scholarly journals Bijections between Lattice Paths and Plane Partitions

2011 ◽  
Vol 01 (03) ◽  
pp. 108-115 ◽  
Author(s):  
Mateus Alegri ◽  
Eduardo Henrique de Mattos Brietzke ◽  
José Plínio de Oliveira Santos ◽  
Robson da Silva
Keyword(s):  
Lattice Paths ◽  
Plane Partitions

scholarly journals Signed Lozenge Tilings

10.37236/5579 ◽  
2017 ◽  
Vol 24 (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.

scholarly journals A triple product formula for plane partitions derived from biorthogonal polynomials

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Shuhei Kamioka
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Laguerre Polynomials ◽  
Product Formula ◽  
Triple Product ◽  
Lattice Paths ◽  
Plane Partitions ◽  
International Audience ◽  
Biorthogonal Polynomials ◽  
Bounded Size

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.

scholarly journals Combinatorics on lattice paths in strips

2021 ◽  
Vol 94 ◽  
pp. 103310
Author(s):  
Nancy S.S. Gu ◽  
Helmut Prodinger
Keyword(s):  
Lattice Paths

scholarly journals Plane partitions of shifted double staircase shape

2021 ◽  
Vol 183 ◽  
pp. 105486
Author(s):  
Sam Hopkins ◽  
Tri Lai
Keyword(s):  
Plane Partitions
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