Intervals of Permutations with a Fixed Number of Descents are Shellable

The set of all permutations, ordered by pattern containment, is a poset. We present an order isomorphism from the poset of permutations with a fixed number of descents to a certain poset of words with subword order. We use this bijection to show that intervals of permutations with a fixed number of descents are shellable, and we present a formula for the M\"obius function of these intervals. We present an alternative proof for a result on the M\"obius function of intervals $[1,\pi]$ such that $\pi$ has exactly one descent. We prove that if $\pi$ has exactly one descent and avoids 456123 and 356124, then the intervals $[1,\pi]$ have no nontrivial disconnected subintervals; we conjecture that these intervals are shellable

On the Möbius function and topology of the permutation poset

A permutation is an ordering of the letters 1, . . . , n. A permutation σ occurs as a pattern in a permutation π if there is a subsequence of π whose letters appear in the same relative order of size as the letters of σ, such a subsequence is called an occurrence. The set of all permutations, ordered by pattern containment, is a poset. In this thesis we study the behaviour of the Möbius function and topology of the permutation poset. The first major result in this thesis is on the Möbius function of intervals [1,π], such that π = π₁π₂. . . πn has exactly one descent, where a descent occurs at position i if πi > π i+1. We show that the Möbius function of these intervals can be computed as a function of the positions and number of adjacencies, where an adjacency is a pair of letters in consecutive positions with consecutive increasing values. We then alter the definition of adjacencies to be a maximal sequence of letters in consecutive positions with consecutive increasing values. An occurrence is normal if it includes all letters except (possibly) the first one of each of all the adjacencies. We show that the absolute value of the Möbius function of an interval [σ, π] of permutations with a fixed number of descents equals the number of normal occurrences of σ in π. Furthermore, we show that these intervals are shellable, which implies many useful topological properties. Finally, we allow adjacencies to be increasing or decreasing and apply the same definition of normal occurrence. We present a result that shows the Möbius function of any interval of permutations equals the number of normal occurrences plus an extra term. Furthermore, we conjecture that this extra term vanishes for a significant proportion of intervals.A permutation is an ordering of the letters 1, . . . , n. A permutation σ occurs as a pattern in a permutation π if there is a subsequence of π whose letters appear in the same relative order of size as the letters of σ, such a subsequence is called an occurrence. The set of all permutations, ordered by pattern containment, is a poset. In this thesis we study the behaviour of the Möbius function and topology of the permutation poset. The first major result in this thesis is on the Möbius function of intervals [1,π], such that π = π₁π₂. . . πn has exactly one descent, where a descent occurs at position i if πi > π i+1. We show that the Möbius function of these intervals can be computed as a function of the positions and number of adjacencies, where an adjacency is a pair of letters in consecutive positions with consecutive increasing values. We then alter the definition of adjacencies to be a maximal sequence of letters in consecutive positions with consecutive increasing values. An occurrence is normal if it includes all letters except (possibly) the first one of each of all the adjacencies. We show that the absolute value of the Möbius function of an interval [σ, π] of permutations with a fixed number of descents equals the number of normal occurrences of σ in π. Furthermore, we show that these intervals are shellable, which implies many useful topological properties. Finally, we allow adjacencies to be increasing or decreasing and apply the same definition of normal occurrence. We present a result that shows the Möbius function of any interval of permutations equals the number of normal occurrences plus an extra term. Furthermore, we conjecture that this extra term vanishes for a significant proportion of intervals

A two-sided analogue of the Coxeter complex

For any Coxeter system $(W,S)$ of rank $n$, we introduce an abstract boolean complex (simplicial poset) of dimension $2n-1$ that contains the Coxeter complex as a relative subcomplex. Faces are indexed by triples $(I,w,J)$, where $I$ and $J$ are subsets of the set $S$ of simple generators, and $w$ is a minimal length representative for the parabolic double coset $W_I w W_J$. There is exactly one maximal face for each element of the group $W$. The complex is shellable and thin, which implies the complex is a sphere for the finite Coxeter groups. In this case, a natural refinement of the $h$-polynomial is given by the "two-sided" $W$-Eulerian polynomial, i.e., the generating function for the joint distribution of left and right descents in $W$.Comment: 26 pages, several large tables and figure

The structure of the consecutive pattern poset

The consecutive pattern poset is the infinite partially ordered set of all permutations where $\sigma\le\tau$ if $\tau$ has a subsequence of adjacent entries in the same relative order as the entries of $\sigma$. We study the structure of the intervals in this poset from topological, poset-theoretic, and enumerative perspectives. In particular, we prove that all intervals are rank-unimodal and strongly Sperner, and we characterize disconnected and shellable intervals. We also show that most intervals are not shellable and have M\"obius function equal to zero.Comment: 29 pages, 7 figures. To appear in IMR