Duplicial algebras and Lagrange inversion

Novelli, Jean-Christophe; Thibon, Jean-Yves

doi:10.48550/arxiv.1209.5959

Cited by 6 publications

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“…The graded dimensions of the various Hopf algebras are computed by means of explicit formulas for the number of congruence classes of a given evaluation. These results are of course new for the new congruences, but also for the sylvester congruence, where it arises as an application of our previous results on noncommutative Lagrange inversion [35].…”

Section: Introductionmentioning

confidence: 56%

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Hopf algebras of m-permutations, (m + 1)-ary trees, and m-parking functions

Novelli

Thibon

2020

Advances in Applied Mathematics

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The m-Tamari lattice of F. Bergeron is an analogue of the clasical Tamari order defined on objects counted by Fuss-Catalan numbers, such as m-Dyck paths or (m + 1)-ary trees. On another hand, the Tamari order is related to the product in the Loday-Ronco Hopf algebra of planar binary trees. We introduce new combinatorial Hopf algebras based on (m + 1)-ary trees, whose structure is described by the m-Tamari lattices.In the same way as planar binary trees can be interpreted as sylvester classes of permutations, we obtain (m + 1)-ary trees as sylvester classes of what we call mpermutations. These objects are no longer in bijection with decreasing (m + 1)-ary trees, and a finer congruence, called metasylvester, allows us to build Hopf algebras based on these decreasing trees. At the opposite, a coarser congruence, called hyposylvester, leads to Hopf algebras of graded dimensions (m+1) n−1 , generalizing noncommutative symmetric functions and quasi-symmetric functions in a natural way. Finally, the algebras of packed words and parking functions also admit such m-analogues, and we present their subalgebras and quotients induced by the various congruences.

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Section: Introductionmentioning

confidence: 56%

“…The hypoplactic Hopf algebras obtained from PQSym have graded dimensions given by the little Schröder numbers, these are the free triduplicial algebra on one generator and its dual [35].…”

Section: 31mentioning

confidence: 99%

Hopf algebras of m-permutations, (m + 1)-ary trees, and m-parking functions

Novelli

Thibon

2020

Advances in Applied Mathematics

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“…Various consequences of this fact, including a noncommutative q-Lagrange formula and generalisations to (k, ℓ)-parking functions have been derived there. Other applications have been given in [18,19,13].…”

Section: Introductionmentioning

confidence: 99%

“…where E is the so-called essential basis of quasi-symmetric functions. It is equal to the number of sylvester classes of words of evaluation 2 Ī [18,19] or alternatively, to the number of parking quasi-ribbons of shape (2I) ∼ [17], and also to the number of nondecreasing parking functions of type 2I + 1 r := (2i 1 + 1, . .…”

Section: Introductionmentioning

confidence: 99%

Noncommutative Symmetric Functions and Lagrange Inversion II: Noncrossing partitions and the Farahat-Higman algebra

Novelli¹,

Thibon²

2021

Preprint

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We introduce a new pair of mutually dual bases of noncommutative symmetric functions and quasi-symmetric functions, and use it to derive generalizations of several results on the reduced incidence algebra of the lattice of noncrossing partitions. As a consequence, we obtain a quasi-symmetric version of the Farahat-Higman algebra.2000 Mathematics Subject Classification. 05E05; 20C08; 05A15. Key words and phrases. Lagrange inversion; Parking functions; Noncommutative symmetric functions; Noncrossing partitions; Incidence Hopf algebras.1 Actually, Macdonald uses the equivalent basis h * µ (X) = g µ (−X).

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“…This approach to functional inversion has been initiated in [16,17,18], and then extended in [13] to deal with the conjugacy equation for formal diffeomorphisms.…”

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confidence: 99%

Free cumulants, Schröder trees, and operads

Josuat-Vergès

Menous

Novelli

et al. 2017

Advances in Applied Mathematics

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The functional equation defining the free cumulants in free probability is lifted successively to the noncommutative Faà di Bruno algebra, and then to the group of a free operad over Schröder trees. This leads to new combinatorial expressions, which remain valid for operator-valued free probability. Specializations of these expressions give back Speicher's formula in terms of noncrossing partitions, and its interpretation in terms of characters due to Ebrahimi-Fard and Patras.

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Duplicial algebras and Lagrange inversion

Cited by 6 publications

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Hopf algebras of m-permutations, (m + 1)-ary trees, and m-parking functions

Hopf algebras of m-permutations, (m + 1)-ary trees, and m-parking functions

Noncommutative Symmetric Functions and Lagrange Inversion II: Noncrossing partitions and the Farahat-Higman algebra

Free cumulants, Schröder trees, and operads

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Duplicial algebras and Lagrange inversion

Cited by 6 publications

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Hopf algebras of m-permutations, (m + 1)-ary trees, and m-parking functions

Hopf algebras of m-permutations, (m + 1)-ary trees, and m-parking functions

Noncommutative Symmetric Functions and Lagrange Inversion II: Noncrossing partitions and the Farahat-Higman algebra

Free cumulants, Schröder trees, and operads

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Hopf algebras of m-permutations, (m + 1)-ary trees, and m-parking functions

Hopf algebras of m-permutations, (m + 1)-ary trees, and m-parking functions