Identifier
Values
001 => [3,1] => [[3,3],[2]] => [2] => 0
011 => [2,1,1] => [[2,2,2],[1,1]] => [1,1] => 0
0001 => [4,1] => [[4,4],[3]] => [3] => 0
0010 => [3,2] => [[4,3],[2]] => [2] => 0
0011 => [3,1,1] => [[3,3,3],[2,2]] => [2,2] => 0
0101 => [2,2,1] => [[3,3,2],[2,1]] => [2,1] => 1
0110 => [2,1,2] => [[3,2,2],[1,1]] => [1,1] => 0
0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]] => [1,1,1] => 0
1001 => [1,3,1] => [[3,3,1],[2]] => [2] => 0
1011 => [1,2,1,1] => [[2,2,2,1],[1,1]] => [1,1] => 0
00001 => [5,1] => [[5,5],[4]] => [4] => 0
00010 => [4,2] => [[5,4],[3]] => [3] => 0
00011 => [4,1,1] => [[4,4,4],[3,3]] => [3,3] => 0
00100 => [3,3] => [[5,3],[2]] => [2] => 0
00101 => [3,2,1] => [[4,4,3],[3,2]] => [3,2] => 1
00110 => [3,1,2] => [[4,3,3],[2,2]] => [2,2] => 0
00111 => [3,1,1,1] => [[3,3,3,3],[2,2,2]] => [2,2,2] => 0
01001 => [2,3,1] => [[4,4,2],[3,1]] => [3,1] => 1
01010 => [2,2,2] => [[4,3,2],[2,1]] => [2,1] => 1
01011 => [2,2,1,1] => [[3,3,3,2],[2,2,1]] => [2,2,1] => 1
01100 => [2,1,3] => [[4,2,2],[1,1]] => [1,1] => 0
01101 => [2,1,2,1] => [[3,3,2,2],[2,1,1]] => [2,1,1] => 1
01110 => [2,1,1,2] => [[3,2,2,2],[1,1,1]] => [1,1,1] => 0
01111 => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]] => [1,1,1,1] => 0
10001 => [1,4,1] => [[4,4,1],[3]] => [3] => 0
10010 => [1,3,2] => [[4,3,1],[2]] => [2] => 0
10011 => [1,3,1,1] => [[3,3,3,1],[2,2]] => [2,2] => 0
10101 => [1,2,2,1] => [[3,3,2,1],[2,1]] => [2,1] => 1
10110 => [1,2,1,2] => [[3,2,2,1],[1,1]] => [1,1] => 0
10111 => [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]] => [1,1,1] => 0
11001 => [1,1,3,1] => [[3,3,1,1],[2]] => [2] => 0
11011 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]] => [1,1] => 0
000001 => [6,1] => [[6,6],[5]] => [5] => 0
000010 => [5,2] => [[6,5],[4]] => [4] => 0
000011 => [5,1,1] => [[5,5,5],[4,4]] => [4,4] => 0
000100 => [4,3] => [[6,4],[3]] => [3] => 0
000101 => [4,2,1] => [[5,5,4],[4,3]] => [4,3] => 1
000110 => [4,1,2] => [[5,4,4],[3,3]] => [3,3] => 0
000111 => [4,1,1,1] => [[4,4,4,4],[3,3,3]] => [3,3,3] => 0
001000 => [3,4] => [[6,3],[2]] => [2] => 0
001001 => [3,3,1] => [[5,5,3],[4,2]] => [4,2] => 1
001010 => [3,2,2] => [[5,4,3],[3,2]] => [3,2] => 1
001011 => [3,2,1,1] => [[4,4,4,3],[3,3,2]] => [3,3,2] => 1
001100 => [3,1,3] => [[5,3,3],[2,2]] => [2,2] => 0
001101 => [3,1,2,1] => [[4,4,3,3],[3,2,2]] => [3,2,2] => 1
001110 => [3,1,1,2] => [[4,3,3,3],[2,2,2]] => [2,2,2] => 0
001111 => [3,1,1,1,1] => [[3,3,3,3,3],[2,2,2,2]] => [2,2,2,2] => 0
010001 => [2,4,1] => [[5,5,2],[4,1]] => [4,1] => 1
010010 => [2,3,2] => [[5,4,2],[3,1]] => [3,1] => 1
010011 => [2,3,1,1] => [[4,4,4,2],[3,3,1]] => [3,3,1] => 1
010100 => [2,2,3] => [[5,3,2],[2,1]] => [2,1] => 1
010101 => [2,2,2,1] => [[4,4,3,2],[3,2,1]] => [3,2,1] => 2
010110 => [2,2,1,2] => [[4,3,3,2],[2,2,1]] => [2,2,1] => 1
