scholarly journals Combinatorial proof of the minimal excludant theorem

2021 ◽  
pp. 1-15
Author(s):  
Cristina Ballantine ◽  
Mircea Merca
Keyword(s):  
Positive Integer ◽  
Generating Functions ◽  
Combinatorial Proof ◽  
Combinatorial Interpretation

The minimal excludant of a partition [Formula: see text], [Formula: see text], is the smallest positive integer that is not a part of [Formula: see text]. For a positive integer [Formula: see text], [Formula: see text] denotes the sum of the minimal excludants of all partitions of [Formula: see text]. Recently, Andrews and Newman obtained a new combinatorial interpretation for [Formula: see text]. They showed, using generating functions, that [Formula: see text] equals the number of partitions of [Formula: see text] into distinct parts using two colors. In this paper, we provide a purely combinatorial proof of this result and new properties of the function [Formula: see text]. We generalize this combinatorial interpretation to [Formula: see text], the sum of least [Formula: see text]-gaps in all partitions of [Formula: see text]. The least [Formula: see text]-gap of a partition [Formula: see text] is the smallest positive integer that does not appear at least [Formula: see text] times as a part of [Formula: see text].

scholarly journals Jacobi Ensemble, Hurwitz Numbers and Wilson Polynomials

2021 ◽  
Vol 111 (3) ◽  
Author(s):  
Massimo Gisonni ◽  
Tamara Grava ◽  
Giulio Ruzza
Keyword(s):  
Generating Functions ◽  
Combinatorial Interpretation ◽  
Hurwitz Numbers ◽  
Jacobi Ensemble ◽  
Matrix Ensembles ◽  
Wilson Polynomials ◽  
Unitary Ensemble ◽  
Topological Expansion

AbstractWe express the topological expansion of the Jacobi Unitary Ensemble in terms of triple monotone Hurwitz numbers. This completes the combinatorial interpretation of the topological expansion of the classical unitary invariant matrix ensembles. We also provide effective formulæ for generating functions of multipoint correlators of the Jacobi Unitary Ensemble in terms of Wilson polynomials, generalizing the known relations between one point correlators and Wilson polynomials.

scholarly journals ON THE NUMBER OF REPRESENTATIONS OF INTEGERS BY CERTAIN QUADRATIC FORMS

2008 ◽  
Vol 78 (1) ◽  
pp. 129-140 ◽  
Author(s):  
SHAUN COOPER
Keyword(s):  
Positive Integer ◽  
Generating Functions ◽  
Quadratic Forms ◽  
Triangular Numbers ◽  
Representations Of Integers

AbstractGenerating functions are used to derive formulas for the number of representations of a positive integer by each of the quadratic forms x12+x22+x32+2x42, x12+2x22+2x32+2x42, x12+x22+2x32+4x42 and x12+2x22+4x32+4x42. The formulas show that the number of representations by each form is always positive. Some of the analogous results involving sums of triangular numbers are also given.

Arithmetic properties for Fu's 9 dots bracelet partitions

2015 ◽  
Vol 11 (04) ◽  
pp. 1063-1072
Author(s):  
Olivia X. M. Yao
Keyword(s):  
Positive Integer ◽  
Generating Functions ◽  
Arithmetic Properties ◽  
Powers Of 2

The notion of Fu's k dots bracelet partitions was introduced by Shishuo Fu. For any positive integer k, let 𝔅k(n) denote the number of Fu's k dots bracelet partitions of n. Fu also proved several congruences modulo primes and modulo powers of 2. Recently, Radu and Sellers extended the set of congruences proven by Fu by proving three congruences modulo squares of primes for 𝔅5(n), 𝔅7(n) and 𝔅11(n). More recently, Cui and Gu, and Xia and the author derived a number of congruences modulo powers of 2 for 𝔅5(n). In this paper, we prove four congruences modulo 2 and two congruences modulo 4 for 𝔅9(n) by establishing the generating functions of 𝔅9(An+B) modulo 4 for some values of A and B.

