Equivalence Classes of Permutations under Various Relations Generated by Constrained Transpositions
Abstract
We consider a large family of equivalence relations on permutations in Sn that generalise those discovered by Knuth in his study of the RobinsonSchensted correspondence. In our most general setting, two permutations are equivalent if one can be obtained from the other by a sequence of patternreplacing moves of prescribed form; however, we limit our focus to patterns where two elements are transposed, subject to the constraint that a third element of a suitable type be in a suitable position. For various instances of the problem, we compute the number of equivalence classes, determine how many npermutations are equivalent to the identity permutation, or characterise this equivalence class. Although our results feature familiar integer sequences (e.g., Catalan, Fibonacci, and Tribonacci numbers) and special classes of permutations (layered, connected, and 123avoiding), some of the sequences that arise appear to be new.
 Publication:

arXiv eprints
 Pub Date:
 November 2011
 DOI:
 10.48550/arXiv.1111.3920
 arXiv:
 arXiv:1111.3920
 Bibcode:
 2011arXiv1111.3920L
 Keywords:

 Mathematics  Combinatorics;
 05A05
 EPrint:
 21 pages, submitted