Differential poset

This latter reformulation makes a differential poset into a combinatorial realization of a Weyl algebra, and in particular explains the name differential: the operators "d/dx" and "multiplication by x" on the vector space of polynomials obey the same commutation relation as U and D/r.

(The Young–Fibonacci lattice is the poset that arises by applying this construction beginning with a single point.)

Stanley (1988) includes a remark that "[David] Wagner described a very general method for constructing differential posets which make it unlikely that [they can be classified]."

Stanley & Zanello (2012) proved an asymptotic version of the lower bound, showing that for every differential poset and some constant a.

In the original paper of Stanley, it was shown (using eigenvalues of the operator DU) that the rank sizes are weakly increasing.

[citation needed] In another direction, Lam & Shimozono (2007) defined dual graded graphs corresponding to any Kac–Moody algebra.

Other variations are possible; Stanley (1990) defined versions in which the number r in the definition varies from rank to rank, while Lam (2008) defined a signed analogue of differential posets in which cover relations may be assigned a "weight" of −1.