Identifier
Values
[[1],[2]] => 1 => 1 => 1 => 1
[[1,2,3]] => 00 => 00 => 10 => 1
[[1,2],[3]] => 01 => 10 => 01 => 1
[[1],[2],[3]] => 11 => 11 => 11 => 2
[[1,2,3,4]] => 000 => 000 => 010 => 1
[[1,3,4],[2]] => 100 => 001 => 110 => 1
[[1,2,4],[3]] => 010 => 010 => 100 => 1
[[1,2,3],[4]] => 001 => 100 => 101 => 2
[[1,3],[2,4]] => 101 => 101 => 001 => 1
[[1,2],[3,4]] => 010 => 010 => 100 => 1
[[1,3],[2],[4]] => 101 => 101 => 001 => 1
[[1,2],[3],[4]] => 011 => 110 => 011 => 1
[[1],[2],[3],[4]] => 111 => 111 => 111 => 3
[[1,2,3,4,5]] => 0000 => 0000 => 1010 => 0
[[1,3,4,5],[2]] => 1000 => 0001 => 0010 => 1
[[1,2,4,5],[3]] => 0100 => 0010 => 0110 => 2
[[1,2,3,5],[4]] => 0010 => 0100 => 0100 => 1
[[1,2,3,4],[5]] => 0001 => 1000 => 0101 => 0
[[1,3,5],[2,4]] => 1010 => 0101 => 1100 => 1
[[1,2,5],[3,4]] => 0100 => 0010 => 0110 => 2
[[1,3,4],[2,5]] => 1001 => 1001 => 1101 => 2
[[1,2,4],[3,5]] => 0101 => 1010 => 1001 => 2
[[1,2,3],[4,5]] => 0010 => 0100 => 0100 => 1
[[1,4,5],[2],[3]] => 1100 => 0011 => 1110 => 2
[[1,3,5],[2],[4]] => 1010 => 0101 => 1100 => 1
[[1,2,5],[3],[4]] => 0110 => 0110 => 1000 => 1
[[1,3,4],[2],[5]] => 1001 => 1001 => 1101 => 2
[[1,2,4],[3],[5]] => 0101 => 1010 => 1001 => 2
[[1,2,3],[4],[5]] => 0011 => 1100 => 1011 => 2
[[1,4],[2,5],[3]] => 1101 => 1011 => 0001 => 1
[[1,3],[2,5],[4]] => 1010 => 0101 => 1100 => 1
[[1,2],[3,5],[4]] => 0110 => 0110 => 1000 => 1
[[1,3],[2,4],[5]] => 1011 => 1101 => 0011 => 1
[[1,2],[3,4],[5]] => 0101 => 1010 => 1001 => 2
[[1,4],[2],[3],[5]] => 1101 => 1011 => 0001 => 1
[[1,3],[2],[4],[5]] => 1011 => 1101 => 0011 => 1
[[1,2],[3],[4],[5]] => 0111 => 1110 => 0111 => 2
[[1],[2],[3],[4],[5]] => 1111 => 1111 => 1111 => 4
search for individual values
searching the database for the individual values of this statistic
Description
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.
Map
reverse
Description
Return the reversal of a binary word.
Map
descent word
Description
The descent word of a standard Young tableau.
For a standard Young tableau of size $n$ we set $w_i=1$ if $i+1$ is in a lower row than $i$, and $0$ otherwise, for $1\leq i < n$.
Map
zeros to flag zeros
Description
Return a binary word of the same length, such that the number of occurrences of $10$ in the word obtained by prepending the reverse of the complement equals the number of $0$s in the original word.
For example, the image of the word $w=1\dots1$ is $1\dots1$, because $w$ has no zeros, and $1\dots1$ is the only word such that prepending the reverse of its complement has no occurrence of the factor $10$.
On the other hand, $0\dots0$ must be mapped to $10\dots10$ if the length is even, and $010\dots10$ if it is odd.