Weakly increasing trees on a multiset

A symmetry on weakly increasing trees and multiset Schett polynomials

“…This enables us to study the unity between plane trees and increasing trees in the framework of weakly increasing trees. The enumerative results obtained in [15] reflect that weakly increasing trees is a natural and nice generalization of plane trees and increasing trees:…”

“…As unified generalization of increasing trees and plane trees, the weakly increasing trees on a multiset were introduced recently in the joint work of the two authors with Ma and Zhou [15]. The objective of this article is to prove both bijectively and algebraically the symmetry of the joint distribution of "even-degree nodes on odd levels" and "odddegree nodes" on weakly increasing trees.…”

“…, n pn u and let p " ř n i"1 p i . Definition 1.1 (Weakly increasing trees [15]). A weakly increasing tree on M is a plane tree such that (i) it contains p `1 nodes that are labeled by elements from the multiset M Y t0u, (ii) each sequence of labels along a path from the root to any leaf is weakly increasing, and (iii) for each node j in the tree, labels of the roots of subtrees of j are weakly increasing from left to right.…”

“…where leafpT q denotes the number of leaves of T . The γ-positivity of A M ptq has an unified group action proof which possesses an unexpected application in interpreting the γ-coefficients of the k-multiset Eulerian polynomials (see [15] for k " 2 and [16] for general k). ‚ There are connections between M-Eulerian-Narayana polynomials for the multiset M " t1 2 , 2 2 , .…”

A Combinatorial Link Between Labelled Graphs and Increasingly Labelled Schröder Trees

“…In the recent years, many links were found between evolution processes in the form of increasing trees and classical combinatorial structures, for instance permutations are known to be in bijection with increasing binary trees [10, p. 143], increasing even trees and alternating permutations are put in bijection in [16,5], plane recursive trees are related to Stirling permutations [15] and more recently increasing Schröder trees have been proved in one-to-one correspondence with even permutations and with weak orderings on sets of n elements (counted by ordered Bell numbers) in [2,3]. By adding some constraint in the increasing labelling of the latter model, Lin et al [17] exhibited closed relationships between various families of polynomials (especially Eulerian, Narayana and Savage and Schuster polynomials).…”