Truncated determinants and the refined enumeration of Alternating Sign Matrices and Descending Plane Partitions
نویسنده
چکیده
In these notes, we will be mainly focussing on the proof of the so-called ASM-DPP conjecture of Mills, Robbins and Rumsey [22] which relates refined enumerations of Alternating Sign Matrices (ASM) and Descending Plane Partitions (DPP). ASMs were introduced by Mills, Robbins and Rumsey [24] in their study of Dodgsons condensation algorithm for the evaluation of determinants. DPPs were introduced by Andrews [1] while attempting to prove a conjectured formula for the generating function of cyclically symmetric plane partitions.
منابع مشابه
On the weighted enumeration of alternating sign matrices and descending plane partitions
We prove a conjecture of Mills, Robbins and Rumsey [Alternating sign matrices and descending plane partitions, J. Combin. Theory Ser. A 34 (1983), 340–359] that, for any n, k, m and p, the number of n × n alternating sign matrices (ASMs) for which the 1 of the first row is in column k + 1 and there are exactly m −1’s and m+ p inversions is equal to the number of descending plane partitions (DPP...
متن کاملAlternating Sign Matrices and Descending Plane Partitions
An alternating sign matrix is a square matrix such that (i) all entries are 1,-1, or 0, (ii) every row and column has sum 1, and (iii) in every row and column the nonzero entries alternate in sign. Striking numerical evidence of a connection between these matrices and the descending plane partitions introduced by Andrews (Invent. Math. 53 (1979), 193-225) have been discovered, but attempts to p...
متن کاملA doubly-refined enumeration of alternating sign matrices and descending plane partitions
It was shown recently by the authors that, for any n, there is equality between the distributions of certain triplets of statistics on n × n alternating sign matrices (ASMs) and descending plane partitions (DPPs) with each part at most n. The statistics for an ASM A are the number of generalized inversions in A, the number of −1’s in A and the number of 0’s to the left of the 1 in the first row...
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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