Combinatorial results for order-preserving partial injective contraction mappings
DOI:
https://doi.org/10.55016/ojs/cdm.v19i2.62669Keywords:
semigroups of transformations, partial injective contraction mappingsAbstract
Let $ \mathcal{I}_n$ be the symmetric inverse semigroup on $X_n = \{1, 2, \ldots , n\}$. Let $\mathcal{OCI}_n$ be the subsemigroup of $\mathcal{I}_n$ consisting of all order-preserving injective partial contraction mappings, and let $\mathcal{ODCI}_n$ be the subsemigroup of $\mathcal{I}_n$ consisting of all order-preserving and order-decreasing injective partial contraction mappings of $X_n$. In this paper, we investigate the cardinalities of some equivalences on $\mathcal{OCI}_n$ and $\mathcal{ODCI}_n$ which lead naturally to obtaining the order of these semigroups. Then, we relate the formulae obtained to Fibonacci numbers. Similar results about $\mathcal{ORCI}_n$, the semigroup of order-preserving or order-reversing injective partial contraction mappings, are deduced.
References
[2] F. Al-Kharousi, R. Kehinde, and A. Umar, Combinatorial results for certain semigroups of partial isometries of a finite chain. Australas. J. Combin. 58 (2014), 365–375.
[3] F. Al-Kharousi, R. Kehinde, and A. Umar, Combinatorial results for certain semigroups of order-decreasing partial isometries of a finite chain. arXiv:1702.04485.
[4] A. D. Adeshola, and A. Umar, Combinatorial results for certain semigroups of order-preserving full contraction mappings of a finite chain. J. Combin. Math. Combin. Comput., (2017) (To appear).
[5] E. Barcucci, R. Pinzani, and R. Sprugnoli, Directed column-convex polyominoes by recurrence relations. Lecture Notes in Comput. Sci. 668 (1993), 282–298.
[6] D. Borwein, S. Rankin, and L. Renner, Enumeration of injective partial transformations. Discrete Math. 73 (1989), 291–296.
[7] S. Epp, Discrete mathematics with applications, 4th Edition. Brooks/Cole, Boston, MA, 2011.
[8] V. H. Fernandes, The monoid of all injective orientation-preserving partial transformations on a finite chain. Comm. Algebra 28 (2000), 3401–3426.
[9] V. H. Fernandes, G. M. S. Gomes, and M. M. Jesus, The cardinal and idempotent number of various monoids of transformations on a finite chain. Bull. Malays. Math. Sci. Soc. 34 (2011), 79–85.
[10] G. U. Garba, Nilpotents in semigroups of partial one-to-one order-preserving mappings. Semigroup Forum 48 (1994), 37–49.
[11] H. W. Gould, Combinatorial identities. A standardized set of tables listing 500 binomial coefficient summations., Morgantown, W.Va., 1972.
[12] R. P. Grimaldi, Fibonacci and Catalan numbers. An introduction., John Wiley & Sons, 2012.
[13] J. M. Howie, Fundamentals of semigroup theory. London Mathematical Society Monographs. New series, 12. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1995.
[14] A. Laradji and A. Umar, Combinatorial results for the symmetric inverse semigroup. Semigroup Forum 75 (2007), 221–236.
[15] S. Lipscomb, Symmetric Inverse Semigroups, Mathematical Surveys and Monographs, 46. American mathematical Society, Providence, R. I., 1996.
[16] S. Rinaldi and D. G. Rogers, How the odd terms in the Fibonacci sequence stack up. Math. Gaz., 90
(2006), 431–442.
[17] J. Riordan, Combinatorial identities. Reprint of the 1968 original. Robert E. Krieger Publishing Co., Huntington, N.Y., 1979.
[18] N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences, published electronically at https://oeis.org.
[19] A. Umar, On the semigroups of partial one-to-one order-decreasing finite transformations, Proc. Roy. Soc. Edinburgh, Sect. A, 123 (1993), 355–363.
[20] A. Umar, Some combinatorial problems in the theory of symmetric inverse semigroups. Algebra Discrete Math. 9 (2010), 115–126.
Downloads
Published
Issue
Section
License
Copyright (c) 2024 Contributions to Discrete Mathematics
This work is licensed under a Creative Commons Attribution-NoDerivatives 4.0 International License.
This copyright statement was adapted from the statement for the University of Calgary Repository and from the statement for the Electronic Journal of Combinatorics (with permission).
The copyright policy for Contributions to Discrete Mathematics (CDM) is changed for all articles appearing in issues of the journal starting from Volume 15 Number 3.
Author(s) retain copyright over submissions published starting from Volume 15 number 3. When the author(s) indicate approval of the finalized version of the article provided by the technical editors of the journal and indicate approval, they grant to Contributions to Discrete Mathematics (CDM) a world-wide, irrevocable, royalty free, non-exclusive license as described below:
The author(s) grant to CDM the right to reproduce, translate (as defined below), and/or distribute the material, including the abstract, in print and electronic format, including but not limited to audio or video.
The author(s) agree that the journal may translate, without changing the content the material, to any medium or format for the purposes of preservation.
The author(s) also agree that the journal may keep more than one copy of the article for the purposes of security, back-up, and preservation.
In granting the journal this license the author(s) warrant that the work is their original work and that they have the right to grant the rights contained in this license.
The authors represent that the work does not, to the best of their knowledge, infringe upon anyone’s copyright.
If the work contains material for which the author(s) do not hold copyright, the author(s) represent that the unrestricted permission of the copyright holder(s) to grant CDM the rights required by this license has been obtained, and that such third-party owned material is clearly identified and acknowledged within the text or content of the work.
The author(s) agree to ensure, to the extent reasonably possible, that further publication of the Work, with the same or substantially the same content, will acknowledge prior publication in CDM.
The journal highly recommends that the work be published with a Creative Commons license. Unless otherwise arranged at the time the finalized version is approved and the licence granted with CDM, the work will appear with the CC-BY-ND logo. Here is the site to get more detail, and an excerpt from the site about the CC-BY-ND. https://creativecommons.org/licenses/
Attribution-NoDerivs
CC BY-ND
This license lets others reuse the work for any purpose, including commercially; however, it cannot be shared with others in adapted form, and credit must be provided to you.