The number of coverings in four catalan lattices

We consider in this note a series of lattices that are enumerated by the well-known Catalan numbers. We give formulas for the number of coverings, i.e. the number of edges in the Hasse diagrams of these lattices.

[1]  N. Sloane.  A Handbook Of Integer Sequences , 1973 .

[2]  Germain Kreweras,et al.  Sur les partitions non croisees d'un cycle , 1972, Discret. Math..

[3]  Thomas Penyngton Kirkman,et al.  XII. On the K-partitions of the R-gon and R-ace , 1857, Philosophical Transactions of the Royal Society of London.

[4]  C. Lanczos.  Applied Analysis , 1961 .

[5]  Jean Marcel Pallo,et al.  A-transformation dans les arbres n-aires , 1983, Discret. Math..

[6]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .

[7]  Nachum Dershowitz,et al.  Enumerations of ordered trees , 1980, Discret. Math..

[8]  Jean Marcel Pallo,et al.  A shortest path metric on unlabeled binary trees , 1992, Pattern Recognit. Lett..

[9]  Frank Ruskey,et al.  Generating Binary Trees by Transpositions , 1988, J. Algorithms.

[10]  Dov Tamari,et al.  Problèmes d'associativité: Une structure de treillis finis induite par une loi demi-associative , 1967 .

[11]  Cayley.  On the Partitions of a Polygon , 1890 .

[12]  George Markowsky,et al.  Primes, irreducibles and extremal lattices , 1992 .

[13]  Paul H. Edelman.  Chain enumeration and non-crossing partitions , 1980, Discret. Math..

[14]  Xavier Gérard Viennot.  Enumerative Combinatorics and Computer Science , 1990, STACS.

[15]  R. Tarjan,et al.  Rotation distance, triangulations, and hyperbolic geometry , 1986, STOC '86.

[16]  Winfried Geyer.  On Tamari lattices , 1994, Discret. Math..

[17]  Dan Rubenstein.  Catalan Numbers Revisited , 1994, J. Comb. Theory, Ser. A.

[18]  Jean Marcel Pallo.  Some Properties of the Rotation Lattice of Binary Trees , 1988, Comput. J..

[19]  Rodica Simion,et al.  On the structure of the lattice of noncrossing partitions , 1991, Discret. Math..

[20]  J. Pallo,et al.  A one-to-one correspondence between binary trees and certain Jacobi matrices , 1992 .

[21]  Jean Marcel Pallo,et al.  An Algorithm to Compute the Möbius Function of the Rotation Lattice of Binary Trees , 1993, RAIRO Theor. Informatics Appl..

[22]  Jean Marcel Pallo,et al.  A note on generating binary trees in A-order and B-order , 1985 .

[23]  Alasdair Urquhart,et al.  A topological representation theory for lattices , 1978 .

[24]  P. Hilton,et al.  Catalan Numbers, Their Generalization, and Their Uses , 1991 .