We develop some tools, of an algebraic and combinatorial nature, which enable us to obtain a detailed description of certain quadratic subgroups of the (outer) reduced Weyl group of the Cuntz algebra 𝒪_n. In particular, for n=4 our findings give a self-contained theoretical interpretation of the groups tabulated in [AJS18], which were obtained with the help of a computer. For each of these groups we provide a set of generators. A prominent role in our analysis is played by a certain family of subgroups of the symmetric group of a discrete square which we call bicompatible.

We study the partial orders induced on Wachs and signed Wachs permutations by the Bruhat and weak orders of the symmetric and hyperoctahedral groups. We show that these orders are graded, determine their rank function, characterize their ordering and covering relations, and compute their characteristic polynomials, when partially ordered by Bruhat order, and determine their structure explicitly when partially ordered by right weak order.

In this paper I present some open problems on Coxeter groups and unimodality, together with the main partial results, and computational evidence, that are known about them.

We characterize the permutative automorphisms of the Cuntz algebra O n (namely, stable permutations) in terms of two sequences of graphs that we associate to any permutation of a discrete hypercube [n](t). As applications we show that in the limit of large t (resp. n) almost all permutations are not stable, thus proving Conj. 12.5 of Brenti and Conti [Adv. Math. 381 (2021), p. 60], characterize (and enumerate) stable quadratic 4 and 5-cycles, as well as a notable class of stable quadratic r-cycles, i.e. those admitting a compatible cyclic factorization by stable transpositions. Some of our results use new combinatorial concepts that may be of independent interest.

We show how the R-polynomials of the symmetric groups can be computed, in a poset-theoretic way, from canonical hypercube decompositions. This involves a new combinatorial concept, which we call a shortcut. We conjecture that the same formula holds for a certain class of combinatorially defined hypercube decompositions. We also study the behavior of these concepts under the operation of taking the direct product of two Bruhat intervals, and characterize the shortcuts of the canonical hypercube decompositions. Our main conjecture implies the Combinatorial Invariance Conjecture.

Permutative automorphisms of the Cuntz algebra On are in bijection with a class of permutations of nk elements, that are called stable, and are further partitioned by rank. In this work we mainly focus on stable cycles in the quadratic case (i.e., k=2). More precisely, in such a quadratic case we provide a characterization of the stable cycles of rank one (so proving Conjecture 12.1 in [3]), exhibit a closed formula for the number of stable r-cycles of rank one (valid for all n and r), and characterize and enumerate the stable 3-cycles of any given rank. We also show that the set of stable permutations is equipped with a natural involution that preserves the cycle-type and the rank, and that there is a map that associates to two stable permutations of nk and mk elements, respectively, a stable permutation of (nm)k elements.

The odd diagram of a permutation is a subset of the classical diagram with additional parity conditions. In this paper, we study classes of permutations with the same odd diagram, which we call odd diagram classes. First, we prove a conjecture relating odd diagram classes and 213- and 312-avoiding permutations. Secondly, we show that each odd diagram class is a Bruhat interval. Instrumental to our proofs is an explicit description of the Bruhat edges that link permutations in a class.

We study the reduced Weyl groups of the Cuntz algebras On from a combinatorial point of view. Their elements correspond bijectively to certain permutations of nr elements, which we call stable. We mostly focus on the case r=2 and general n. A notion of rank is introduced, which is subadditive in a suitable sense. Being of rank 1 corresponds to solving an equation which is reminiscent of the Yang-Baxter equation. Symmetries of stable permutations are also investigated, along with an immersion procedure that allows to obtain stable permutations of (n+1)2 objects starting from stable permutations of n2 objects. A complete description of stable transpositions and of stable 3-cycles of rank 1 is obtained, leading to closed formulas for their number. Other enumerative results are also presented which yield lower and upper bounds for the number of stable permutations.

We define and study odd analogues of classical geometric and combinatorial objects associated to permutations, namely odd Schubert varieties, odd diagrams, and odd inversion sets. We show that there is a bijection between odd inversion sets of permutations and acyclic orientations of the Turán graph, that the dimension of the odd Schubert variety associated to a permutation is the odd length of the permutation, and give several necessary conditions for a subset of [n]×[n] to be the odd diagram of a permutation. We also study the sign-twisted generating function of the odd length over descent classes of the symmetric groups.

We define and study odd and even analogues of the major index statistics for the classical Weyl groups. More precisely, we show that the generating functions of these statistics, twisted by the one-dimensional characters of the corresponding groups, always factor in an explicit way. In particular, we obtain odd and even analogues of Carlitz's identity, of the Gessel-Simion Theorem, and a parabolic extension, and refinement, of a result of Wachs.

We define a new statistic on any Weyl group which we call the odd length and which reduces, for Weyl groups of types $A$, $B$, and $D$, the the statistics by the same name that have already been defined and studied in [10], [13], [14], and [3]. We show that the signed (by length) generating function of the odd length always factors nicely except possibly in type $E_8$, and we obtain multivariate analogues of these factorizations in types $B$ and $D$.

We characterize the classical complex reflection groups for which a recent symmetric group equidistribution result studied by Diaconis, Evans, and Graham holds. This leads to some refinements of the original result, which seem to be new even in the symmetric group case.

We prove a duality result for the parabolic Kazhdan–Lusztig R-polynomials of a finite Coxeter system. This duality is similar to, but different from, the one obtained in [9]. As a consequence of our duality we obtain an identity between the parabolic Kazhdan–Lusztig and inverse Kazhdan–Lusztig polynomials of a finite Coxeter system. We also obtain applications to certain modules defined by Deodhar and derive a result that gives evidence in favor of Marietti's combinatorial invariance conjecture for parabolic Kazhdan–Lusztig polynomials.

We define a new statistic on the even hyperoctahedral groups which is a natural analogue of the odd length statistic recently defined and studied on Coxeter groups of types $A$ and $B$. We compute the signed (by length) generating function of this statistic over the whole group and over its maximal and some other quotients and show that it always factors nicely. We also present some conjectures.

We give a new characterization of the peak subalgebra of the algebra of quasisymmetric functions and use this to construct a new basis for this subalgebra. As an application of these results we obtain a combinatorial formula for the Kazhdan–Lusztig polynomials which holds in complete generality and is simpler and more explicit than any existing one. We point out that, in a certain sense, this formula cannot be simplified.

We prove a conjecture of Klopsch-Voll on the signed generating function of a new statistic on the quotients of the symmetric groups. As a consequence of our results we also prove a conjecture of Stasinski-Voll in type B.

We study the signed generating function for the number of odd inversions on descent classes of the symmetric group. We present operations on the descent class that leave the corresponding generating function unchanged, give sufficient conditions for it to be zero, and compute it explicitly for the alternating permutations and for a family of descent classes that includes all quotients.

problems that arise naturally from his work in this area.

We give explicit combinatorial formulas for the parabolic Kazhdan–Lusztig R-polynomials of the quasi-minuscule quotients of the classical Weyl groups. As an application of our results we obtain explicit combinatorial formulas for certain sums and alternating sums of ordinary Kazhdan–Lusztig R-polynomials.

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.