Journal of Applied Mathematics and Computation

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Article http://dx.doi.org/10.26855/jamc.2023.03.004

Three Open Problems in Enumerative Combinatorics

Firdous Ahmad Mala

1Department of Mathematics, Government Degree College Sopore, Baramulla, Jammu and Kashmir, India.

2Department of Mathematics, Chandigarh University, Gharuan Mohali, Punjab, India.

*Corresponding author: Firdous Ahmad Mala

Published: February 14,2023

Abstract

Enumerative Combinatorics is the study of counting problems and counting techniques. Counting elements of various sets is a primary concern in Enumerative Combinatorics. An interesting observation about “counting problems” is the fact that they are somewhat easier to understand but hard to solve. This means that no specilaised or sophisticated knowledge is required to understand the subject-matter of Enumerative Combinatorics. However, an in-depth study of various counting techniques is often one of the several requirements for being able to solve these problems. Owing to this nature, this branch of mathematics has a plethora of open problems. Among them there are the problems of counting transitive relations, counting partial orders and counting quasiorders on a finite set. In this paper, we briefly revisit three closely-knit open problems in Enumerative Combinatorics. We also show how, in the light of the available literature, the solution of one of these three problems would lead to the solutions of the other two.

References

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[8] Mala, F. A. Counting Transitive Relations with Two Ordered Pairs.  Journal of Applied Mathematics and Computation, 5(4), 247-251. 

[9] Mala, F. A. (2021). Interesting observations on the numbers of partial orders and transitive relations. Cape Comorin Trust, India, 215.

[10] OEIS Foundation Inc. (2022). Number of transitive relations on n nodes, Entry A006905 in The On-Line Encyclopedia of Integer Sequences, http://oeis.org/A006905.

[11] OEIS Foundation Inc. (2022). Number of partially ordered sets with n labeled elements, Entry A001035 in The On-Line Encyclopedia of Integer Sequences, http://oeis.org/A001035.

[12] OEIS Foundation Inc. (2022). Number of different quasi-orders with n labeled elements, Entry A000798 in The On-Line Encyclopedia of Integer Sequences, http://oeis.org/A000798.

How to cite this paper

Three Open Problems in Enumerative Combinatorics

How to cite this paper:  Firdous Ahmad Mala. (2023) Three Open Problems in Enumerative Combinatorics. Journal of Applied Mathematics and Computation7(1), 24-27.

DOI: http://dx.doi.org/10.26855/jamc.2023.03.004