The Sorting Index and Permutation Codes

In the combinatorial study of the coefficients of a bivariate polynomial that generalizes both the length and the reflection length generating functions for finite Coxeter groups, Petersen introduced a new Mahonian statistic $sor$, called the sorting index. Petersen proved that the pairs of statistics $(sor,cyc)$ and $(inv,rl\textrm{-}min)$ have the same joint distribution over the symmetric group, and asked for a combinatorial proof of this fact. In answer to the question of Petersen, we observe a connection between the sorting index and the B-code of a permutation defined by Foata and Han, and we show that the bijection of Foata and Han serves the purpose of mapping $(inv,rl\textrm{-}min)$ to $(sor,cyc)$. We also give a type $B$ analogue of the Foata-Han bijection, and we derive the quidistribution of $(inv_B,{\rm Lmap_B},{\rm Rmil_B})$ and $(sor_B,{\rm Lmap_B},{\rm Cyc_B})$ over signed permutations. So we get a combinatorial interpretation of Petersen's equidistribution of $(inv_B,nmin_B)$ and $(sor_B,l_B')$. Moreover, we show that the six pairs of set-valued statistics $\rm (Cyc_B,Rmil_B)$, $\rm(Cyc_B,Lmap_B)$, $\rm(Rmil_B,Lmap_B)$, $\rm(Lmap_B,Rmil_B)$, $\rm(Lmap_B,Cyc_B)$ and $\rm(Rmil_B,Cyc_B)$ are equidistributed over signed permutations. For Coxeter groups of type $D$, Petersen showed that the two statistics $inv_D$ and $sor_D$ are equidistributed. We introduce two statistics $nmin_D$ and $\tilde{l}_D'$ for elements of $D_n$ and we prove that the two pairs of statistics $(inv_D,nmin_D)$ and $(sor_D,\tilde{l}_D')$ are equidistributed.Comment: 25 page

Fixed Point Polynomials of Permutation Groups

In this paper we study, given a group $G$ of permutations of a finite set, the so-called fixed point polynomial $\sum_{i=0}^{n}f_{i}x^{i}$, where $f_{i}$ is the number of permutations in $G$ which have exactly $i$ fixed points. In particular, we investigate how root location relates to properties of the permutation group. We show that for a large family of such groups most roots are close to the unit circle and roughly uniformly distributed round it. We prove that many families of such polynomials have few real roots. We show that many of these polynomials are irreducible when the group acts transitively. We close by indicating some future directions of this research. A corrigendum was appended to this paper on 10th October 2014. </jats:p

Cubical rectangles and rectangular lattices

Cubical rectangles are being defined and explored here over the $n-$dimensional geometric cube $Q_n.$ They form a new class of geometric objects that includes all the edges and all the squares of the $n-$cube. We enumerate and characterize them here in order to construct new posets, transforming into special lattices that will be called rectangular lattices. We show that rectangular lattices are closely related to the class of cubical lattices, that is, the face lattice of the $n-$cube

The poset of Specht ideals for hyperoctahedral groups

Specht polynomials classically realize the irreducible representations of the symmetric group. The ideals defined by these polynomials provide a strong connection with the combinatorics of Young tableaux and have been intensively studied by several authors. We initiate similar investigations for the ideals defined by the Specht polynomials associated to the hyperoctahedral group Bn. We introduce a bidominance order on bipartitions which describes the poset of inclusions of these ideals and study algebraic consequences on general Bn-invariant ideals and varieties, which can lead to computational simplifications

Jack Derangements

For each integer partition $\lambda \vdash n$ we give a simple combinatorial expression for the sum of the Jack character $\theta^\lambda_\alpha$ over the integer partitions of $n$ with no singleton parts. For $\alpha = 1,2$ this gives closed forms for the eigenvalues of the permutation and perfect matching derangement graphs, resolving an open question in algebraic graph theory. A byproduct of the latter is a simple combinatorial formula for the immanants of the matrix $J-I$ where $J$ is the all-ones matrix, which might be of independent interest. Our proofs center around a Jack analogue of a hook product related to Cayley's $\Omega$--process in classical invariant theory, which we call the principal lower hook product

Hypercubes, Peak Patterns and Universal Positive Epistasis

Genes and their interactions with one another crucially affect reproductive success, commonly referred to as fitness. Biallelic models have been used in the past as a mathematical framework to model and explain these interactions. One approach is to represent L biallelic loci as hypercubic graphs, known as L-cubes. On these $L$-cubes, vertices model genotypes and the edges connect the vertices that differ by a single locus value. Assigning fitness values to genotypes gives edges a direction towards higher fitness. Local optimal genotypes, called peaks, then have a higher fitness than all their direct neighbors. Recently, researchers have introduced the notion of peak patterns, referring to the set of peaks that are unique up to relabeling of vertices. However, a complete characterization of all possible peak patterns has not yet been performed for L => 4. This work concerns itself with an analysis for L=4 regarding peak patterns and all possible instances of sign and reciprocal sign epistasis, substantiating the importance of peak patterns. Additionally a lower bound proportional to $2^{2^{L-1}}$ is provided for the set containing all possible peak patterns for a given L. Informed by this, all peak patterns up to L=6 are computed and joined with a variant of Fishers Geometric Model having a one dimensional phenotype. Moreover peak patterns are used to calculate the maximal number of peaks for the staircase triangulation up to L=8

Derangements on the n-cube

Let Q. be the n-dimensional cube represented by a graph whose vertices are sequences of O’s and l’s of length n, where two vertices are adjacent if and only if they differ only at one position. A k-dimensional subcube or a k-face of Q. is a subgraph of Q. spanned by all the vertices u1 u2 u, with constant entries on n-k positions. For a k-face Gx of Q. and a symmetry w of Q., we say that w fixes Gt if w induces a symmetry of Gt; in other words, the image of any vertex of G,, is still a vertex in Gk. A symmetry w of Q. is said to be a k-dimensional derangement if w does not fix any k-dimensional subcube of Q.; otherwise, w is said to be a k-dimensional rearrangement. In this paper, we find a necessary and sufficient condition for a symmetry of Q. to have a fixed k-dimensional subcube. We find a way to compute the generating function for the number of k-dimensional rearrangements on Q.. This makes it possible to compute explicitly such generating functions for small k. Especially, for k =O, we have that there are 1.3. (2n- 1) symmetries of Q. with at least one fixed vertex. A combinatorial proof of this formula is also given