Order quasisymmetric functions distinguish rooted trees

Tsujie, Shuhei

doi:10.1007/s10801-017-0761-7

Cited by 16 publications

(35 citation statements)

References 17 publications

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“…Hasebe and Tsujie [HT17] have shown the case when all the relations are weak (or all strict), and we generalize their result by establishing Conjecture 1.3 for a class of labeled rooted trees that we call fair trees. We conclude in Section 5 with some directions for further study.…”

Section: Introductionmentioning

confidence: 58%

“…• Rooted trees [HT17, Zho20], i.e, posets that are trees with a single minimal element. • More generally, posets that are both bowtie-free and N-free [HT17]. As one would expect, N is the poset consisting of elements a 1 , a 2 , b 1 , b 2 whose cover relations are a 1 < b 1 > a 2 < b 2 , and a poset is N-free if it does not contain N as an induced subposet.…”

Section: Consequences Of the Poset Viewpointmentioning

confidence: 99%

“…On the other hand, Hasebe and Tsujie [HT17] have shown that K (P,ω) distinguishes rooted trees with all strict (equivalently all weak) edges. In Conjecture 1.3, we propose that their result also holds for arbitrary labeled posets that are rooted trees.…”

Section: Adding Strict Edgesmentioning

confidence: 99%

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Quasisymmetric functions distinguishing trees

Aval¹,

Djenabou²,

McNamara³

2022

Preprint

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A famous conjecture of Stanley states that his chromatic symmetric function distinguishes trees. As a quasisymmetric analogue, we conjecture that the chromatic quasisymmetric function of Shareshian and Wachs and of Ellzey distinguishes directed trees. This latter conjecture would be implied by an affirmative answer to a question of Hasebe and Tsujie about the Ppartition enumerator distinguishing posets whose Hasse diagrams are trees. They proved the case of rooted trees and our results include a generalization of their result.

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Section: Introductionmentioning

confidence: 58%

Section: Consequences Of the Poset Viewpointmentioning

confidence: 99%

See 1 more Smart Citation

Quasisymmetric functions distinguishing trees

Aval¹,

Djenabou²,

McNamara³

2022

Preprint

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“…In [HT17] (and later in [MW18]) the authors gave a simple description of posets that are both induced N -free and induced bowtie-free which we include here as Definition 10 and Theorem 2.…”

Section: Resultsmentioning

confidence: 99%

A New Characterization of $\mathcal {V}$-Posets

Cooper

Gartland

Whitlatch

2019

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In 2016, Hasebe and Tsujie gave a recursive characterization of the set of induced N -free and bowtie-free posets; Misanantenaina and Wagner studied these orders further, naming them "V-posets". Here we offer a new characterization of V-posets by introducing a property we refer to as autonomy. A poset P is said to be autonomous if there exists a directed acyclic graph D (with adjacency matrix U ) whose transitive closure is P, with the property that any total ordering of the vertices of D so that Gaussian elimination of U T U proceeds without row swaps is a linear extension of P. Autonomous posets arise from the theory of pressing sequences in graphs, a problem with origins in phylogenetics. The pressing sequences of a graph can be partitioned into families corresponding to posets; because of the interest in enumerating pressing sequences, we investigate when this partition has only one block, that is, when the pressing sequences are all linear extensions of a single autonomous poset. We also provide an efficient algorithm for recognition of autonomy using structural information and the forbidden subposet characterization, and we discuss a few open questions that arise in connection with these posets.arXiv:1810.07276v2 [math.CO]

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“…The order quasisymmetric functions are considered to be analogs of the chromatic symmetric function. A recent study [HT17] by Hasebe and the author showed that the order quasisymmetric functions distinguish rooted trees (with the natural poset structures). The proof is based on algebraic structures of the ring of quasisymmetric functions.…”

Section: Introductionmentioning

confidence: 99%

The Chromatic Symmetric Functions of Trivially Perfect Graphs and Cographs

Tsujie¹

2018

Graphs and Combinatorics

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defined the chromatic symmetric function of a simple graph and has conjectured that every tree is determined by its chromatic symmetric function. Recently, Takahiro Hasebe and the author proved that the order quasisymmetric functions, which are analogs of the chromatic symmetric functions, distinguish rooted trees. In this paper, using a similar method, we prove that the chromatic symmetric functions distinguish trivially perfect graphs. Moreover, we also prove that claw-free cographs, that is, {K 1,3 , P 4 }-free graphs belong to a known class of e-positive graphs.

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Order quasisymmetric functions distinguish rooted trees

Cited by 16 publications

References 17 publications

Quasisymmetric functions distinguishing trees

Quasisymmetric functions distinguishing trees

A New Characterization of $\mathcal {V}$-Posets

The Chromatic Symmetric Functions of Trivially Perfect Graphs and Cographs

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