Dual Immaculate Quasisymmetric Functions Expand Positively into Young Quasisymmetric Schur Functions

Allen, Edward E.; Hallam, Joshua; Mason, Sarah

doi:10.46298/dmtcs.6410

Cited by 4 publications

(18 citation statements)

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“…And w ≡ plax r(P (w )) by Section 3, and finally r(P (w )) ≡ styl r(N (w)), by Lemma 5.1 and Lemma 6.1. Thus (1) w ≡ styl r(N (w)), which proves injectivity.…”

Section: A Bijectionmentioning

confidence: 61%

“…Skew-partitions with a hole. Comparison of the definitions below with Ferrers diagram, lower poset ideals in N 2 , Young tableaux, skew Young (1,4)…”

Section: Cardinality and Presentation Of The Stylic Monoidmentioning

confidence: 99%

“…If S = I \ J is a skew ideal, then a point H ∈ S such that S \ H is still a skew ideal is called a corner of S. We call it a lower corner if J ∪ H is an ideal, and an upper corner if I \ H is an ideal. For example, in Figure 11, with S the set of labelled points, the lower corners are (1, 3), (2, 2), (3, 1) and the upper corners are (1,4), (3,3), (3,2), (4,1).…”

Section: Cardinality and Presentation Of The Stylic Monoidmentioning

confidence: 99%

“…Suppose first that we are in Case 1, hence x ≤ p 1 . Then x → T is obtained by replacing R 1 by R 1 ∪ x, and, since x ≤ p i for each i, it follows from Lemma 11.4 (1), that R i = S i .…”

Section: Ordering Columnsmentioning

confidence: 99%

“…The proof is nontrivial, but we followed the classical case (skew diagrams with a hole [23]), as is shown in Sagan's book [21], with the help of Fomin's growth diagrams, which may be extended to our case: partitions are replaced by compositions, appropriately ordered. We use a notion that appeared previously in the literature: composition tableaux of [10,14] (with one condition removed), and more precisely, standard immaculate tableaux [2] (see also [3], [7], [9], [1], and [18]).…”

Section: Introductionmentioning

confidence: 99%

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The Stylic Monoid

Abram¹,

Reutenauer²

2021

Preprint

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Contents1. Introduction 2. Schensted insertions 3. The plactic monoid 4. An action on columns 5. The stylic monoid 6. A variant of Schensted row insertion 6.1. N -tableaux and insertion 6.2. Inflation and simulation by Schensted insertion 6.3. The mapping δ 7. A bijection 8. Cardinality and presentation of the stylic monoid 9. Evacuation of partitions 9.1. An involution 9.2. Skew-partitions with a hole 9.3. Jeu de taquin on skew partitions 9.4. Properties of the mappings ∆ and π. 9.5. Growth diagram 9.6. Proof of the evacuation theorem 10. Ordering columns 11. J-relations on the stylic monoid 11.1. J-triviality 11.2. Left insertion 11.3. Grading of the J-order 12. Fixpoints and idempotents 12.1. Applications to the plactic monoid: a confluent rewriting system 13. Syntacticity 14. Appendix: a theorem of Lascoux and Schützenberger

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“…And w ≡ plax r(P (w )) by Section 3, and finally r(P (w )) ≡ styl r(N (w)), by Lemma 5.1 and Lemma 6.1. Thus (1) w ≡ styl r(N (w)), which proves injectivity.…”

Section: A Bijectionmentioning

confidence: 61%

“…Skew-partitions with a hole. Comparison of the definitions below with Ferrers diagram, lower poset ideals in N 2 , Young tableaux, skew Young (1,4)…”

Section: Cardinality and Presentation Of The Stylic Monoidmentioning

confidence: 99%

Section: Cardinality and Presentation Of The Stylic Monoidmentioning

confidence: 99%

Section: Ordering Columnsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The Stylic Monoid

Abram¹,

Reutenauer²

2021

Preprint

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Permuted composition tableaux, 0-Hecke algebra and labeled binary trees

Tewari

Willigenburg

2019

Journal of Combinatorial Theory, Series A

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We introduce a generalization of semistandard composition tableaux called permuted composition tableaux. These tableaux are intimately related to permuted basement semistandard augmented fillings studied by Haglund, Mason and Remmel. Our primary motivation for studying permuted composition tableaux is to enumerate all possible ordered pairs of permutations (σ 1 , σ 2 ) that can be obtained by standardizing the entries in two adjacent columns of an arbitrary composition tableau. We refer to such pairs as compatible pairs. To study compatible pairs in depth, we define a 0-Hecke action on permuted composition tableaux. This action naturally defines an equivalence relation on these tableaux. Certain distinguished representatives of the resulting equivalence classes in the special case of two-columned tableaux are in bijection with compatible pairs. We provide a bijection between two-columned tableaux and labeled binary trees. This bijection maps a quadruple of descent statistics for 2-columned tableaux to left and right ascent-descent statistics on labeled binary trees introduced by Gessel, and we use it to prove that the number of compatible pairs is (n + 1) n−1 . 14 4.2. Bijection between LD n and SPCT((2 n )) 15 4.3. Plane binary trees, unlabeled and labeled 162010 Mathematics Subject Classification. Primary 05E10, 20C08; Secondary 05A05, 05A19, 05C20, 05E05, 05E15.

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The "Young" and "Reverse" Dichotomy of Polynomials

Mason

Searles

2022

Electron. J. Combin.

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A ``flip-and-reversal" involution arising in the study of quasisymmetric Schur functions provides a passage between what we term ``Young" and ``reverse" variants of bases of polynomials or quasisymmetric functions. Building on this perspective, which has found recent application in the study of q-analogues of combinatorial Hopf algebras and generalizations of dual immaculate functions, we develop and explore Young analogues of well-known bases for polynomials. We prove several combinatorial formulas for the Young analogue of the key polynomials, show that they form the generating functions for left keys, and provide a representation-theoretic interpretation of Young key polynomials as traces on certain modules. We also give combinatorial formulas for the Young analogues of Schubert polynomials, including their crystal graph structure. We moreover determine the intersections of (reverse) bases and their Young counterparts, further clarifying their relationships to one another.

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Dual Immaculate Quasisymmetric Functions Expand Positively into Young Quasisymmetric Schur Functions

Cited by 4 publications

References 0 publications

The Stylic Monoid

The Stylic Monoid

Permuted composition tableaux, 0-Hecke algebra and labeled binary trees

The "Young" and "Reverse" Dichotomy of Polynomials

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