A. K. Kwasniewski
Combinatorial interpretation of the fibonomial coefficients as a number of choices of specific finite subsets of an infinite partially ordered set of not binomial type is proposed. This partially ordered set is here defined via characteristic matrix of the corresponding partial order relation . Relevance of the proposal to more general unification treatment is indicated.
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Fibonomial coefficients count the number of specific finite birth self-similar subposets of an infinite non-tree poset naturally related to the Fibonacci tree of rabbits growth process.
Combinatorial interpretation of the fibonomial coefficients recently proposed by the present author results here in combinatorial interpretation of the recurrence relation for fibonomial coefficients . The presentation is provided with quite an exhaustive context where apart from plane grid coordinate system used several figures illustrate the exposition of statements and the derivation of the recurrence itself.
Up to our knowledge -since about 126 years we were lacking of classical type combinatorial interpretation of Fibonomial coefficients as it was Lukas \cite{1} - to our knowledge -who was the first who had defined Finonomial coefficients and derived a recurrence for them (see Historical Note in \cite{2,3}). Here we inform that a join combinatorial interpretation was found \cite{4} for all binomial-type coefficient - Fibonomial coefficients included.
In 2009, Sagan and Savage introduced a combinatorial model for the Fibonomial numbers, integer numbers that are obtained from the binomial coefficients by replacing each term by its corresponding Fibonacci number. In this paper, we present a combinatorial description for the $q$-analog and elliptic analog of the Fibonomial numbers. This is achieved by introducing some $q$-weights and elliptic weights to a slight modification of the combinatorial model of Sagan and Savage.
Combinatorial Interpretations of $q$-Fibonacci Numbers and Their Binomial Analogues
本文提出一个推广Bergeron方法的q-Fibonacci数新模型，为q-Fibonomial系数的阶梯模型提供非双射证明，并展示了涉及该系数的新恒等式，增强了对相关数学对象的组合理解。
Unimodality of $q$-Fibonomial coefficients for small cases
论文研究了$q$-Fibonomial系数的单峰性，通过组合和代数方法证明了$n\leq3$时的单峰性，并对$n=3$情况提出更一般的结论与猜想。
Following Lucas and then other Fibonacci people Kwasniewski had introduced and had started ten years ago the open investigation of the overall F-nomial coefficients which encompass among others Binomial, Gaussian and Fibonomial coefficients with a new unified combinatorial interpretation expressed in terms of cobweb posets' partitions and tilings of discrete hyperboxes. In this paper, we deal with special subfamily of T-nomial coefficients. The main aim of this note is to develop the theory of T-nomial coefficients with the help of generating functions. The binomial-like theorem for T-nomials …
It is well known that Pascal's triangle exhibits fractal behavior when reduced modulo a prime. We show that the triangle of Fibonomial coefficients has a similar nature modulo two. Specifically, for any $m \ge 0$, the subtriangle consisting of the first $3 \cdot 2^m$ rows is duplicated on the left and right sides of the next $3 \cdot 2^m$ rows, with an inverted triangle of zeros in between. We give three proofs of this fact. The first uses a combinatorial interpretation of the Fibonomials due to Sagan and Savage. The second employs an analogue of Lucas' congruence for the parity of binomial co…
A research problem for undergraduates and graduates is being posed as a cap for the prior antecedent regular discrete mathematics exercises. [Here cap is not necessarily CAP=Competitive Access Provider, though nevertheless ...] The object of the cap problem of final interest i.e. array of fibonomial coefficients and the issue of its combinatorial meaning is to be found in A.K.Kwaśniewski's source papers. The cap problem number seven - still opened for students has been placed on Mathemagics page of the first author [http://ii.uwb.edu.pl/akk/dydaktyka/dyskr/dyskretna.htm]. The indicatory refere…
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…
In a series of papers, P. Blasiak et al. developed a wide-ranging generalization of Bell numbers (and of Stirling numbers of the second kind) that appears to be relevant to the so-called Boson normal ordering problem. They provided a recurrence and, more recently, also offered a (fairly complex) combinatorial interpretation of these numbers. We show that by restricting the numbers somewhat (but still widely generalizing Bell and Stirling numbers), one can supply a much more natural combinatorial interpretation. In fact, we offer two different such interpretations, one in terms of graph colouri…
Signed combinatorial interpretations in algebraic combinatorics
本文证明了代数组合学中几大类结构常数（包括对称、拟对称多项式基及Schubert理论基）存在带符号组合解释，结果以计算复杂性语言表述，证明基于有效的Möbius反演。
In 2017 Aguiar and Ardila provided a generic way to construct polynomial invariants of combinatorial objects using the notions of Hopf monoids and characters of Hopf monoids. They show that it is possible to find a combinatorial interpretation of these polynomials over negative integers using the antipode and give a cancellation-free grouping-free formula for the antipode on generalized permutahedra. In this work, we give a combinatorial interpretation of these polynomials over both positive integers and negative integers for the Hopf monoids of generalized permutahedra and hypergraphs and for…
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.
