scholarly journals Records in geometrically distributed words: Sum of positions

2008 ◽  
Vol 2 (2) ◽  
pp. 234-240 ◽  
Author(s):  
Helmut Prodinger
Keyword(s):  
Generating Function ◽  
Random Permutations ◽  
Bivariate Generating Function

Kortchemski introduced a new parameter for random permutations: the sum of the positions of the records. We investigate this parameter in the context of random words, where the letters are obtained by geometric probabilities. We find a relation for a bivariate generating function, from which we can obtain the expectation, exactly and asymptotically. In principle, one could get all moments from it, but the computations would be huge.

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.

The Bivariate Generating Function and Two Problems in Discrete Stochastic Processes

SIAM Review ◽  
1962 ◽  
Vol 4 (2) ◽  
pp. 105-114 ◽  
Author(s):  
Irving Gerst
Keyword(s):  
Stochastic Processes ◽  
Generating Function ◽  
Bivariate Generating Function

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.

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 Towards the Distribution of the Size of a Largest Planar Matching and Largest Planar Subgraph in Random Bipartite Graphs

10.37236/859 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Marcos Kiwi ◽  
Martin Loebl
Keyword(s):  
Generating Function ◽  
Maximum Size ◽  
Pattern Avoidance ◽  
Combinatorial Identities ◽  
Young Tableaux ◽  
Color Class ◽  
Lattice Walks ◽  
Type Property ◽  
Bivariate Generating Function ◽  
Standard Young Tableaux

We address the following question: When a randomly chosen regular bipartite multi–graph is drawn in the plane in the "standard way", what is the distribution of its maximum size planar matching (set of non–crossing disjoint edges) and maximum size planar subgraph (set of non–crossing edges which may share endpoints)? The problem is a generalization of the Longest Increasing Sequence (LIS) problem (also called Ulam's problem). We present combinatorial identities which relate the number of $r$-regular bipartite multi–graphs with maximum planar matching (maximum planar subgraph) of at most $d$ edges to a signed sum of restricted lattice walks in ${\Bbb Z}^d$, and to the number of pairs of standard Young tableaux of the same shape and with a "descend–type" property. Our results are derived via generalizations of two combinatorial proofs through which Gessel's identity can be obtained (an identity that is crucial in the derivation of a bivariate generating function associated to the distribution of the length of LISs, and key to the analytic attack on Ulam's problem). Finally, we generalize Gessel's identity. This enables us to count, for small values of $d$ and $r$, the number of $r$-regular bipartite multi-graphs on $n$ nodes per color class with maximum planar matchings of size $d$.Our work can also be viewed as a first step in the study of pattern avoidance in ordered bipartite multi-graphs.

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.

scholarly journals Analysis of the asymmetrical shortest two-server queueing model

1998 ◽  
Vol 11 (2) ◽  
pp. 115-162 ◽  
Author(s):  
J. W. Cohen
Keyword(s):  
Generating Function ◽  
Analytic Solution ◽  
Model Parameters ◽  
Queueing Model ◽  
Starting Point ◽  
Poisson Arrivals ◽  
Queue Lengths ◽  
Bivariate Generating Function ◽  
Queueing Processes

This study presents the analytic solution for an asymmetrical two-server queueing model for arriving customers joining the shorter queue for the case of Poisson arrivals and negative exponentially distributed service times. The bivariate generating function of the stationary joint distribution of the queue lengths is explicitly determined.The determination of this bivariate generating function requires a construction of four generating functions. It is shown that each of these functions is the sum of a polynomial and a meromorphic function. The poles and residues at the poles of the meromorphic functions can be simply calculated recursively; the coefficients of the polynomials are easily found, in particular, if the asymmetry in the model parameters is not excessively large. The starting point for the asymptotic analysis for the queue lengths is obtained. The approach developed in the present study is applicable to a larger class of random walks modeling asymmetrical two-dimensional queueing processes.

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.

scholarly journals The expected variation of random bounded integer sequences of finite length

2005 ◽  
Vol 2005 (14) ◽  
pp. 2277-2285
Author(s):  
Rudolfo Angeles ◽  
Don Rawlings ◽  
Lawrence Sze ◽  
Mark Tiefenbruck
Keyword(s):  
Generating Function ◽  
Finite Length ◽  
Random Permutations ◽  
Integer Sequences ◽  
Mean And Variance ◽  
The Mean

From the enumerative generating function of an abstract adjacency statistic, we deduce the mean and variance of the variation on random permutations, rearrangements, compositions, and bounded integer sequences of finite length.

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