Enumeration of lattice paths and generating functions for skew plane partitions

International audience A new triple product formulae for plane partitions with bounded size of parts is derived from a combinato- rial interpretation of biorthogonal polynomials in terms of lattice paths. Biorthogonal polynomials which generalize the little q-Laguerre polynomials are introduced to derive a new triple product formula which recovers the classical generating function in a triple product by MacMahon and generalizes the trace-type generating functions in double products by Stanley and Gansner.

Lattice Paths, Sampling Without Replacement, and Limiting Distributions

The Electronic Journal of Combinatorics ◽

10.37236/156 ◽

2009 ◽

Vol 16 (1) ◽

Author(s):

M. Kuba ◽

A. Panholzer ◽

H. Prodinger

Keyword(s):

Generating Functions ◽

Probability Distributions ◽

Random Variable ◽

Phase Changes ◽

Lattice Paths ◽

Sampling Without Replacement ◽

Sample Paths ◽

Quarter Plane ◽

Explicit Formulae ◽

Weighted Lattice Paths

In this work we consider weighted lattice paths in the quarter plane ${\Bbb N}_0\times{\Bbb N}_0$. The steps are given by $(m,n)\to(m-1,n)$, $(m,n)\to(m,n-1)$ and are weighted as follows: $(m,n)\to(m-1,n)$ by $m/(m+n)$ and step $(m,n)\to(m,n-1)$ by $n/(m+n)$. The considered lattice paths are absorbed at lines $y=x/t -s/t$ with $t\in{\Bbb N}$ and $s\in{\Bbb N}_0$. We provide explicit formulæ for the sum of the weights of paths, starting at $(m,n)$, which are absorbed at a certain height $k$ at lines $y=x/t -s/t$ with $t\in{\Bbb N}$ and $s\in{\Bbb N}_0$, using a generating functions approach. Furthermore these weighted lattice paths can be interpreted as probability distributions arising in the context of Pólya-Eggenberger urn models, more precisely, the lattice paths are sample paths of the well known sampling without replacement urn. We provide limiting distribution results for the underlying random variable, obtaining a total of five phase changes.

An extension to overpartitions of Rogers-Ramanujan identities for even moduli

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.3498 ◽

2006 ◽

Vol DMTCS Proceedings vol. AG,... (Proceedings) ◽

Author(s):

Sylvie Corteel ◽

Jeremy Lovejoy ◽

Olivier Mallet

Keyword(s):

Generating Functions ◽

Hypergeometric Series ◽

Basic Hypergeometric Series ◽

Lattice Paths ◽

Infinite Products ◽

International Audience

International audience We investigate class of well-poised basic hypergeometric series $\tilde{J}_{k,i}(a;x;q)$, interpreting these series as generating functions for overpartitions defined by multiplicity conditions. We also show how to interpret the $\tilde{J}_{k,i}(a;1;q)$ as generating functions for overpartitions whose successive ranks are bounded, for overpartitions that are invariant under a certain class of conjugations, and for special restricted lattice paths. We highlight the cases $(a,q) \to (1/q,q)$, $(1/q,q^2)$, and $(0,q)$, where some of the functions $\tilde{J}_{k,i}(a;x;q)$ become infinite products. The latter case corresponds to Bressoud's family of Rogers-Ramanujan identities for even moduli.

Slicings of Parallelogram Polyominoes: Catalan, Schröder, Baxter, and Other Sequences

The Electronic Journal of Combinatorics ◽

10.37236/7375 ◽

2019 ◽

Vol 26 (3) ◽

Author(s):

Nicholas R. Beaton ◽

Mathilde Bouvel ◽

Veronica Guerrini ◽

Simone Rinaldi

Keyword(s):

Generating Functions ◽

Functional Equations ◽

Kernel Method ◽

Lattice Paths ◽

Schröder Numbers ◽

Special Cases ◽

Generating Tree

We provide a new succession rule (i.e. generating tree) associated with Schröder numbers, that interpolates between the known succession rules for Catalan and Baxter numbers. We define Schröder and Baxter generalizations of parallelogram polyominoes, called slicings, which grow according to these succession rules. In passing, we also exhibit Schröder subclasses of Baxter classes, namely a Schröder subset of triples of non-intersecting lattice paths, a new Schröder subset of Baxter permutations, and a new Schröder subset of mosaic floorplans. Finally, we define two families of subclasses of Baxter slicings: the $m$-skinny slicings and the $m$-row-restricted slicings, for $m \in \mathbb{N}$. Using functional equations and the kernel method, their generating functions are computed in some special cases, and we conjecture that they are algebraic for any $m$.