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.
Since 1950s, mathematicians have successfully interpreted the traditional Eulerian numbers and $q-$Eulerian numbers combinatorially. In this paper, the authors give a combinatorial interpretation to the general Eulerian numbers defined on general arithmetic progressions { a, a+d, a+2d,...}.
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.
Recently, Guo and Zeng discovered q-analogues of Faulhaber's formulas for the sums of powers. They left it as an open problem to extend the combinatorial interpretation of Faulhaber's formulas as given by Gessel and Viennot to the q case. In this note we will provide such an interpretation.
Let $w$ be a word in alphabet $\{x,D\}$ with $m$ $x$'s and $n$ $D$'s. Interpreting "$x$" as multiplication by $x$, and "$D$" as differentiation with respect to $x$, the identity $wf(x) = x^{m-n}\sum_k S_w(k) x^k D^k f(x)$, valid for any smooth function $f(x)$, defines a sequence $(S_w(k))_k$, the terms of which we refer to as the {\em Stirling numbers (of the second kind)} of $w$. The nomenclature comes from the fact that when $w=(xD)^n$, we have $S_w(k)={n \brace k}$, the ordinary Stirling number of the second kind. Explicit expressions for, and identities satisfied by, the $S_w(k)$ have been…
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.
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.
The Jacobi-Stirling numbers and the Legendre-Stirling numbers of the first and second kind were first introduced in [6], [7]. In this paper we note that Jacobi-Stirling numbers and Legendre-Stirling numbers are specializations of elementary and complete symmetric functions. We then study combinatorial interpretations of this specialization and obtain new combinatorial interpretations of the Jacobi-Stirling and Legendre-Stirling numbers.
Two Combinatorial Interpretations of Rascal Numbers
该研究为Angorro等人定义的Rascal三角形元素给出两种组合解释，分别对应二进制字上升计数与Duncan等研究的上升序列模式避免，后者还关联Baxter和Pudwell的最新成果，丰富了Rascal数的组合内涵。
We give a combinatorial interpretation of a classical meta-Fibonacci sequence defined by G(n) = n - G(G(n-1)) with the initial condition G(1) = 1, which appears in Hofstadter's 'Godel, Escher, Bach: An Eternal Golden Braid'. The interpretation is in terms of an infinite labelled tree. We then show a few corollaries about the behaviour of the sequence G(n) directly from the interpretation.
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).
Nicholas Pippenger and Kristin Schleich have recently given a combinatorial interpretation for the second-order super-Catalan numbers (u_{n})_{n>=0}=(3,2,3,6,14,36,...): they count "aligned cubic trees" on n internal vertices. Here we give a combinatorial interpretation of the recurrence u_{n} = Sum_{k=0}^{n/2-1} ({n-2}choose{2k} 2^{n-2-2k} u_{k}): it counts these trees by number of deep interior vertices where deep interior means "neither a leaf nor adjacent to a leaf".
The title identity appeared as Problem 75-4, proposed by P. Barrucand, in Siam Review in 1975. The published solution equated constant terms in a suitable polynomial identity. Here we give a combinatorial interpretation in terms of card deals.
The identity j/n {kn}\choose{n+j} =(k-1) {kn-1}\choose{n+j-1}- {kn-1}\choose{n+j} shows that j/n {kn}\choose{n+j} is always an integer. Here we give a combinatorial interpretation of this integer in terms of lattice paths, using a uniformly distributed statistic. In particular, the case j=1,k=2 gives yet another manifestation of the Catalan numbers.
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.
In math.CO/0109093 the author obtained a formula for the value of an irreducible symmetric group character indexed by a partition of rectangular shape. In the present paper this formula is (conjecturally) generalized to arbitrary shapes.
