A Tile with Nested Chain Abacus
Main Article Content
Abstract
This study had succeeded in producing a new graphical representation of James abacus called nested chain abacus. Nested chain abacus provides a unique mathematical expression to encode each tile (image) using a partition theory where each form or shape of tile will be associated with exactly one partition.Furthermore, an algorithm of nested chain abacus movement will be constructed, which can be applied in tiling theory.
Received 10/5/2020, Accepted 24/2/2021, Published Online First 20/11/2021
Article Details
This work is licensed under a Creative Commons Attribution 4.0 International License.
How to Cite
References
James G. Some combinatorial results involving young diagrams. In Mathematical proceedings of the cambridge philosophical society. 1978; 83: p 1–10.
James G, Sinéad L, Andrew M. Rouquier blocks. M. Zeitschrift. 2006 Mar 1;252(3):511. DOI: 10.1007/s00209-005-0863-0.
Fayers, M. Another runner removal theorem for v-decomposition numbers of Iwahori–Hecke algebras and q-Schur algebras. J. of Alge. 2007 Jan; 310(1): 396-404.
Chuang, J, Miyachi H, TanK M. Parallelotopetilings and q-decomposition numbers. Adv. Math. 2017 Jul; 321: 80-159.
Fayers, M. On the irreducible representations of the alternating group which remain irreducible in characteristic. Represent. Theory of the AMS. 2010 Sep; 14(16): 601–626.
Wildon, M. Counting partitions on the abacus. The Ramanujan J. 2008 Sep; 17(3): 355-367.
Loehr, N. Abacus proofs of schur function identities. SIDMA. 2010 May; 24(4): 1356–1370.
Wildon M. A combinatorial proof of a plethystic Murnaghan--Nakayama rule. SIDMA. 2016 Aug ;30(3):1526-33..
Tingley P. Three combinatorial models for sln crystals, with applications to cylindric plane partitions. IMRN. 2011 Jan; 2011(10): 2374–2375.https://doi.org/10.1093/imrn/rnq095
King O. The representation theory of diagram algebras. PhD [Doctoral dissertation]. City University London; 2014.
Mahoommed E F, Ahmad N, Ibrahim H, Mahmood A S. Nested chain movement of length 1 of beta number in James abacus diagram. GJPAM. 2016; 12(4): 2953-2969.
Mohommed, E F. Constructing a Nested Chain in James Abacus Diagram. JPCS ( Vol. 1294. No. 3.) IOP Publishing, 2019.
Mohommed, E F, Haslinda I, Nazihah A. Enumeration of N-connected Ominoes Inscribed in an Abacus. JPJournal of Algebra Number Theory and Appl. 2017 Dec 1;39(6):843-74.
Fayers M. Irreducible projective representations of the symmetric group which remain irreducible in characteristic 2. IMS. 2018; 116(4): 878-928..
Yamzon N. Algebraic and combinatorial aspects of polytopes and domino tilings . PhD [dissertation], San Francisco State University; 2019.
Carney M R. Computability and Tiling Problems. PhD [dissertation], University of Leeds; (2019).
Mohommed E F. Topological Structure of Nested Chain Abacus. IJS. 2020; 153-160.