scholarly journals Euler–Catalan’s Number Triangle and Its Application

Symmetry ◽  
2020 ◽  
Vol 12 (4) ◽  
pp. 600 ◽  
Author(s):  
Yuriy Shablya ◽  
Dmitry Kruchinin
Keyword(s):  
Explicit Formula ◽  
Generating Function ◽  
Binary Trees ◽  
Combinatorial Objects ◽  
Left Branch ◽  
Bivariate Generating Function

In this paper, we study such combinatorial objects as labeled binary trees of size n with m ascents on the left branch and labeled Dyck n-paths with m ascents on return steps. For these combinatorial objects, we present the relation of the generated number triangle to Catalan’s and Euler’s triangles. On the basis of properties of Catalan’s and Euler’s triangles, we obtain an explicit formula that counts the total number of such combinatorial objects and a bivariate generating function. Combining the properties of these two number triangles allows us to obtain different combinatorial objects that may have a symmetry, for example, in their form or in their formulas.

scholarly journals Counting Nondecreasing Integer Sequences that Lie Below a Barrier

10.37236/149 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Robin Pemantle ◽  
Herbert S. Wilf
Keyword(s):  
Explicit Formula ◽  
Generating Function ◽  
Binomial Coefficients ◽  
Probabilistic Interpretation ◽  
Integer Sequences ◽  
Bivariate Generating Function ◽  
Linear Recursion ◽  
Special Case

Given a barrier $0 \leq b_0 \leq b_1 \leq \cdots$, let $f(n)$ be the number of nondecreasing integer sequences $0 \leq a_0 \leq a_1 \leq \cdots \leq a_n$ for which $a_j \leq b_j$ for all $0 \leq j \leq n$. Known formulæ for $f(n)$ include an $n \times n$ determinant whose entries are binomial coefficients (Kreweras, 1965) and, in the special case of $b_j = rj+s$, a short explicit formula (Proctor, 1988, p.320). A relatively easy bivariate recursion, decomposing all sequences according to $n$ and $a_n$, leads to a bivariate generating function, then a univariate generating function, then a linear recursion for $\{ f(n) \}$. Moreover, the coefficients of the bivariate generating function have a probabilistic interpretation, leading to an analytic inequality which is an identity for certain values of its argument.

A note on the time-dependent and the stationary behaviour of a semi-infinite reservoir subject to a combination of Markovian inflows

1971 ◽  
Vol 8 (04) ◽  
pp. 708-715 ◽  
Author(s):  
Emlyn H. Lloyd
Keyword(s):  
Functional Equation ◽  
Explicit Formula ◽  
Generating Function ◽  
Present Theory ◽  
The Other ◽  
Time Dependent ◽  
Independent Sequence ◽  
Time Dependent Solution ◽  
Infinite Reservoir ◽  
Stationary Behaviour

The present theory of finite reservoirs is not rich in general theorems even when of the simple Moran type, with unit draft and stationary discrete independent-sequence inflows. For the corresponding systems with unbounded capacity however there are two classes of results which have been known for some time. One of these classes is concerned with the time-dependent solution, where the theory provides a functional equation for the generating function of the time to first emptiness (Kendall (1957)), and the other with the asymptotic stationary distribution of reservoir contents, for which an explicit formula for the generating function is available (Moran (1959)).

scholarly journals The Number of Chains of Subgroups in the Lattice of Subgroups of the Dicyclic Group

2012 ◽  
Vol 2012 ◽  
pp. 1-17
Author(s):  
Ju-Mok Oh ◽  
Yunjae Kim ◽  
Kyung-Won Hwang
Keyword(s):  
Explicit Formula ◽  
Generating Function ◽  
Lattice Of Subgroups

We give an explicit formula for the number of chains of subgroups in the lattice of subgroups of the dicyclic groupB4nof order4nby finding its generating function of multivariables.

scholarly journals Explicit formula of a new class of q-Hermite-based Apostol-type polynomials and generalization

2020 ◽  
Vol 26 (4) ◽  
pp. 93-102
Author(s):  
Mouloud Goubi ◽  
Keyword(s):  
Explicit Formula ◽  
Present Article ◽  
Generating Function ◽  
New Class

The present article deals with a recent study of a new class of q-Hermite-based Apostol-type polynomials introduced by Waseem A. Khan and Divesh Srivastava. We give their explicit formula and study a generalized class depending in any q-analog generating function.

scholarly journals Special Cases of the Parking Functions Conjecture and Upper-Triangular Matrices

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Paul Levande
Keyword(s):  
Generating Function ◽  
Hilbert Series ◽  
Parking Functions ◽  
Combinatorial Interpretation ◽  
Forthcoming Article ◽  
Triangular Matrices ◽  
Upper Triangular Matrices ◽  
Diagonal Harmonics ◽  
Special Cases ◽  
Bivariate Generating Function

