Classifying ascents and descents with specified equivalences mod k

Given a permutationof length j, we say that a permutationhas a �-match starting at position i, if the elements in position i,i + 1,...,i + j 1 inhave the same relative order as the elements of �. Ifis set of permutations of length j, then we say that a permutationhas an �-match starting at position j if it has a �-match at position j for some � 2 �. A number of recent papers have studied the distribution of �-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 general, when one includes parity conditions or conditions involving equivalence mod k, then the problem of counting the number of pattern matchings becomes more complicated. In this paper, we prove explicit formulas for the number of permutations of n which have s �- equivalence mod k matches whenis of length 2. We also show that similar formulas hold for �-equivalence mod k matches for certain subsets of permutations of length two.