Permutation Pattern Avoidance in the Alternating Sign Matrices
DOI:
https://doi.org/10.56919/usci.2434.033Keywords:
Alternating Sign Matrix, Restricted permutation, Pattern avoidanceAbstract
Study’s Excerpt
- Aunu permutations (permutations of prime length with their first entry as unity) are explored with a specific focus on involutions.
- The concept of pattern avoidance is examined using Aunu permutations that avoid the 213 pattern.
- The results reveal that Catalan numbers count the set of alternating sign matrices.
- Both bijective and algebraic proofs are provided to support the findings.
- The study transforms classical ideas, first presented by Abdullahi bin Fodiyo in his book called Da'ulmusalli (light to a worshipper), into modern mathematical science.
Full Abstract
We represented permutations using matrices. In the alternating sign matrices, the idea of pattern avoidance in permutation matrices with reference to Aunu Permutations is examined. The primary outcome is that the Catalan numbers count the set of alternating sign matrices related to Aunu Permutations that do not follow pattern 213. Both a bijective and an algebraic proof are given.
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