Bruhat intervals as rooks on skew Ferrers boards

We characterise the permutations pi such that the elements in the closed lower Bruhat interval [id,pi] of the symmetric group correspond to non-taking rook configurations on a skew Ferrers board. It turns out that these are exactly the permutations pi such that [id,pi] corresponds to a flag manifold defined by inclusions, studied by Gasharov and Reiner. Our characterisation connects the Poincare polynomials (rank-generating function) of Bruhat intervals with q-rook polynomials, and we are able to compute the Poincare polynomial of some particularly interesting intervals in the finite Weyl groups A_n and B_n. The expressions involve q-Stirling numbers of the second kind. As a by-product of our method, we present a new Stirling number identity connected to both Bruhat intervals and the poly-Bernoulli numbers defined by Kaneko.Comment: 16 pages, 9 figure

First Observations on Prefab Posets Whitney Numbers

We introduce a natural partial order in structurally natural finite subsets of the cobweb prefabs sets recently constructed by the present author. Whitney numbers of the second kind of the corresponding subposet which constitute Stirling like numbers triangular array are then calculated and the explicit formula for them is provided. Next, in the second construction we endow the set sums of prefabiants with such an another partial order that their Bell like numbers include Fibonacci triad sequences introduced recently by the present author in order to extend famous relation between binomial Newton coefficients and Fibonacci numbers onto the infinity of their relatives among whom there are also the Fibonacci triad sequences and binomial like coefficients (incidence coefficients included). The first partial order is F sequence independent while the second partial order is F sequence dependent where F is the so called admissible sequence determining cobweb poset by construction. An F determined cobweb posets Hasse diagram becomes Fibonacci tree sheathed with specific cobweb if the sequence F is chosen to be just the Fibonacci sequence. From the stand-point of linear algebra of formal series these are generating functions which stay for the so called extended coherent states of quantum physics. This information is delivered in the last section.Comment: 14 page

Heisenberg–Weyl algebra revisited: Combinatorics of words and paths

The Heisenberg–Weyl algebra, which underlies virtually all physical representations of Quantum Theory, is considered from the combinatorial point of view. We provide a concrete model of the algebra in terms of paths on a lattice with some decomposition rules. We also discuss the rook problem on the associated Ferrers board; this is related to the calculus in the normally ordered basis. From this starting point we explore a combinatorial underpinning of the Heisenberg–Weyl algebra, which offers novel perspectives, methods and applications

Closed expressions for averages of set partition statistics

In studying the enumerative theory of super characters' of the group of upper triangular matrices over a finite field we found that the moments (mean, variance and higher moments) of novel statistics on set partitions have simple closed expressions as linear combinations of shifted bell numbers. It is shown here that families of other statistics have similar moments. The coefficients in the linear combinations are polynomials in $n$. This allows exact enumeration of the moments for small $n$ to determine exact formulae for all $n$

Flag weak order on wreath products

A generating set for the wreath product \ZZ_r \wr S_n which leads to a nicely behaved weak order is presented, and properties of the resulting order are studied.Comment: 20 pages, 2 figures; corrected and added proofs and explanation