Abstract
A barcode is a finite multiset of intervals on the real line. Jaramillo-Rodriguez (2023) previously defined a map from the space of barcodes with a fixed number of bars to a set of multipermutations, which presented new combinatorial invariants on the space of barcodes. A partial order can be defined on these multipermutations, resulting in a class of posets known as combinatorial barcode lattices. In this paper, we provide a number of equivalent definitions for the combinatorial barcode lattice, show that its Möbius function is a restriction of the Möbius function of the symmetric group under the weak Bruhat order, and show its ground set is the Jordan-Hölder set of a labeled poset. Furthermore, we obtain formulas for the number of join-irreducible elements, the rank-generating function, and the number of maximal chains of combinatorial barcode lattices. Lastly, we make connections between intervals in the combinatorial barcode lattice and certain classes of matchings.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Data Availability
No datasets were generated or analysed during the current study.
References
Bennett, M.K., Birkhoff, G.: Two families of Newman lattices. Algebra Universalis 32(1), 115–144 (1994)
Björner, A., Wachs, M.L.: Permutation statistics and linear extensions of posets. J. Comb. Theory Ser. A 58(1), 85–114 (1991)
Flath, S.: The order dimension of multinomial lattices. Order 10(3), 201–219 (1993)
Jaramillo-Rodriguez, E.: Combinatorial analysis of barcodes and interval graphs for applications in data science. Doctoral dissertation, UC Davis (2023)
Jaramillo-Rodriguez, E.: Combinatorial methods for barcode analysis. J. Appl. Comput. Topol., 1–32 (2023). https://doi.org/10.1007/s41468-023-00143-8
Lascoux, A., Schützenberger, M.-P.: Schubert polynomials and the Littlewood-Richardson rule. Lett. Math. Phys. 10(2–3), 111–124 (1985)
Lekkerkerker, C.G., Boland, J.Ch.: Representation of a finite graph by a set of intervals on the real line. Fund. Math. 51, 45–64 (1962/63)
Lovász, L., Plummer, M.D.: Matching theory. AMS Chelsea Publishing, Providence, RI. Corrected reprint of the 1986 original [MR0859549] (2009)
Stanley, R.P.: On the number of reduced decompositions of elements of Coxeter groups. Eur. J. Comb. 5(4), 359–372 (1984)
Stanley, R.P.: Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, vol. 62. Cambridge University Press, Cambridge. With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin (1999)
Stanley, R.P.: Enumerative combinatorics. Volume 1,2nd ed., Cambridge Studies in Advanced Mathematics, vol. 49. Cambridge University Press, Cambridge (2012)
Stanley, R.P.: Catalan numbers. Cambridge University Press, New York (2015)
Tenner, B.E.: Pattern avoidance and the Bruhat order. J. Combin. Theory Ser. A 114(5), 888–905 (2007)
Tenner, B.E.: Interval structures in the Bruhat and weak orders. J. Comb. 13(1), 135–165 (2022)
Zomorodian, A., Carlsson, G.: Computing persistent homology. Discrete Comput. Geom. 33(2), 249–274 (2005)
Acknowledgements
The authors thank Elena Jaramillo-Rodriguez for her fruitful correspondence.
Funding
ARVM was partially supported by the NSF under Award DMS-2102921. This material was also based in part upon work supported by the NSF Grant DMS-1928930 and the Alfred P. Sloan Foundation under grant G-2021-16778, while ARVM was in residence at the Simons Laufer Mathematical Sciences Institute in Berkeley, California, during the Fall 2023 semester.
Author information
Authors and Affiliations
Contributions
AB and ARVM equally prepared and reviewed the manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Bouquet, A., Vindas-Meléndez, A.R. Combinatorial Results on Barcode Lattices. Order (2024). https://doi.org/10.1007/s11083-024-09670-0
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11083-024-09670-0