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Your data matches 4 different statistics following compositions of up to 3 maps.
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Matching statistic: St001607
St001607: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 1
[2]
=> 2
[1,1]
=> 2
[3]
=> 4
[2,1]
=> 6
[1,1,1]
=> 8
[4]
=> 11
[3,1]
=> 20
[2,2]
=> 28
[2,1,1]
=> 40
[1,1,1,1]
=> 64
[5]
=> 34
[4,1]
=> 90
[3,2]
=> 148
[3,1,1]
=> 240
[2,2,1]
=> 336
[2,1,1,1]
=> 576
[1,1,1,1,1]
=> 1024
[6]
=> 156
[5,1]
=> 544
[4,2]
=> 1144
[4,1,1]
=> 1992
[3,3]
=> 1408
[3,2,1]
=> 3568
[3,1,1,1]
=> 6528
[2,2,2]
=> 5120
[2,2,1,1]
=> 9344
[2,1,1,1,1]
=> 17408
[1,1,1,1,1,1]
=> 32768
[7]
=> 1044
[6,1]
=> 5096
[5,2]
=> 13128
[5,1,1]
=> 24416
[4,3]
=> 20364
[4,2,1]
=> 55472
[4,1,1,1]
=> 105536
[3,3,1]
=> 71552
[3,2,2]
=> 104160
[3,2,1,1]
=> 199040
[3,1,1,1,1]
=> 382976
[2,2,2,1]
=> 290304
[2,2,1,1,1]
=> 559104
[2,1,1,1,1,1]
=> 1081344
[1,1,1,1,1,1,1]
=> 2097152
Description
The number of coloured graphs such that the multiplicities of colours are given by a partition.
In particular, the value on the partition $(n)$ is the number of unlabelled graphs on $n$ vertices, [[oeis:A000088]], whereas the value on the partition $(1^n)$ is the number of labelled graphs [[oeis:A006125]].
Matching statistic: St001313
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001313: Binary words ⟶ ℤResult quality: 7% ●values known / values provided: 9%●distinct values known / distinct values provided: 7%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001313: Binary words ⟶ ℤResult quality: 7% ●values known / values provided: 9%●distinct values known / distinct values provided: 7%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 110100 => 2 = 1 + 1
[2]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 3 = 2 + 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 11011000 => 3 = 2 + 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1111000100 => ? ∊ {6,8} + 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 11010100 => 5 = 4 + 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1101110000 => ? ∊ {6,8} + 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 111110000100 => ? ∊ {11,20,28,40,64} + 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1110100100 => ? ∊ {11,20,28,40,64} + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1110011000 => ? ∊ {11,20,28,40,64} + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1101101000 => ? ∊ {11,20,28,40,64} + 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 110111100000 => ? ∊ {11,20,28,40,64} + 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> 11111100000100 => ? ∊ {34,90,148,240,336,576,1024} + 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> 111101000100 => ? ∊ {34,90,148,240,336,576,1024} + 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1110010100 => ? ∊ {34,90,148,240,336,576,1024} + 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1101100100 => ? ∊ {34,90,148,240,336,576,1024} + 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1101011000 => ? ∊ {34,90,148,240,336,576,1024} + 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 110111010000 => ? ∊ {34,90,148,240,336,576,1024} + 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> 11011111000000 => ? ∊ {34,90,148,240,336,576,1024} + 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> 1111111000000100 => ? ∊ {156,544,1144,1408,1992,3568,5120,6528,9344,17408,32768} + 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> 11111010000100 => ? ∊ {156,544,1144,1408,1992,3568,5120,6528,9344,17408,32768} + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> 111100100100 => ? ∊ {156,544,1144,1408,1992,3568,5120,6528,9344,17408,32768} + 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 111011000100 => ? ∊ {156,544,1144,1408,1992,3568,5120,6528,9344,17408,32768} + 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 111100011000 => ? ∊ {156,544,1144,1408,1992,3568,5120,6528,9344,17408,32768} + 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1101010100 => ? ∊ {156,544,1144,1408,1992,3568,5120,6528,9344,17408,32768} + 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 110111001000 => ? ∊ {156,544,1144,1408,1992,3568,5120,6528,9344,17408,32768} + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 111001110000 => ? ∊ {156,544,1144,1408,1992,3568,5120,6528,9344,17408,32768} + 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> 110110110000 => ? ∊ {156,544,1144,1408,1992,3568,5120,6528,9344,17408,32768} + 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> 11011110100000 => ? ∊ {156,544,1144,1408,1992,3568,5120,6528,9344,17408,32768} + 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 1101111110000000 => ? ∊ {156,544,1144,1408,1992,3568,5120,6528,9344,17408,32768} + 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> 111111110000000100 => ? ∊ {1044,5096,13128,20364,24416,55472,71552,104160,105536,199040,290304,382976,559104,1081344,2097152} + 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
=> 1111110100000100 => ? ∊ {1044,5096,13128,20364,24416,55472,71552,104160,105536,199040,290304,382976,559104,1081344,2097152} + 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> 11111001000100 => ? ∊ {1044,5096,13128,20364,24416,55472,71552,104160,105536,199040,290304,382976,559104,1081344,2097152} + 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> 11110110000100 => ? ∊ {1044,5096,13128,20364,24416,55472,71552,104160,105536,199040,290304,382976,559104,1081344,2097152} + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 111100010100 => ? ∊ {1044,5096,13128,20364,24416,55472,71552,104160,105536,199040,290304,382976,559104,1081344,2097152} + 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 111010100100 => ? ∊ {1044,5096,13128,20364,24416,55472,71552,104160,105536,199040,290304,382976,559104,1081344,2097152} + 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 110111000100 => ? ∊ {1044,5096,13128,20364,24416,55472,71552,104160,105536,199040,290304,382976,559104,1081344,2097152} + 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> 111010011000 => ? ∊ {1044,5096,13128,20364,24416,55472,71552,104160,105536,199040,290304,382976,559104,1081344,2097152} + 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> 111001101000 => ? ∊ {1044,5096,13128,20364,24416,55472,71552,104160,105536,199040,290304,382976,559104,1081344,2097152} + 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 110110101000 => ? ∊ {1044,5096,13128,20364,24416,55472,71552,104160,105536,199040,290304,382976,559104,1081344,2097152} + 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> 11011110010000 => ? ∊ {1044,5096,13128,20364,24416,55472,71552,104160,105536,199040,290304,382976,559104,1081344,2097152} + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 110101110000 => ? ∊ {1044,5096,13128,20364,24416,55472,71552,104160,105536,199040,290304,382976,559104,1081344,2097152} + 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> 11011101100000 => ? ∊ {1044,5096,13128,20364,24416,55472,71552,104160,105536,199040,290304,382976,559104,1081344,2097152} + 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 1101111101000000 => ? ∊ {1044,5096,13128,20364,24416,55472,71552,104160,105536,199040,290304,382976,559104,1081344,2097152} + 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> 110111111100000000 => ? ∊ {1044,5096,13128,20364,24416,55472,71552,104160,105536,199040,290304,382976,559104,1081344,2097152} + 1
Description
The number of Dyck paths above the lattice path given by a binary word.
One may treat a binary word as a lattice path starting at the origin and treating $1$'s as steps $(1,0)$ and $0$'s as steps $(0,1)$. Given a binary word $w$, this statistic counts the number of lattice paths from the origin to the same endpoint as $w$ that stay weakly above $w$.
See [[St001312]] for this statistic on compositions treated as bounce paths.
