Enumeration of bigrassmannian permutations below a permutation in Bruhat order

In theory of Coxeter groups, bigrassmannian elements are well known as elements which have precisely one left descent and precisely one right descent. In this article, we prove formulas on enumeration of bigrassmannian permutations weakly below a permutation in Bruhat order in the symmetric groups. For the proof, we use equivalent characterizations of bigrassmannian permutations by Lascoux-Schutzenberger and Reading.Comment: 7 pages

ENUMERATIVE COMBINATORICS ON DETERMINANTS AND SIGNED BIGRASSMANNIAN POLYNOMIALS

As an application of linear algebra for enumerative combinatorics, we introduce two new ideas, signed bigrassmannian polynomials and bigrassmannian determinant. First, a signed bigrassmannian polynomial is a variant of the statistic given by the number of bigrassmannian permutations below a permutation in Bruhat order as Reading suggested (2002) and afterward the author developed (2011). Second, bigrassmannian determinant is a q-analog of the determinant with respect to our statistic. It plays a key role for a determinantal expression of those polynomials. We further show that bigrassmannian determinant satisfies weighted condensation as a generalization of Dodgson, Jacobi-Desnanot and Robbins-Rumsey (1986)

On comparability of bigrassmannian permutations

Let Sn and Gn denote the respective sets of ordinary and bigrassmannian (BG) permutations of order n, and let (Gn,≤) denote the Bruhat ordering permutation poset. We study the restricted poset (Bn,≤), first providing a simple criterion for comparability. This criterion is used to show that that the poset is connected, to enumerate the saturated chains between elements, and to enumerate the number of maximal elements below r fixed elements. It also quickly produces formulas for β(ω) (α(ω), respectively), the number of BG permutations weakly below (weakly above, respectively) a fixed ω ∈ Bn, and is used to compute the Mo¨bius function on any interval in Bn. We then turn to a probabilistic study of β = β(ω) (α = α(ω) respectively) for the uniformly random ω ∈ Bn. We show that α and β are equidistributed, and that β is of the same order as its expectation with high probability, but fails to concentrate about its mean. This latter fact derives from the limiting distribution of β/n3. We also compute the probability that randomly chosen BG permutations form a 2- or 3-element multichain

Prism tableaux and alternating sign matrices

A. Lascoux and M.-P. Schutzenberger introduced Schubert polynomials to study the cohomology ring of the complete flag variety Fl(C^n). Each Schubert polynomial corresponds to the class defined by a Schubert variety X_w in Fl(C^n). Schubert polynomials are indexed by elements of the symmetric group and form a basis of the ring Z[x1,x2,...]. The expansion of the product of two Schubert polynomials in the Schubert basis has been of particular interest. The structure coefficients are known to be nonnegative integers. As of yet, there are only combinatorial formulas for these coefficients in special cases, such as the Littlewood-Richardson rule for multiplying Schur polynomials. Schur polynomials form a basis of the ring of symmetric polynomials. They have a combinatorial formula as a weighted sum over semistandard tableaux. In joint work with A. Yong, the author introduced prism tableaux. A prism tableau consists of a tuple of tableaux, positioned within an ambient grid. With A. Yong, the author gave a formula for Schubert polynomials using prism tableaux. We continue the study of prism tableaux, detailing their connection to the poset of alternating sign matrices (ASMs). Schubert polynomials can be interpreted as multidegrees of the matrix Schubert varieties of Fulton. We study a more general class of determinantal varieties, indexed by ASMs. More generally, one can consider subvarieties of the space of n by n matrices cut out by imposing rank conditions on maximal northwest submatrices. We show that, up to an affine factor, such a variety is isomorphic to an ASM variety. The multidegrees of ASM varieties can be expressed as a sum over prism tableaux. In joint work with A. Yong and R. Rimanyi, the author studies representations of quivers and their connection to the dilogarithm identities of M. Reineke. We give a bijective proof to establish an identity of generating series. This bijection uses a generalization of Durfee squares. From this identity, we give a new proof of M. Reineke's identities in type A

Schubert Numbers

This thesis discusses intersections of the Schubert varieties in the flag variety associated to a vector space of dimension n. The Schubert number is the number of irreducible components of an intersection of Schubert varieties. Our main result gives the lower bound on the maximum of Schubert numbers. This lower bound varies quadratically with n. The known lower bound varied only linearly with n. We also establish a few technical results of independent interest in the combinatorics of strong Bruhat orders

Convex (0,1)(0,1)-Matrices and Their Epitopes

We investigate $(0,1)$-matrices that are {\em convex}, which means that the ones are consecutive in every row and column. These matrices occur in discrete tomography. The notion of ranked essential sets, known for permutation matrices, is extended to convex sets. We show a number of results for the class \mc{C}(R,S) of convex matrices with given row and column sum vectors $R$ and $S$. Also, it is shown that the ranked essential set uniquely determines a matrix in \mc{C}(R,S)