Wilf-equivalence for singleton classes

Abstract

Write p 1 p 2 … p m for the permutation matrix ( δ p i , j ) m × m . Let S n ( M ) be the set of n × n permutation matrices which do not contain the m × m permutation matrix M as a submatrix. In [R. Simion, F.W. Schmidt, Restricted permutations, European J. Combin. 6 (1985) 383–406] Simion and Schmidt show bijectively that | S n ( 123 ) | = | S n ( 213 ) | . In the present work, we give a bijection from S n ( 12 … t p t + 1 … p m ) to S n ( t … 21 p t + 1 … p m ) . This result was established for t = 2 in [J. West, Permutations with forbidden subsequences and stack-sortable permutations, PhD thesis, MIT, Cambridge, MA, 1990] and for t = 3 in [E. Babson, J. West, The permutations 123 p 4 … p t and 321 p 4 … p t are Wilf-equivalent, Graphs Combin. 16 (2001) 373–380]. Moreover, if we think of n × n permutation matrices as transversals of the n by n square diagram, then we generalise this result to transversals of Young diagrams.