Staircase tilings and k-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.

Colored Tensor Models - a Review

“…The condition that at leading order in the large N limit the 1PI two point functions factors into contributions from the connected two point functions is at the core of all the studies of the critical behavior of tensor models performed so far. [92]. Similar equations appear in the study of branched polymers.…”

Critical behavior of colored tensor models in the large N limit

“…At order (λ λ) p we obtain contributions (with combinatorial weight 1) from all rooted colored trees with p vertices of degree D + 2 and with Dp + 1 leaves. Such trees go under the name of (D + 1)-ary trees in the mathematical literature [39,40]. Basically, any vertex has either (D + 1) or 0 children (with respect to the natural order starting from the root).…”

“…This equation is well-known in the literature ( [42] exercise 2.7.1, [43] pp. 200, [44] proposition 6.2.2), specifically in various problems of enumeration [40]. The solution which goes to 1 when (λ λ) goes to zero can be written as a power series in (λ λ) with coefficients the (D + 1)-Catalan numbers.…”

Generation, Ranking and Unranking of Ordered Trees with Degree Bounds

“…Many papers have been published earlier in the literature for generating different classes of trees. For example we can mention the generation of binary trees in [26,22,30,2,3,4,36], k-ary trees in [27,10,31,19,37,18,17,1,35], rooted trees in [7,32], trees with n nodes and m leaves in [24,28], neuronal trees in [25], and AVL trees in [21]. On the other hand, many papers have thoroughly investigated basic combinatorial features of chemical trees [12,14,13,9,8,20,33].…”