10 research outputs found
Gog and Magog triangles, and the Schutzenberger involution
We describe an approach to finding a bijection between Alternating Sign
Matrices and Totally Symmetric Self-Complementary Plane Partitions, which is
based on the Schutzenberger involution. In particular we give an explicit
bijection between Gog and Magog trapezoids with two diagonals.Comment: 17 page
Combinatorial Hopf algebra structure on packed square matrices
We construct a new bigraded Hopf algebra whose bases are indexed by square
matrices with entries in the alphabet , , without
null rows or columns. This Hopf algebra generalizes the one of permutations of
Malvenuto and Reutenauer, the one of -colored permutations of Novelli and
Thibon, and the one of uniform block permutations of Aguiar and Orellana. We
study the algebraic structure of our Hopf algebra and show, by exhibiting
multiplicative bases, that it is free. We moreover show that it is self-dual
and admits a bidendriform bialgebra structure. Besides, as a Hopf subalgebra,
we obtain a new one indexed by alternating sign matrices. We study some of its
properties and algebraic quotients defined through alternating sign matrices
statistics.Comment: 35 page
Gog and GOGAm pentagons
International audienceWe consider the problem of finding a bijection between the sets of alternating sign matrices and of totally symmetric self complementary plane partitions, which can be reformulated using Gog and Magog triangles. In a previous work we introduced GOGAm triangles , which are images of Magog triangles by the SchĂĽtzenberger involution. In this paper we introduce Gog and GOGAm pentagons. We conjecture that they are equienumerated. We provide some numerical evidence as well as an explicit bijection in the case when they have one or two diagonals. We also consider some interesting statistics on Gog and Magog triangles
Gog, Magog and SchĂĽtzenberger II: left trapezoids
International audienceWe are interested in finding an explicit bijection between two families of combinatorial objects: Gog and Magog triangles. These two families are particular classes of Gelfand-Tsetlin triangles and are respectively in bijection with alternating sign matrices (ASM) and totally symmetric self complementary plane partitions (TSSCPP). For this purpose, we introduce left Gog and GOGAm trapezoids. We conjecture that these two families of trapezoids are equienumerated and we give an explicit bijection between the trapezoids with one or two diagonals.Nous nous intéressons ici à trouver une bijection explicite entre deux familles d’objets combinatoire: les triangles Gog et Magog. Ces deux familles d’objets sont des classes particulières des triangles de Gelfand-Tsetlin et sont respectivement en bijection avec les matrices à signes alternants (ASMs) et les partitions planes totalement symétriques auto-complémentaires (TSSCPPs). Pour ce faire, nous introduisons les Gog et les GOGAm trapèzes gauches. Nous conjecturons que ces deux familles de trapèzes sont équipotents et nous donnons une bijection explicite entre ces trapèzes à une et deux lignes
Gog, Magog and SchĂĽtzenberger II: left trapezoids
We are interested in finding an explicit bijection between two families of combinatorial objects: Gog and Magog triangles. These two families are particular classes of Gelfand-Tsetlin triangles and are respectively in bijection with alternating sign matrices (ASM) and totally symmetric self complementary plane partitions (TSSCPP). For this purpose, we introduce left Gog and GOGAm trapezoids. We conjecture that these two families of trapezoids are equienumerated and we give an explicit bijection between the trapezoids with one or two diagonals
On the computation of the Möbius transform
International audienceThe Möbius transform is a crucial transformation into the Boolean world; it allows to change the Boolean representation between the True Table and Algebraic Normal Form. In this work, we introduce a new algebraic point of view of this transformation based on the polynomial form of Boolean functions. It appears that we can perform a new notion: the Möbius computation variable by variable and new computation properties. As a consequence, we propose new algorithms which can produce a huge speed up of the Möbius computation for sub-families of Boolean function. Furthermore we compute directly the Möbius transformation of some particular Boolean functions. Finally, we show that for some of them the Hamming weight is directly related to the algebraic degree of specific factors