On the sparseness of the downsets of permutations via their number of separators

Conventionally, a pair (σi, σi+1) is a bond in a permutation σ = σ1σ2 · · ·σn if σi− σi+1 = ±1. The number of bonds in a permutation σ ∈ Sn has a direct influence on the number of distinct patterns of order n − 1 contained in σ, affecting the structure of the downset of σ in the containment poset ⋃ n∈N Sn. Thus, to characterize the sparseness of the downset of a permutation σ ∈ Sn, we aim not only to find the number of bonds in σ, but also to predict the number of bonds contained in its patterns. To this end, we introduce a new statistic, separator number, as a significant factor in measuring the sparseness of this poset. An element σj in a permutation σ = σ1 · · ·σn ∈ Sn is defined to be a separator of σ if we can obtain a new bond by omitting it from σ. We also present some enumerative and asymptotic results regarding this new statistic.