F$ of nonempty sets which is closed under unions ($X,Y\in\mathcal F\implies X\cup Y\in\mathcal F$) and is such that no ...
- 13.4k
7
votes
0
answers
206
views
Classification of posets that are quotient posets of the Boolean lattice
Quotient posets of the Boolean lattice $B_n$ have interesting properties and are for example discussed in chapter 5 of Stanley's book on algebraic combinatorics.
$B_n/G$ for a subgroup $G$ of the ...
- 26.5k
7
votes
0
answers
74
views
Graphs all of whose cuts are positive
Let $(V, E, w)$ a weighted graph, with vertices $V$, edges $E$, and signed weight $w:E\to \mathbb R$.
I am interested to know other popular properties that are known to imply, or are equivalent to, ...
- 2,041
7
votes
0
answers
174
views
A diagonal generating function for Fibonacci: Part II
In my earlier MO question, I mentioned although we have for the Fibonacci numbers that
$$F_n=[x^n]\left(\frac1{1-x-x^2}\right),$$
is there a function $F(x)$ such that $F_n=[x^n]\left(F(x)\right)^n$?
...
- 43.2k
7
votes
0
answers
171
views
What is known about the distribution of lengths of the cycle you get by adding an edge to a uniform spanning tree?
Let $G$ be a finite, connected graph. Let $T$ be a uniform spanning tree, and let $e$ be a uniformly random edge not in $T$. When we add $e$ to $T$, we get a subgraph with a unique cycle, $C$. I am ...
- 1,462
7
votes
0
answers
284
views
What are $(m,n)$-pseudoplanes?
An incidence geometry is a set $P$ (the "points"), a set $L$ (the "lines"), and a relation $I\subseteq P\times L$ ("incidence"). Equivalently, a bipartite graph with the halves of the partition ...
- 5,031
7
votes
0
answers
220
views
A funny kind of Ramsey number
A shorter version of this question was posted on Math Stack Exchange.
Let $V$ be a nonempty set. $(V,S)$ is a graph if $S\subseteq\binom V2,$ a triple system if $S\subseteq\binom V3,$
a quadruple ...
- 13.4k
7
votes
0
answers
279
views
Relations between Betti numbers for clique complex
Given a clique complex $K$ constructed from a discrete set of vertices (i.e. its faces are isomorphic to the set of cliques in the 1-skeleton of $K$.), it seems that the Betti numbers $\beta_k$ ...
- 8,071
7
votes
0
answers
188
views
Reference for an "elementary" combinatorial fact
This is a question I've been meaning to ask for quite some time.
Fact. For $n\in\mathbb N$ consider the set of segments $R=\{[i,j], 1\le i<j\le n\}$. Let a subset $E\subset R$ be nice iff $E$ is ...
- 3,513
7
votes
0
answers
298
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 (wikipedia):
However I can not find this paper (...
- 2,576
7
votes
0
answers
229
views
Has anyone seen these binary trees (Catalan-type related to the Gegenbauer polynomials and Motzkin paths)?
The OEIS entry A121448 enumerates binary trees with $n$ edges and $k$ vertices with outdegree 1.
Has anyone seen these trees?
The o.g.f. for this entry, $G(x,t)$, is essentially a discriminant ...
- 10.5k