2015/10/14 by Quang T. Bach, Bach, Quang T., Jeffrey B. Remmel +1 · 1 citation
Computer Science · Mathematics · #05A15 #05E05 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Algorithms and Data Compression #Bayesian Methods and Mixture Models #Combinatorics (math.CO) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.1510.04319
openalex publication_date 2015/10/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We extend the reciprocity method of Jones and Remmel to study generating functions of the form ∑n ≥ 0 (tn)/(n!) ∑σ∈ NMn(Γ)xLRmin(σ)y1+des(σ) where Γ is a set of permutations which start with 1 and have at most one descent, NMn(Γ) is the set of permutations σ in the symmetric group \mathfrakSn which have no Γ-matches, des(σ) is the number of descents of σ and LRmin(σ) is the number of left-to-right minima of σ. We show that this generating function is of the form ( (1)/(UΓ(t,y)))x where UΓ(t,y) = ∑n≥ 0UΓ,n(y) (tn)/(n!) and the coefficients UΓ,n(y) satisfy some simple recursions in the case where Γ equals \1324,123\, \1324 ⋯ p,12 ⋯ (p-1)\ for p ≥ 5, or Γ is the set of permutations σ= σ1 ⋯ σn of length n=k1+k2 where k1,k2 ≥ 2, σ1 =1, σk1+1=2, and des(σ) =1.