Matching statistic: St001583
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St001583: Permutations ⟶ ℤResult quality: 7% ●values known / values provided: 9%●distinct values known / distinct values provided: 7%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St001583: Permutations ⟶ ℤResult quality: 7% ●values known / values provided: 9%●distinct values known / distinct values provided: 7%
Values
[1]
=> [1,0,1,0]
=> [3,1,2] => [1,3,2] => 2 = 1 + 1
[2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => [2,4,1,3] => 3 = 2 + 1
[1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => [3,1,4,2] => 3 = 2 + 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [2,5,1,3,4] => ? ∊ {6,8} + 1
[2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => [1,2,4,3] => 5 = 4 + 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [3,1,4,5,2] => ? ∊ {6,8} + 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => [2,6,1,3,4,5] => ? ∊ {11,20,28,40,64} + 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,3,5,4,2] => ? ∊ {11,20,28,40,64} + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [2,1,4,3,5] => ? ∊ {11,20,28,40,64} + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [1,5,2,4,3] => ? ∊ {11,20,28,40,64} + 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [3,1,4,5,6,2] => ? ∊ {11,20,28,40,64} + 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => [2,7,1,3,4,5,6] => ? ∊ {34,90,148,240,336,576,1024} + 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => [3,4,6,1,5,2] => ? ∊ {34,90,148,240,336,576,1024} + 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [2,3,5,1,4] => ? ∊ {34,90,148,240,336,576,1024} + 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [1,3,2,5,4] => ? ∊ {34,90,148,240,336,576,1024} + 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [4,1,2,5,3] => ? ∊ {34,90,148,240,336,576,1024} + 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [4,6,1,2,5,3] => ? ∊ {34,90,148,240,336,576,1024} + 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => [3,1,4,5,6,7,2] => ? ∊ {34,90,148,240,336,576,1024} + 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => [2,8,1,3,4,5,6,7] => ? ∊ {156,544,1144,1408,1992,3568,5120,6528,9344,17408,32768} + 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => [3,4,7,1,5,6,2] => ? ∊ {156,544,1144,1408,1992,3568,5120,6528,9344,17408,32768} + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => [2,4,6,5,1,3] => ? ∊ {156,544,1144,1408,1992,3568,5120,6528,9344,17408,32768} + 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [1,4,3,6,5,2] => ? ∊ {156,544,1144,1408,1992,3568,5120,6528,9344,17408,32768} + 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [2,1,5,3,4,6] => ? ∊ {156,544,1144,1408,1992,3568,5120,6528,9344,17408,32768} + 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [1,2,3,5,4] => ? ∊ {156,544,1144,1408,1992,3568,5120,6528,9344,17408,32768} + 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [1,6,3,2,5,4] => ? ∊ {156,544,1144,1408,1992,3568,5120,6528,9344,17408,32768} + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [2,1,4,5,3,6] => ? ∊ {156,544,1144,1408,1992,3568,5120,6528,9344,17408,32768} + 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [5,1,6,2,4,3] => ? ∊ {156,544,1144,1408,1992,3568,5120,6528,9344,17408,32768} + 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => [4,7,1,2,5,6,3] => ? ∊ {156,544,1144,1408,1992,3568,5120,6528,9344,17408,32768} + 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,2] => [3,1,4,5,6,7,8,2] => ? ∊ {156,544,1144,1408,1992,3568,5120,6528,9344,17408,32768} + 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,7,9,1,8] => [2,9,1,3,4,5,6,7,8] => ? ∊ {1044,5096,13128,20364,24416,55472,71552,104160,105536,199040,290304,382976,559104,1081344,2097152} + 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [8,3,4,5,6,1,2,7] => [3,4,8,1,5,6,7,2] => ? ∊ {1044,5096,13128,20364,24416,55472,71552,104160,105536,199040,290304,382976,559104,1081344,2097152} + 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,7,4,5,1,3,6] => [2,4,7,1,5,3,6] => ? ∊ {1044,5096,13128,20364,24416,55472,71552,104160,105536,199040,290304,382976,559104,1081344,2097152} + 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,3,4,1,7,2,6] => [3,5,1,7,4,6,2] => ? ∊ {1044,5096,13128,20364,24416,55472,71552,104160,105536,199040,290304,382976,559104,1081344,2097152} + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [2,3,6,1,4,5] => ? ∊ {1044,5096,13128,20364,24416,55472,71552,104160,105536,199040,290304,382976,559104,1081344,2097152} + 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [1,2,4,6,5,3] => ? ∊ {1044,5096,13128,20364,24416,55472,71552,104160,105536,199040,290304,382976,559104,1081344,2097152} + 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [3,4,1,2,6,5] => ? ∊ {1044,5096,13128,20364,24416,55472,71552,104160,105536,199040,290304,382976,559104,1081344,2097152} + 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [5,1,3,6,4,2] => ? ∊ {1044,5096,13128,20364,24416,55472,71552,104160,105536,199040,290304,382976,559104,1081344,2097152} + 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [2,6,3,5,1,4] => ? ∊ {1044,5096,13128,20364,24416,55472,71552,104160,105536,199040,290304,382976,559104,1081344,2097152} + 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [1,2,6,3,5,4] => ? ∊ {1044,5096,13128,20364,24416,55472,71552,104160,105536,199040,290304,382976,559104,1081344,2097152} + 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => [3,7,1,5,2,6,4] => ? ∊ {1044,5096,13128,20364,24416,55472,71552,104160,105536,199040,290304,382976,559104,1081344,2097152} + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [4,1,2,5,6,3] => ? ∊ {1044,5096,13128,20364,24416,55472,71552,104160,105536,199040,290304,382976,559104,1081344,2097152} + 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => [4,1,6,2,5,7,3] => ? ∊ {1044,5096,13128,20364,24416,55472,71552,104160,105536,199040,290304,382976,559104,1081344,2097152} + 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [8,1,4,5,6,7,2,3] => [4,8,1,2,5,6,7,3] => ? ∊ {1044,5096,13128,20364,24416,55472,71552,104160,105536,199040,290304,382976,559104,1081344,2097152} + 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,2] => [3,1,4,5,6,7,8,9,2] => ? ∊ {1044,5096,13128,20364,24416,55472,71552,104160,105536,199040,290304,382976,559104,1081344,2097152} + 1
Description
The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order.
