Free Cumulants and Enumeration of Connected Partitions

Lehner, Franz

doi:10.1006/eujc.2002.0619

Cited by 39 publications

(38 citation statements)

References 23 publications

(26 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…As an added bonus, we will obtain a new combinatorial formula for converting from free to classical cumulants in noncommutative probability theory that involves quasicanonical trees (see Corollary 4.2). This result adds to the recent surge of interest in combinatorial cumulant conversion formulas [1,3,11,24,28,32,38].…”

Section: Introductionmentioning

confidence: 66%

Fertilitopes

Defant¹

2021

Preprint

View full text Add to dashboard Cite

We introduce tools from discrete convexity theory and polyhedral geometry into the theory of West's stack-sorting map s. Associated to each permutation π is a particular set V(π) of integer compositions that appears in a formula for the fertility of π, which is defined to be |s −1 (π)|. These compositions also feature prominently in more general formulas involving families of colored binary plane trees called troupes and in a formula that converts from free to classical cumulants in noncommutative probability theory. We show that V(π) is a transversal discrete polymatroid when it is nonempty. We define the fertilitope of π to be the convex hull of V(π), and we prove a surprisingly simple characterization of fertilitopes as nestohedra arising from full binary plane trees. Using known facts about nestohedra, we provide a procedure for describing the structure of the fertilitope of π directly from π using Bousquet-Mélou's notion of the canonical tree of π. As a byproduct, we obtain a new combinatorial cumulant conversion formula in terms of generalizations of canonical trees that we call quasicanonical trees. We also apply our results on fertilitopes to study combinatorial properties of the stack-sorting map. In particular, we show that the set of fertility numbers has density 1, and we determine all infertility numbers of size at most 126. Finally, we reformulate the conjecture that σ∈s −1 (π) x des(σ)+1 is always real-rooted in terms of nestohedra, and we propose natural ways in which this new version of the conjecture could be extended.

show abstract

Section: Introductionmentioning

confidence: 66%

Fertilitopes

Defant¹

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…In the special case when s = 1 and t = 0, Equation (7.16) becomes the transition formula from free cumulants to Boolean cumulants, which is well-known since the work of Lehner [18]. When swapping the role of the parameters and putting s = 0 and t = 1, one finds the inverse transition formula which writes free cumulants in terms of Boolean cumulants, and is also well-known (cf.…”

Section: An Interpolation Between Free and Boolean: T-boolean Cumulantsmentioning

confidence: 98%

“…This action captures the transitions between moment functionals and the brands of cumulants mentioned above, and as a consequence it also captures the formulas for direct transitions between two such brands of cumulants. We mention that the study of direct transitions between different brands of cumulants goes back to the work of Lehner [18], and was thoroughly pursued in [1]. The benefit of using the group G is that it offers an efficient framework for streamlining calculations related to various moment-cumulant and inter-cumulant formulas.…”

mentioning

confidence: 99%

Multiplicative and semi-multiplicative functions on non-crossing partitions, and relations to cumulants

Celestino¹,

Kurusch²,

Nica³

et al. 2021

Preprint

View full text Add to dashboard Cite

We consider the group (G, * ) of unitized multiplicative functions in the incidence algebra of non-crossing partitions, where " * " denotes the convolution operation. We introduce a larger group ( G, * ) of unitized functions from the same incidence algebra, which satisfy a weaker semi-multiplicativity condition. The natural action of G on sequences of multilinear functionals of a non-commutative probability space captures the combinatorics of transitions between moments and some brands of cumulants that are studied in the non-commutative probability literature. We use the framework of G in order to explain why the multiplication of free random variables can be very nicely described in terms of Boolean cumulants and more generally in terms of t-Boolean cumulants, a oneparameter interpolation between free and Boolean cumulants arising from work of Bożejko and Wysoczanski.It is known that the group G can be naturally identified as the group of characters of the Hopf algebra Sym of symmetric functions. We show that G can also be identified as group of characters of a Hopf algebra T , which is an incidence Hopf algebra in the sense of Schmitt. Moreover, the inclusion G ⊆ G turns out to be the dual of a natural bialgebra homomorphism from T onto Sym.

show abstract

“…Often, one requests a monic condition: ω n (1 n ) = 1, for all n ≥ 1 (see [15]), so that the n-th moment cumulant formula is monic on the n-th cumulant, and hence, in particular, cumulants can always be solved inductively.…”

Section: The Sets Of Partitions Non-crossing Partitions and Interval ...mentioning

confidence: 99%

Combinatorics of NC-Probability Spaces with Independent Constants

Carlos¹,

Gaxiola²,

Santos³

et al. 2021

Preprint

View full text Add to dashboard Cite

Unlike classical and free independence, the boolean and monotone notions of independence lack of the property of independent constants.In the scalar case, this leads to restrictions for the central limit theorems, as observed by F. Oravecz.We characterize the property of independent constants from a combinatorial point of view, based on cumulants and set partitions. This characterization also holds for the operator-valued extension.Our considerations lead rather directly to very mild variations of boolean and monotone cumulants, where constants are now independent. These alternative probability theories are closely related to the usual notions. Hence, an important part of the boolean/monotone probability theories can be imported directly.We describe some standard combinatorial aspects of these variations (and their cyclic versions), such as their Möbius functions, which feature well-known combinatorial integer sequences.The new notions with independent constants seem also more strongly related to the operator-valued extension of c-free probability.

show abstract

Free Cumulants and Enumeration of Connected Partitions

Cited by 39 publications

References 23 publications

Fertilitopes

Fertilitopes

Multiplicative and semi-multiplicative functions on non-crossing partitions, and relations to cumulants

Combinatorics of NC-Probability Spaces with Independent Constants

Contact Info

Product

Resources

About