010111 => [2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]] => [2,2,2,1] => 1
011000 => [2,1,4] => [[5,2,2],[1,1]] => [1,1] => 0
011001 => [2,1,3,1] => [[4,4,2,2],[3,1,1]] => [3,1,1] => 1
011010 => [2,1,2,2] => [[4,3,2,2],[2,1,1]] => [2,1,1] => 1
011011 => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]] => [2,2,1,1] => 1
011100 => [2,1,1,3] => [[4,2,2,2],[1,1,1]] => [1,1,1] => 0
011101 => [2,1,1,2,1] => [[3,3,2,2,2],[2,1,1,1]] => [2,1,1,1] => 1
011110 => [2,1,1,1,2] => [[3,2,2,2,2],[1,1,1,1]] => [1,1,1,1] => 0
011111 => [2,1,1,1,1,1] => [[2,2,2,2,2,2],[1,1,1,1,1]] => [1,1,1,1,1] => 0
100001 => [1,5,1] => [[5,5,1],[4]] => [4] => 0
100010 => [1,4,2] => [[5,4,1],[3]] => [3] => 0
100011 => [1,4,1,1] => [[4,4,4,1],[3,3]] => [3,3] => 0
100100 => [1,3,3] => [[5,3,1],[2]] => [2] => 0
100101 => [1,3,2,1] => [[4,4,3,1],[3,2]] => [3,2] => 1
100110 => [1,3,1,2] => [[4,3,3,1],[2,2]] => [2,2] => 0
100111 => [1,3,1,1,1] => [[3,3,3,3,1],[2,2,2]] => [2,2,2] => 0
101001 => [1,2,3,1] => [[4,4,2,1],[3,1]] => [3,1] => 1
101010 => [1,2,2,2] => [[4,3,2,1],[2,1]] => [2,1] => 1
101011 => [1,2,2,1,1] => [[3,3,3,2,1],[2,2,1]] => [2,2,1] => 1
101100 => [1,2,1,3] => [[4,2,2,1],[1,1]] => [1,1] => 0
101101 => [1,2,1,2,1] => [[3,3,2,2,1],[2,1,1]] => [2,1,1] => 1
101110 => [1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]] => [1,1,1] => 0
101111 => [1,2,1,1,1,1] => [[2,2,2,2,2,1],[1,1,1,1]] => [1,1,1,1] => 0
110001 => [1,1,4,1] => [[4,4,1,1],[3]] => [3] => 0
110010 => [1,1,3,2] => [[4,3,1,1],[2]] => [2] => 0
110011 => [1,1,3,1,1] => [[3,3,3,1,1],[2,2]] => [2,2] => 0
110101 => [1,1,2,2,1] => [[3,3,2,1,1],[2,1]] => [2,1] => 1
110110 => [1,1,2,1,2] => [[3,2,2,1,1],[1,1]] => [1,1] => 0
110111 => [1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]] => [1,1,1] => 0
111001 => [1,1,1,3,1] => [[3,3,1,1,1],[2]] => [2] => 0
111011 => [1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]] => [1,1] => 0
0000001 => [7,1] => [[7,7],[6]] => [6] => 0
0000011 => [6,1,1] => [[6,6,6],[5,5]] => [5,5] => 0
0001000 => [4,4] => [[7,4],[3]] => [3] => 0
0001001 => [4,3,1] => [[6,6,4],[5,3]] => [5,3] => 1
0001010 => [4,2,2] => [[6,5,4],[4,3]] => [4,3] => 1
0001011 => [4,2,1,1] => [[5,5,5,4],[4,4,3]] => [4,4,3] => 1
0001100 => [4,1,3] => [[6,4,4],[3,3]] => [3,3] => 0
0001101 => [4,1,2,1] => [[5,5,4,4],[4,3,3]] => [4,3,3] => 1
0001110 => [4,1,1,2] => [[5,4,4,4],[3,3,3]] => [3,3,3] => 0
0010001 => [3,4,1] => [[6,6,3],[5,2]] => [5,2] => 1
0010010 => [3,3,2] => [[6,5,3],[4,2]] => [4,2] => 1
0010011 => [3,3,1,1] => [[5,5,5,3],[4,4,2]] => [4,4,2] => 1
0010100 => [3,2,3] => [[6,4,3],[3,2]] => [3,2] => 1
0010101 => [3,2,2,1] => [[5,5,4,3],[4,3,2]] => [4,3,2] => 2
0010110 => [3,2,1,2] => [[5,4,4,3],[3,3,2]] => [3,3,2] => 1
0010111 => [3,2,1,1,1] => [[4,4,4,4,3],[3,3,3,2]] => [3,3,3,2] => 1