Matching statistic: St001821
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001821: Signed permutations ⟶ ℤResult quality: 7% ●values known / values provided: 9%●distinct values known / distinct values provided: 7%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001821: Signed permutations ⟶ ℤResult quality: 7% ●values known / values provided: 9%●distinct values known / distinct values provided: 7%
Values
[1]
=> [1,0,1,0]
=> [3,1,2] => [3,1,2] => 3 = 1 + 2
[2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => [2,4,1,3] => 4 = 2 + 2
[1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => [3,1,4,2] => 4 = 2 + 2
[3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [2,3,5,1,4] => ? ∊ {6,8} + 2
[2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => [4,1,2,3] => 6 = 4 + 2
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [3,1,4,5,2] => ? ∊ {6,8} + 2
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => [2,3,4,6,1,5] => ? ∊ {11,20,28,40,64} + 2
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [5,3,1,2,4] => ? ∊ {11,20,28,40,64} + 2
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [2,4,1,5,3] => ? ∊ {11,20,28,40,64} + 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [5,1,4,2,3] => ? ∊ {11,20,28,40,64} + 2
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [3,1,4,5,6,2] => ? ∊ {11,20,28,40,64} + 2
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => [2,3,4,5,7,1,6] => ? ∊ {34,90,148,240,336,576,1024} + 2
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => [6,3,4,1,2,5] => ? ∊ {34,90,148,240,336,576,1024} + 2
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [2,5,1,3,4] => ? ∊ {34,90,148,240,336,576,1024} + 2
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [3,1,5,2,4] => ? ∊ {34,90,148,240,336,576,1024} + 2
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [4,1,2,5,3] => ? ∊ {34,90,148,240,336,576,1024} + 2
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [6,1,4,5,2,3] => ? ∊ {34,90,148,240,336,576,1024} + 2
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => [3,1,4,5,6,7,2] => ? ∊ {34,90,148,240,336,576,1024} + 2
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => [2,3,4,5,6,8,1,7] => ? ∊ {156,544,1144,1408,1992,3568,5120,6528,9344,17408,32768} + 2
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => [7,3,4,5,1,2,6] => ? ∊ {156,544,1144,1408,1992,3568,5120,6528,9344,17408,32768} + 2
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => [2,6,4,1,3,5] => ? ∊ {156,544,1144,1408,1992,3568,5120,6528,9344,17408,32768} + 2
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [4,3,1,6,2,5] => ? ∊ {156,544,1144,1408,1992,3568,5120,6528,9344,17408,32768} + 2
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [2,3,5,1,6,4] => ? ∊ {156,544,1144,1408,1992,3568,5120,6528,9344,17408,32768} + 2
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [5,1,2,3,4] => ? ∊ {156,544,1144,1408,1992,3568,5120,6528,9344,17408,32768} + 2
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [3,1,6,5,2,4] => ? ∊ {156,544,1144,1408,1992,3568,5120,6528,9344,17408,32768} + 2
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [2,4,1,5,6,3] => ? ∊ {156,544,1144,1408,1992,3568,5120,6528,9344,17408,32768} + 2
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [5,1,4,2,6,3] => ? ∊ {156,544,1144,1408,1992,3568,5120,6528,9344,17408,32768} + 2
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => [7,1,4,5,6,2,3] => ? ∊ {156,544,1144,1408,1992,3568,5120,6528,9344,17408,32768} + 2
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,2] => [3,1,4,5,6,7,8,2] => ? ∊ {156,544,1144,1408,1992,3568,5120,6528,9344,17408,32768} + 2
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,7,9,1,8] => [2,3,4,5,6,7,9,1,8] => ? ∊ {1044,5096,13128,20364,24416,55472,71552,104160,105536,199040,290304,382976,559104,1081344,2097152} + 2
