Related papers: Omitting parentheses from the cyclic notation
We reformulate Milgram's model of a double loop suspension in terms of a preoperad of posets, each stage of which is the poset of all ordered partitions of a finite set. Using this model, we give a combinatorial model for the evaluation map…
In this paper, the first in a projected two-part series, we describe an organizing framework for the study of infinitary combinatorics. This framework is \v{C}ech cohomology. We show in particular that the \v{C}ech cohomology groups of the…
Graded posets frequently arise throughout combinatorics, where it is natural to try to count the number of elements of a fixed rank. These counting problems are often $\#\textbf{P}$-complete, so we consider approximation algorithms for…
We give a non-negative combinatorial formula, in terms of Littlewood-Richardson numbers, for the homology of the unitary representations of the cyclotomic rational Cherednik algebra, and as a consequence, for the graded Betti numbers for…
Following the classical results of Stong, we introduce a cohomological analogue of a core of a finite sheaved topological space and propose an algorithm for simplification in this category. In particular we generalize the notion of beat…
We introduce and study a family of operators which act in the span of a Weyl group $W$ and provide a multi-parameter solution to the quantum Yang-Baxter equations of the corresponding type. Our operators generalize the "quantum Bruhat…
The theory of shellable simplicial complexes brings together combinatorics, algebra, and topology in a remarkable way. Initially introduced by Alder for $q$-simplicial complexes, recent work of Ghorpade, Pratihar, and Randrianarisoa extends…
In this paper, we deal with reversing and extended symmetries of shifts generated by bijective substitutions. We provide equivalent conditions for a permutation on the alphabet to generate a reversing/extended symmetry, and algorithms how…
We investigate the poset of strata of a Schubert-like stratification of certain natural compactification of the space of hermitian $n\times n$ matrices. We prove that this poset is a modular ortholattice, we compute its M\"{o}bius function…
Shuffle algebras are monoids for an unconvential monoidal category structure on graded vector spaces. We present two homological results on shuffle algebras with monomial relations, and use them to prove exact and asymptotic results on…
We provide a systematic enumerative and combinatorial study of geometric cellular diagonals on the permutahedra. In the first part of the paper, we study the combinatorics of certain hyperplane arrangements obtained as the union of $\ell$…
We prove several combinatorial properties of suffix arrays, including a characterization of suffix arrays through a bijection with a certain well-defined class of permutations. Our approach is based on the characterization of…
A poset has the non-messing-up property if it has two covering sets of disjoint saturated chains so that for any labeling of the poset, sorting the labels along one set of chains and then sorting the labels along the other set yields a…
We establish a hierarchy of weighted majorization relations for the singularities of generalized Lam\'e equations and the zeros of their Van Vleck and Heine-Stieltjes polynomials as well as for multiparameter spectral polynomials of higher…
Let $B$ be the group of invertible upper-triangular complex $n\times n$ matrices, $\mathfrak{u}$ the space of upper-triangular complex matrices with zeroes on the diagonal and $\mathfrak{u}^*$ its dual space. The group $B$ acts on…
We describe a family of shellings for the canonical triangulation of the order polytope of the zig-zag poset. This gives a new combinatorial interpretation for the coefficients in the numerator of the Ehrhart series of this order polytopein…
Orbits of automorphism groups of partially ordered sets are not necessarily congruence classes, i.e. images of an order homomorphism. Based on so-called orbit categories a framework of factorisations and unfoldings is developed that…
In this article we give a classification of the sub-groups in PSL(2,Z) and of the conjugacy classes of these sub-groups by the mean of an combinatorial invariant: some trivalent diagrams (dotted or not). We give explicit formulae enabling…
A grid class consists of permutations whose pictorial depiction can be partitioned into increasing and decreasing parts as determined by a given matrix. In this paper, we introduce a method for enumerating cyclic permutations in vector grid…
The aim of this paper is to build a theory of commutative and noncommutative {\it injective} valuations of various algebras (including algebras with zero divisors). The targets of our valuations are (well-)ordered commutative and…