The major counting of nonintersecting lattice paths and generating functions for tableaux

Memoirs of the American Mathematical Society ◽

10.1090/memo/0552 ◽

1995 ◽

Vol 115 (552) ◽

pp. 0-0 ◽

Cited By ~ 12

Author(s):

C. Krattenthaler

Keyword(s):

Generating Functions ◽

Lattice Paths ◽

Nonintersecting Lattice Paths

Some Restricted Plane partitions and Associated Lattice Paths

International Journal of Mathematics Trends and Technology ◽

10.14445/22315373/ijmtt-v51p525 ◽

2017 ◽

Vol 51 (3) ◽

pp. 190-196

Author(s):

S.Be di ◽

Keyword(s):

Lattice Paths ◽

Plane Partitions

Vertically Constrained Motzkin-Like Paths Inspired by Bobbin Lace

The Electronic Journal of Combinatorics ◽

10.37236/7799 ◽

2019 ◽

Vol 26 (2) ◽

Cited By ~ 1

Author(s):

Veronika Irvine ◽

Stephen Melczer ◽

Frank Ruskey

Keyword(s):

Mathematical Model ◽

Generating Functions ◽

Finite Lattice ◽

Lattice Paths ◽

Change Direction ◽

Quarter Plane ◽

Directed Paths ◽

Schröder Paths ◽

Motzkin Paths

Inspired by a new mathematical model for bobbin lace, this paper considers finite lattice paths formed from the set of step vectors $\mathfrak{A}=$$\{\rightarrow,$ $\nearrow,$ $\searrow,$ $\uparrow,$ $\downarrow\}$ with the restriction that vertical steps $(\uparrow, \downarrow)$ cannot be consecutive. The set $\mathfrak{A}$ is the union of the well known Motzkin step vectors $\mathfrak{M}=$$\{\rightarrow,$ $\nearrow,$ $\searrow\}$ with the vertical steps $\{\uparrow, \downarrow\}$. An explicit bijection $\phi$ between the exhaustive set of vertically constrained paths formed from $\mathfrak{A}$ and a bisection of the paths generated by $\mathfrak{M}S$ is presented. In a similar manner, paths with the step vectors $\mathfrak{B}=$$\{\nearrow,$ $\searrow,$ $\uparrow,$ $\downarrow\}$, the union of Dyck step vectors and constrained vertical steps, are examined. We show, using the same $\phi$ mapping, that there is a bijection between vertically constrained $\mathfrak{B}$ paths and the subset of Motzkin paths avoiding horizontal steps at even indices. Generating functions are derived to enumerate these vertically constrained, partially directed paths when restricted to the half and quarter-plane. Finally, we extend Schröder and Delannoy step sets in a similar manner and find a bijection between these paths and a subset of Schröder paths that are smooth (do not change direction) at a regular horizontal interval.

Signed Lozenge Tilings

The Electronic Journal of Combinatorics ◽

10.37236/5579 ◽

2017 ◽

Vol 24 (1) ◽

Cited By ~ 1

Author(s):

D. Cook II ◽

Uwe Nagel

Keyword(s):

Perfect Matchings ◽

Lattice Paths ◽

New Method ◽

Plane Partitions ◽

Triangular Region ◽

Lozenge Tilings

It is well-known that plane partitions, lozenge tilings of a hexagon, perfect matchings on a honeycomb graph, and families of non-intersecting lattice paths in a hexagon are all in bijection. In this work we consider regions that are more general than hexagons. They are obtained by further removing upward-pointing triangles. We call the resulting shapes triangular regions. We establish signed versions of the latter three bijections for triangular regions. We first investigate the tileability of triangular regions by lozenges. Then we use perfect matchings and families of non-intersecting lattice paths to define two signs of a lozenge tiling. Using a new method that we call resolution of a puncture, we show that the two signs are in fact equivalent. As a consequence, we obtain the equality of determinants, up to sign, that enumerate signed perfect matchings and signed families of lattice paths of a triangular region, respectively. We also describe triangular regions, for which the signed enumerations agree with the unsigned enumerations.

$n$-color overpartitions, lattice paths, and multiple basic hypergeometric series

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.3643 ◽

2008 ◽

Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽

Author(s):

Olivier Mallet

Keyword(s):

Generating Functions ◽

Hypergeometric Series ◽

Basic Hypergeometric Series ◽

Combinatorial Identities ◽

Lattice Paths ◽

Multiple Series ◽

Infinite Products ◽

Nous Montrons ◽

International Audience ◽

Multiple Basic Hypergeometric Series

International audience We define two classes of multiple basic hypergeometric series $V_{k,t}(a,q)$ and $W_{k,t}(a,q)$ which generalize multiple series studied by Agarwal, Andrews, and Bressoud. We show how to interpret these series as generating functions for special restricted lattice paths and for $n$-color overpartitions with weighted difference conditions. We also point out that some specializations of our series can be written as infinite products, which leads to combinatorial identities linking $n$-color overpartitions with ordinary partitions or overpartitions. Nous définissons deux classes de séries hypergéométriques basiques multiples $V_{k,t}(a,q)$ et $W_{k,t}(a,q)$ qui généralisent des séries multiples étudiées par Agarwal, Andrews et Bressoud. Nous montrons comment interpréter ces séries comme les fonctions génératrices de chemins avec certaines restrictions et de surpartitions $n$-colorées vérifiant des conditions de différences pondérées. Nous remarquons aussi que certaines spécialisations de nos séries peuvent s'écrire comme des produits infinis, ce qui conduit à des identités combinatoires reliant les surpartitions $n$-colorées aux partitions ou surpartitions ordinaires.

Lattice Paths and Plane Partitions

Proofs and Confirmations ◽

10.1017/cbo9780511613449.004 ◽

1999 ◽

pp. 73-118

Keyword(s):

Lattice Paths ◽

Plane Partitions