Stanley has studied a symmetric function generalization X_G of the chromatic polynomial of a graph G. The innocent-looking Stanley-Stembridge Poset Chain Conjecture states that the expansion of X_G in terms of elementary symmetric functions has nonnegative coefficients if G is a clawfree incomparability graph. Here we give a combinatorial interpretation of these coefficients by combining Gasharov's work on the conjecture with Egecioglu and Remmel's combinatorial interpretation of the inverse Kostka matrix. This gives a new proof of a partial nonnegativity result of Stanley. As an interesting b…
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.
Combinatorial interpretation of a truncated identity of Gauss
本文针对高斯恒等式的截断版本，利用过划分中最小排除非划线部分的概念，给出了新的组合解释，丰富了截断theta级数的组合研究。
In this paper we give a combinatorial interpretation for the coefficient of $s_ν$ in the Kronecker product $s_{(n-p,p)}\ast s_λ$, where $λ=(λ_1, ..., λ_{\ell(λ)})\vdash n$, if $\ell(λ)\geq 2p-1$ or $λ_1\geq 2p-1$; that is, if $λ$ is not a partition inside the $2(p-1)\times 2(p-1)$ square. For $λ$ inside the square our combinatorial interpretation provides an upper bound for the coefficients. In general, we are able to combinatorially compute these coefficients for all $λ$ when $n>(2p-2)^2$. We use this combinatorial interpretation to give characterizations for multiplicity free Kronecker produ…
The monic sequence that shifts left under convolution with itself is the Catalan numbers with 130+ combinatorial interpretations. Here we establish a combinatorial interpretation for the monic sequence that shifts left under composition: it counts permutations that contain a 3241 pattern only as part of a 35241 pattern. We give two recurrences, the first allowing relatively fast computation, the second similar to one for the Catalan numbers. Among the 4 times 4! = 96 similarly restricted patterns involving 4 letters (such as 4\underline{2}31: a 431 pattern only occurs as part of a 4231), four …
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 Catalan transform of a sequence (a_{n})_{n>=0} is the sequence (b_{n})_{n>=0} with b_{n} = Sum[k/(2n-k) (2n-k)-choose-(n-k) a_{k},k=0..n]. Here we show that the Catalan transform of the Catalan numbers has a simple interpretation: it counts functions f:[1,n] -> [1,n] satisfying the condition that, for all i<j, f(j)-(j-i) is not in the interval [1,f(i)-1].
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.
The Jacobi-Stirling numbers of the first and second kinds were introduced in 2006 in the spectral theory and are polynomial refinements of the Legendre-Stirling numbers. Andrews and Littlejohn have recently given a combinatorial interpretation for the second kind of the latter numbers. Noticing that these numbers are very similar to the classical central factorial numbers, we give combinatorial interpretations for the Jacobi-Stirling numbers of both kinds, which provide a unified treatment of the combinatorial theories for the two previous sequences and also for the Stirling numbers of both ki…
We find an explicit combinatorial interpretation of the coefficients of Kerov character polynomials which express the value of normalized irreducible characters of the symmetric groups S(n) in terms of free cumulants R_2,R_3,... of the corresponding Young diagram. Our interpretation is based on counting certain factorizations of a given permutation.
In this paper, we apply the combinatorial proof technique of Description, Involution, Exceptions (DIE) to prove various known identities for the joint cumulant. Consider a set of random variables $S = \{X_1,..., X_n\} $. Motivated by the definition of the joint cumulant, we define $ \sC(S) $ as the set of cyclically arranged partitions of $S$, allowing us to express the joint cumulant of $ S $ as a weighted, alternating sum over $\sC(S)$. We continue to define other combinatorial objects that allow us to rewrite expressions originally in terms of the joint cumulant as weighted sums over the se…
Combinatorial interpretation of the coefficients of the order polynomial of fence posets
研究针对围栏偏序集定义新统计量blP，为其序多项式系数提供组合解释，还借此解释舒伯特拟阵基多面体埃尔哈特多项式系数，建立序多面体埃尔哈特多项式线性系数下界，并猜想该统计量对其他偏序集的推广。
A combinatorial interpretation of the noncommutative inverse Kostka matrix
该研究提出无瑕非交换对称函数展开为完全齐次非交换对称函数的组合公式，引入GBPR图与隧道钩，推广至斜形状及整数序列索引的无瑕函数，并拓展了Campbell的带状分解结果。
A combinatorial interpretation for certain plethysm and Kronecker coefficients
本文给出了当 λ 至多两行时 plethysm 系数 ⟨s_μ[s_ν], s_λ⟩ 的显式正组合解释，通过计数特定标记树；特例 μ=(n) 也给出了矩形 Kronecker 系数 g(λ, (n^k), (n^k)) 的组合解释。
For any integer s >= 0, we derive a combinatorial interpretation for the family of sequences generated by the recursion (parameterized by s) h_s(n) = h_s(n - s - h_s(n - 1)) + h_s(n - 2 - s - h_s(n - 3)), n > s + 3, with the initial conditions h_s(1) = h_s(2) = ... = h_s(s+2) = 1 and h_s(s+3) = 2. We show how these sequences count the number of leaves of a certain infinite tree structure. Using this interpretation we prove that h_s sequences are "slowly growing", that is, h_s sequences are monotone nondecreasing, with successive terms increasing by 0 or 1, so each sequence hits every positive …
Recently, Guo and Zeng discovered two families of polynomials featuring in a q-analogue of Faulhaber's formula for the sums of powers and a q-analogue of Gessel-Viennot's formula involving Salie's coefficients for the alternating sums of powers. In this paper, we show that these are polynomials with symmetric, nonnegative integral coefficients by refining Gessel-Viennot's combinatorial interpretations.
