Some multivariate master polynomials for permutations, set partitions, and perfect matchings, and their continued fractions

Sokal, Alan D.; Zeng, Jiang

doi:10.1016/j.aam.2022.102341

Cited by 21 publications

(37 citation statements)

References 67 publications

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“…We then state our generalizations, which are T-fractions for super-augmented perfect matchings. Let us stress that none of our results in this paper depend on the results of [40]; we give here self-contained proofs of all our generalizations.…”

Section: Statement Of Resultsmentioning

confidence: 97%

“…We begin by reviewing some S-fractions for perfect matchings that were recently obtained by Zeng and one of us [40]. We then state our generalizations, which are T-fractions for super-augmented perfect matchings.…”

Section: Statement Of Resultsmentioning

confidence: 99%

“…Inspired by (2.1), let us introduce [40] the polynomials M n (x, y, u, v) defined by the continued fraction…”

Section: S-fractions For Perfect Matchingsmentioning

confidence: 99%

“…according to some natural trivariate statistic. Let us now explain, following [40], what this statistic is. Let π be a perfect matching of [2n].…”

Section: S-fractions For Perfect Matchingsmentioning

confidence: 99%

“…Then Sokal and Zeng [40] proved that the polynomials M n (x, y, u, v) defined by (2.2)/(2.3) have the combinatorial interpretation…”

Section: S-fractions For Perfect Matchingsmentioning

confidence: 99%

See 4 more Smart Citations

Phylogenetic trees, augmented perfect matchings, and a Thron-type continued fraction (T-fraction) for the Ward polynomials

Price,

Sokal

2020

Preprint

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We find a Thron-type continued fraction (T-fraction) for the ordinary generating function of the Ward polynomials, as well as for some generalizations employing a large (indeed infinite) family of independent indeterminates. Our proof is based on a bijection between super-augmented perfect matchings and labeled Schröder paths, which generalizes Flajolet's bijection between perfect matchings and labeled Dyck paths.

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Section: Statement Of Resultsmentioning

confidence: 97%

Section: Statement Of Resultsmentioning

confidence: 99%

“…Inspired by (2.1), let us introduce [40] the polynomials M n (x, y, u, v) defined by the continued fraction…”

Section: S-fractions For Perfect Matchingsmentioning

confidence: 99%

“…according to some natural trivariate statistic. Let us now explain, following [40], what this statistic is. Let π be a perfect matching of [2n].…”

Section: S-fractions For Perfect Matchingsmentioning

confidence: 99%

“…Then Sokal and Zeng [40] proved that the polynomials M n (x, y, u, v) defined by (2.2)/(2.3) have the combinatorial interpretation…”

Section: S-fractions For Perfect Matchingsmentioning

confidence: 99%

See 3 more Smart Citations

Phylogenetic trees, augmented perfect matchings, and a Thron-type continued fraction (T-fraction) for the Ward polynomials

Price,

Sokal

2020

Preprint

Self Cite

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Total positivity of some polynomial matrices that enumerate labeled trees and forests I: forests of rooted labeled trees

Sokal

2022

Monatsh Math

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We consider the lower-triangular matrix of generating polynomials that enumerate k-component forests of rooted trees on the vertex set [n] according to the number of improper edges (generalizations of the Ramanujan polynomials). We show that this matrix is coefficientwise totally positive and that the sequence of its row-generating polynomials is coefficientwise Hankel-totally positive. More generally, we define the generic rooted-forest polynomials by introducing also a weight $$m! \, \phi _m$$ m ! ϕ m for each vertex with m proper children. We show that if the weight sequence $$\varvec{\phi }$$ ϕ is Toeplitz-totally positive, then the two foregoing total-positivity results continue to hold. Our proofs use production matrices and exponential Riordan arrays.

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Continued fractions for cycle-alternating permutations

Deb,

Sokal

2024

Ramanujan J

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A permutation is said to be cycle-alternating if it has no cycle double rises, cycle double falls or fixed points; thus each index i is either a cycle valley ($$\sigma ^{-1}(i)>i<\sigma (i)$$ σ - 1 ( i ) > i < σ ( i ) ) or a cycle peak ($$\sigma ^{-1}(i)<i>\sigma (i)$$ σ - 1 ( i ) < i > σ ( i ) ). We find Stieltjes-type continued fractions for some multivariate polynomials that enumerate cycle-alternating permutations with respect to a large (sometimes infinite) number of simultaneous statistics that measure cycle status, record status, crossings and nestings along with the parity of the indices. Our continued fractions are specializations of more general continued fractions of Sokal and Zeng. We then introduce alternating Laguerre digraphs, which are generalization of cycle-alternating permutations, and find exponential generating functions for some polynomials enumerating them. We interpret the Stieltjes–Rogers and Jacobi–Rogers matrices associated to some of our continued fractions in terms of alternating Laguerre digraphs.

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Some multivariate master polynomials for permutations, set partitions, and perfect matchings, and their continued fractions

Cited by 21 publications

References 67 publications

Phylogenetic trees, augmented perfect matchings, and a Thron-type continued fraction (T-fraction) for the Ward polynomials

Phylogenetic trees, augmented perfect matchings, and a Thron-type continued fraction (T-fraction) for the Ward polynomials

Total positivity of some polynomial matrices that enumerate labeled trees and forests I: forests of rooted labeled trees

Continued fractions for cycle-alternating permutations

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