On the M\"obius Function and Topology of General Pattern Posets

We introduce a formal definition of a pattern poset which encompasses several previously studied posets in the literature. Using this definition we present some general results on the M\"obius function and topology of such pattern posets. We prove our results using a poset fibration based on the embeddings of the poset, where embeddings are representations of occurrences. We show that the M\"obius function of these posets is intrinsically linked to the number of embeddings, and in particular to so called normal embeddings. We present results on when topological properties such as Cohen-Macaulayness and shellability are preserved by this fibration. Furthermore, we apply these results to some pattern posets and derive alternative proofs of existing results, such as Bj\"orner's results on subword order.Comment: 28 Page

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 M\"obius Function of Permutations With One Descent

The set of all permutations, ordered by pattern containment, is a poset. We give a formula for the M\"obius function of intervals $[1,\pi]$ in this poset, for any permutation $\pi$ with at most one descent. We compute the M\"obius function as a function of the number and positions of pairs of consecutive letters in $\pi$ that are consecutive in value. As a result of this we show that the M\"obius function is unbounded on the poset of all permutations. We show that the M\"obius function is zero on any interval $[1,\pi]$ where $\pi$ has a triple of consecutive letters whose values are consecutive and monotone. We also conjecture values of the M\"obius function on some other intervals of permutations with at most one descent

The poset of graphs ordered by induced containment

I would like to express my gratitude to the anonymous referees for their extremely useful comments and corrections which greatly improved the paper.Peer reviewedPreprintPostprin

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

Statistical Complexity of Heterogeneous Geometric Networks

Heterogeneity and geometry are key explanatory components underlying the structure of real-world networks. The relationship between these components and the statistical complexity of networks is not well understood. We introduce a parsimonious normalised measure of statistical complexity for networks -- normalised hierarchical complexity. The measure is trivially 0 in regular graphs and we prove that this measure tends to 0 in Erd\"os-R\'enyi random graphs in the thermodynamic limit. We go on to demonstrate that greater complexity arises from the combination of hierarchical and geometric components to the network structure than either on their own. Further, the levels of complexity achieved are similar to those found in many real-world networks. We also find that real world networks establish connections in a way which increases hierarchical complexity and which our null models and a range of attachment mechanisms fail to explain. This underlines the non-trivial nature of statistical complexity in real-world networks and provides foundations for the comparative analysis of network complexity within and across disciplines.Comment: 12 pages, 6 figure

On Distance Preserving and Sequentially Distance Preserving Graphs

Preprin