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [8,3,4,5,6,1,2,7] => [8,3,4,5,6,1,2,7] => ? ∊ {1044,5096,13128,20364,24416,55472,71552,104160,105536,199040,290304,382976,559104,1081344,2097152} + 2
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,7,4,5,1,3,6] => [2,7,4,5,1,3,6] => ? ∊ {1044,5096,13128,20364,24416,55472,71552,104160,105536,199040,290304,382976,559104,1081344,2097152} + 2
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,3,4,1,7,2,6] => [5,3,4,1,7,2,6] => ? ∊ {1044,5096,13128,20364,24416,55472,71552,104160,105536,199040,290304,382976,559104,1081344,2097152} + 2
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [2,3,6,1,4,5] => ? ∊ {1044,5096,13128,20364,24416,55472,71552,104160,105536,199040,290304,382976,559104,1081344,2097152} + 2
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [6,4,1,2,3,5] => ? ∊ {1044,5096,13128,20364,24416,55472,71552,104160,105536,199040,290304,382976,559104,1081344,2097152} + 2
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [3,1,4,6,2,5] => ? ∊ {1044,5096,13128,20364,24416,55472,71552,104160,105536,199040,290304,382976,559104,1081344,2097152} + 2
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [5,3,1,2,6,4] => ? ∊ {1044,5096,13128,20364,24416,55472,71552,104160,105536,199040,290304,382976,559104,1081344,2097152} + 2
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [2,6,1,5,3,4] => ? ∊ {1044,5096,13128,20364,24416,55472,71552,104160,105536,199040,290304,382976,559104,1081344,2097152} + 2
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [6,1,5,2,3,4] => ? ∊ {1044,5096,13128,20364,24416,55472,71552,104160,105536,199040,290304,382976,559104,1081344,2097152} + 2
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => [3,1,7,5,6,2,4] => ? ∊ {1044,5096,13128,20364,24416,55472,71552,104160,105536,199040,290304,382976,559104,1081344,2097152} + 2
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [4,1,2,5,6,3] => ? ∊ {1044,5096,13128,20364,24416,55472,71552,104160,105536,199040,290304,382976,559104,1081344,2097152} + 2
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => [6,1,4,5,2,7,3] => ? ∊ {1044,5096,13128,20364,24416,55472,71552,104160,105536,199040,290304,382976,559104,1081344,2097152} + 2
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [8,1,4,5,6,7,2,3] => [8,1,4,5,6,7,2,3] => ? ∊ {1044,5096,13128,20364,24416,55472,71552,104160,105536,199040,290304,382976,559104,1081344,2097152} + 2
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,2] => [3,1,4,5,6,7,8,9,2] => ? ∊ {1044,5096,13128,20364,24416,55472,71552,104160,105536,199040,290304,382976,559104,1081344,2097152} + 2
Description
The sorting index of a signed permutation.
A signed permutation $\sigma = [\sigma(1),\ldots,\sigma(n)]$ can be sorted $[1,\ldots,n]$ by signed transpositions in the following way:
First move $\pm n$ to its position and swap the sign if needed, then $\pm (n-1), \pm (n-2)$ and so on.
For example for $[2,-4,5,-1,-3]$ we have the swaps
$$
[2,-4,5,-1,-3] \rightarrow [2,-4,-3,-1,5] \rightarrow [2,1,-3,4,5] \rightarrow [2,1,3,4,5] \rightarrow [1,2,3,4,5]
$$
given by the signed transpositions $(3,5), (-2,4), (-3,3), (1,2)$.
If $(i_1,j_1),\ldots,(i_n,j_n)$ is the decomposition of $\sigma$ obtained this way (including trivial transpositions) then the sorting index of $\sigma$ is defined as
$$
\operatorname{sor}_B(\sigma) = \sum_{k=1}^{n-1} j_k - i_k - \chi(i_k < 0),
$$
where $\chi(i_k < 0)$ is 1 if $i_k$ is negative and 0 otherwise.
For $\sigma = [2,-4,5,-1,-3]$ we have
$$
\operatorname{sor}_B(\sigma) = (5-3) + (4-(-2)-1) + (3-(-3)-1) + (2-1) = 13.
$$
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