Identifier
Values
0101 => [2,2,1] => [[3,3,2],[2,1]] => ([(0,2),(2,1)],3) => 0
1010 => [1,2,2] => [[3,2,1],[1]] => ([(0,2),(2,1)],3) => 0
00100 => [3,3] => [[5,3],[2]] => ([(0,2),(2,1)],3) => 0
00101 => [3,2,1] => [[4,4,3],[3,2]] => ([(0,2),(2,1)],3) => 0
01001 => [2,3,1] => [[4,4,2],[3,1]] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
01010 => [2,2,2] => [[4,3,2],[2,1]] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 1
01011 => [2,2,1,1] => [[3,3,3,2],[2,2,1]] => ([(0,2),(2,1)],3) => 0
01101 => [2,1,2,1] => [[3,3,2,2],[2,1,1]] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
10010 => [1,3,2] => [[4,3,1],[2]] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
10100 => [1,2,3] => [[4,2,1],[1]] => ([(0,2),(2,1)],3) => 0
10101 => [1,2,2,1] => [[3,3,2,1],[2,1]] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 1
10110 => [1,2,1,2] => [[3,2,2,1],[1,1]] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
11010 => [1,1,2,2] => [[3,2,1,1],[1]] => ([(0,2),(2,1)],3) => 0
11011 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]] => ([(0,2),(2,1)],3) => 0
000100 => [4,3] => [[6,4],[3]] => ([(0,2),(2,1)],3) => 0
000101 => [4,2,1] => [[5,5,4],[4,3]] => ([(0,2),(2,1)],3) => 0
001000 => [3,4] => [[6,3],[2]] => ([(0,2),(2,1)],3) => 0
001001 => [3,3,1] => [[5,5,3],[4,2]] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 1
001010 => [3,2,2] => [[5,4,3],[3,2]] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => 1
001011 => [3,2,1,1] => [[4,4,4,3],[3,3,2]] => ([(0,2),(2,1)],3) => 0
001100 => [3,1,3] => [[5,3,3],[2,2]] => ([(0,2),(2,1)],3) => 0
001101 => [3,1,2,1] => [[4,4,3,3],[3,2,2]] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
010001 => [2,4,1] => [[5,5,2],[4,1]] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
010010 => [2,3,2] => [[5,4,2],[3,1]] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => 1
010011 => [2,3,1,1] => [[4,4,4,2],[3,3,1]] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
010100 => [2,2,3] => [[5,3,2],[2,1]] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => 1
010111 => [2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]] => ([(0,2),(2,1)],3) => 0
011011 => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 1
011101 => [2,1,1,2,1] => [[3,3,2,2,2],[2,1,1,1]] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
100010 => [1,4,2] => [[5,4,1],[3]] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
100100 => [1,3,3] => [[5,3,1],[2]] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 1
101000 => [1,2,4] => [[5,2,1],[1]] => ([(0,2),(2,1)],3) => 0
101011 => [1,2,2,1,1] => [[3,3,3,2,1],[2,2,1]] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => 1
101100 => [1,2,1,3] => [[4,2,2,1],[1,1]] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
101101 => [1,2,1,2,1] => [[3,3,2,2,1],[2,1,1]] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => 1
101110 => [1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
110010 => [1,1,3,2] => [[4,3,1,1],[2]] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
110011 => [1,1,3,1,1] => [[3,3,3,1,1],[2,2]] => ([(0,2),(2,1)],3) => 0
110100 => [1,1,2,3] => [[4,2,1,1],[1]] => ([(0,2),(2,1)],3) => 0
110101 => [1,1,2,2,1] => [[3,3,2,1,1],[2,1]] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => 1
110110 => [1,1,2,1,2] => [[3,2,2,1,1],[1,1]] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 1
110111 => [1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]] => ([(0,2),(2,1)],3) => 0
111010 => [1,1,1,2,2] => [[3,2,1,1,1],[1]] => ([(0,2),(2,1)],3) => 0
111011 => [1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]] => ([(0,2),(2,1)],3) => 0
0001000 => [4,4] => [[7,4],[3]] => ([(0,3),(2,1),(3,2)],4) => 0
0001001 => [4,3,1] => [[6,6,4],[5,3]] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 1
0001010 => [4,2,2] => [[6,5,4],[4,3]] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => 1
0001011 => [4,2,1,1] => [[5,5,5,4],[4,4,3]] => ([(0,2),(2,1)],3) => 0
0001100 => [4,1,3] => [[6,4,4],[3,3]] => ([(0,2),(2,1)],3) => 0
0001101 => [4,1,2,1] => [[5,5,4,4],[4,3,3]] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
0010001 => [3,4,1] => [[6,6,3],[5,2]] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 2
0010010 => [3,3,2] => [[6,5,3],[4,2]] => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8) => 2