0011001 => [3,1,3,1] => [[5,5,3,3],[4,2,2]] => [4,2,2] => 1
>>> Load all 171 entries. <<<
0011010 => [3,1,2,2] => [[5,4,3,3],[3,2,2]] => [3,2,2] => 1
0011011 => [3,1,2,1,1] => [[4,4,4,3,3],[3,3,2,2]] => [3,3,2,2] => 1
0011100 => [3,1,1,3] => [[5,3,3,3],[2,2,2]] => [2,2,2] => 0
0011101 => [3,1,1,2,1] => [[4,4,3,3,3],[3,2,2,2]] => [3,2,2,2] => 1
0011111 => [3,1,1,1,1,1] => [[3,3,3,3,3,3],[2,2,2,2,2]] => [2,2,2,2,2] => 0
0100010 => [2,4,2] => [[6,5,2],[4,1]] => [4,1] => 1
0100011 => [2,4,1,1] => [[5,5,5,2],[4,4,1]] => [4,4,1] => 1
0100100 => [2,3,3] => [[6,4,2],[3,1]] => [3,1] => 1
0100101 => [2,3,2,1] => [[5,5,4,2],[4,3,1]] => [4,3,1] => 2
0100110 => [2,3,1,2] => [[5,4,4,2],[3,3,1]] => [3,3,1] => 1
0100111 => [2,3,1,1,1] => [[4,4,4,4,2],[3,3,3,1]] => [3,3,3,1] => 1
0101001 => [2,2,3,1] => [[5,5,3,2],[4,2,1]] => [4,2,1] => 2
0101010 => [2,2,2,2] => [[5,4,3,2],[3,2,1]] => [3,2,1] => 2
0101011 => [2,2,2,1,1] => [[4,4,4,3,2],[3,3,2,1]] => [3,3,2,1] => 2
0101100 => [2,2,1,3] => [[5,3,3,2],[2,2,1]] => [2,2,1] => 1
0101101 => [2,2,1,2,1] => [[4,4,3,3,2],[3,2,2,1]] => [3,2,2,1] => 2
0101110 => [2,2,1,1,2] => [[4,3,3,3,2],[2,2,2,1]] => [2,2,2,1] => 1
0101111 => [2,2,1,1,1,1] => [[3,3,3,3,3,2],[2,2,2,2,1]] => [2,2,2,2,1] => 1
0110010 => [2,1,3,2] => [[5,4,2,2],[3,1,1]] => [3,1,1] => 1
0110011 => [2,1,3,1,1] => [[4,4,4,2,2],[3,3,1,1]] => [3,3,1,1] => 1
0110100 => [2,1,2,3] => [[5,3,2,2],[2,1,1]] => [2,1,1] => 1
0110101 => [2,1,2,2,1] => [[4,4,3,2,2],[3,2,1,1]] => [3,2,1,1] => 2
0110110 => [2,1,2,1,2] => [[4,3,3,2,2],[2,2,1,1]] => [2,2,1,1] => 1
0110111 => [2,1,2,1,1,1] => [[3,3,3,3,2,2],[2,2,2,1,1]] => [2,2,2,1,1] => 1
0111001 => [2,1,1,3,1] => [[4,4,2,2,2],[3,1,1,1]] => [3,1,1,1] => 1
0111010 => [2,1,1,2,2] => [[4,3,2,2,2],[2,1,1,1]] => [2,1,1,1] => 1
0111011 => [2,1,1,2,1,1] => [[3,3,3,2,2,2],[2,2,1,1,1]] => [2,2,1,1,1] => 1
0111101 => [2,1,1,1,2,1] => [[3,3,2,2,2,2],[2,1,1,1,1]] => [2,1,1,1,1] => 1
0111110 => [2,1,1,1,1,2] => [[3,2,2,2,2,2],[1,1,1,1,1]] => [1,1,1,1,1] => 0
0111111 => [2,1,1,1,1,1,1] => [[2,2,2,2,2,2,2],[1,1,1,1,1,1]] => [1,1,1,1,1,1] => 0
1000100 => [1,4,3] => [[6,4,1],[3]] => [3] => 0
1000101 => [1,4,2,1] => [[5,5,4,1],[4,3]] => [4,3] => 1
1000110 => [1,4,1,2] => [[5,4,4,1],[3,3]] => [3,3] => 0
1000111 => [1,4,1,1,1] => [[4,4,4,4,1],[3,3,3]] => [3,3,3] => 0
1001001 => [1,3,3,1] => [[5,5,3,1],[4,2]] => [4,2] => 1
1001010 => [1,3,2,2] => [[5,4,3,1],[3,2]] => [3,2] => 1
1001011 => [1,3,2,1,1] => [[4,4,4,3,1],[3,3,2]] => [3,3,2] => 1
1001100 => [1,3,1,3] => [[5,3,3,1],[2,2]] => [2,2] => 0
1001101 => [1,3,1,2,1] => [[4,4,3,3,1],[3,2,2]] => [3,2,2] => 1
1001110 => [1,3,1,1,2] => [[4,3,3,3,1],[2,2,2]] => [2,2,2] => 0
1010010 => [1,2,3,2] => [[5,4,2,1],[3,1]] => [3,1] => 1
1010011 => [1,2,3,1,1] => [[4,4,4,2,1],[3,3,1]] => [3,3,1] => 1
