The length of the shortest palindromic decomposition of a binary word. A palindromic decomposition (paldec for short) of a word $w=a_1,\dots,a_n$ is any list of factors $p_1,\dots,p_k$ such that $w=p_1\dots p_k$ and each $p_i$ is a palindrome, i.e. coincides with itself read backwards.

Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. Graphically, this is the number of returns to the main diagonal of the monotone lattice path of a binary word.

Half the length of the longest even length palindromic prefix of a binary word.

The degree of symmetry of a binary word. For a binary word $w$ of length $n$, this is the number of positions $i\leq n/2$ such that $w_i = w_{n+1-i}$.

The index of the maximal parabolic seaweed algebra associated with the composition. Let $a_1,\dots,a_m$ and $b_1,\dots,b_t$ be a pair of compositions of $n$. The meander associated to this pair is obtained as follows: * place $n$ dots on a horizontal line * subdivide the dots into $m$ blocks of sizes $a_1, a_2,\dots$ * within each block, connect the first and the last dot, the second and the next to last, and so on, with an arc above the line * subdivide the dots into $t$ blocks of sizes $b_1, b_2,\dots$ * within each block, connect the first and the last dot, the second and the next to last, and so on, with an arc below the line By [1, thm.5.1], the index of the seaweed algebra associated to the pair of compositions is $$\operatorname{ind}\displaystyle\frac{b_1|b_2|...|b_t}{a_1|a_2|...|a_m} = 2C+P-1,$$ where $C$ is the number of cycles (of length at least $2$) and P is the number of paths in the meander. This statistic is $\operatorname{ind}\displaystyle\frac{b_1|b_2|...|b_t}{n}$.

The number of distinct squares in a binary word. A factor of a word is a sequence of consecutive letters. This statistic records the number of distinct non-empty words $u$ such that $uu$ is a factor of the word. Note that every word of length at least four contains a square.

The degree of asymmetry of an integer composition. This is the number of pairs of symmetrically positioned distinct entries.

The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. Consider the recurrence $$f(n)=\sum_{p\in\lambda} f(n-p).$$ This statistic returns the number of distinct real roots of the associated characteristic polynomial. For example, the partition $(2,1)$ corresponds to the recurrence $f(n)=f(n-1)+f(n-2)$ with associated characteristic polynomial $x^2-x-1$, which has two real roots.

The length of the longest constant subword.

The distinguishing number of a poset. This is the minimal number of colours needed to colour the vertices of a poset, such that only the trivial automorphism of the poset preserves the colouring. See also [[St000469]], which is the same concept for graphs.