arXiv subjects
Publications and source records attributed to Eli Bagno.
Using Reiner's definition of Stirling numbers of type B of the second kind, we provide a 'balls into urns' approach for proving a generalization of a well-known identity concerning the classical Stirling numbers of the second kind: $x^n=\sum\limits_{k=0}^n{S(n,k)[x]_k}.$
Using Reiner's definition of Stirling numbers of the second kind in types $B$ and $D$, we generalize two well-known identities concerning the classical Stirling numbers of the second kind. The first identity relates them with Eulerian numbers and the second identity interprets them as entries in a transition matrix between the elements of two standard bases of the polynomial ring $R[x]$. Finally, we generalize these identities to the group of colored permutations $G_{m,n}$.
Gessel conjectured that the two-sided Eulerian polynomial, recording the common distribution of the descent number of a permutation and that of its inverse, has non-negative integer coefficients when expanded in terms of the gamma basis. This conjecture has been proved recently by Lin. We conjecture that an analogous statement holds for simple permutations, and use the substitution decomposition tree of a permutation (by repeated inflation) to show that this would imply the Gessel-Lin result. We provide supporting evidence for this stronger conjecture.
The block number of a permutation is the maximal number of components in its expression as a direct sum. We show that, for $321$-avoiding permutations, the set of left-to-right maxima has the same distribution when the block number is assumed to be $k$ as when the last descent of the inverse is assumed to be at position $n - k$. This result is analogous to the Foata-Schützenberger equi-distribution theorem, and implies that the quasi-symmetric generating function of descent set over $321$-avoiding permutations with a prescribed number of blocks is Schur-positive.
The depth statistic was defined by Petersen and Tenner for an element of an arbitrary Coxeter group in terms of factorizations of the element into a product of reflections. It can also be defined as the minimal cost, given certain prescribed edge weights, for a path in the Bruhat graph from the identity to an element. We present algorithms for calculating the depth of an element of a classical Coxeter group that yield simple formulas for this statistic. We use our algorithms to characterize elements having depth equal to length. These are the short-braid-avoiding elements. We also give a characterization of the elements for which the reflection length coincides with both the depth and the length. These are the boolean elements.
The group of alternating colored permutations is the natural analogue of the classical alternating group, inside the wreath product $\mathbb{Z}_r \wr S_n$. We present a 'Coxeter-like' presentation for this group and compute the length function with respect to that presentation. Then, we present this group as a covering of $\mathbb{Z}_{\frac{r}{2}} \wr S_n$ and use this point of view to give another expression for the length function. We also use this covering to lift several known parameters of $\mathbb{Z}_{\frac{r}{2}} \wr S_n$ to the group of alternating colored permutations.
We study the poset of Borel congruence classes of symmetric matrices ordered by containment of closures. We give a combinatorial description of this poset and calculate its rank function. We discuss the relation between this poset and the Bruhat poset of involutions of the symmetric group.
We define the excedence set and the excedance word on $G_{r,n}$, generalizing a work of Ehrenborg and Steingrimsson and use the inclusion-exclusion principle to calculate the number of colored permutations having a prescribed excedance word. We show some symmetric properties as Log concavity and unimodality of a specific sequence of excedance words.
The excedance number for S_n is known to have an Eulerian distribution. Nevertheless, the classical proof uses descents rather than excedances. We present a direct recursive proof which seems to be folklore and extend it to the colored permutation groups G_r,n. The generalized recursion yields some interesting connection to Stirling numbers of the second kind. We also show some logconcavity result concerning a variant of the excedance number. Finally, we show that the generating function of the excedance number defined on G_r,n is symmetric.
We define new statistics, (c, d)-descents, on the colored permutation groups Z_r \wr S_n and compute the distribution of these statistics on the elements in these groups. We use some combinatorial approaches, recurrences, and generating functions manipulations to obtain our results.
We define an excedance number for the multi-colored permutation group, i.e. the wreath product of Z_{r_1} x ... x Z_{r_k} with S_n, and calculate its multi-distribution with some natural parameters. We also compute the multi-distribution of the parameters exc(pi) and fix(pi) over the sets of involutions in the multi-colored permutation group. Using this, we count the number of involutions in this group having a fixed number of excedances and absolute fixed points.
We define the excedance number on the complex reflection groups and compute its multidistribution with the number of fixed points on the set of involutions in these groups. We use some recurrence formulas and generating functions manipulations to obtain our results.
We generalize the results of Ksavrelof and Zeng about the multidistribution of the excedance number of $S_n$ with some natural parameters to the colored permutation group and to the Coxeter group of type $D$. We define two different orders on these groups which induce two different excedance numbers. Surprisingly, in the case of the colored permutation group, we get the same generalized formulas for both orders.
We study the complex reflection groups G(r,p,n). By considering these groups as subgroups of the wreath products Z_r wr S_n, and by using Clifford theory, we define combinatorial parameters and descent representations of G(r,p,n), previously known for classical Weyl groups. One of these parameters is the flag major index, which also has an important role in the decomposition of these representations into irreducibles. A Carlitz type identity relating the combinatorial parameters with the degrees of the group, is presented.
A product formula for the parity generating function of the number of 1's in invertible matrices over Z_2 is given. The computation is based on algebraic tools such as the Bruhat decomposition. The same technique is used to obtain a parity generating function also for symplectic matrices over Z_2. We present also a generating function for the sum of entries of matrices over an arbitrary finite field F_q calculated in F_q. These formulas are new appearances of the Mahonian distribution.
We discuss permutation representations which are obtained by the natural action of $S_n \times S_n$ on some special sets of invertible matrices, defined by simple combinatorial attributes. We decompose these representations into irreducibles. The multiplicities involved have a nice combinatorial interpretation. We also generalize known results on asymptotic behavior of the conjugacy representation of $S_n$.