Two injective proofs of a conjecture of Simion

Simion (J. Combin. Theory Ser. A 94 (1994) 270) conjectured the unimodality of a sequence counting lattice paths in a grid with a Ferrers diagram removed from the northwest corner. Recently, Hildebrand (J. Combin. Theory Ser. A 97 (2002) 108) and then Wang (A simple proof of a conjecture of Simion, J. Combin. Theory Ser. A 100 (2002) 399) proved the stronger result that this sequence is actually log concave. Both proofs were mainly algebraic in nature. We give two combinatorial proofs of this theorem.