Potential polynomials and Motzkin paths

A Motzkin path of length n is a lattice path from (0,0) to (n,0) in the plane integer lattice ZxZ consisting of horizontal-steps (1,0), up-steps (1,1), and down-steps (1,-1), which never passes below the x-axis. A u-segment (resp. h-segment) of a Motzkin path is a maximal sequence of consecutive up-steps (resp. horizontal-steps). The present paper studies two kinds of statistics on Motzkin paths: ''number of u-segments'' and ''number of h-segments''. The Lagrange inversion formula is utilized to represent the weighted generating function for the number of Motzkin paths according to the two statistics as a sum of the partial Bell polynomials or the potential polynomials. As an application, a general framework for studying compositions are also provided.

[1]  Toufik Mansour,et al.  Dyck paths and partial Bell polynomials , 2008, Australas. J Comb..

[2]  I. Goulden,et al.  Combinatorial Enumeration , 2004 .

[3]  W. J. Thron,et al.  Encyclopedia of Mathematics and its Applications. , 1982 .

[4]  O. Kuzmin,et al.  On partition polynomials , 2001 .

[5]  R. Stanley,et al.  An Analogue of Young's Lattice for Compositions , 2005, math/0508043.

[6]  Arnold Knopfmacher,et al.  On Carlitz Compositions , 1998, Eur. J. Comb..

[7]  Charalambos A. Charalambides,et al.  Enumerative combinatorics , 2018, SIGA.

[8]  D. Klarner,et al.  Correspondences between plane trees and binary sequences , 1970 .

[9]  Toufik Mansour,et al.  Bell polynomials and k-generalized Dyck paths , 2008, Discret. Appl. Math..

[10]  M. C. Verri,et al.  A unified approach to the study of general and palindromic compositions , 2004 .

[11]  W Y Chen,et al.  A general bijective algorithm for trees. , 1990, Proceedings of the National Academy of Sciences of the United States of America.

[12]  Toufik Mansour,et al.  Counting rises, levels, and drops in compositions. , 2003 .

[13]  Emil Grosswald,et al.  The Theory of Partitions , 1984 .

[14]  H. Wilf.  generatingfunctionology: Third Edition , 1990 .

[15]  Phyllis Z. Chinn,et al.  Compositions of n with no occurrence of k , 2003 .

[16]  Sadek Bouroubi,et al.  On new identities for Bell's polynomials , 2005, Discret. Math..

[17]  Toufik Mansour,et al.  Compositions of n with parts in a set, Thirty-Fifth Southeastern International Conference on Combina , 2003 .

[18]  Herbert S. Wilf,et al.  Generating functionology , 1990 .