Processing math: 14%

Your data matches 12 different statistics following compositions of up to 3 maps.
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Matching statistic: St000277
(load all 4 compositions to match this statistic)
Mapping
Mp00071: Permutations —descent composition⟶ Integer compositions
St000277: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1
[1,2] => [2] => 1
[2,1] => [1,1] => 1
[1,2,3] => [3] => 1
[1,3,2] => [2,1] => 2
[2,1,3] => [1,2] => 2
[2,3,1] => [2,1] => 2
[3,1,2] => [1,2] => 2
[3,2,1] => [1,1,1] => 1
[1,2,3,4] => [4] => 1
[1,2,4,3] => [3,1] => 3
[1,3,2,4] => [2,2] => 5
[1,3,4,2] => [3,1] => 3
[1,4,2,3] => [2,2] => 5
[1,4,3,2] => [2,1,1] => 3
[2,1,3,4] => [1,3] => 3
[2,1,4,3] => [1,2,1] => 5
[2,3,1,4] => [2,2] => 5
[2,3,4,1] => [3,1] => 3
[2,4,1,3] => [2,2] => 5
[2,4,3,1] => [2,1,1] => 3
[3,1,2,4] => [1,3] => 3
[3,1,4,2] => [1,2,1] => 5
[3,2,1,4] => [1,1,2] => 3
[3,2,4,1] => [1,2,1] => 5
[3,4,1,2] => [2,2] => 5
[3,4,2,1] => [2,1,1] => 3
[4,1,2,3] => [1,3] => 3
[4,1,3,2] => [1,2,1] => 5
[4,2,1,3] => [1,1,2] => 3
[4,2,3,1] => [1,2,1] => 5
[4,3,1,2] => [1,1,2] => 3
[4,3,2,1] => [1,1,1,1] => 1
[1,2,3,4,5] => [5] => 1
[1,2,3,5,4] => [4,1] => 4
[1,2,4,3,5] => [3,2] => 9
[1,2,4,5,3] => [4,1] => 4
[1,2,5,3,4] => [3,2] => 9
[1,2,5,4,3] => [3,1,1] => 6
[1,3,2,4,5] => [2,3] => 9
[1,3,2,5,4] => [2,2,1] => 16
[1,3,4,2,5] => [3,2] => 9
[1,3,4,5,2] => [4,1] => 4
[1,3,5,2,4] => [3,2] => 9
[1,3,5,4,2] => [3,1,1] => 6
[1,4,2,3,5] => [2,3] => 9
[1,4,2,5,3] => [2,2,1] => 16
[1,4,3,2,5] => [2,1,2] => 11
[1,4,3,5,2] => [2,2,1] => 16
[1,4,5,2,3] => [3,2] => 9
Description
The number of ribbon shaped standard tableaux. A ribbon is a connected skew shape which does not contain a 2×2 square. The set of ribbon shapes are therefore in bijection with integer compositons, the parts of the composition specify the row lengths. This statistic records the number of standard tableaux of the given shape. This is also the size of the preimage of the map 'descent composition' [[Mp00071]] from permutations to integer compositions: reading a tableau from bottom to top we obtain a permutation whose descent set is as prescribed. For a composition c=c1,…,ck of n, the number of ribbon shaped standard tableaux equals \sum_d (-1)^{k-\ell} \binom{n}{d_1, d_2, \dots, d_\ell}, where the sum is over all coarsenings of c obtained by replacing consecutive summands by their sum, see [sec 14.4, 1]
Matching statistic: St000529
(load all 4 compositions to match this statistic)
Mapping
Mp00109: Permutations —descent word⟶ Binary words
St000529: Binary words ⟶ ℤResult quality: 27% ●values known / values provided: 70%●distinct values known / distinct values provided: 27%
Values
[1] => => ? = 1
[1,2] => 0 => 1
[2,1] => 1 => 1
[1,2,3] => 00 => 1
[1,3,2] => 01 => 2
[2,1,3] => 10 => 2
[2,3,1] => 01 => 2
[3,1,2] => 10 => 2
[3,2,1] => 11 => 1
[1,2,3,4] => 000 => 1
[1,2,4,3] => 001 => 3
[1,3,2,4] => 010 => 5
[1,3,4,2] => 001 => 3
[1,4,2,3] => 010 => 5
[1,4,3,2] => 011 => 3
[2,1,3,4] => 100 => 3
[2,1,4,3] => 101 => 5
[2,3,1,4] => 010 => 5
[2,3,4,1] => 001 => 3
[2,4,1,3] => 010 => 5
[2,4,3,1] => 011 => 3
[3,1,2,4] => 100 => 3
[3,1,4,2] => 101 => 5
[3,2,1,4] => 110 => 3
