Staircase tilings and [formula omitted]-Catalan structures

Abstract

Many interesting combinatorial objects are enumerated by the k -Catalan numbers, one possible generalization of the Catalan numbers. We will present a new combinatorial object that is enumerated by the k -Catalan numbers, staircase tilings. We give a bijection between staircase tilings and k -good paths, and between k -good paths and k -ary trees. In addition, we enumerate k -ary paths according to DD , UDU , and UU , and connect these statistics for k -ary paths to statistics for the staircase tilings. Using the given bijections, we enumerate statistics on the staircase tilings, and obtain connections with Catalan numbers for special values of k . The second part of the paper lists a sampling of other combinatorial structures that are enumerated by the k -Catalan numbers. Many of the proofs generalize from those for the Catalan structures that are being generalized, but we provide one proof that is not a straightforward generalization. We propose a web site repository for these structures, similar to those maintained by Richard Stanley for the Catalan numbers [R.P. Stanley, Catalan addendum. Available at: http://www-math.mit.edu/~rstan/ec/] and by Robert Sulanke for the Delannoy numbers [R. Sulanke, Objects counted by the central Delannoy numbers, J. Integer Seq. 6 (1) (2003), Article 03, 1, 5, 19 pp. Available also at: math.boisestate.edu/~sulanke/infowhowasdelannoy.html]. On the website, we list additional combinatorial objects, together with hints on how to show that they are indeed enumerated by the k -Catalan numbers.