scholarly journals Eulerian Numbers Associated with Arithmetical Progressions

10.37236/7182 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
José L. Ramírez ◽  
Sergio N. Villamarin ◽  
Diego Villamizar
Keyword(s):  
Generating Functions ◽  
Combinatorial Identities ◽  
Combinatorial Interpretation ◽  
Eulerian Numbers ◽  
Signed Permutations ◽  
Whitney Numbers

In this paper, we give a combinatorial interpretation of the $r$-Whitney-Eulerian numbers by means of coloured signed permutations. This sequence is a generalization of the well-known Eulerian numbers and it is connected to $r$-Whitney numbers of the second kind. Using generating functions, we provide some combinatorial identities and the log-concavity property. Finally, we show some basic congruences involving the $r$-Whitney-Eulerian numbers.

scholarly journals Counting words with Laguerre polynomials

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Jair Taylor
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Laguerre Polynomials ◽  
Equal Length ◽  
Combinatorial Interpretation ◽  
International Audience ◽  
Nous Utilisons

International audience We develop a method for counting words subject to various restrictions by finding a combinatorial interpretation for a product of formal sums of Laguerre polynomials. We use this method to find the generating function for $k$-ary words avoiding any vincular pattern that has only ones. We also give generating functions for $k$-ary words cyclically avoiding vincular patterns with only ones whose runs of ones between dashes are all of equal length, as well as the analogous results for compositions. Nous développons une méthode pour compter des mots satisfaisants certaines restrictions en établissant une interprétation combinatoire utile d’un produit de sommes formelles de polynômes de Laguerre. Nous utilisons cette méthode pour trouver la série génératrice pour les mots $k$-aires évitant les motifs vinculars consistant uniquement de uns. Nous présentons en suite les séries génératrices pour les mots $k$-aires évitant de façon cyclique les motifs vinculars consistant uniquement de uns et dont chaque série de uns entre deux tirets est de la même longueur. Nous présentons aussi les résultats analogues pour les compositions.

scholarly journals Distribution of generalized mex-related integer partitions

2021 ◽  
Vol Volume 43 - Special... ◽  
Author(s):  
Kalyan Chakraborty ◽  
Chiranjit Ray
Keyword(s):  
Positive Integer ◽  
Integer Partitions ◽  
Combinatorial Interpretation ◽  
Integer Partition ◽  
International Audience

International audience The minimal excludant or "mex" function for an integer partition π of a positive integer n, mex(π), is the smallest positive integer that is not a part of π. Andrews and Newman introduced σmex(n) to be the sum of mex(π) taken over all partitions π of n. Ballantine and Merca generalized this combinatorial interpretation to σrmex(n), as the sum of least r-gaps in all partitions of n. In this article, we study the arithmetic density of σ_2 mex(n) and σ_3 mex(n) modulo 2^k for any positive integer k.

scholarly journals A Triple Lacunary Generating Function for Hermite Polynomials

10.37236/1927 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
Ira M. Gessel ◽  
Pallavi Jayawant
Keyword(s):  
Orthogonal Polynomials ◽  
Generating Function ◽  
Generating Functions ◽  
Hermite Polynomials ◽  
Combinatorial Proof ◽  
Classical Orthogonal Polynomials ◽  
Combinatorial Objects ◽  
Combinatorial Interpretations

Some of the classical orthogonal polynomials such as Hermite, Laguerre, Charlier, etc. have been shown to be the generating polynomials for certain combinatorial objects. These combinatorial interpretations are used to prove new identities and generating functions involving these polynomials. In this paper we apply Foata's approach to generating functions for the Hermite polynomials to obtain a triple lacunary generating function. We define renormalized Hermite polynomials $h_n(u)$ by $$\sum_{n= 0}^\infty h_n(u) {z^n\over n!}=e^{uz+{z^2\!/2}}.$$ and give a combinatorial proof of the following generating function: $$ \sum_{n= 0}^\infty h_{3n}(u) {{z^n\over n!}}= {e^{(w-u)(3u-w)/6}\over\sqrt{1-6wz}} \sum_{n= 0}^\infty {{(6n)!\over 2^{3n}(3n)!(1-6wz)^{3n}} {z^{2n}\over(2n)!}}, $$ where $w=(1-\sqrt{1-12uz})/6z=uC(3uz)$ and $C(x)=(1-\sqrt{1-4x})/(2x)$ is the Catalan generating function. We also give an umbral proof of this generating function.

