The number of twos in an integer partition. The total number of twos in all partitions of $n$ is equal to the total number of singletons [[St001484]] in all partitions of $n-1$, see [1].

The number of distinct parts of a partition that occur at least twice. See Section 3.3.1 of [2].

The balance of a binary word. The balance of a word is the smallest number $q$ such that the word is $q$-balanced [1]. A binary word $w$ is $q$-balanced if for any two factors $u$, $v$ of $w$ of the same length, the difference between the number of ones in $u$ and $v$ is at most $q$.

The Grundy value of Welter's game on a binary word. Two players take turns moving a $1$ to the left. The loosing positions are the words $1\dots 10\dots 0$.

Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra.

The minimal period of a binary word. This is the smallest natural number $p$ such that $w_i=w_{i+p}$ for all $i\in\{1,\dots,|w|-p\}$.

The number of singletons of an integer partition. A singleton in an integer partition is a part that appear precisely once.

The number of invariant subsets of size 2 when acting with a permutation of given cycle type.

The largest repeated part of a partition. If the parts of the partition are all distinct, the value of the statistic is defined to be zero.

The number of parts from which one can substract 2 and still get an integer partition.