Identifier
-
Mp00178:
Binary words
—to composition⟶
Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000621: Integer partitions ⟶ ℤ
Values
001 => [3,1] => [[3,3],[2]] => [2] => 1
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] => 1
0011 => [3,1,1] => [[3,3,3],[2,2]] => [2,2] => 1
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] => 1
1011 => [1,2,1,1] => [[2,2,2,1],[1,1]] => [1,1] => 0
00001 => [5,1] => [[5,5],[4]] => [4] => 1
00010 => [4,2] => [[5,4],[3]] => [3] => 0
00011 => [4,1,1] => [[4,4,4],[3,3]] => [3,3] => 2
00100 => [3,3] => [[5,3],[2]] => [2] => 1
00101 => [3,2,1] => [[4,4,3],[3,2]] => [3,2] => 2
00110 => [3,1,2] => [[4,3,3],[2,2]] => [2,2] => 1
00111 => [3,1,1,1] => [[3,3,3,3],[2,2,2]] => [2,2,2] => 2
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] => 2
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] => 1
10011 => [1,3,1,1] => [[3,3,3,1],[2,2]] => [2,2] => 1
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] => 1
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] => 1
000011 => [5,1,1] => [[5,5,5],[4,4]] => [4,4] => 6
000100 => [4,3] => [[6,4],[3]] => [3] => 0
000101 => [4,2,1] => [[5,5,4],[4,3]] => [4,3] => 6
000110 => [4,1,2] => [[5,4,4],[3,3]] => [3,3] => 2
000111 => [4,1,1,1] => [[4,4,4,4],[3,3,3]] => [3,3,3] => 16
001000 => [3,4] => [[6,3],[2]] => [2] => 1
001001 => [3,3,1] => [[5,5,3],[4,2]] => [4,2] => 4
001010 => [3,2,2] => [[5,4,3],[3,2]] => [3,2] => 2
001011 => [3,2,1,1] => [[4,4,4,3],[3,3,2]] => [3,3,2] => 16
001100 => [3,1,3] => [[5,3,3],[2,2]] => [2,2] => 1
001101 => [3,1,2,1] => [[4,4,3,3],[3,2,2]] => [3,2,2] => 8
001110 => [3,1,1,2] => [[4,3,3,3],[2,2,2]] => [2,2,2] => 2
001111 => [3,1,1,1,1] => [[3,3,3,3,3],[2,2,2,2]] => [2,2,2,2] => 5
010001 => [2,4,1] => [[5,5,2],[4,1]] => [4,1] => 2
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] => 8
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] => 6
010110 => [2,2,1,2] => [[4,3,3,2],[2,2,1]] => [2,2,1] => 2
010111 => [2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]] => [2,2,2,1] => 5
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] => 2
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] => 3
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] => 1
100010 => [1,4,2] => [[5,4,1],[3]] => [3] => 0
100011 => [1,4,1,1] => [[4,4,4,1],[3,3]] => [3,3] => 2
100100 => [1,3,3] => [[5,3,1],[2]] => [2] => 1
100101 => [1,3,2,1] => [[4,4,3,1],[3,2]] => [3,2] => 2
100110 => [1,3,1,2] => [[4,3,3,1],[2,2]] => [2,2] => 1
100111 => [1,3,1,1,1] => [[3,3,3,3,1],[2,2,2]] => [2,2,2] => 2
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] => 2
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] => 1
110011 => [1,1,3,1,1] => [[3,3,3,1,1],[2,2]] => [2,2] => 1
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] => 1
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] => 1
0000011 => [6,1,1] => [[6,6,6],[5,5]] => [5,5] => 18
0000111 => [5,1,1,1] => [[5,5,5,5],[4,4,4]] => [4,4,4] => 182
0001000 => [4,4] => [[7,4],[3]] => [3] => 0
0001001 => [4,3,1] => [[6,6,4],[5,3]] => [5,3] => 12
0001010 => [4,2,2] => [[6,5,4],[4,3]] => [4,3] => 6
0001011 => [4,2,1,1] => [[5,5,5,4],[4,4,3]] => [4,4,3] => 182
0001100 => [4,1,3] => [[6,4,4],[3,3]] => [3,3] => 2
0001101 => [4,1,2,1] => [[5,5,4,4],[4,3,3]] => [4,3,3] => 82
0001110 => [4,1,1,2] => [[5,4,4,4],[3,3,3]] => [3,3,3] => 16
0001111 => [4,1,1,1,1] => [[4,4,4,4,4],[3,3,3,3]] => [3,3,3,3] => 168
0010001 => [3,4,1] => [[6,6,3],[5,2]] => [5,2] => 6
0010010 => [3,3,2] => [[6,5,3],[4,2]] => [4,2] => 4
0010011 => [3,3,1,1] => [[5,5,5,3],[4,4,2]] => [4,4,2] => 100
0010100 => [3,2,3] => [[6,4,3],[3,2]] => [3,2] => 2
0010101 => [3,2,2,1] => [[5,5,4,3],[4,3,2]] => [4,3,2] => 66
0010110 => [3,2,1,2] => [[5,4,4,3],[3,3,2]] => [3,3,2] => 16
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Description
The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even.
To be precise, this is given for a partition $\lambda \vdash n$ by the number of standard tableaux $T$ of shape $\lambda$ such that $\min\big( \operatorname{Des}(T) \cup \{n\} \big)$ is even.
This notion was used in [1, Proposition 2.3], see also [2, Theorem 1.1].
The case of an odd minimum is St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd..
To be precise, this is given for a partition $\lambda \vdash n$ by the number of standard tableaux $T$ of shape $\lambda$ such that $\min\big( \operatorname{Des}(T) \cup \{n\} \big)$ is even.
This notion was used in [1, Proposition 2.3], see also [2, Theorem 1.1].
The case of an odd minimum is St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd..
Map
inner shape
Description
The inner shape of a skew 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.
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
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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