It is well known that the numbers $(2m)! (2n)!/m! n! (m+n)!$ are integers, but in general there is no known combinatorial interpretation for them. When $m=0$ these numbers are the middle binomial coefficients $\binom{2n}{n}$, and when $m=1$ they are twice the Catalan numbers. In this paper, we give combinatorial interpretations for these numbers when $m=2$ or 3.
In this paper we present several natural $q$-analogues of the poly-Bernoulli numbers arising in combinatorial contexts. We also recall some relating analytical results and ask for combinatorial interpretations.
Gaussian blur is a commonly-used method to filter image data. This paper introduces the collapsing sum, a new operator on matrices that provides a combinatorial interpretation of Gaussian blur. We study the combinatorial properties of this operator and prove the explicit relation between Gaussian blur and the collapsing sum.
We show how the combinatorial interpretation of the normalized median Genocchi numbers in terms of multiset tuples, defined by Hetyei in his study of the alternation acyclic tournaments, is bijectively equivalent to previous models like the normalized Dumont permutations or the Dellac configurations, and we extend the interpretation to the Kreweras triangle.
We give a combinatorial interpretation in terms of bicolored ordered trees for the sequence (a_n)_{n>=1}=(1, 1, 1, 2, 3, 6, 10, 20, 36, 73,... ), A345973 in OEIS, whose generating function satisfies the defining identity Sum_{n>=1}a_n x^n = x + x^2/Product_{n>=1}(1 - a_n x^n).
The restricted partitions in which the largest part is less than or equal to $N$ and the number of parts is less than or equal to $k$ were investigated by Andrews in \cite{Andrews76}. These partitions were extended recently by the author to the partitions into parts of two kinds. In this paper, we use a new class of restricted partitions into parts of two kinds to provide new combinatorial interpretations for two identities of Guo and Yang.
We study the combinatorial description of the LDU decomposition of totally positive matrices. We give a description of the lower triangular L, the diagonal D, and the upper triangular U matrices of the LDU decomposition of totally positive matrices in terms of the combinatorial structure of essential planar networks described by Zelvinsky and Fomin. Similarly, we find a combinatorial description of the inverses of these matrices. In addition, we provide recursive formulae for computing the L, D, and U matrices of a totally positive matrix.
A Combinatorial Interpretation for the coefficients in the Kronecker Product $s_{(n-p,p)}\ast s_λ$ (Multiplicities in the Kronecker Product $s_{(n-p,p)}\ast s_λ$)
Combinatorial interpretations of binomial coefficient analogues related to Lucas sequences
Let s and t be variables. Define polynomials {n} in s, t by {0}=0, {1}=1, and {n}=s{n-1}+t{n-2} for n >= 2. If s, t are integers then the corresponding sequence of integers is called a Lucas sequence. Define an analogue of the binomial coefficients by C{n,k}={n}!/({k}!{n-k}!) where {n}!={1}{2}...{n}. It is easy to see that C{n,k} is a polynomial in s and t. The purpose of this note is to give two combinatorial interpretations for this polynomial in terms of statistics on integer partitions inside a k by n-k rectangle. When s=t=1 we obtain combinatorial interpretations of the fibonomial coeffic…
Using generating functions and some trivial bijections, we show in this paper that the binomial coefficients count the set of (123,132) and (123,213)-avoiding permutations according to the number of crossings. We also define a q-tableau of power of two and prove that it counts the set of (213,312) and (132,312)-avoiding permutations according to the number of crossings.
We use a q-series identity by Ramanujan to give a combinatorial interpretation of Ramanujan's tau function which involves t-cores and a new class of partitions which we call (m,k)-capsids. The same method can be applied in conjunction with other related identities yielding alternative combinatorial interpretations of the tau function.
The super Catalan numbers $T(m,n)=(2m)!(2n)!/2m!n!(m+n)!$ are integers which generalize the Catalan numbers. With the exception of a few values of $m$, no combinatorial interpretation in known for $T(m,n)$. We give a weighted interpretation for $T(m,n)$ and develop a technique that converts this weighted interpretation into a conventional combinatorial interpretation in the case $m=2$.
We consider the tiling of an $n$-board (a $1\times n$ array of square cells of unit width) with half-squares ($\frac12\times1$ tiles) and $(\frac12,\frac12)$-fence tiles. A $(\frac12,\frac12)$-fence tile is composed of two half-squares separated by a gap of width $\frac12$. We show that the number of ways to tile an $n$-board using these types of tiles equals $F_{n+1}^2$ where $F_n$ is the $n$th Fibonacci number. We use these tilings to devise combinatorial proofs of identities relating the Fibonacci numbers squared to one another and to other number sequences. Some of these identities appear …
Let G be a graph, and let $χ$G be its chromatic polynomial. For any non-negative integers i, j, we give an interpretation for the evaluation $χ$ (i) G (--j) in terms of acyclic orientations. This recovers the classical interpretations due to Stanley and to Green and Zaslavsky respectively in the cases i = 0 and j = 0. We also give symmetric function refinements of our interpretations, and some extensions. The proofs use heap theory in the spirit of a 1999 paper of Gessel.
In this survey we discuss the notion of combinatorial interpretation in the context of Algebraic Combinatorics and related areas. We approach the subject from the Computational Complexity perspective. We review many examples, state a workable definition, discuss many open problems, and present recent results on the subject.
The Lucas sequence is a sequence of polynomials in s, and t defined recursively by {0}=0, {1}=1, and {n}=s{n-1}+t{n-2} for n >= 2. On specialization of s and t one can recover the Fibonacci numbers, the nonnegative integers, and the q-integers [n]_q. Given a quantity which is expressed in terms of products and quotients of nonnegative integers, one obtains a Lucas analogue by replacing each factor of n in the expression with {n}. It is then natural to ask if the resulting rational function is actually a polynomial in s and t with nonnegative integer coefficients and, if so, what it counts. The…
Schreyer has proved that the graded Betti numbers of a canonical tetragonal curve are determined by two integers $b_1$ and $b_2$, associated to the curve through a certain geometric construction. In this article we prove that in the case of a smooth projective tetragonal curve on a toric surface, these integers have easy interpretations in terms of the Newton polygon of its defining Laurent polynomial. We can use this to prove an intrinsicness result on Newton polygons of small lattice width.
In this paper, we will investigate a harmonic cycle (discrete harmonic form). With a CW-complex, we can construct the combinatorial Laplacian operator. The kernel of the operator is the harmonic space, the set of harmonic cycles, and is isomorphic to its homology due to combinatorial Hodge theory. We will introduce four concepts; cycletree, unicyclization, winding number map, and standard harmonic cycle. A cycletree is the disjoint union of a spanning tree and an edge on a given graph. A unicyclization consists of a graph and information substituting for faces. Then, we will define the winding…
