The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition. In other words, it is the sum of the coefficients in $$(-1)^{|\lambda|-\ell(\lambda)}\nabla p_\lambda \vert_{q=1,t=1},$$ when expanded in the monomial basis. Here, $\nabla$ is the linear operator on symmetric functions where the modified Macdonald polynomials are eigenvectors. See the Sage documentation for definition and references [[http://doc.sagemath.org/html/en/reference/combinat/sage/combinat/sf/sfa.html]]

The sum of the entries of a semistandard tableau.

The reduced word complexity of a permutation. For a permutation $\pi$, this is the smallest length of a word in simple transpositions that contains all reduced expressions of $\pi$. For example, the permutation $[3,2,1] = (12)(23)(12) = (23)(12)(23)$ and the reduced word complexity is $4$ since the smallest words containing those two reduced words as subwords are $(12),(23),(12),(23)$ and also $(23),(12),(23),(12)$. This statistic appears in [1, Question 6.1].

The number of posets with the same zeta polynomial. The zeta polynomial $Z$ is the polynomial such that $Z(m)$ is the number of weakly increasing sequences $x_1\leq x_2\leq\dots\leq x_{m−1}$ of elements of the poset. See section 3.12 of [1]. Since $$ Z(q) = \sum_{k\geq 1} \binom{q-2}{k-1} c_k, $$ where $c_k$ is the number of chains of length $k$, this statistic is the same as the number of posets with the same chain polynomial.

The number of nonzero entries in a Gelfand Tsetlin pattern.

The number of facets in the order polytope of this poset.

The number of facets in the chain polytope of the poset.