Refined Enumerations of Totally Symmetric Self-Complementary Plane Partitions and Lattice Path Combinatorics
نویسنده
چکیده
This article is a short explanation of some of the results obtained in my papers “On refined enumerations of totally symmetric self-complementary plane partitions I, II”. We give Pfaffian expressions for some of the conjectures in the paper “Self-complementary totally symmetric plane partitions” (J. Combin. Theory Ser. A 42, 277–292) by Mills, Robbins and Rumsey, using the lattice path method.
منابع مشابه
A Connection between Alternating Sign Matrices and Totally Symmetric Self-Complementary Plane Partitions
We give a lattice path interpretation for totally symmetric self-complementary plane partitions. This is a first step in solving the long standing problem of enumerating such plane partitions. Another outstanding problem in enumerative combinatorics is the search for a bijection between alternating sign matrices and totally symmetric self-complementary plane partitions. From the lattice path in...
متن کاملOn refined enumerations of totally symmetric self-complementary plane partitions I
Abstract In this paper we give Pfaffian expressions and constant term identities for three conjectures (i.e. Conjecture 2, Conjecture 3 and Conjecture 7) by Mills, Robbins and Rumsey in the paper “Self-complementary totally symmetric plane partitions” J. Combin. Theory Ser. A 42, 277–292) concerning the refined enumeration problems of totally symmetric self-complementary plane partitions. We al...
متن کاملOn the Doubly Refined Enumeration of Alternating Sign Matrices and Totally Symmetric Self-Complementary Plane Partitions
We prove the equality of doubly refined enumerations of Alternating Sign Matrices and of Totally Symmetric Self-Complementary Plane Partitions using integral formulae originating from certain solutions of the quantum Knizhnik– Zamolodchikov equation. The authors thank N. Kitanine for discussions, and J.-B. Zuber for a careful reading of the manuscript. PZJ was supported by EU Marie Curie Resear...
متن کاملOn refined enumerations of totally symmetric self-complementary plane partitions II
In this paper we settle a weak version of a conjecture (i.e. Conjecture 6) by Mills, Robbins and Rumsey in the paper “Self-complementary totally symmetric plane partitions” J. Combin. Theory Ser. A 42, 277–292. In other words we show that the number of shifted plane partitions invariant under the involution γ is equal to the number of alternating sign matrices invariant under the vertical flip....
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