The Generating Functions Enumerating 12..d-Avoiding Words with r occurrences of each of 1,2, ... , n are D-finite for all d and all r

Abstract

In this article, dedicated with admiration and gratitude to guru Neil Sloane on his 75-th birthday, we observe that the generating functions for multi-set permutations that do not contain an increasing subsequence of length d, and where every letter appears the same number of times, say r, are always D-finite, (for every d and every r), and we actually crank out the first few terms of quite a few of them, many of whom are not yet in the OEIS. We also state a conjectured asymptotic formula for these sequences, that reduces to Amitai Regev's famous formula when r=1, and pledge a 100 dollar donation to the OEIS in honor of the first one to prove our conjecture. We pledge another 100 dollars for extending Ira Gessel's spectacular Bessel determinant, from the r=1 case to general r.