Operads of (noncrossing) partitions, interacting bialgebras, and moment-cumulant relations

Shuffle Algebras and Non-Commutative Probability for Pairs of Faces

“…In this work, we extended the dendriform algebraic perspective on the free, boolean and monotone moment-cumulant relations to bifree, biBoolean and bimonotone moment-cumulant relations. To achieve this, we leverage an idea latent in [EFFKP20] abridged as follows: there exists a certain coDendriform structure on noncrossing partitions which, when restricted to "sticks" (partitions into singletons) yields the coDendriform coalgebra introduced in [EFP15]. This former coDendriform structure stems from a certain composition rule, formalized using the theory of operads, between noncrossing partitions, the gap insertion operad in which a partitition is inserted into an other by "placing" blocks inbetween two consecutive elements of the latter.…”

“…Subordination products, Bercovici-Pata bijection, and formulas relating the various families of cumulants can be cast in the dendriform (and the accompanying preLie) algebraic realm [EFP19]. It was recognized later that the discussed coDendriform structure is the restriction of a coDendriform structure on (polynomials on) noncrossing partitions, drawing a tight connection with the theory of operads, see [EFFKP20]. The present work elaborates on the construction described in [Gil20] designed to extend the work [EFP15] to operator-valued probability spaces which was itself inspired by [EFFKP20].…”

“…It was recognized later that the discussed coDendriform structure is the restriction of a coDendriform structure on (polynomials on) noncrossing partitions, drawing a tight connection with the theory of operads, see [EFFKP20]. The present work elaborates on the construction described in [Gil20] designed to extend the work [EFP15] to operator-valued probability spaces which was itself inspired by [EFFKP20]. In particular, we use notions originating from higher category theory, duoidal categories namely.…”

“…Remark. We compare the restriction of the product m ⦶ W to (one-sided) noncrossing partitions and the gap-insertion operad introduced in[EFFKP20]. First, the monoidal product m ⦶ W restricts to the sub-TW-collection of NC comprising all shaded noncrossing partitions; the shaded noncrossing bipartitions with type (L n , b), n ≥ 1 and b a Boolean word with length n. Pick shaded noncrossing bipartition π with type t = (L n , b) and denote by I 0 , .…”

“…Pick a noncrossing partition π ′ with type (L p , 1). Denote by ρ the gap-insertion operadic product introduced in[EFFKP20]. Then ρ(π ′ , π 0 , .…”