TY - JOUR

T1 - Hopf algebras of parking functions and decorated planar trees

AU - Bergeron, Nantel

AU - González D'León, Rafael S.

AU - Li, Shu Xiao

AU - Pang, C. Y. Amy

AU - Vargas, Yannic

N1 - Funding Information:
Bergeron is partially supported by NSERC and the York Research Chair in applied algebra.Pang and Li are partially supported by the Hong Kong Research Grants Council grant ECS 22300017.
Publisher Copyright:
© 2022 The Authors

PY - 2023/2

Y1 - 2023/2

N2 - We construct three new combinatorial Hopf algebras based on the Loday-Ronco operations on planar binary trees. The first and second algebras are defined on planar trees and labeled planar trees extending the Loday-Ronco and Malvenuto-Reutenauer Hopf algebras respectively. We show that the latter is bidendriform which implies that it is also free, cofree, and self-dual. The third algebra involves a new visualization of parking functions as decorated binary trees; it is also bidendriform, free, cofree, and self-dual, and therefore abstractly isomorphic to the algebra PQSym of Novelli and Thibon. We define partial orders on the objects indexing each of these three Hopf algebras, one of which, when restricting to (m+1)-ary trees, coarsens the m-Tamari order of Bergeron and Préville-Ratelle. We show that multiplication of dual fundamental basis elements is given by intervals in each of these orders. Finally, we use an axiomatized version of the techniques of Aguiar and Sottile on the Malvenuto-Reutenauer Hopf algebra to define a monomial basis on each of our Hopf algebras, and to show that comultiplication is cofree on the monomial elements. This in particular, implies the cofreeness of the Hopf algebra on planar trees. We also find explicit positive formulas for the multiplication on monomial basis and a cancellation-free and grouping-free formula for the antipode of monomial elements.

AB - We construct three new combinatorial Hopf algebras based on the Loday-Ronco operations on planar binary trees. The first and second algebras are defined on planar trees and labeled planar trees extending the Loday-Ronco and Malvenuto-Reutenauer Hopf algebras respectively. We show that the latter is bidendriform which implies that it is also free, cofree, and self-dual. The third algebra involves a new visualization of parking functions as decorated binary trees; it is also bidendriform, free, cofree, and self-dual, and therefore abstractly isomorphic to the algebra PQSym of Novelli and Thibon. We define partial orders on the objects indexing each of these three Hopf algebras, one of which, when restricting to (m+1)-ary trees, coarsens the m-Tamari order of Bergeron and Préville-Ratelle. We show that multiplication of dual fundamental basis elements is given by intervals in each of these orders. Finally, we use an axiomatized version of the techniques of Aguiar and Sottile on the Malvenuto-Reutenauer Hopf algebra to define a monomial basis on each of our Hopf algebras, and to show that comultiplication is cofree on the monomial elements. This in particular, implies the cofreeness of the Hopf algebra on planar trees. We also find explicit positive formulas for the multiplication on monomial basis and a cancellation-free and grouping-free formula for the antipode of monomial elements.

UR - http://www.scopus.com/inward/record.url?scp=85138815803&partnerID=8YFLogxK

U2 - 10.1016/j.aam.2022.102436

DO - 10.1016/j.aam.2022.102436

M3 - Journal article

AN - SCOPUS:85138815803

SN - 0196-8858

VL - 143

JO - Advances in Applied Mathematics

JF - Advances in Applied Mathematics

M1 - 102436

ER -