Narayana polynomials and Hall–Littlewood symmetric functions

Lassalle, Michel

doi:10.1016/j.aam.2012.06.002

Cited by 6 publications

(6 citation statements)

References 23 publications

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“…It is known that the generalized Narayana numbers associated with finite Coxeter groups have only real roots, see [21, §5.2]. Remark 6.13 The polynomials z p,p+1 and z p,p , as well as the polynomials F for forks seem to appear in the article [14], which deals with symmetric functions. The relationship with the present work is not clear to us.…”

Section: Conjecture 611mentioning

confidence: 99%

Flows on Rooted Trees and the Menous-Novelli-Thibon Idempotents

Chapoton

2014

MATH. SCAND.

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Several generating series for flows on rooted trees are introduced, as elements in the group of series associated with the Pre-Lie operad. By combinatorial arguments, one proves identities that characterise these series. One then gives a complete description of the image of these series in the group of series associated with the Dendriform operad. This allows to recover the Lie idempotents in the descent algebras recently introduced by Menous, Novelli and Thibon. Moreover, one defines new Lie idempotents and conjecture the existence of some others.

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Section: Conjecture 611mentioning

confidence: 99%

Flows on Rooted Trees and the Menous-Novelli-Thibon Idempotents

Chapoton

2014

MATH. SCAND.

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“…Using known results about generating functions [37,38], we have For more details we refer to [21] and references therein.…”

Section: Leading Termsmentioning

confidence: 99%

Class expansion of some symmetric functions in Jucys–Murphy elements

Lassalle

2013

Journal of Algebra

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“…The formula of Lassalle for the Narayana polynomials. It has been observed by M. Lassalle [10] that the Narayana polynomials c n (q) could be expressed in the form (85) qc n (q) = 1 n + 1 h n ((n + 1)q) in λ-ring notation, with the assumption that x = 1 − q is of rank 1. Otherwise said,…”

Section: 1mentioning

confidence: 99%

“…Finally, we illustrate the idea that combinatorial series derived by means of Lagrange inversion should come from some character of the algebra of parking functions. First, we generalize Lassalle's expression [10] of the Narayana polynomials c n (t) to super-Narayana polynomials P n (t, q) counting signed parking functions according to certain statistics, and such that P n (t, 0) = (1 + t)c n (1 + t). This last polynomial is known to count Schröder paths according to the number of horizontal steps [26, A060693].…”

Section: Introductionmentioning

confidence: 99%

Duplicial algebras, parking functions, and Lagrange inversion

Novelli¹,

Thibon²

2020

IRMA Lectures in Mathematics and Theoretical Physics

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We provide operadic interpretations for two Hopf subalgebras of the algebra of parking functions. The Catalan subalgebra is identified with the free duplicial algebra on one generator, and the Schröder subalgebra is interpreted by means of a new operad, which we call triduplicial.The noncommutative Lagrange inversion formula is then interpreted in terms of duplicial structures. The generic solution of the noncommutative inversion problem appears as the formal sum of all parking functions. This suggests that combinatorial generating functions derived by functional inversion should be obtainable by evaluating a suitable character on this generic solution. This idea is illustrated by means of the Narayana polynomials, of which we obtain bivariate "super-analogues" by lifting to parking functions a classical character of the algebra of symmetric functions. Other characters, such as evaluation of symmetric functions on a binomial element, are also discussed.1991 Mathematics Subject Classification. 18D50,05E05,16T30.

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Narayana polynomials and Hall–Littlewood symmetric functions

Cited by 6 publications

References 23 publications

Flows on Rooted Trees and the Menous-Novelli-Thibon Idempotents

Flows on Rooted Trees and the Menous-Novelli-Thibon Idempotents

Class expansion of some symmetric functions in Jucys–Murphy elements

Duplicial algebras, parking functions, and Lagrange inversion

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