Identifier
-
Mp00097:
Binary words
—delta morphism⟶
Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
St001597: Skew partitions ⟶ ℤ
Values
0 => [1] => [[1],[]] => 1
1 => [1] => [[1],[]] => 1
00 => [2] => [[2],[]] => 1
01 => [1,1] => [[1,1],[]] => 1
10 => [1,1] => [[1,1],[]] => 1
11 => [2] => [[2],[]] => 1
000 => [3] => [[3],[]] => 1
001 => [2,1] => [[2,2],[1]] => 1
010 => [1,1,1] => [[1,1,1],[]] => 1
011 => [1,2] => [[2,1],[]] => 1
100 => [1,2] => [[2,1],[]] => 1
101 => [1,1,1] => [[1,1,1],[]] => 1
110 => [2,1] => [[2,2],[1]] => 1
111 => [3] => [[3],[]] => 1
0000 => [4] => [[4],[]] => 1
0001 => [3,1] => [[3,3],[2]] => 1
0010 => [2,1,1] => [[2,2,2],[1,1]] => 1
0011 => [2,2] => [[3,2],[1]] => 1
0100 => [1,1,2] => [[2,1,1],[]] => 1
0101 => [1,1,1,1] => [[1,1,1,1],[]] => 1
0110 => [1,2,1] => [[2,2,1],[1]] => 1
0111 => [1,3] => [[3,1],[]] => 1
1000 => [1,3] => [[3,1],[]] => 1
1001 => [1,2,1] => [[2,2,1],[1]] => 1
1010 => [1,1,1,1] => [[1,1,1,1],[]] => 1
1011 => [1,1,2] => [[2,1,1],[]] => 1
1100 => [2,2] => [[3,2],[1]] => 1
1101 => [2,1,1] => [[2,2,2],[1,1]] => 1
1110 => [3,1] => [[3,3],[2]] => 1
1111 => [4] => [[4],[]] => 1
00000 => [5] => [[5],[]] => 1
00001 => [4,1] => [[4,4],[3]] => 1
00010 => [3,1,1] => [[3,3,3],[2,2]] => 1
00011 => [3,2] => [[4,3],[2]] => 1
00100 => [2,1,2] => [[3,2,2],[1,1]] => 1
00101 => [2,1,1,1] => [[2,2,2,2],[1,1,1]] => 1
00110 => [2,2,1] => [[3,3,2],[2,1]] => 1
00111 => [2,3] => [[4,2],[1]] => 1
01000 => [1,1,3] => [[3,1,1],[]] => 1
01001 => [1,1,2,1] => [[2,2,1,1],[1]] => 1
01010 => [1,1,1,1,1] => [[1,1,1,1,1],[]] => 1
01011 => [1,1,1,2] => [[2,1,1,1],[]] => 1
01100 => [1,2,2] => [[3,2,1],[1]] => 1
01101 => [1,2,1,1] => [[2,2,2,1],[1,1]] => 1
01110 => [1,3,1] => [[3,3,1],[2]] => 1
01111 => [1,4] => [[4,1],[]] => 1
10000 => [1,4] => [[4,1],[]] => 1
10001 => [1,3,1] => [[3,3,1],[2]] => 1
10010 => [1,2,1,1] => [[2,2,2,1],[1,1]] => 1
10011 => [1,2,2] => [[3,2,1],[1]] => 1
10100 => [1,1,1,2] => [[2,1,1,1],[]] => 1
10101 => [1,1,1,1,1] => [[1,1,1,1,1],[]] => 1
10110 => [1,1,2,1] => [[2,2,1,1],[1]] => 1
10111 => [1,1,3] => [[3,1,1],[]] => 1
11000 => [2,3] => [[4,2],[1]] => 1
11001 => [2,2,1] => [[3,3,2],[2,1]] => 1
11010 => [2,1,1,1] => [[2,2,2,2],[1,1,1]] => 1
11011 => [2,1,2] => [[3,2,2],[1,1]] => 1
11100 => [3,2] => [[4,3],[2]] => 1
11101 => [3,1,1] => [[3,3,3],[2,2]] => 1
11110 => [4,1] => [[4,4],[3]] => 1
11111 => [5] => [[5],[]] => 1
000000 => [6] => [[6],[]] => 1
000001 => [5,1] => [[5,5],[4]] => 1
000010 => [4,1,1] => [[4,4,4],[3,3]] => 1
000011 => [4,2] => [[5,4],[3]] => 1
000100 => [3,1,2] => [[4,3,3],[2,2]] => 1
000101 => [3,1,1,1] => [[3,3,3,3],[2,2,2]] => 1
000110 => [3,2,1] => [[4,4,3],[3,2]] => 1
000111 => [3,3] => [[5,3],[2]] => 1
001000 => [2,1,3] => [[4,2,2],[1,1]] => 1
001001 => [2,1,2,1] => [[3,3,2,2],[2,1,1]] => 1
001010 => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]] => 1
001011 => [2,1,1,2] => [[3,2,2,2],[1,1,1]] => 1
001100 => [2,2,2] => [[4,3,2],[2,1]] => 1
001101 => [2,2,1,1] => [[3,3,3,2],[2,2,1]] => 1
001110 => [2,3,1] => [[4,4,2],[3,1]] => 1
001111 => [2,4] => [[5,2],[1]] => 1
010000 => [1,1,4] => [[4,1,1],[]] => 1
010001 => [1,1,3,1] => [[3,3,1,1],[2]] => 1
010010 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]] => 1
010011 => [1,1,2,2] => [[3,2,1,1],[1]] => 1
010100 => [1,1,1,1,2] => [[2,1,1,1,1],[]] => 1
010101 => [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]] => 1
010110 => [1,1,1,2,1] => [[2,2,1,1,1],[1]] => 1
010111 => [1,1,1,3] => [[3,1,1,1],[]] => 1
011000 => [1,2,3] => [[4,2,1],[1]] => 1
011001 => [1,2,2,1] => [[3,3,2,1],[2,1]] => 1
011010 => [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]] => 1
011011 => [1,2,1,2] => [[3,2,2,1],[1,1]] => 1
011100 => [1,3,2] => [[4,3,1],[2]] => 1
011101 => [1,3,1,1] => [[3,3,3,1],[2,2]] => 1
011110 => [1,4,1] => [[4,4,1],[3]] => 1
011111 => [1,5] => [[5,1],[]] => 1
100000 => [1,5] => [[5,1],[]] => 1
100001 => [1,4,1] => [[4,4,1],[3]] => 1
100010 => [1,3,1,1] => [[3,3,3,1],[2,2]] => 1
100011 => [1,3,2] => [[4,3,1],[2]] => 1
100100 => [1,2,1,2] => [[3,2,2,1],[1,1]] => 1
100101 => [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]] => 1
100110 => [1,2,2,1] => [[3,3,2,1],[2,1]] => 1
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Description
The Frobenius rank of a skew partition.
This is the minimal number of border strips in a border strip decomposition of the skew partition.
This is the minimal number of border strips in a border strip decomposition of the skew partition.
Map
delta morphism
Description
Applies the delta morphism to a binary word.
The delta morphism of a finite word $w$ is the integer compositions composed of the lengths of consecutive runs of the same letter in $w$.
The delta morphism of a finite word $w$ is the integer compositions composed of the lengths of consecutive runs of the same letter in $w$.
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.
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.
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