Chromatic Polynomial and Heaps of Pieces

Deb, Bishal

doi:10.48550/arxiv.1902.02240

Cited by 3 publications

(6 citation statements)

References 2 publications

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“…2, 3, 4) is 16 (resp. 8,4,3). This gives a total of 31 pairs which, as predicted by Theorem 1.1, is equal to −χ G (−1).…”

Section: Introductionmentioning

confidence: 84%

“…Our proof is based on the theory of heaps, which takes its root in the work of Cartier and Foata [1], and has been popularized by Viennot [13]. In fact, our proof is in the same spirit as the one used by Gessel in [6], and subsequently by Lass in [9] (see also the recent preprint [3]). It consists in showing that well-known counting lemmas for heaps imply a relation between proper colorings and acyclic orientations.…”

Section: Introductionmentioning

confidence: 91%

“…To prove Theorem 2.4, observe first that (2) is the special case S = [n] of (4). It remains to prove (3). Let t be an indeterminate.…”

Section: Heaps: Definition and Counting Lemmasmentioning

confidence: 98%

“…Left: A graph G on 4 vertices, having chromatic polynomial χ G (q) = q 4 − 4q 3 + 6q 2 − 3q. Right: The graph G (3,2,0,1) .…”

Section: Introductionmentioning

confidence: 99%

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Combinatorial reciprocity for the chromatic polynomial and the chromatic symmetric function

Bernardi¹,

Nadeau²

2019

Preprint

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Let G be a graph, and let χG be its chromatic polynomial. For any nonnegative integers i, j, we give an interpretation for the evaluation χ (i) G (−j) in terms of acyclic orientations. This recovers the classical interpretations due to Stanley and to Greene and Zaslavsky respectively in the cases i = 0 and j = 0. We also give symmetric function refinements of our interpretations, and some extensions. The proofs use heap theory in the spirit of a 1999 paper of Gessel.

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“…2, 3, 4) is 16 (resp. 8,4,3). This gives a total of 31 pairs which, as predicted by Theorem 1.1, is equal to −χ G (−1).…”

Section: Introductionmentioning

confidence: 84%

Section: Introductionmentioning

confidence: 91%

“…To prove Theorem 2.4, observe first that (2) is the special case S = [n] of (4). It remains to prove (3). Let t be an indeterminate.…”

Section: Heaps: Definition and Counting Lemmasmentioning

confidence: 98%

“…Left: A graph G on 4 vertices, having chromatic polynomial χ G (q) = q 4 − 4q 3 + 6q 2 − 3q. Right: The graph G (3,2,0,1) .…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Combinatorial reciprocity for the chromatic polynomial and the chromatic symmetric function

Bernardi¹,

Nadeau²

2019

Preprint

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“…The following results of Stanley [12], and Greene and Zaslavsky [5] respectively are classical and follow immediately from Corollary 1.5. In [3], these results are proved using the method of involution on heaps. In this section, we will denote the chromatic polynomial by χ G (q) for notational convenience.…”

Section: Introductionmentioning

confidence: 92%

Generalized chromatic polynomials of graphs from Heaps of pieces

Arunkumar

2019

Preprint

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Let G be a simple graph and let L(G) be the free partially commutative Lie algebra associated to G. In this paper, using heaps of pieces, we prove an expression for the generalized k-chromatic polynomial of G in terms of dimensions of the grade spaces of L(G). This will give us a new interpretation for the chromatic polynomials in terms of multilinear heaps and Lyndon length. The classical results of Stanley, and Greene and Zaslavsky regarding the acyclic orientations of G are obtained as corollaries. A heap with a unique minimal piece is said to be a pyramid and our main theorem is proved using the properties of pyramids. For this reason, we prove two important properties of pyramids namely the pyramid proportionality lemma, and the pyramid and Lyndon heap lemma. In the last section, we will introduce the (m, λ)-labelling on acyclic orientations which is similar to Stanley's λ-compatible pairs. As an application of our main theorem, using this (m, λ)-labelled acyclic orientations, we will prove the reciprocity theorem for the derivatives of the chromatic polynomials.

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Bialgebras in cointeraction, the antipode and the eulerian idempotent

Foissy¹

2022

Preprint

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We give here a review of results about double bialgebras, that is to say bialgebras with two coproducts, the first one being a comodule morphism for the coaction induced by the second one. An accent is put on the case of connected bialgebras. The subjects of these results are the monoid of characters and their actions, polynomial invariants, the antipode and the eulerian idempotent. As examples, they are applied on a double bialgebra of graphs and on quasishuffle bialgebras. This includes a new proof of a combinatorial interpretation of the coefficients of the chromatic polynomial due to Greene and Zaslavsky.

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Chromatic Polynomial and Heaps of Pieces

Cited by 3 publications

References 2 publications

Combinatorial reciprocity for the chromatic polynomial and the chromatic symmetric function

Combinatorial reciprocity for the chromatic polynomial and the chromatic symmetric function

Generalized chromatic polynomials of graphs from Heaps of pieces

Bialgebras in cointeraction, the antipode and the eulerian idempotent

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