Generalized Schröder paths arising from a combinatorial interpretation of generalized Laurent bi-orthogonal polynomials
本文定义并研究ℓ-Schröder路径，推广了多种已知路径，并揭示其与广义Laurent双正交多项式矩的组合联系，且给出非相交路径生成函数的闭式分解。
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 $m=2$.
Combinatorial Interpretations of Cranks of Overpartitions and Partitions without Repeated Odd Parts
本文为过划分的两个剩余曲柄及无重复奇数部分划分的 $M_2$-曲柄提供组合解释，给出无调整权重的新定义，并揭示其与秩统计量和 Ramanujan 十阶伪 theta 函数的联系。
Combinatorial interpretation of the coefficients of the causal set d'Alembertian
针对因果集理论d'Alembertian的有理系数，研究给出其组合解释，补充了该系数已知交错表达式之外的组合层面认知。
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.
Boij-Söderberg theory shows that the Betti table of a graded module can be written as a liner combination of pure diagrams with integer coefficients. Using Ferrers hypergraphs and simplicial polytopes, we provide interpretations of these coefficients for ideals with a d-linear resolution, their quotient rings, and for Gorenstein rings whose resolution has essentially at most two linear strands. We also establish a structural result on the decomposition in the case of quasi-Gorenstein modules.
We describe a method for constructing characters of combinatorial Hopf algebras by means of integrals over certain polyhedral cones. This is based on ideas from resurgence theory, in particular on the construction of well-behaved averages induced by diffusion processes on the real line. We give several interpretations and proofs of the main result in terms of noncommutative symmetric and quasisymmetric functions, as well as generalizations involving matrix quasi-symmetric functions. The interpretation of noncommutative symmetric functions as alien operators in resurgence theory is also discuss…
The Alladi-Gordon identity plays an important role for the Alladi-Gordon generalization of Schur's partition theorem. By using Joichi-Stanton's insertion algorithm, we present an overpartition interpretation for the Alladi-Gordon key identity. Based on this interpretation, we further obtain a combinatorial proof of the Alladi-Gordon key identity by establishing an involution on the underlying set of overpartitions.
Fomin and Zelevinsky show that a certain two-parameter family of rational recurrence relations, here called the (b,c) family, possesses the Laurentness property: for all b,c, each term of the (b,c) sequence can be expressed as a Laurent polynomial in the two initial terms. In the case where the positive integers b,c satisfy bc<4, the recurrence is related to the root systems of finite-dimensional rank 2 Lie algebras; when bc>4, the recurrence is related to Kac-Moody rank 2 Lie algebras of hyperbolic type. Here we investigate the borderline cases bc=4, corresponding to Kac-Moody Lie algebras of…
Kerov's polynomials give irreducible character values in term of the free cumulants of the associated Young diagram. We prove in this article a positivity result on their coefficients, which extends a conjecture of S. Kerov. Our method, through decomposition of maps, gives a description of the coefficients of the k-th Kerov's polynomials using permutations in S(k). We also obtain explicit formulas or combinatorial interpretations for some coefficients. In particular, we are able to compute the subdominant term for character values on any fixed permutation (it was known for cycles).
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.
We give a purely combinatorial proof of the Glaisher-Crofton identity which derives from the analysis of discrete structures generated by iterated second derivative. The argument illustrates utility of symbolic and generating function methodology of modern enumerative combinatorics and their applications to computational problems.
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…
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