We use categories to recast the combinatorial theory of full heaps, which are certain labelled partially ordered sets that we introduced in previous work. This gives rise to a far simpler set of definitions, which we use to outline a combinatorial construction of the so-called loop algebras associated to affine untwisted Kac--Moody algebras. The finite convex subsets of full heaps are equipped with a statistic called parity, and this naturally gives rise to Kac's asymmetry function. The latter is...

We maximize the total height of order ideals in direct products of finitely many finite chains. We also consider several order ideals simultaneously. As a corollary, a shifting property of some integer sequences, including digit sum sequences, is derived.

Let $Q$ be the lexicographic sum of finite ordered sets $Q_x$ over a finite ordered set $P$. For some $P$ we can give a formula for the jump number of $Q$ in terms of the jump numbers of $Q_x$ and $P$, that is, $s\left(Q\right)=s\left(P\right)+{\sum }_{x\in P}s\left(Q_x\right)$, where $s\left(X\right)$ denotes the jump number of an ordered set $X$. We first show that $w\left(P\right)-1+{\sum }_{x\in P}s\left(Q_x\right)\le s\left(Q\right)\le s\left(P\right)+{\sum }_{x\in P}s\left(Q_x\right)$, where $w\left(X\right)$ denotes the width of an ordered set $X$. Consequently, if $P$ is a Dilworth ordered set, that is, $s\left(P\right)=w\left(P\right)-1$, then the formula holds. We also show that it holds again if $P$ is bipartite. Finally, we prove that the lexicographic sum of...

The main result of the paper is that for a circular element c in a C*-probability space, $(cⁿ,c^{n*})$ is an R-diagonal pair in the sense of Nica and Speicher for every n = 1,2,... The coefficients of the R-series are found to be the generalized Catalan numbers of parameter n-1.

L’espace des configurations de $p$ points distincts de ${\mathbf{R}}^{\infty }$ admet une filtration naturelle qui est induite par les inclusions des ${\mathbf{R}}^n$ dans ${\mathbf{R}}^{\infty }$. Nous caractérisons le type d’homotopie de cette filtration par les propriétés combinatoires d’une structure cellulaire sous-jacente, étroitement liée à la théorie des $E_n$-opérades de May. Cela donne une approche unifiée des différents modèles combinatoires d’espaces de lacets itérés et redémontre les théorèmes d’approximation de Milgram, Smith et Kashiwabara.

It is shown that if α,ζ are ordinals such that 1 ≤ ζ < α < ζω, then there is an operator from $C\left({\omega }^{{\omega }^{\alpha }}\right)$ onto itself such that if Y is a subspace of $C\left({\omega }^{{\omega }^{\alpha }}\right)$ which is isomorphic to $C\left({\omega }^{{\omega }^{\alpha }}\right)$, then the operator is not an isomorphism on Y. This contrasts with a result of J. Bourgain that implies that there are uncountably many ordinals α for which for any operator from $C\left({\omega }^{{\omega }^{\alpha }}\right)$ onto itself there is a subspace of $C\left({\omega }^{{\omega }^{\alpha }}\right)$ which is isomorphic to $C\left({\omega }^{{\omega }^{\alpha }}\right)$ on which the operator is an isomorphism.

Order complex is an important object associated to a partially ordered set. Following a suggestion from V. A. Vassiliev (1994), we investigate an order complex associated to the partially ordered set of nontrivial ideals in a commutative ring with identity. We determine the homotopy type of the geometric realization for the order complex associated to a general commutative ring with identity. We show that this complex is contractible except for semilocal rings with trivial Jacobson radical when...