Identifier
-
Mp00097:
Binary words
—delta morphism⟶
Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000704: Integer partitions ⟶ ℤ
Values
0001 => [3,1] => [[3,3],[2]] => [2] => 1
0010 => [2,1,1] => [[2,2,2],[1,1]] => [1,1] => 1
1101 => [2,1,1] => [[2,2,2],[1,1]] => [1,1] => 1
1110 => [3,1] => [[3,3],[2]] => [2] => 1
00001 => [4,1] => [[4,4],[3]] => [3] => 1
00010 => [3,1,1] => [[3,3,3],[2,2]] => [2,2] => 1
00011 => [3,2] => [[4,3],[2]] => [2] => 1
00100 => [2,1,2] => [[3,2,2],[1,1]] => [1,1] => 1
00101 => [2,1,1,1] => [[2,2,2,2],[1,1,1]] => [1,1,1] => 1
00110 => [2,2,1] => [[3,3,2],[2,1]] => [2,1] => 2
01101 => [1,2,1,1] => [[2,2,2,1],[1,1]] => [1,1] => 1
01110 => [1,3,1] => [[3,3,1],[2]] => [2] => 1
10001 => [1,3,1] => [[3,3,1],[2]] => [2] => 1
10010 => [1,2,1,1] => [[2,2,2,1],[1,1]] => [1,1] => 1
11001 => [2,2,1] => [[3,3,2],[2,1]] => [2,1] => 2
11010 => [2,1,1,1] => [[2,2,2,2],[1,1,1]] => [1,1,1] => 1
11011 => [2,1,2] => [[3,2,2],[1,1]] => [1,1] => 1
11100 => [3,2] => [[4,3],[2]] => [2] => 1
11101 => [3,1,1] => [[3,3,3],[2,2]] => [2,2] => 1
11110 => [4,1] => [[4,4],[3]] => [3] => 1
000001 => [5,1] => [[5,5],[4]] => [4] => 1
000010 => [4,1,1] => [[4,4,4],[3,3]] => [3,3] => 1
000011 => [4,2] => [[5,4],[3]] => [3] => 1
000100 => [3,1,2] => [[4,3,3],[2,2]] => [2,2] => 1
000101 => [3,1,1,1] => [[3,3,3,3],[2,2,2]] => [2,2,2] => 1
000110 => [3,2,1] => [[4,4,3],[3,2]] => [3,2] => 2
000111 => [3,3] => [[5,3],[2]] => [2] => 1
001000 => [2,1,3] => [[4,2,2],[1,1]] => [1,1] => 1
001001 => [2,1,2,1] => [[3,3,2,2],[2,1,1]] => [2,1,1] => 3
001010 => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]] => [1,1,1,1] => 1
001011 => [2,1,1,2] => [[3,2,2,2],[1,1,1]] => [1,1,1] => 1
001100 => [2,2,2] => [[4,3,2],[2,1]] => [2,1] => 2
001101 => [2,2,1,1] => [[3,3,3,2],[2,2,1]] => [2,2,1] => 3
001110 => [2,3,1] => [[4,4,2],[3,1]] => [3,1] => 3
010001 => [1,1,3,1] => [[3,3,1,1],[2]] => [2] => 1
010010 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]] => [1,1] => 1
011001 => [1,2,2,1] => [[3,3,2,1],[2,1]] => [2,1] => 2
011010 => [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]] => [1,1,1] => 1
011011 => [1,2,1,2] => [[3,2,2,1],[1,1]] => [1,1] => 1
011100 => [1,3,2] => [[4,3,1],[2]] => [2] => 1
011101 => [1,3,1,1] => [[3,3,3,1],[2,2]] => [2,2] => 1
011110 => [1,4,1] => [[4,4,1],[3]] => [3] => 1
100001 => [1,4,1] => [[4,4,1],[3]] => [3] => 1
100010 => [1,3,1,1] => [[3,3,3,1],[2,2]] => [2,2] => 1
100011 => [1,3,2] => [[4,3,1],[2]] => [2] => 1
100100 => [1,2,1,2] => [[3,2,2,1],[1,1]] => [1,1] => 1
100101 => [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]] => [1,1,1] => 1
100110 => [1,2,2,1] => [[3,3,2,1],[2,1]] => [2,1] => 2
101101 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]] => [1,1] => 1
101110 => [1,1,3,1] => [[3,3,1,1],[2]] => [2] => 1
110001 => [2,3,1] => [[4,4,2],[3,1]] => [3,1] => 3
110010 => [2,2,1,1] => [[3,3,3,2],[2,2,1]] => [2,2,1] => 3
110011 => [2,2,2] => [[4,3,2],[2,1]] => [2,1] => 2
110100 => [2,1,1,2] => [[3,2,2,2],[1,1,1]] => [1,1,1] => 1
110101 => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]] => [1,1,1,1] => 1
110110 => [2,1,2,1] => [[3,3,2,2],[2,1,1]] => [2,1,1] => 3
110111 => [2,1,3] => [[4,2,2],[1,1]] => [1,1] => 1
111000 => [3,3] => [[5,3],[2]] => [2] => 1
111001 => [3,2,1] => [[4,4,3],[3,2]] => [3,2] => 2
111010 => [3,1,1,1] => [[3,3,3,3],[2,2,2]] => [2,2,2] => 1
111011 => [3,1,2] => [[4,3,3],[2,2]] => [2,2] => 1
111100 => [4,2] => [[5,4],[3]] => [3] => 1
111101 => [4,1,1] => [[4,4,4],[3,3]] => [3,3] => 1
111110 => [5,1] => [[5,5],[4]] => [4] => 1
