Skip to content
DraconiticResearch
Search Sources API
182,460 studies31 sourcesUpdated 38 min ago
PreprintCC BY 4.0HAL

Tree-indexed sums of Catalan numbers

  • Alin Bostan
  • Valentin Féray
  • Paul Thévenin
Open at sourcePDFCite

Abstract

We consider a family of infinite sums of products of Catalan numbers, indexed by trees. We show that these sums are polynomials in 1/π with rational coefficients; the proof is effective and provides an algorithm to explicitly compute these sums. Along the way we introduce parametric liftings of our sums, and show that they are polynomials in the complete elliptic integrals of the first and second kind. Moreover, the degrees of these polynomials are at most half of the number of vertices of the tree. The computation of these tree-indexed sums is motivated by the study of large meandric systems, which are non-crossing configurations of loops in the plane.

Related

  1. Rational Dyck Excursion Counts as Sums of Generalized Fuss-Catalan Products
    Brecht Verbeken, Arne Vanhoyweghen, Vincent Ginis
    arXivPreprintCC BY 4.0Full textCombinatorics & discrete mathematicsarXiv
  2. A Matrix Model for Higher-Genus Fuss--Catalan Numbers
    Anatol Kirillov, Ivan Kostov
    arXivPreprintarXiv licenceFull textMathematical physicsHigh-energy & particle physicsarXiv
  3. Asymptotic q,t-Fuss--Catalan numbers for type B
    Alexei Oblomkov
    arXivPreprintarXiv licenceFull textAlgebra & number theoryCombinatorics & discrete mathematicsarXiv
  4. Schreier Sets of Intervals, Super-Schreier Sets, and Catalan Numbers
    Hung Viet Chu, Mariam Khaduri, Moiz M. Khokhar +1
    arXivPreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv
  5. Finite-sum Identities and Supercongruences for Cubes of Catalan Numbers
    Yuqi Liu
    arXivPreprintarXiv licenceFull textAlgebra & number theoryCombinatorics & discrete mathematicsarXiv

Cite

BibTeXRIS
BibTeX
@misc{bostan2026treeindexed,
  title = {{Tree-indexed sums of Catalan numbers}},
  author = {Alin Bostan and Valentin Féray and Paul Thévenin},
  howpublished = {HAL},
  year = {2026},
  month = sep,
  eprint = {2507.23557},
  archivePrefix = {arXiv},
  url = {https://hal.science/hal-05195303v2},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Tree-indexed sums of Catalan numbers
AU  - Alin Bostan
AU  - Valentin Féray
AU  - Paul Thévenin
PY  - 2026
DA  - 2026/09/11
PB  - HAL
UR  - https://hal.science/hal-05195303v2
AB  - We consider a family of infinite sums of products of Catalan numbers, indexed by trees. We show that these sums are polynomials in 1/π with rational coefficients; the proof is effective and provides an algorithm to explicitly compute these sums. Along the way we introduce parametric liftings of our sums, and show that they are polynomials in the complete elliptic integrals of the first and second kind. Moreover, the degrees of these polynomials are at most half of the number of vertices of the tree. The computation of these tree-indexed sums is motivated by the study of large meandric systems, which are non-crossing configurations of loops in the plane.
ER  -