Cain, Alan J., and António Malheiro. "
Combinatorics of Cyclic Shifts in Plactic, Hypoplactic, Sylvester, and Related Monoids."
Combinatorics on Words: 11th International Conference, WORDS 2017, Montréal, QC, Canada, September 11-15, 2017, Proceedings. Eds. Srečko Brlek, Francesco Dolce, Christophe Reutenauer, and Élise Vandomme. Cham: Springer International Publishing, 2017. 190-202.
AbstractThe cyclic shift graph of a monoid is the graph whose vertices are elements of the monoid and whose edges link elements that differ by a cyclic shift. For certain monoids connected with combinatorics, such as the plactic monoid (the monoid of Young tableaux) and the sylvester monoid (the monoid of binary search trees), connected components consist of elements that have the same evaluation (that is, contain the same number of each generating symbol). This paper discusses new results on the diameters of connected components of the cyclic shift graphs of the finite-rank analogues of these monoids, showing that the maximum diameter of a connected component is dependent only on the rank. The proof techniques are explained in the case of the sylvester monoid.
Cain, Alan J., António Malheiro, and Fábio M. Silva. "
Combinatorics of patience sorting monoids."
Discrete Mathematics. 342.9 (2019): 2590-2611.
AbstractThis paper makes a combinatorial study of the two monoids and the two types of tableaux that arise from the two possible generalizations of the Patience Sorting algorithm from permutations (or standard words) to words. For both types of tableaux, we present Robinson--Schensted--Knuth-type correspondences (that is, bijective correspondences between word arrays and certain pairs of semistandard tableaux of the same shape), generalizing two known correspondences: a bijective correspondence between standard words and certain pairs of standard tableaux, and an injective correspondence between words and pairs of tableaux.
We also exhibit formulas to count both the number of each type of tableaux with given evaluations (that is, containing a given number of each symbol). Observing that for any natural number $n$, the $n$-th Bell number is given by the number of standard tableaux containing $n$ symbols, we restrict the previous formulas to standard words and extract a formula for the Bell numbers. Finally, we present a `hook length formula' that gives the number of standard tableaux of a given shape and deduce some consequences.