The order of rowmotion on the set of order ideals of a poset.

The least common multiple of the parts of the partition.

The major index of an integer partition when read from bottom to top. This is the sum of the positions of the corners of the shape of an integer partition when reading from bottom to top. For example, the partition $\lambda = (8,6,6,4,3,3)$ has corners at positions 3,6,9, and 13, giving a major index of 31.

Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. Given $\lambda$ count how many ''integer compositions'' $w$ (weight) there are, such that $P_{\lambda,w}$ is integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has all vertices in integer lattice points.

The number of words with multiplicities of the letters given by the composition, avoiding the consecutive pattern 132. The total number of words with letter multiplicities given by an integer partition is [[St000048]]. For example, there are twelve words with letters $0,0,1,2$ corresponding to the partition $[2,1,1]$. Two of these contain the pattern $132$: $0,0,2,1$ and $0,2,1,0$. Note that this statistic is not constant on compositions having the same parts. The number of words of length $n$ with letters in an alphabet of size $k$ avoiding the consecutive pattern $132$ is determined in [1].

The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. The total number of words with letter multiplicities given by an integer partition is [[St000048]]. For example, there are twelve words with letters $0,0,1,2$ corresponding to the partition $[2,1,1]$. Two of these contain an increasing factor of length three: $0012$ and $0120$. Note that prescribing the multiplicities for different letters yields the same number. For example, there are also two words with letters $0,1,1,2$ containing an increasing factor of length three: $1012$ and $0121$. The number of words of length $n$ with letters in an alphabet of size $k$ avoiding the consecutive pattern $123$ is determined in [1].

Number of parabolic noncrossing partitions indexed by the composition. Also the number of elements in the $\nu$-Tamari lattice with $\nu = \nu_\alpha = 1^{\alpha_1} 0^{\alpha_1} \cdots 1^{\alpha_k} 0^{\alpha_k}$, the bounce path indexed by the composition $\alpha$. These elements are Dyck paths weakly above the bounce path $\nu_\alpha$.

The Gini index of an integer partition. As discussed in [1], this statistic is equal to [[St000567]] applied to the conjugate partition.

The major index of a binary word. This is the sum of the positions of descents, i.e., a one followed by a zero. For words of length $n$ with $a$ zeros, the generating function for the major index is the $q$-binomial coefficient $\binom{n}{a}_q$.

Half of MacMahon's equal index of a Dyck path. This is half the sum of the positions of double (up- or down-)steps of a Dyck path, see [1, p. 135].