We show that the bistatistic of right nestings and right crossings in matchings without left nestings is equidistributed with the number of occurrences of two certain patterns in permutations, and furthermore that this equidistribution holds when refined to positions of these statistics in matchings and permutations. For this distribution we obtain a non-commutative generating function which specializes to Zagier's generating function for the Fishburn numbers after abelianization.As a special case we obtain proofs of two conjectures of Claesson and Linusson.Finally, we conjecture that our results can be generalized to involving left crossings of matchings too.
@article{10_37236_3981,
author = {Niklas Eriksen and Jonas Sj\"ostrand},
title = {Equidistributed statistics on matchings and permutations},
journal = {The electronic journal of combinatorics},
year = {2014},
volume = {21},
number = {4},
doi = {10.37236/3981},
zbl = {1305.05017},
url = {http://geodesic.mathdoc.fr/articles/10.37236/3981/}
}
TY - JOUR
AU - Niklas Eriksen
AU - Jonas Sjöstrand
TI - Equidistributed statistics on matchings and permutations
JO - The electronic journal of combinatorics
PY - 2014
VL - 21
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.37236/3981/
DO - 10.37236/3981
ID - 10_37236_3981
ER -
%0 Journal Article
%A Niklas Eriksen
%A Jonas Sjöstrand
%T Equidistributed statistics on matchings and permutations
%J The electronic journal of combinatorics
%D 2014
%V 21
%N 4
%U http://geodesic.mathdoc.fr/articles/10.37236/3981/
%R 10.37236/3981
%F 10_37236_3981
Niklas Eriksen; Jonas Sjöstrand. Equidistributed statistics on matchings and permutations. The electronic journal of combinatorics, Tome 21 (2014) no. 4. doi: 10.37236/3981
