116 research outputs found
Quadrant marked mesh patterns in 123-avoiding permutations
Given a permutation in the symmetric
group , we say that matches the quadrant marked
mesh pattern in if there are at least
points to the right of in which are greater than
, at least points to the left of in which are
greater than , at least points to the left of in
which are smaller than , and at least points to the
right of in which are smaller than . Kitaev,
Remmel, and Tiefenbruck systematically studied the distribution of the number
of matches of in 132-avoiding permutations. The
operation of reverse and complement on permutations allow one to translate
their results to find the distribution of the number of
matches in 231-avoiding, 213-avoiding, and 312-avoiding permutations. In this
paper, we study the distribution of the number of matches of
in 123-avoiding permutations. We provide explicit
recurrence relations to enumerate our objects which can be used to give closed
forms for the generating functions associated with such distributions. In many
cases, we provide combinatorial explanations of the coefficients that appear in
our generating functions
Descent c-Wilf Equivalence
Let denote the symmetric group. For any , we let
denote the number of descents of ,
denote the number of inversions of , and
denote the number of left-to-right minima of .
For any sequence of statistics on
permutations, we say two permutations and in are
-c-Wilf equivalent if the generating
function of over all permutations which
have no consecutive occurrences of equals the generating function of
over all permutations which have no
consecutive occurrences of . We give many examples of pairs of
permutations and in which are -c-Wilf
equivalent, -c-Wilf equivalent, and
-c-Wilf equivalent. For example, we
will show that if and are minimally overlapping permutations
in which start with 1 and end with the same element and
and , then and are
-c-Wilf equivalent.Comment: arXiv admin note: text overlap with arXiv:1510.0431
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