Walk Generating Functions, Christoffel-Darboux Identities and the Adjacency Matrix of a Graph

1992 ◽  
Vol 1 (1) ◽  
pp. 13-25 ◽  
Author(s):  
C. D. Godsil
Keyword(s):  
Orthogonal Polynomials ◽  
Characteristic Polynomial ◽  
Generating Functions ◽  
Adjacency Matrix ◽  
Characteristic Polynomials ◽  
Combinatorial Interpretation ◽  
Determinantal Identity

In this work we show that that many of the basic results concerning the theory of the characteristic polynomial of a graph can be derived as easy consequences of a determinantal identity due to Jacobi. As well as improving known results, we are also able to derive a number of new ones. A combinatorial interpretation of the Christoffel-Darboux identity from the theory of orthogonal polynomials is also presented. Finally, we extend some work of Tutte on the reconstructibility of graphs with irreducible characteristic polynomials.

scholarly journals The Borwein Conjecture and Partitions with Prescribed Hook Differences

10.37236/1262 ◽  
1995 ◽  
Vol 3 (2) ◽  
Author(s):  
David A. Bressoud
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Combinatorial Proof

Peter Borwein has conjectured that certain polynomials have non-negative coefficients. In this paper we look at some generalizations of this conjecture and observe how they relate to the study of generating functions for partitions with prescribed hook differences. A combinatorial proof of the generating function for partitions with prescribed hook differences is given.

scholarly journals Top Degree Part in $b$-Conjecture for Unicellular Bipartite Maps

10.37236/6130 ◽  
2017 ◽  
Vol 24 (3) ◽  
Author(s):  
Maciej Dołęga
Keyword(s):  
Positive Integer ◽  
Symmetric Functions ◽  
White Vertex ◽  
Additional Parameter ◽  
Combinatorial Interpretation ◽  
Integer Coefficients ◽  
Special Values ◽  
Generating Series ◽  
Special Value ◽  
Vertex Degrees

Goulden and Jackson (1996) introduced, using Jack symmetric functions, some multivariate generating series $\psi(\boldsymbol{x}, \boldsymbol{y},\boldsymbol{z}; 1, 1+\beta)$ with an additional parameter $\beta$ that might be interpreted as a continuous deformation of the rooted bipartite maps generating series. Indeed, it has a property that for $\beta \in \{0,1\}$, it specializes to the rooted, orientable (general, i.e. orientable or not, respectively) bipartite maps generating series. They made the following conjecture: coefficients of $\psi$ are polynomials in $\beta$ with positive integer coefficients that can be written as a multivariate generating series of rooted, general bipartite maps, where the exponent of $\beta$ is an integer-valued statistics that in some sense "measures the non-orientability" of the corresponding bipartite map.We show that except two special values of $\beta = 0,1$ for which the combinatorial interpretation of the coefficients of $\psi$ is known, there exists a third special value $\beta = -1$ for which the coefficients of $\psi$ indexed by two partitions $\mu,\nu$, and one partition with only one part are given by rooted, orientable bipartite maps with arbitrary face degrees and black/white vertex degrees given by $\mu$/$\nu$, respectively. We show that this evaluation corresponds, up to a sign, to a top-degree part of the coefficients of $\psi$. As a consequence, we introduce a collection of integer-valued statistics of maps $(\eta)$ such that the top-degree of the multivariate generating series of rooted, bipartite maps with only one face (called unicellular) with respect to $\eta$ gives the top degree of the appropriate coefficients of $\psi$. Finally, we show that $b$ conjecture holds true for all rooted, unicellular bipartite maps of genus at most $2$.

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