The proof of a conjecture of Simion for certain partitions

Simion has a conjecture concerning the number of lattice paths in a rectangular grid with the Ferrers diagram of a partition removed. The conjecture concerns the unimodality of a sequence of these numbers where the sum of the length and width of each rectangle is a constant and where the partition is constant. This paper demonstrates this unimodality if the partition is self-conjugate or if the Ferrers diagram of the partition has precisely one column or one row. This paper also shows log concavity for partitions of "staircase" shape via a Reflection Principle argument.