[Paper Review] Equivalence Classes of Permutations under Various Relations Generated by Constrained Transpositions

Steven J. Linton, James Propp+2 more|arXiv (Cornell University)|Nov 16, 2011

Advanced Combinatorial Mathematics15 references11 citations

TL;DR

This paper introduces a framework for studying equivalence classes of permutations under constrained transposition relations, generalizing Knuth's work on the Robinson-Schensted correspondence. By defining equivalence via replaceable 3-element patterns (e.g., 123 ↔ 321), the authors derive enumerative results, revealing connections to Catalan, Fibonacci, and Tribonacci numbers, and identify new integer sequences arising from specific equivalence classes.

ABSTRACT

We consider a large family of equivalence relations on permutations in Sn that generalise those discovered by Knuth in his study of the Robinson-Schensted correspondence. In our most general setting, two permutations are equivalent if one can be obtained from the other by a sequence of pattern-replacing 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 n-permutations 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 123-avoiding), some of the sequences that arise appear to be new.

Motivation & Objective

• To generalize Knuth's equivalence relations in the context of the Robinson-Schensted correspondence by introducing broader classes of pattern-replacing transpositions.
• To systematically analyze equivalence relations on permutations in $ S_n $ generated by constrained transpositions of 3-element subsequences.
• To determine the number of equivalence classes under various pattern-replacement rules, particularly focusing on cases where transpositions are restricted by position or value adjacency.
• To characterize the size and structure of equivalence classes, especially the class containing the identity permutation.
• To identify and explore connections between these equivalence classes and well-known integer sequences, including Catalan, Fibonacci, and Tribonacci numbers, as well as novel sequences.

Proposed method

• Define equivalence relations on $ S_n $ via set partitions of $ S_3 $, where elements in the same block can be interchanged in any subsequence of the same relative order.
• Introduce three distinct equivalence types: general (no constraints), adjacent positions ($ P^{f ext{vrule}} $), and both adjacent positions and values ($ P^{lacksquare} $), with the latter being symmetric under inverse map.
• Use recursive decomposition based on block structures (e.g., $ ho_1, ho_2, ho_3 $) representing 1-, 2-, and 3-cycles in the permutation's direct sum decomposition.
• Model the number of permutations equivalent to the identity using recurrence relations derived from block appendages: e.g., $ a_n = a_{n-1} + a_{n-2} $ for Fibonacci-like growth.
• Apply regular expression notation to describe allowed block size sequences (e.g., $ [12^*]^* $, $ \{13\}^* $) to characterize the structure of equivalence classes.
• Verify base cases and recurrence relations by induction, using the fact that any non-identity permutation in a class can be reduced to the identity via valid moves.

Experimental results

Research questions

• RQ1 How many equivalence classes arise under different constrained transposition rules defined by partitions of $ S_3 $, and how do these numbers relate to known integer sequences?
• RQ2 What is the size of the equivalence class containing the identity permutation under various pattern-replacement rules, and how can it be characterized via recurrence relations?
• RQ3 Which permutations are equivalent to the identity under rules like $ 123 \leftrightarrow 321 $, and what structural properties do they share?
• RQ4 How do the equivalence classes differ when transpositions are restricted to adjacent positions, adjacent values, or both?
• RQ5 Are there new integer sequences generated by these equivalence relations, and if so, what combinatorial structures do they enumerate?

Key findings

• For the relation $ \{123, 321\} $, the number of permutations equivalent to the identity follows the Fibonacci recurrence: $ a_n = a_{n-1} + a_{n-2} $, with $ a_1 = a_2 = 1 $.
• Under the rule $ \{123, 132, 321\} $, the number of identity-equivalent permutations satisfies the Tribonacci recurrence $ a_n = a_{n-1} + a_{n-2} + a_{n-3} $, yielding sequence A000073.
• For the rule $ \{123, 321\} $ with adjacent positions and values constrained, the number of identity-equivalent permutations is given by $ a_n = a_{n-2} + U_n $, where $ U_n $ follows the $ P_5 $ recurrence.
• The equivalence class of the identity under $ \{123, 321\} $ with no constraints consists of all permutations whose block decomposition avoids the pattern $ 2^* $ (i.e., no all-2-blocks), and the count is Fibonacci minus 1 for even $ n $.
• The authors identify new integer sequences not previously documented in the OEIS, arising from equivalence classes under certain constrained transposition rules.
• The structure of equivalence classes is fully characterized via regular expressions over block types (e.g., $ \{13\}^* $), enabling precise enumeration and recurrence derivation.