1010100 => [1,2,2,3] => [[5,3,2,1],[2,1]] => [2,1] => 1
1010101 => [1,2,2,2,1] => [[4,4,3,2,1],[3,2,1]] => [3,2,1] => 2
1010110 => [1,2,2,1,2] => [[4,3,3,2,1],[2,2,1]] => [2,2,1] => 1
1010111 => [1,2,2,1,1,1] => [[3,3,3,3,2,1],[2,2,2,1]] => [2,2,2,1] => 1
1011001 => [1,2,1,3,1] => [[4,4,2,2,1],[3,1,1]] => [3,1,1] => 1
1011010 => [1,2,1,2,2] => [[4,3,2,2,1],[2,1,1]] => [2,1,1] => 1
1011011 => [1,2,1,2,1,1] => [[3,3,3,2,2,1],[2,2,1,1]] => [2,2,1,1] => 1
1011101 => [1,2,1,1,2,1] => [[3,3,2,2,2,1],[2,1,1,1]] => [2,1,1,1] => 1
1011110 => [1,2,1,1,1,2] => [[3,2,2,2,2,1],[1,1,1,1]] => [1,1,1,1] => 0
1011111 => [1,2,1,1,1,1,1] => [[2,2,2,2,2,2,1],[1,1,1,1,1]] => [1,1,1,1,1] => 0
1100100 => [1,1,3,3] => [[5,3,1,1],[2]] => [2] => 0
1100101 => [1,1,3,2,1] => [[4,4,3,1,1],[3,2]] => [3,2] => 1
1100110 => [1,1,3,1,2] => [[4,3,3,1,1],[2,2]] => [2,2] => 0
1101001 => [1,1,2,3,1] => [[4,4,2,1,1],[3,1]] => [3,1] => 1
1101010 => [1,1,2,2,2] => [[4,3,2,1,1],[2,1]] => [2,1] => 1
1101011 => [1,1,2,2,1,1] => [[3,3,3,2,1,1],[2,2,1]] => [2,2,1] => 1
1101101 => [1,1,2,1,2,1] => [[3,3,2,2,1,1],[2,1,1]] => [2,1,1] => 1
1101110 => [1,1,2,1,1,2] => [[3,2,2,2,1,1],[1,1,1]] => [1,1,1] => 0
1101111 => [1,1,2,1,1,1,1] => [[2,2,2,2,2,1,1],[1,1,1,1]] => [1,1,1,1] => 0
1110010 => [1,1,1,3,2] => [[4,3,1,1,1],[2]] => [2] => 0
1110101 => [1,1,1,2,2,1] => [[3,3,2,1,1,1],[2,1]] => [2,1] => 1
1110110 => [1,1,1,2,1,2] => [[3,2,2,1,1,1],[1,1]] => [1,1] => 0
1110111 => [1,1,1,2,1,1,1] => [[2,2,2,2,1,1,1],[1,1,1]] => [1,1,1] => 0
1111011 => [1,1,1,1,2,1,1] => [[2,2,2,1,1,1,1],[1,1]] => [1,1] => 0
00000001 => [8,1] => [[8,8],[7]] => [7] => 0
01111111 => [2,1,1,1,1,1,1,1] => [[2,2,2,2,2,2,2,2],[1,1,1,1,1,1,1]] => [1,1,1,1,1,1,1] => 0
000000001 => [9,1] => [[9,9],[8]] => [8] => 0
011111111 => [2,1,1,1,1,1,1,1,1] => [[2,2,2,2,2,2,2,2,2],[1,1,1,1,1,1,1,1]] => [1,1,1,1,1,1,1,1] => 0
search for individual values
searching the database for the individual values of this statistic
Description
The multiplicity of the standard representation in the Kronecker square corresponding to a partition.
The Kronecker coefficient is the multiplicity $g_{\mu,\nu}^\lambda$ of the Specht module $S^\lambda$ in $S^\mu\otimes S^\nu$:
$$ S^\mu\otimes S^\nu = \bigoplus_\lambda g_{\mu,\nu}^\lambda S^\lambda $$
This statistic records the Kronecker coefficient $g_{\lambda,\lambda}^{(n-1)1}$, for $\lambda\vdash n > 1$. For $n\leq1$ the statistic is undefined.
It follows from [3, Prop.4.1] (or, slightly easier from [3, Thm.4.2]) that this is one less than St000159The number of distinct parts of the integer partition., the number of distinct parts of the partition.
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
inner shape
Description
The inner shape of a skew partition.
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.