Séminaire Lotharingien de Combinatoire, 86B.2 (2022), 12 pp.
Refined Consecutive Pattern Enumeration Via a Generalized Cluster Method
We present a general approach for counting permutations by occurrences
of prescribed consecutive patterns together with various
inverse statistics. We first lift the Goulden-Jackson cluster method
for permutations - a standard tool
in the study of consecutive
patterns - to the Malvenuto-Reutenauer algebra. Upon applying
standard homomorphisms, our result specializes to both the cluster
for permutations as well as a q-analogue which keeps track of the
inversion number statistic. We construct additional homomorphisms
which lead to further specializations for keeping track of inverses of
shuffle-compatible descent statistics; these include the
inverse descent number, inverse peak number, and inverse left peak
number. To illustrate this approach, we present new formulas that
count permutations by occurrences of the monotone consecutive pattern
12...m while also keeping track of these inverse statistics.
Received: November 25, 2021.
Accepted: March 4, 2022.
Final version: April 1, 2022.
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