0010011 => [3,3,1,1] => [[5,5,5,3],[4,4,2]] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 1
0010100 => [3,2,3] => [[6,4,3],[3,2]] => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8) => 2
0010111 => [3,2,1,1,1] => [[4,4,4,4,3],[3,3,3,2]] => ([(0,2),(2,1)],3) => 0
0011011 => [3,1,2,1,1] => [[4,4,4,3,3],[3,3,2,2]] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 1
0011100 => [3,1,1,3] => [[5,3,3,3],[2,2,2]] => ([(0,2),(2,1)],3) => 0
0011101 => [3,1,1,2,1] => [[4,4,3,3,3],[3,2,2,2]] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
0100010 => [2,4,2] => [[6,5,2],[4,1]] => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7) => 2
0100011 => [2,4,1,1] => [[5,5,5,2],[4,4,1]] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
0100100 => [2,3,3] => [[6,4,2],[3,1]] => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8) => 2
0100111 => [2,3,1,1,1] => [[4,4,4,4,2],[3,3,3,1]] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
0101111 => [2,2,1,1,1,1] => [[3,3,3,3,3,2],[2,2,2,2,1]] => ([(0,2),(2,1)],3) => 0
0110111 => [2,1,2,1,1,1] => [[3,3,3,3,2,2],[2,2,2,1,1]] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 1
0111011 => [2,1,1,2,1,1] => [[3,3,3,2,2,2],[2,2,1,1,1]] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 2
0111101 => [2,1,1,1,2,1] => [[3,3,2,2,2,2],[2,1,1,1,1]] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
1000100 => [1,4,3] => [[6,4,1],[3]] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 2
1010111 => [1,2,2,1,1,1] => [[3,3,3,3,2,1],[2,2,2,1]] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => 1
1011011 => [1,2,1,2,1,1] => [[3,3,3,2,2,1],[2,2,1,1]] => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8) => 2
1011101 => [1,2,1,1,2,1] => [[3,3,2,2,2,1],[2,1,1,1]] => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7) => 2
1011110 => [1,2,1,1,1,2] => [[3,2,2,2,2,1],[1,1,1,1]] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
1100100 => [1,1,3,3] => [[5,3,1,1],[2]] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 1
1101011 => [1,1,2,2,1,1] => [[3,3,3,2,1,1],[2,2,1]] => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8) => 2
1101101 => [1,1,2,1,2,1] => [[3,3,2,2,1,1],[2,1,1]] => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8) => 2
1101110 => [1,1,2,1,1,2] => [[3,2,2,2,1,1],[1,1,1]] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 2
1101111 => [1,1,2,1,1,1,1] => [[2,2,2,2,2,1,1],[1,1,1,1]] => ([(0,2),(2,1)],3) => 0
1110010 => [1,1,1,3,2] => [[4,3,1,1,1],[2]] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
1110101 => [1,1,1,2,2,1] => [[3,3,2,1,1,1],[2,1]] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => 1
1110110 => [1,1,1,2,1,2] => [[3,2,2,1,1,1],[1,1]] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 1
1110111 => [1,1,1,2,1,1,1] => [[2,2,2,2,1,1,1],[1,1,1]] => ([(0,3),(2,1),(3,2)],4) => 0
1111010 => [1,1,1,1,2,2] => [[3,2,1,1,1,1],[1]] => ([(0,2),(2,1)],3) => 0
1111011 => [1,1,1,1,2,1,1] => [[2,2,2,1,1,1,1],[1,1]] => ([(0,2),(2,1)],3) => 0
search for individual values
searching the database for the individual values of this statistic
Description
The number of 2-regular simple modules in the incidence algebra of the lattice.
Map
to composition
Description
The composition corresponding to a binary word.
Prepending $1$ to a binary word $w$, the $i$-th part of the composition equals $1$ plus the number of zeros after the $i$-th $1$ in $w$.
This map is not surjective, since the empty composition does not have a preimage.
Map
dominating sublattice
Description
Return the sublattice of the dominance order induced by the support of the expansion of the skew Schur function into Schur functions.
Consider the expansion of the skew Schur function $s_{\lambda/\mu}=\sum_\nu c^\lambda_{\mu, \nu} s_\nu$ as a linear combination of straight Schur functions.
It is shown in [1] that the subposet of the dominance order whose elements are the partitions $\nu$ with $c^\lambda_{\mu, \nu} > 0$ form a lattice.
The example $\lambda = (5^2,4^2,1)$ and $\mu=(3,2)$ shows that this lattice is not a sublattice of the dominance order.
Map
to ribbon
Description
The ribbon shape corresponding to an integer composition.
For an integer composition $(a_1, \dots, a_n)$, this is the ribbon shape whose $i$th row from the bottom has $a_i$ cells.