The Rank and Minimal Border Strip Decompositions of a Skew Partition

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Nazarov and Tarasov recently generalized the notion of the rank of a partition to skew partitions. We give several characterizations of the rank of a skew partition and one possible characterization that remains open. One of the characterizations involves the decomposition of a skew shape into a minimal number of border strips, and we develop a theory of these MBSD's as well as of the closely related minimal border strip tableaux. An application is given to the value of a character of the symmetric group Sₙ indexed by a skew shape z at a permutation whose number of cycles is the rank of z.