Shuffle series

TL;DR

The paper develops a operadic framework for counting shuffles between series-parallel posets and chains via shuffle series $\mathcal{SH}(P)$, showing that the shuffle series form a ${\mathcal{SP}}$-algebra generated by $\mathcal{SH}(1)=\frac{1}{(1-x)^2}$ and that chain-shuffles obey explicit generator actions: $\widetilde{\{c,d\}}$ acts by the Hadamard product and $\widetilde{\{c<d\}}$ by the deformed product with $1-x$. It introduces three algebras of order-type series (strict, weak, and weak-surjective) and proves they are linked by operadic change of basis to the corresponding shuffle-series algebras, unifying shuffles with Ehrhart theory via order polytopes and their triangulations. The work connects shuffle-series to tree operads and their tensor products, and provides explicit combinatorial interpretations through vector data $d_{P,i}$, Doppelgänger posets, and colimit- or deck-divider shuffle constructions. Open questions address extending the theory to the full finite poset operad, geometric interpretations of polytope recurrences, and the limits of current explicit formulas for complex posets.

Abstract

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, two of them noncommutative, for example a left deck-divider shuffle $A$ between $P$ and $Q$ is a shuffle of the posets in which, on every maximal chain $m\subset A$, the minimum and maximum elements belong to $P$ and no two consecutive points of $Q$ appear consecutively on $m$. The number of left deck-divider shuffles of $P$ and $Q$ differ from the number of left deck-divider shuffles of $Q$ and $P$. The generating functions whose $n$ coefficient counts shuffles between a poset $P$ and $1<2<\cdots<n$ are called shuffle series. We explain how shuffle series are isomorphic to order series as algebras over the operad of series parallel posets. The weak and strict order polynomials are well known in the literature. At the level of generating series, 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 free trees operads.

Figures (9)

• Figure 1: The Hasse diagram of the poset $\{r<s,r<t, q<s,q<t\}$.
• Figure 2: A factorization for the poset $\{r<s,r<t, q<s,q<t\}$.
• Figure 3:
• Figure 4: The four-ary operation $\widetilde{\{c<d\}}(\widetilde{\{c,d\}},\widetilde{\{c<d\}})$ evaluated in four points $(p,q,r,s)$ returns the poset $\{p,q,r,s|p<r,q<r,r<s\}.$
• Figure 5: Reduced tree.

Theorems & Definitions (77)

• Definition 1.1
• Definition 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Definition 2.4
• Remark 2.5
• Example 2.6
• Definition 2.7