Identifier
-
Mp00097:
Binary words
—delta morphism⟶
Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
St001596: Skew partitions ⟶ ℤ
Values
0 => [1] => [[1],[]] => 0
1 => [1] => [[1],[]] => 0
00 => [2] => [[2],[]] => 0
01 => [1,1] => [[1,1],[]] => 0
10 => [1,1] => [[1,1],[]] => 0
11 => [2] => [[2],[]] => 0
000 => [3] => [[3],[]] => 0
001 => [2,1] => [[2,2],[1]] => 0
010 => [1,1,1] => [[1,1,1],[]] => 0
011 => [1,2] => [[2,1],[]] => 0
100 => [1,2] => [[2,1],[]] => 0
101 => [1,1,1] => [[1,1,1],[]] => 0
110 => [2,1] => [[2,2],[1]] => 0
111 => [3] => [[3],[]] => 0
0000 => [4] => [[4],[]] => 0
0001 => [3,1] => [[3,3],[2]] => 0
0010 => [2,1,1] => [[2,2,2],[1,1]] => 0
0011 => [2,2] => [[3,2],[1]] => 0
0100 => [1,1,2] => [[2,1,1],[]] => 0
0101 => [1,1,1,1] => [[1,1,1,1],[]] => 0
0110 => [1,2,1] => [[2,2,1],[1]] => 0
0111 => [1,3] => [[3,1],[]] => 0
1000 => [1,3] => [[3,1],[]] => 0
1001 => [1,2,1] => [[2,2,1],[1]] => 0
1010 => [1,1,1,1] => [[1,1,1,1],[]] => 0
1011 => [1,1,2] => [[2,1,1],[]] => 0
1100 => [2,2] => [[3,2],[1]] => 0
1101 => [2,1,1] => [[2,2,2],[1,1]] => 0
1110 => [3,1] => [[3,3],[2]] => 0
1111 => [4] => [[4],[]] => 0
00000 => [5] => [[5],[]] => 0
00001 => [4,1] => [[4,4],[3]] => 0
00010 => [3,1,1] => [[3,3,3],[2,2]] => 0
00011 => [3,2] => [[4,3],[2]] => 0
00100 => [2,1,2] => [[3,2,2],[1,1]] => 0
00101 => [2,1,1,1] => [[2,2,2,2],[1,1,1]] => 0
00110 => [2,2,1] => [[3,3,2],[2,1]] => 0
00111 => [2,3] => [[4,2],[1]] => 0
01000 => [1,1,3] => [[3,1,1],[]] => 0
01001 => [1,1,2,1] => [[2,2,1,1],[1]] => 0
01010 => [1,1,1,1,1] => [[1,1,1,1,1],[]] => 0
01011 => [1,1,1,2] => [[2,1,1,1],[]] => 0
01100 => [1,2,2] => [[3,2,1],[1]] => 0
01101 => [1,2,1,1] => [[2,2,2,1],[1,1]] => 0
01110 => [1,3,1] => [[3,3,1],[2]] => 0
01111 => [1,4] => [[4,1],[]] => 0
10000 => [1,4] => [[4,1],[]] => 0
10001 => [1,3,1] => [[3,3,1],[2]] => 0
10010 => [1,2,1,1] => [[2,2,2,1],[1,1]] => 0
10011 => [1,2,2] => [[3,2,1],[1]] => 0
10100 => [1,1,1,2] => [[2,1,1,1],[]] => 0
10101 => [1,1,1,1,1] => [[1,1,1,1,1],[]] => 0
10110 => [1,1,2,1] => [[2,2,1,1],[1]] => 0
10111 => [1,1,3] => [[3,1,1],[]] => 0
11000 => [2,3] => [[4,2],[1]] => 0
11001 => [2,2,1] => [[3,3,2],[2,1]] => 0
11010 => [2,1,1,1] => [[2,2,2,2],[1,1,1]] => 0
11011 => [2,1,2] => [[3,2,2],[1,1]] => 0
11100 => [3,2] => [[4,3],[2]] => 0
11101 => [3,1,1] => [[3,3,3],[2,2]] => 0
11110 => [4,1] => [[4,4],[3]] => 0
11111 => [5] => [[5],[]] => 0
000000 => [6] => [[6],[]] => 0
000001 => [5,1] => [[5,5],[4]] => 0
000010 => [4,1,1] => [[4,4,4],[3,3]] => 0
000011 => [4,2] => [[5,4],[3]] => 0
000100 => [3,1,2] => [[4,3,3],[2,2]] => 0
000101 => [3,1,1,1] => [[3,3,3,3],[2,2,2]] => 0
000110 => [3,2,1] => [[4,4,3],[3,2]] => 0
000111 => [3,3] => [[5,3],[2]] => 0
001000 => [2,1,3] => [[4,2,2],[1,1]] => 0
001001 => [2,1,2,1] => [[3,3,2,2],[2,1,1]] => 0
001010 => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]] => 0
001011 => [2,1,1,2] => [[3,2,2,2],[1,1,1]] => 0
001100 => [2,2,2] => [[4,3,2],[2,1]] => 0
001101 => [2,2,1,1] => [[3,3,3,2],[2,2,1]] => 0
001110 => [2,3,1] => [[4,4,2],[3,1]] => 0
001111 => [2,4] => [[5,2],[1]] => 0
010000 => [1,1,4] => [[4,1,1],[]] => 0
010001 => [1,1,3,1] => [[3,3,1,1],[2]] => 0
010010 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]] => 0
010011 => [1,1,2,2] => [[3,2,1,1],[1]] => 0
010100 => [1,1,1,1,2] => [[2,1,1,1,1],[]] => 0
010101 => [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]] => 0
010110 => [1,1,1,2,1] => [[2,2,1,1,1],[1]] => 0
010111 => [1,1,1,3] => [[3,1,1,1],[]] => 0
011000 => [1,2,3] => [[4,2,1],[1]] => 0
011001 => [1,2,2,1] => [[3,3,2,1],[2,1]] => 0
011010 => [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]] => 0
011011 => [1,2,1,2] => [[3,2,2,1],[1,1]] => 0
011100 => [1,3,2] => [[4,3,1],[2]] => 0
011101 => [1,3,1,1] => [[3,3,3,1],[2,2]] => 0
011110 => [1,4,1] => [[4,4,1],[3]] => 0
011111 => [1,5] => [[5,1],[]] => 0
100000 => [1,5] => [[5,1],[]] => 0
100001 => [1,4,1] => [[4,4,1],[3]] => 0
100010 => [1,3,1,1] => [[3,3,3,1],[2,2]] => 0
100011 => [1,3,2] => [[4,3,1],[2]] => 0
100100 => [1,2,1,2] => [[3,2,2,1],[1,1]] => 0
100101 => [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]] => 0
100110 => [1,2,2,1] => [[3,3,2,1],[2,1]] => 0
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Description
The number of two-by-two squares inside a skew partition.
This is, the number of cells (i,j) in a skew partition for which the box (i+1,j+1) is also a cell inside the skew partition.
This is, the number of cells (i,j) in a skew partition for which the box (i+1,j+1) is also a cell inside the skew partition.
Map
to ribbon
Description
The ribbon shape corresponding to an integer composition.
For an integer composition (a1,…,an), this is the ribbon shape whose ith row from the bottom has ai cells.
For an integer composition (a1,…,an), this is the ribbon shape whose ith row from the bottom has ai cells.
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.
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