Enriched toric $[\vec{D}]$-partitions

Liang, Jinting

doi:10.5802/alco.314

Enriched toric

[\vec{D}]

-partitions

Liang, Jinting ¹

¹ Michigan State University Department of Mathematics 619 Red Cedar Road C212 Wells Hall East Lansing MI 48824-1027 (USA)

Algebraic Combinatorics, Volume 6 (2023) no. 6, pp. 1491-1518

Abstract

This paper develops the theory of enriched toric $[\vec{D}]$ -partitions. Whereas Stembridge’s enriched $P$ -partitions give rise to the peak algebra which is a subring of the ring of quasi-symmetric functions $QSym$ , our enriched toric $[\vec{D}]$ -partitions generate the cyclic peak algebra which is a subring of the ring of cyclic quasi-symmetric functions $cQSym$ . In the same manner as the peak set of linear permutations appears when considering enriched $P$ -partitions, the cyclic peak set of cyclic permutations plays an important role in our theory. The associated order polynomial is discussed based on this framework.

Received: 2022-09-12
Accepted: 2023-05-01
Published online: 2024-01-08

DOI: 10.5802/alco.314

Classification: 05A05, 05E05, 06A11
Keywords: Cyclic peak, cyclic permutation, cyclic quasi-symmetric function, enriched $P$-partition, toric poset, order polynomial

Author's affiliations:

Liang, Jinting ¹

¹ Michigan State University Department of Mathematics 619 Red Cedar Road C212 Wells Hall East Lansing MI 48824-1027 (USA)

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

@article{ALCO_2023__6_6_1491_0,
     author = {Liang, Jinting},
     title = {Enriched toric $[\vec{D}]$-partitions},
     journal = {Algebraic Combinatorics},
     pages = {1491--1518},
     year = {2023},
     publisher = {The Combinatorics Consortium},
     volume = {6},
     number = {6},
     doi = {10.5802/alco.314},
     language = {en},
     url = {https://alco.centre-mersenne.org/articles/10.5802/alco.314/}
}

TY  - JOUR
AU  - Liang, Jinting
TI  - Enriched toric $[\vec{D}]$-partitions
JO  - Algebraic Combinatorics
PY  - 2023
SP  - 1491
EP  - 1518
VL  - 6
IS  - 6
PB  - The Combinatorics Consortium
UR  - https://alco.centre-mersenne.org/articles/10.5802/alco.314/
DO  - 10.5802/alco.314
LA  - en
ID  - ALCO_2023__6_6_1491_0
ER  -

%0 Journal Article
%A Liang, Jinting
%T Enriched toric $[\vec{D}]$-partitions
%J Algebraic Combinatorics
%D 2023
%P 1491-1518
%V 6
%N 6
%I The Combinatorics Consortium
%U https://alco.centre-mersenne.org/articles/10.5802/alco.314/
%R 10.5802/alco.314
%G en
%F ALCO_2023__6_6_1491_0

Liang, Jinting. Enriched toric $[\vec{D}]$-partitions. Algebraic Combinatorics, Volume 6 (2023) no. 6, pp. 1491-1518. doi: 10.5802/alco.314

References
Cited by

[1] Adin, Ron M.; Gessel, Ira M.; Reiner, Victor; Roichman, Yuval Cyclic quasi-symmetric functions, Israel J. Math., Volume 243 (2021) no. 1, pp. 437-500 | MR | Zbl | DOI

[2] Aguiar, Marcelo; Bergeron, Nantel; Sottile, Frank Combinatorial Hopf algebras and generalized Dehn-Sommerville relations, Compos. Math., Volume 142 (2006) no. 1, pp. 1-30 | DOI | MR | Zbl

[3] Billera, Louis J.; Hsiao, Samuel K.; van Willigenburg, Stephanie Peak quasisymmetric functions and Eulerian enumeration, Adv. Math., Volume 176 (2003) no. 2, pp. 248-276 | DOI | MR | Zbl

[4] Develin, Mike; Macauley, Matthew; Reiner, Victor Toric partial orders, Trans. Amer. Math. Soc., Volume 368 (2016) no. 4, pp. 2263-2287 | DOI | MR | Zbl

[5] Gessel, Ira M. Multipartite $P$ -partitions and inner products of skew Schur functions, Combinatorics and algebra (Boulder, Colo., 1983) (Contemp. Math.), Volume 34, Amer. Math. Soc., Providence, RI, 1984, pp. 289-317 | DOI | MR | Zbl

[6] Gessel, Ira M.; Zhuang, Yan Shuffle-compatible permutation statistics, Adv. Math., Volume 332 (2018), pp. 85-141 | DOI | MR | Zbl

[7] Hivert, Florent Hecke algebras, difference operators, and quasi-symmetric functions, Adv. Math., Volume 155 (2000) no. 2, pp. 181-238 | DOI | MR | Zbl

[8] Shareshian, John; Wachs, Michelle L. Chromatic quasisymmetric functions and Hessenberg varieties, Configuration spaces (CRM Series), Volume 14, Ed. Norm., Pisa, 2012, pp. 433-460 | DOI | MR | Zbl

[9] Stanley, Richard P. Ordered structures and partitions, Memoirs of the American Mathematical Society, No. 119, American Mathematical Society, Providence, R.I., 1972, iii+104 pages

[10] Stanley, Richard P. Generalized riffle shuffles and quasisymmetric functions, Volume 5, 2001 no. 3-4, pp. 479-491 Dedicated to the memory of Gian-Carlo Rota (Tianjin, 1999) | MR | Zbl

[11] Stembridge, John R. Enriched $P$ -partitions, Trans. Amer. Math. Soc., Volume 349 (1997) no. 2, pp. 763-788 | DOI | MR | Zbl

Cited by Sources: