... ,aj} be the elements of [k] which are mapped by π into [k], with a1<a2< ···<ajif A = ∅.Now assume that j = 0. Because π(ai) ∈ [k] by definition and π(π(ai)) = ai∈ [k],π(ai) ... on ST.Ofthefµtableaux Qso obtained by letting Qrange over ST, exactly one is the order preserving relabeling of TSTwith the elements of[j]. By the observation above, this choice of ... 5, 6, 7} 8 1 2 5 14{0, 1, 2, 3, 4, 5, 6, 7} 64 1 2 5 14Table 5: Classifying S7 by subsequence containment by involutions.the electronic journal of combinatorics 12 (2005), #R14 9Before proving...