Abstract
We present a bijection between permutation matrices and descending plane partitions without special parts, which respects two of the statistics considered by Mills, Robbins and Rumsey, and an additional statistic considered by Behrend, Di Francesco and Zinn–Justin.
Highlights
IntroductionOf descending plane partitions with parts not exceeding n (let us denote the set of these objects by Dn)
It is a well–known fact [11, 5, 4] that the enumeration of descending plane partitions with parts not exceeding n and of alternating sign matrices of dimension n gives the same number: n∏−1 (3j + 1)! |An| = |Dn| = . (n + j)! (1) j=0the electronic journal of combinatorics 27(1) (2020), #P1.391.1 Search for a bijection It appears to be quite difficult to find some “natural” bijection Φ : An → Dn.there are two additional informations which might help in the search for such bijection:
Of descending plane partitions with parts not exceeding n
Summary
Of descending plane partitions with parts not exceeding n (let us denote the set of these objects by Dn). Of alternating sign matrices of dimension n (let us denote the set of these objects by An) gives the same number: n∏−1 (3j + 1)!. – descending plane partitions with statistic s = 0 (let us denote this set by Dn0), which are much simpler to understand and for which it is, quite easy to give “natural” bijections (see below). The purpose of this note is to present a simple bijection Ψ which respects the triplet (p, i, q) of statistics in the sense of (2): The construction of Ψ relies on the representation of descending plane partitions as families of non–intersecting lattice paths and on a certain “visualization” of the statistic i (as number of certain entries) for matrices)
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