Related papers: Bruhat intervals as rooks on skew Ferrers boards
We define $q$-poly-Bernoulli polynomials $B_{n,\rho,q}^{(k)}(z)$ with a parameter $\rho$, $q$-poly-Cauchy polynomials of the first kind $c_{n,\rho,q}^{(k)}(z)$ and of the second kind $\widehat c_{n,\rho,q}^{(k)}(z)$ with a parameter $\rho$…
Generalized flag manifolds endowed with the Bruhat-Poisson bracket are compact Poisson homogeneous spaces, whose decompositions in symplectic leaves coincide with their stratifications in Schubert cells. In this note it is proved that the…
For words in the variables $X$ and $Y$ satisfying the commutation relation of the $q$-deformed generalized Ore algebra, $XY-qYX= \mu I + \nu Y$, we show that the corresponding normal ordering coefficients can be given an interpretation in…
We give an explicit and simple construction of the incidence graph for the integral cohomology of real Grassmann manifold Gr(k,n) in terms of the Young diagrams filled with the letter q in checkered pattern. It turns out that there are two…
The paper concerns the coarse flag Hilbert-Poincar\'e series of Maglione-Voll in the case of the braid arrangement associated to the symmetric group. We explicitly construct a companion statistic $\operatorname{ino} : \mathfrak{S}_{n+1}…
Two well-known polytopes whose vertices are indexed by permutations in the symmetric group $\mathfrak{S}_n$ are the permutohedron $P_n$ and the Birkhoff polytope $B_n$. We consider polytopes $P_n(\Pi)$ and $B_n(\Pi)$, whose vertices…
In 2011, Barot and Marsh provided an explicit construction of presentation of a finite Weyl group $W$ by any quiver mutation-equivalent to an orientation of a Dynkin diagram with Weyl group $W$. The construction was extended by the authors…
We study the algebraic aspects of equivariant quantum cohomology algebra of the flag manifold. We introduce and study the quantum double Schubert polynomials, which are the Lascoux-Schutzenberger type representatives of the equivariant…
In our previous work we have introduced an analogue of Robinson-Schensted-Knuth correspondence for Schubert calculus of the complete flag varieties. The objects inserted are certain biwords, the outcomes of insertion are bumpless pipe…
In this paper we study those generic intervals in the Bruhat order of the symmetric group that are isomorphic to the principal order ideal of a permutation w, and consider when the minimum and maximum elements of those intervals are related…
Poincar\'e profiles are a family of analytically defined coarse invariants, which can be used as obstructions to the existence of coarse embeddings between metric spaces. In this paper we calculate the Poincar\'e profiles of all connected…
The Stirling numbers of type $B$ of the second kind count signed set partitions. In this paper we provide new combinatorial and analytical identities regarding these numbers as well as Broder's $r$-version of these numbers. Among these…
Motivated by recent analytical and numerical work on two- and three-dimensional convection with imposed spatial periodicity, we analyse three examples of bifurcations from a continuous group orbit of spatio-temporally symmetric periodic…
Recently, Naruse discovered a hook length formula for the number of standard Young tableaux of a skew shape. Morales, Pak and Panova found two $q$-analogs of Naruse's hook length formula over semistandard Young tableaux (SSYTs) and reverse…
In this study, we investigate the binomial edge ring associated with the skew Ferrers diagram. By employing Sagbi basis theory, we construct a quadratic Gr\"{o}bner basis for its defining ideal. As an application, we prove that this ring is…
The introduction of bars in-between blocks of an ordered set partition(preferential arrangement) results in a barred ordered set partition(barred preferential arrange- ment). Having the restriction that some blocks of barred preferential…
We study the semiclassical asymptotics of twisted algebras induced by symbol correspondences for quark systems ($SU(3)$-symmetric mechanical systems) as defined in our previous paper [3]. The linear span of harmonic functions on (co)adjoint…
We characterize those strings whose suffix arrays are based on arithmetic progressions, in particular, arithmetically progressed permutations where all pairs of successive entries of the permutation have the same difference modulo the…
We access the edge of Gaussian beta ensembles with one spike by analyzing high powers of the associated tridiagonal matrix models. In the classical cases beta=1, 2, 4, this corresponds to studying the fluctuations of the largest eigenvalues…
We propose and develop a theory of Ferrers diagrams and their $q$-rook polynomials solely based on their diagonals. We show that the cardinalities of the diagonals of a Ferrers diagram are equivalent information to their rook numbers,…