Minimal Permutations and 2-Regular Skew Tableaux

Bouvel and Pergola introduced the notion of minimal permutations in the study of the whole genome duplication-random loss model for genome rearrangements. Let $\mathcal{F}_d(n)$ denote the set of minimal permutations of length $n$ with $d$ descents, and let $f_d(n)= |\mathcal{F}_d(n)|$. They derived that $f_{n-2}(n)=2^{n}-(n-1)n-2$ and $f_n(2n)=C_n$, where $C_n$ is the $n$-th Catalan number. Mansour and Yan proved that $f_{n+1}(2n+1)=2^{n-2}nC_{n+1}$. In this paper, we consider the problem of counting minimal permutations in $\mathcal{F}_d(n)$ with a prescribed set of ascents. We show that such structures are in one-to-one correspondence with a class of skew Young tableaux, which we call $2$-regular skew tableaux. Using the determinantal formula for the number of skew Young tableaux of a given shape, we find an explicit formula for $f_{n-3}(n)$. Furthermore, by using the Knuth equivalence, we give a combinatorial interpretation of a formula for a refinement of the number $f_{n+1}(2n+1)$.Comment: 19 page

A Construction of Coxeter Group Representations (II)

An axiomatic approach to the representation theory of Coxeter groups and their Hecke algebras was presented in [1]. Combinatorial aspects of this construction are studied in this paper. In particular, the symmetric group case is investigated in detail. The resulting representations are completely classified and include the irreducible ones.Comment: 24 pages, shorter background; to appear in J. Algebr

On the eigenvalues of Cayley graphs on the symmetric group generated by a complete multipartite set of transpositions

Given a finite simple graph \cG with $n$ vertices, we can construct the Cayley graph on the symmetric group $S_n$ generated by the edges of \cG, interpreted as transpositions. We show that, if \cG is complete multipartite, the eigenvalues of the Laplacian of \Cay(\cG) have a simple expression in terms of the irreducible characters of transpositions, and of the Littlewood-Richardson coefficients. As a consequence we can prove that the Laplacians of \cG and of \Cay(\cG) have the same first nontrivial eigenvalue. This is equivalent to saying that Aldous's conjecture, asserting that the random walk and the interchange process have the same spectral gap, holds for complete multipartite graphs.Comment: 29 pages. Includes modification which appear on the published version in J. Algebraic Combi

Crystal approach to affine Schubert calculus

We apply crystal theory to affine Schubert calculus, Gromov-Witten invariants for the complete flag manifold, and the positroid stratification of the positive Grassmannian. We introduce operators on decompositions of elements in the type-$A$ affine Weyl group and produce a crystal reflecting the internal structure of the generalized Young modules whose Frobenius image is represented by stable Schubert polynomials. We apply the crystal framework to products of a Schur function with a $k$-Schur function, consequently proving that a subclass of 3-point Gromov-Witten invariants of complete flag varieties for $\mathbb C^n$ enumerate the highest weight elements under these operators. Included in this class are the Schubert structure constants in the (quantum) product of a Schubert polynomial with a Schur function $s_\lambda$ for all $|\lambda^\vee|< n$. Another by-product gives a highest weight formulation for various fusion coefficients of the Verlinde algebra and for the Schubert decomposition of certain positroid classes.Comment: 42 pages; version to appear in IMR

Schur polynomials, banded Toeplitz matrices and Widom's formula

We prove that for arbitrary partitions $\mathbf{\lambda} \subseteq \mathbf{\kappa},$ and integers $0\leq c<r\leq n,$ the sequence of Schur polynomials $S_{(\mathbf{\kappa} + k\cdot \mathbf{1}^c)/(\mathbf{\lambda} + k\cdot \mathbf{1}^r)}(x_1,...,x_n)$ for $k$ sufficiently large, satisfy a linear recurrence. The roots of the characteristic equation are given explicitly. These recurrences are also valid for certain sequences of minors of banded Toeplitz matrices. In addition, we show that Widom's determinant formula from 1958 is a special case of a well-known identity for Schur polynomials

Some remarks on sign-balanced and maj-balanced posets

Let P be a poset with elements 1,2,...,n. We say that P is sign-balanced if exactly half the linear extensions of P (regarded as permutations of 1,2,...,n) are even permutations, i.e., have an even number of inversions. This concept first arose in the work of Frank Ruskey, who was interested in the efficient generation of all linear extensions of P. We survey a number of techniques for showing that posets are sign-balanced, and more generally, computing their "imbalance." There are close connections with domino tilings and, for certain posets, a "domino generalization" of Schur functions due to Carre and Leclerc. We also say that P is maj-balanced if exactly half the linear extensions of P have even major index. We discuss some similarities and some differences between sign-balanced and maj-balanced posets.Comment: 30 pages. Some inaccuracies in Section 3 have been corrected, and Conjecture 3.6 has been adde