Minor summation formula and a proof of Stanley’s open problem

Abstract

In the open problem session of the FPSAC'03, R.P. Stanley gave an open problem about a certain sum of the Schur functions (See [20]). The purpose of this paper is to give a proof of this open problem. The proof consists of three steps. At the first step we express the sum by a Pfaffian as an application of our minor summation formula ([8]

The Andrews–Stanley partition function and Al-Salam–Chihara polynomials

“…Especially, if we put i = 1 in (2.12), then we obtain the expansion formula along the first row: . It is essential that the weight ω(λ) can be expressed by a Pfaffian, which is a fact proved in [6]:…”

“…For the case when n is odd, perform the same operation on (5.17). To prove Theorems 5.2 and 5.3, we need to cite a lemma from [6]. (See Corollary 3.3 of [6] and Theorem 3.2 of [7].)…”

Pfaffians and representations of the symmetric group

“…where the product is the concatenation product, and where v \{v i , v n } means suppressing the components v i , v n inside v. Indeed, let S n/2 be the symmetric group which permutes the blocks [1,2], [3,4], . .…”

“…Thus, the evaluation of the Pfaffian of order 2m has been reduced to the evaluation of a determinant of order m. Ishikawa [4], Okada [18], and Ishikawa-Okada-Tagawa-Zeng [5] have given many generalizations of Sundquist's Pfaffian. Instead of using Plücker coordinates, they use specific determinants (which, of course, satisfy built-in Plücker relations).…”

“…Ishikawa [4], Okada [18], M. Ishikawa, S. Okada, H. Tagawa and J. Zeng [5] have given different generalizations of Sundquist's Pfaffian. We show how to connect their results to Th.6 and Th.7.…”