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Counting Descents and Ascents Relative to Equivalence Classes mod k
Jeffrey Liese
Department of Mathematics, University of California, San Diego, La Jolla, CA 92093-0112, USA
jliese@math.ucsd.edu
Annals of Combinatorics 11 (3-4) p.481-506 September, 2007
AMS Subject Classification:05A05, 05A15, 05A19
Abstract:
Given a permutation τ of length j, we say that a permutation б has a τ-match starting at position i, if the elements in position i, i+1, … ,i+ j-1 in б have the same relative order as the elements of τ. If is the set of permutations of length j, then we say that a permutation б has a -match starting at position j if it has a τ-match at position j for some τє. A number of recent papers have studied the distribution of τ-matches and -matches in permutations. In this paper, we consider a more refined pattern matching condition where we take into account conditions involving the equivalence classes of the elements mod k for some integer k≥2. In this paper, we prove explicit formulas for the number of permutations of n which have s τ-equivalence mod k matches when τ is of length 2. We also show that similar formulas hold for -equivalence mod k matches for certain subsets of permutations of length 2.
Keywords: permutations, restricted descent sets, rook placements

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