Shuffle series

Khushdil Ahmad, Eric Rubiel Dolores-Cuenca, Khurram Shabbir

Published 2023-11-15Version 1

We apply operad theory to enumerative combinatorics in order to count the number of shuffles between series-parallel posets and chains. We work with three types of shuffles, at least one noncommutative: a left shuffle between P and Q is a shuffle of the posets in which the minimum and maximum elements belong to P and no two elements of Q appear consecutively. The number of left shuffles of P and Q differ from the number of left shuffles of Q and P. We explain how shuffle series are isomorphic to order series as algebras over the operad of series parallel posets. Concerning order series, the weak order series and strict order series are well known in the literature. With the theory of sets with a negative number of elements, we introduce a third order series and prove a theorem in the style of Stanley's Reciprocity Theorem compatible with the structure of algebras over the operad of finite posets. We conclude by describing the relationship of our work with the combinatorial properties of the operadic tensor product of trees.