Wilf-equivalence for singleton classes

Write p1p2…pm for the permutation matrix (δpi,j)m×m. Let Sn(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 |Sn(123)|=|Sn(213)|. In the present work, we give a bijection from Sn(12…tpt+1…pm) to Sn(t…21pt+1…pm). 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 123p4…pt and 321p4…pt 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.