0000001 => [6,1] => [[6,6],[5]] => [5] => 1
0000010 => [5,1,1] => [[5,5,5],[4,4]] => [4,4] => 1
0000011 => [5,2] => [[6,5],[4]] => [4] => 1
0000100 => [4,1,2] => [[5,4,4],[3,3]] => [3,3] => 1
0000101 => [4,1,1,1] => [[4,4,4,4],[3,3,3]] => [3,3,3] => 1
0000110 => [4,2,1] => [[5,5,4],[4,3]] => [4,3] => 2
0000111 => [4,3] => [[6,4],[3]] => [3] => 1
0001000 => [3,1,3] => [[5,3,3],[2,2]] => [2,2] => 1
0001001 => [3,1,2,1] => [[4,4,3,3],[3,2,2]] => [3,2,2] => 3
0001010 => [3,1,1,1,1] => [[3,3,3,3,3],[2,2,2,2]] => [2,2,2,2] => 1
0001011 => [3,1,1,2] => [[4,3,3,3],[2,2,2]] => [2,2,2] => 1
0001100 => [3,2,2] => [[5,4,3],[3,2]] => [3,2] => 2
0001101 => [3,2,1,1] => [[4,4,4,3],[3,3,2]] => [3,3,2] => 3
0001110 => [3,3,1] => [[5,5,3],[4,2]] => [4,2] => 3
0001111 => [3,4] => [[6,3],[2]] => [2] => 1
0010000 => [2,1,4] => [[5,2,2],[1,1]] => [1,1] => 1
0010001 => [2,1,3,1] => [[4,4,2,2],[3,1,1]] => [3,1,1] => 6
0010010 => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]] => [2,2,1,1] => 6
0010011 => [2,1,2,2] => [[4,3,2,2],[2,1,1]] => [2,1,1] => 3
0010100 => [2,1,1,1,2] => [[3,2,2,2,2],[1,1,1,1]] => [1,1,1,1] => 1
0010101 => [2,1,1,1,1,1] => [[2,2,2,2,2,2],[1,1,1,1,1]] => [1,1,1,1,1] => 1
0010110 => [2,1,1,2,1] => [[3,3,2,2,2],[2,1,1,1]] => [2,1,1,1] => 4
0010111 => [2,1,1,3] => [[4,2,2,2],[1,1,1]] => [1,1,1] => 1
0011000 => [2,2,3] => [[5,3,2],[2,1]] => [2,1] => 2
0011001 => [2,2,2,1] => [[4,4,3,2],[3,2,1]] => [3,2,1] => 8
0011010 => [2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]] => [2,2,2,1] => 4
0011011 => [2,2,1,2] => [[4,3,3,2],[2,2,1]] => [2,2,1] => 3
0011100 => [2,3,2] => [[5,4,2],[3,1]] => [3,1] => 3
0011101 => [2,3,1,1] => [[4,4,4,2],[3,3,1]] => [3,3,1] => 6
0011110 => [2,4,1] => [[5,5,2],[4,1]] => [4,1] => 4
0100001 => [1,1,4,1] => [[4,4,1,1],[3]] => [3] => 1
0100010 => [1,1,3,1,1] => [[3,3,3,1,1],[2,2]] => [2,2] => 1
0100011 => [1,1,3,2] => [[4,3,1,1],[2]] => [2] => 1
0100100 => [1,1,2,1,2] => [[3,2,2,1,1],[1,1]] => [1,1] => 1
0100101 => [1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]] => [1,1,1] => 1
0100110 => [1,1,2,2,1] => [[3,3,2,1,1],[2,1]] => [2,1] => 2
0101101 => [1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]] => [1,1] => 1
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Description
The number of semistandard tableaux on a given integer partition with minimal maximal entry.
This is, for an integer partition $\lambda = (\lambda_1 > \cdots > \lambda_k > 0)$, the number of semistandard tableaux of shape $\lambda$ with maximal entry $k$.
Equivalently, this is the evaluation $s_\lambda(1,\ldots,1)$ of the Schur function $s_\lambda$ in $k$ variables, or, explicitly,
$$ \prod_{(i,j) \in L} \frac{k + j - i}{ \operatorname{hook}(i,j) }$$
where the product is over all cells $(i,j) \in L$ and $\operatorname{hook}(i,j)$ is the hook length of a cell.
See [Theorem 6.3, 1] for details.
This is, for an integer partition $\lambda = (\lambda_1 > \cdots > \lambda_k > 0)$, the number of semistandard tableaux of shape $\lambda$ with maximal entry $k$.
Equivalently, this is the evaluation $s_\lambda(1,\ldots,1)$ of the Schur function $s_\lambda$ in $k$ variables, or, explicitly,
$$ \prod_{(i,j) \in L} \frac{k + j - i}{ \operatorname{hook}(i,j) }$$
where the product is over all cells $(i,j) \in L$ and $\operatorname{hook}(i,j)$ is the hook length of a cell.
See [Theorem 6.3, 1] for details.
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.
Map
inner shape
Description
The inner shape of a 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 $w$.
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