Some applications arising from the interactions between the theory of Catalan-like numbers and the ECO method

In [FP] the ECO method and Aigner’s theory of Catalan-like numbers are compared, showing that it is often possible to translate a combinatorial situation from one theory into the other by means of a standard change of basis in a suitable vector space. In the present work we emphasize the soundness of such an approach by finding some applications suggested by the above mentioned translation. More precisely, we describe a presumably new bijection between two classes of lattice paths and we give a combinatorial interpretation to an integer sequence not appearing in [Sl].

[1]  George Polya,et al.  On the number of certain lattice polygons , 1969 .

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

[3]  M. Aigner.  Catalan and other numbers: a recurrent theme , 2001 .

[4]  Andrea Frosini,et al.  On directed-convex polyominoes in a rectangle , 2005, Discret. Math..

[6]  Gérard Viennot,et al.  Algebraic Languages and Polyominoes Enumeration , 1983, Theor. Comput. Sci..

[7]  Mireille Bousquet-Mélou,et al.  Generating functions for generating trees , 2002, Discret. Math..

[8]  S. Golomb.  Polyominoes: Puzzles, Patterns, Problems, and Packings , 1994 .

[9]  Alberto Del Lungo,et al.  ECO:a methodology for the enumeration of combinatorial objects , 1999 .

[10]  Elena Barcucci,et al.  ECO method and hill-free generalized Motzkin paths , 2001 .

[11]  Luca Ferrari,et al.  Production matrices , 2005, Adv. Appl. Math..

[12]  Alberto Del Lungo,et al.  Steep polyominoes, q-Motzkin numbers and q-Bessel functions , 1998, Discret. Math..

[13]  N. J. A. Sloane,et al.  The On-Line Encyclopedia of Integer Sequences , 2003, Electron. J. Comb..

[14]  G. Rota,et al.  Finite operator calculus , 1975 .

[15]  Martin Aigner,et al.  A Characterization of the bell numbers , 1999, Discret. Math..

[16]  Renzo Pinzani,et al.  Approximating algebraic functions by means of rational ones , 2002, Theor. Comput. Sci..

[17]  Frank Harary,et al.  The enumeration of tree-like polyhexes † , 1970 .

[18]  Martin Aigner,et al.  Catalan-like Numbers and Determinants , 1999, J. Comb. Theory, Ser. A.