On the $r$-Derangements of type B

Derangements and Relative Derangements of Type $B$

By introducing the notion of relative derangements of type $B$, also called signed relative derangements, which are defined in terms of signed permutations, we obtain a type $B$ analogue of the well-known relation between relative derangements and the classical derangements. While this fact can be proved by using the principle of inclusion and exclusion, we present a combinatorial interpretation with the aid of the intermediate structure of signed skew derangements.

Derangement Polynomials and Excedances of Type B

Adopting the definition of excedances of type B due to Brenti, we give a type B analogue of the q-derangement polynomials. The connection between q-derangement polynomials and Eulerian polynomials naturally extends to the type B case. Based on this relation, we derive some basic properties of the q-derangement polynomials of type B, including the generating function formula, the Sturm sequence property, and the asymptotic normal distribution. We also show that the q-derangement polynomials are almost symmetric in the sense that the coefficients possess the spiral property.

Derangement Frequency in the Boolean Complex

In previous work, we associated to any finite simple graph a particular set of derangements of its vertices. These derangements are in bijection with the spheres in the wedge sum describing the homotopy type of the boolean complex for this graph. Here we study the frequency with which a given derangement appears in this set.

A symmetric unimodal decomposition of the derangement polynomial of type $B$

The derangement polynomial $d_n (x)$ for the symmetric group enumerates derangements by the number of excedances. The derangement polynomial $d^B_n(x)$ for the hyperoctahedral group is a natural type $B$ analogue. A new combinatorial formula for this polynomial is given in this paper. This formula implies that $d^B_n (x)$ decomposes as a sum of two nonnegative, symmetric and unimodal polynomials whose centers of symmetry differ by a half and thus provides a new transparent proof of its unimodality. A geometric interpretation, analogous to Stanley's interpretation of $d_n (x)$ as the local $h$-…

带符号的从右到左极小值与超位数的错位排列计数

Counting derangements with signed right-to-left minima and excedances

该研究将Alexandersson和Getachew提出的错位排列带符号超位数多元推广公式与排列连分数关联，证实部分观察结果，给出两种多元恒等式细化形式以解释公式项含义，还探索了B型排列的类似公式。

The 1/k-Eulerian polynomials of type B

In this paper, we give a type B analogue of the 1/k-Eulerian polynomials. Properties of this kind of polynomials, including combinatorial interpretations, recurrence relations and gamma-positivity are studied. In particular, we show that the 1/k-Eulerian polynomials of type B are gamma-positive when $k>0$. Moreover, we obtain the corresponding results for derangements of type B. We show that a type B 1/k-derangement polynomials $d_n^B(x;k)$ are bi-gamma-positive when $k\geq 1/2$. In particular, we get a symmetric decomposition of $d_n^B(x;1/2)$ in terms of the classical derangement polynomials…