Identifier
Values
[1] => 10 => 00 => [3] => 1
[2] => 100 => 000 => [4] => 1
[1,1] => 110 => 010 => [2,2] => 3
[3] => 1000 => 0000 => [5] => 1
[2,1] => 1010 => 0010 => [3,2] => 6
[1,1,1] => 1110 => 0110 => [2,1,2] => 4
[4] => 10000 => 00000 => [6] => 1
[3,1] => 10010 => 00010 => [4,2] => 10
[2,2] => 1100 => 0100 => [2,3] => 4
[2,1,1] => 10110 => 00110 => [3,1,2] => 10
[1,1,1,1] => 11110 => 01110 => [2,1,1,2] => 5
[5] => 100000 => 000000 => [7] => 1
[4,1] => 100010 => 000010 => [5,2] => 15
[3,2] => 10100 => 00100 => [3,3] => 10
[3,1,1] => 100110 => 000110 => [4,1,2] => 20
[2,2,1] => 11010 => 01010 => [2,2,2] => 15
[2,1,1,1] => 101110 => 001110 => [3,1,1,2] => 15
[1,1,1,1,1] => 111110 => 011110 => [2,1,1,1,2] => 6
[6] => 1000000 => 0000000 => [8] => 1
[5,1] => 1000010 => 0000010 => [6,2] => 21
[4,2] => 100100 => 000100 => [4,3] => 20
[3,3] => 11000 => 01000 => [2,4] => 5
[3,2,1] => 101010 => 001010 => [3,2,2] => 45
[2,2,2] => 11100 => 01100 => [2,1,3] => 5
[2,2,1,1] => 110110 => 010110 => [2,2,1,2] => 24
[1,1,1,1,1,1] => 1111110 => 0111110 => [2,1,1,1,1,2] => 7
[7] => 10000000 => 00000000 => [9] => 1
[6,1] => 10000010 => 00000010 => [7,2] => 28
[5,2] => 1000100 => 0000100 => [5,3] => 35
[4,3] => 101000 => 001000 => [3,4] => 15
[4,2,1] => 1001010 => 0001010 => [4,2,2] => 105
[3,3,1] => 110010 => 010010 => [2,3,2] => 36
[3,2,2] => 101100 => 001100 => [3,1,3] => 15
[3,2,1,1] => 1010110 => 0010110 => [3,2,1,2] => 84
[2,2,2,1] => 111010 => 011010 => [2,1,2,2] => 18
[2,2,1,1,1] => 1101110 => 0101110 => [2,2,1,1,2] => 35
[6,2] => 10000100 => 00000100 => [6,3] => 56
[5,3] => 1001000 => 0001000 => [4,4] => 35
[5,2,1] => 10001010 => 00001010 => [5,2,2] => 210
[4,4] => 110000 => 010000 => [2,5] => 6
[4,3,1] => 1010010 => 0010010 => [3,3,2] => 126
[3,3,2] => 110100 => 010100 => [2,2,3] => 24
[3,3,1,1] => 1100110 => 0100110 => [2,3,1,2] => 70
[3,2,2,1] => 1011010 => 0011010 => [3,1,2,2] => 63
[2,2,2,2] => 111100 => 011100 => [2,1,1,3] => 6
[2,2,2,1,1] => 1110110 => 0110110 => [2,1,2,1,2] => 28
[6,3] => 10001000 => 00001000 => [5,4] => 70
[5,4] => 1010000 => 0010000 => [3,5] => 21
[5,3,1] => 10010010 => 00010010 => [4,3,2] => 336
[4,4,1] => 1100010 => 0100010 => [2,4,2] => 70
[4,3,2] => 1010100 => 0010100 => [3,2,3] => 84
[3,3,3] => 111000 => 011000 => [2,1,4] => 6
[3,3,2,1] => 1101010 => 0101010 => [2,2,2,2] => 105
[3,2,2,2] => 1011100 => 0011100 => [3,1,1,3] => 21
[2,2,2,2,1] => 1111010 => 0111010 => [2,1,1,2,2] => 21
[6,4] => 10010000 => 00010000 => [4,5] => 56
[5,5] => 1100000 => 0100000 => [2,6] => 7
[5,4,1] => 10100010 => 00100010 => [3,4,2] => 280
[4,4,2] => 1100100 => 0100100 => [2,3,3] => 70
[4,3,2,1] => 10101010 => 00101010 => [3,2,2,2] => 420
[3,3,3,1] => 1110010 => 0110010 => [2,1,3,2] => 42
[3,3,2,2] => 1101100 => 0101100 => [2,2,1,3] => 35
[6,5] => 10100000 => 00100000 => [3,6] => 28
[5,4,2] => 10100100 => 00100100 => [3,3,3] => 280
[4,4,3] => 1101000 => 0101000 => [2,2,4] => 35
[4,4,2,1] => 11001010 => 01001010 => [2,3,2,2] => 360
[3,3,3,2] => 1110100 => 0110100 => [2,1,2,3] => 28
[6,6] => 11000000 => 01000000 => [2,7] => 8
[5,5,2] => 11000100 => 01000100 => [2,4,3] => 160
[4,4,3,1] => 11010010 => 01010010 => [2,2,3,2] => 288
[5,5,3] => 11001000 => 01001000 => [2,3,4] => 120
[4,4,3,2] => 11010100 => 01010100 => [2,2,2,3] => 192
[5,5,4] => 11010000 => 01010000 => [2,2,5] => 48
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Description
The number of standard immaculate tableaux of a given shape.
See Proposition 3.13 of [2] for a hook-length counting formula of these tableaux.
Map
twist
Description
Return the binary word with first letter inverted.
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
to binary word
Description
Return the partition as binary word, by traversing its shape from the first row to the last row, down steps as 1 and left steps as 0.