Gustafson and Milne proved an identity on the Schur function indexed by a partition of the form (λ 1 − n + k, λ 2 − n + k, . . . , λ k − n + k). On the other hand, Fehér, Némethi and Rimányi found an identity on the Schur function indexed by a partition of the form (m − k, . . . , m − k, λ 1 , . . . , λ k ). Fehér, Némethi and Rimányi gave a geometric explanation of their identity, and they raised the question of finding a combinatorial proof. In this paper, we establish a Gustafson-Milne type identity as well as a Fehér-Némethi-Rimányi type identity for factorial Grothendieck polynomials. Specializing a factorial Grothendieck polynomial to a Schur function, we obtain a combinatorial proof of the Fehér-Némethi-Rimányi identity.

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The notion of a barely set-valued semistandard Young tableau was introduced by Reiner, Tenner and Yong in their study of the probability distribution of edges in the Young lattice of partitions. Given a partition λ and a positive integer k, let BSSYT(λ, k) (respectively, SYT(λ, k)) denote the set of barely setvalued semistandard Young tableaux (respectively, ordinary semistandard Young tableaux) of shape λ with entries in row i not exceeding k + i. In the case when λ is a rectangular staircase partition δ d (b a ), Reiner, Tenner and Yong conjectured that |BSSYT(λ, k)| = kab(d−1) (a+b) |SYT(λ, k)|. In this paper, we establish a connection between barely set-valued tableaux and reverse plane partitions with designated corners. We show that for any shape λ, the expected jaggedness of a subshape of λ under the weak probability distribution can be expressed as 2|BSSYT(λ,k)| k|SYT(λ,k)| . On the other hand, when λ is a balanced shape with r rows and c columns, Chan, Haddadan, Hopkins and Moci proved that the expected jaggedness of a subshape in λ under the weak distribution equals 2rc/(r + c). Hence, for a balanced shape λ with r rows and c columns, we establish the relation that |BSSYT(λ, k)| = krc (r+c) |SYT(λ, k)|. Since a rectangular staircase shape δ d (b a ) is a balanced shape, we confirm the conjecture of Reiner, Tenner and Yong.

Lam, Lee and Shimozono introduced the structure of bumpless pipedreams in their study of back stable Schubert calculus. They found that a specific family of bumpless pipedreams, called EG-pipedreams, can be used to interpret the Edelman-Greene coefficients appearing in the expansion of a Stanley symmetric function in the basis of Schur functions. It is well known that the Edelman-Greene coefficients can also be interpreted in terms of reduced word tableaux for permutations. Lam, Lee and Shimozono proposed the problem of finding a shape preserving bijection between reduced word tableaux for a permutation w and EG-pipedreams of w. In this paper, we construct such a bijection. The key ingredients are two new developed isomorphic tree structures associated to w: the modified Lascoux-Schützenberger tree of w and the Edelman-Greene tree of w. Using the Little map, we show that the leaves in the modified Lascoux-Schützenberger of w are in bijection with the reduced word tableaux for w. On the other hand, applying the droop operation on bumpless pipedreams also introduced by Lam, Lee and Shimozono, we show that the leaves in the Edelman-Greene tree of w are in bijection with the EG-pipedreams of w. This allows us to establish a shape preserving one-to-one correspondence between reduced word tableaux for w and EG-pipedreams of w.

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