Signed differential posets and sign-imbalance

We study signed differential posets, a signed version of differential posets. These posets satisfy enumerative identities which are signed analogues of those satisfied by differential posets. Our main motivations are the sign-imbalance identities for partition shapes originally conjectured by Stanley, now proven in [T. Lam, Growth diagrams, domino insertion and sign-imbalance, J. Combin. Theory Ser. A 107 (2004) 87–115; A. Reifergerste, Permutation sign under the Robinson–Schensted–Knuth correspondence, Ann. Comb. 8 (2004) 103–112; J. Sjöstrand, On the sign-imbalance of partition shapes, J. Combin. Theory Ser. A 111 (2005) 190–203]. We show that these identities result from a signed differential poset structure on Youngʹs lattice, and explain similar identities for Fibonacci shapes.