The number of indecomposable projective-injective modules in the algebra corresponding to a subset.
Let $A_n=K[x]/(x^n)$.
We associate to a nonempty subset S of an (n-1)-set the module $M_S$, which is the direct sum of $A_n$-modules with indecomposable non-projective direct summands of dimension $i$ when $i$ is in $S$ (note that such modules have vector space dimension at most n-1). Then the corresponding algebra associated to S is the stable endomorphism ring of $M_S$. We decode the subset as a binary word so that for example the subset $S=\{1,3 \} $ of $\{1,2,3 \}$ is decoded as 101.

Return the partition as binary word, by traversing its shape from the first row to the last row, down steps as 1 and left steps as 0.

Sends a binary word $w_1\cdots w_m$ to the binary word $v_1 \cdots v_m$ with $v_i = w_i$ if $i$ is odd and $v_i = 1 - w_i$ if $i$ is even.
This map is used in [1], see Definitions 3.2 and 5.1.

The previous element in the Gray code for binary words.
Let $w$ be a binary word. If the number of ones in $w$ is odd, return the binary word obtained by flipping the first bit of $w$. If $w$ has only zeros, return the word with the final bit flipped. Otherwise, return the word obtained by flipping the bit after the first one.