International audience We examine the $q=1$ and $t=0$ special cases of the parking functions conjecture. The parking functions conjecture states that the Hilbert series for the space of diagonal harmonics is equal to the bivariate generating function of $area$ and $dinv$ over the set of parking functions. Haglund recently proved that the Hilbert series for the space of diagonal harmonics is equal to a bivariate generating function over the set of Tesler matrices–upper-triangular matrices with every hook sum equal to one. We give a combinatorial interpretation of the Haglund generating function at $q=1$ and prove the corresponding case of the parking functions conjecture (first proven by Garsia and Haiman). We also discuss a possible proof of the $t = 0$ case consistent with this combinatorial interpretation. We conclude by briefly discussing possible refinements of the parking functions conjecture arising from this research and point of view. $\textbf{Note added in proof}$: We have since found such a proof of the $t = 0$ case and conjectured more detailed refinements. This research will most likely be presented in full in a forthcoming article. On examine les cas spéciaux $q=1$ et $t=0$ de la conjecture des fonctions de stationnement. Cette conjecture déclare que la série de Hilbert pour l'espace des harmoniques diagonaux est égale à la fonction génératrice bivariée (paramètres $area$ et $dinv$) sur l'ensemble des fonctions de stationnement. Haglund a prouvé récemment que la série de Hilbert pour l'espace des harmoniques diagonaux est égale à une fonction génératrice bivariée sur l'ensemble des matrices de Tesler triangulaires supérieures dont la somme de chaque équerre vaut un. On donne une interprétation combinatoire de la fonction génératrice de Haglund pour $q=1$ et on prouve le cas correspondant de la conjecture dans le cas des fonctions de stationnement (prouvé d'abord par Garsia et Haiman). On discute aussi d'une preuve possible du cas $t=0$, cohérente avec cette interprétation combinatoire. On conclut en discutant brièvement les raffinements possibles de la conjecture des fonctions de stationnement de ce point de vue. $\textbf{Note ajoutée sur épreuve}$: j'ai trouvé depuis cet article une preuve du cas $t=0$ et conjecturé des raffinements possibles. Ces résultats seront probablement présentés dans un article ultérieur.

PARTITIONS WITH AN ARBITRARY NUMBER OF SPECIFIED DISTANCES

2019 ◽  
Vol 101 (1) ◽  
pp. 35-39 ◽  
Author(s):  
BERNARD L. S. LIN
Keyword(s):  
Explicit Formula ◽  
Generating Function ◽  
Generating Functions ◽  
Arbitrary Number ◽  
Positive Integers

For positive integers $t_{1},\ldots ,t_{k}$, let $\tilde{p}(n,t_{1},t_{2},\ldots ,t_{k})$ (respectively $p(n,t_{1},t_{2},\ldots ,t_{k})$) be the number of partitions of $n$ such that, if $m$ is the smallest part, then each of $m+t_{1},m+t_{1}+t_{2},\ldots ,m+t_{1}+t_{2}+\cdots +t_{k-1}$ appears as a part and the largest part is at most (respectively equal to) $m+t_{1}+t_{2}+\cdots +t_{k}$. Andrews et al. [‘Partitions with fixed differences between largest and smallest parts’, Proc. Amer. Math. Soc.143 (2015), 4283–4289] found an explicit formula for the generating function of $p(n,t_{1},t_{2},\ldots ,t_{k})$. We establish a $q$-series identity from which the formulae for the generating functions of $\tilde{p}(n,t_{1},t_{2},\ldots ,t_{k})$ and $p(n,t_{1},t_{2},\ldots ,t_{k})$ can be obtained.

INTERSECTION THEORY ON SYMPLECTIC QUOTIENTS OF PRODUCTS OF SPHERES

2001 ◽  
Vol 12 (01) ◽  
pp. 97-111 ◽  
Author(s):  
TATSURU TAKAKURA
Keyword(s):  
Explicit Formula ◽  
Generating Function ◽  
Cohomology Ring ◽  
Geometric Quantization ◽  
Intersection Theory ◽  
Symplectic Reduction ◽  
Poincare Duality ◽  
Diagonal Action ◽  
Symplectic Quotients ◽  
Products Of Spheres

We present an explicit formula for cohomology intersection pairings on an arbitrary smooth symplectic quotient of products of 2-spheres, by the standard diagonal action of SO3, without using known results on relations in the cohomology ring. By the Poincaré duality, it contains all the information enough to recover the structure of the cohomology ring. Our method is based on the commutativity of geometric quantization and symplectic reduction, originating from a conjecture of Guillemin-Sternberg. In fact, it enables us to derive a formula for the generating function of the intersection pairings.

On a class of two-dimensional nearest-neighbour random walks

1994 ◽  
Vol 31 (A) ◽  
pp. 207-237 ◽  
Author(s):  
J. W. Cohen
Keyword(s):  
Random Walks ◽  
State Space ◽  
Generating Function ◽  
The State ◽  
Nearest Neighbour ◽  
North East ◽  
The North ◽  
One Step ◽  
Bivariate Generating Function ◽  
Positive Recurrent

For positive recurrent nearest-neighbour, semi-homogeneous random walks on the lattice {0, 1, 2, …} X {0, 1, 2, …} the bivariate generating function of the stationary distribution is analysed for the case where one-step transitions to the north, north-east and east at interior points of the state space all have zero probability. It is shown that this generating function can be represented by meromorphic functions. The construction of this representation is exposed for a variety of one-step transition vectors at the boundary points of the state space.

Sign in / Sign up

Export Citation Format

Share Document