# Code and Data

### J. Han, K. Lee, L. Li, N. Loehr, Chain decompositions of q, t-catalan numbers via local chains, https://arxiv.org/abs/2003.03896

Here is the Extended Appendix.

In this folder you can find the SageMath code to check that our global chains indeed satisfy the desired properties. To run it, put files "k=1.txt",...,"k=11.txt" to the appropriate folder, and paste the code in "main_code.txt" to a Sage cell. The output is in "outpu.txt".

### K. Lee, L. Li, N. Loehr, A Combinatorial Approach to the Symmetry of $q,t$-Catalan Numbers, SIAM J. Discrete Math (SIDMA). 32 (2018) no.1, 191--232.

SageMath is a free open-source mathematics software system, which can be downloaded here. You may also try the online version CoCalc (SageMathCloud).

First input this code and you will see this output. It defines relevant functions in the paper and computes some examples in the paper.

Run this this code, and you will see this output. It shows that there is a unique solution of Conjecture 4.8 for k<=4 and two solutions for k=5.

Run this this code, and you will see this output. It finds a solution for k=6.

List_of_C_mu.pdf (previously available in www.oakland.edu/~li2345/List_of_C_mu.pdf as stated in section 6 of the paper)

Run this code. Then run code(k=7), code(k=8), code(k=9). It checks that the solutions for k=7,8,9 given in the above pdf file satisfy Conjecture 4.8.

### Higher (Haiman/Hagland)-Catalan numbers

It is conjectured that the higher q,t-Catalan number C^m_n(q,t) defined by Garsia and Haiman coincides with the combinatorially defined (Haiman/Haglund-)Catalan number. (cf. N.Loehr's paper "Conjectured Statistics for the Higher q,t-Catalan Sequences"). Here is a list of Catalan numbers, q-Catalan numbers and higher q,t-Catalan numbers.