2 research outputs found
Exterior Pairs and Up Step Statistics on Dyck Paths
Let \C_n be the set of Dyck paths of length . In this paper, by a new
automorphism of ordered trees, we prove that the statistic `number of exterior
pairs', introduced by A. Denise and R. Simion, on the set \C_n is
equidistributed with the statistic `number of up steps at height with
(mod 3)'. Moreover, for , we prove that the two statistics
`number of up steps at height with (mod )' and `number of up
steps at height with (mod )' on the set \C_n are `almost
equidistributed'. Both results are proved combinatorially.Comment: 13 page
Exterior Pairs and Up Step Statistics on Dyck Paths
Let Cn be the set of Dyck paths of length n. In this paper, by a new automorphism of ordered trees, we prove that the statistic ‘number of exterior pairs’, introduced by A. Denise and R. Simion, on the set Cn is equidistributed with the statistic ‘number of up steps at height h with h ≡ 0 (mod 3)’. Moreover, for m ≥ 3, we prove that the two statistics ‘number of up steps at height h with h ≡ 0 (mod m) ’ and ‘number of up steps at height h with h ≡ m − 1 (mod m) ’ on the set Cn are ‘almost equidistributed’. Both results are proved combinatorially