[3,2,4,1] => 101 => 5
[3,4,1,2] => 010 => 5
[3,4,2,1] => 011 => 3
[4,1,2,3] => 100 => 3
[4,1,3,2] => 101 => 5
[4,2,1,3] => 110 => 3
[4,2,3,1] => 101 => 5
[4,3,1,2] => 110 => 3
[4,3,2,1] => 111 => 1
[1,2,3,4,5] => 0000 => 1
[1,2,3,5,4] => 0001 => 4
[1,2,4,3,5] => 0010 => 9
[1,2,4,5,3] => 0001 => 4
[1,2,5,3,4] => 0010 => 9
[1,2,5,4,3] => 0011 => 6
[1,3,2,4,5] => 0100 => 9
[1,3,2,5,4] => 0101 => 16
[1,3,4,2,5] => 0010 => 9
[1,3,4,5,2] => 0001 => 4
[1,3,5,2,4] => 0010 => 9
[1,3,5,4,2] => 0011 => 6
[1,4,2,3,5] => 0100 => 9
[1,4,2,5,3] => 0101 => 16
[1,4,3,2,5] => 0110 => 11
[1,4,3,5,2] => 0101 => 16
[1,4,5,2,3] => 0010 => 9
[1,4,5,3,2] => 0011 => 6
[6,7,8,4,5,3,2,1] => ? => ? = 189
[7,8,5,4,6,3,2,1] => ? => ? = 203
[7,8,4,5,6,3,2,1] => ? => ? = 245
[6,7,8,5,3,4,2,1] => ? => ? = 315
[8,6,5,7,3,4,2,1] => ? => ? = 449
[7,8,4,3,5,6,2,1] => ? => ? = 413
[8,5,6,4,3,7,2,1] => ? => ? = 477
[8,5,4,6,3,7,2,1] => ? => ? = 449
[8,5,6,3,4,7,2,1] => ? => ? = 573
[8,6,3,4,5,7,2,1] => ? => ? = 181
[7,6,4,5,3,8,2,1] => ? => ? = 449
[5,6,4,7,3,8,2,1] => ? => ? = 791
[6,7,5,3,4,8,2,1] => ? => ? = 413
[6,5,7,3,4,8,2,1] => ? => ? = 573
[5,6,7,3,4,8,2,1] => ? => ? = 315
[5,6,4,3,7,8,2,1] => ? => ? = 413
[5,4,6,3,7,8,2,1] => ? => ? = 573
[6,4,3,5,7,8,2,1] => ? => ? = 181
[4,5,3,6,7,8,2,1] => ? => ? = 259
[6,5,7,8,4,2,3,1] => ? => ? = 643
[7,8,6,4,5,2,3,1] => ? => ? = 917
[6,7,8,4,5,2,3,1] => ? => ? = 791
[6,7,5,4,8,2,3,1] => ? => ? = 917
[5,6,7,4,8,2,3,1] => ? => ? = 791
[7,6,4,5,8,2,3,1] => ? => ? = 573
[7,5,4,6,8,2,3,1] => ? => ? = 573
[7,4,5,6,8,2,3,1] => ? => ? = 407
[6,5,4,7,8,2,3,1] => ? => ? = 573
[5,6,4,7,8,2,3,1] => ? => ? = 875
[6,4,5,7,8,2,3,1] => ? => ? = 407
[5,4,6,7,8,2,3,1] => ? => ? = 407
[7,8,5,6,3,2,4,1] => ? => ? = 875
[7,5,6,8,3,2,4,1] => ? => ? = 643
[6,5,7,8,3,2,4,1] => ? => ? = 643
[7,8,6,4,2,3,5,1] => ? => ? = 315
[7,8,4,3,5,2,6,1] => ? => ? = 917
[7,8,3,4,5,2,6,1] => ? => ? = 875
[8,7,5,4,2,3,6,1] => ? => ? = 85
[7,8,5,4,2,3,6,1] => ? => ? = 315
[7,8,4,5,2,3,6,1] => ? => ? = 917
[7,8,5,2,3,4,6,1] => ? => ? = 315
[8,7,3,4,2,5,6,1] => ? => ? = 531
[8,5,3,4,6,2,7,1] => ? => ? = 573
[8,5,4,6,2,3,7,1] => ? => ? = 531
[8,4,3,5,2,6,7,1] => ? => ? = 531
[7,5,4,6,3,2,8,1] => ? => ? = 477
[5,6,4,7,3,2,8,1] => ? => ? = 875
[5,4,6,7,3,2,8,1] => ? => ? = 643
[6,7,4,3,5,2,8,1] => ? => ? = 917
Description
The number of permutations whose descent word is the given binary word. This is the sizes of the preimages of the map [[Mp00109]].
Matching statistic: St000100
Mapping
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St000100: Posets ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 17%
Values
[1] => [1] => [[1],[]]
 => ([],1)
 => ? = 1
[1,2] => [2] => [[2],[]]
 => ([(0,1)],2)
 => 1
[2,1] => [1,1] => [[1,1],[]]
 => ([(0,1)],2)
 => 1
[1,2,3] => [3] => [[3],[]]
 => ([(0,2),(2,1)],3)
 => 1
[1,3,2] => [2,1] => [[2,2],[1]]
 => ([(0,2),(1,2)],3)
 => 2
[2,1,3] => [1,2] => [[2,1],[]]
 => ([(0,1),(0,2)],3)
 => 2
[2,3,1] => [2,1] => [[2,2],[1]]
 => ([(0,2),(1,2)],3)
 => 2
[3,1,2] => [1,2] => [[2,1],[]]
 => ([(0,1),(0,2)],3)
 => 2
[3,2,1] => [1,1,1] => [[1,1,1],[]]
 => ([(0,2),(2,1)],3)
 => 1
[1,2,3,4] => [4] => [[4],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[1,2,4,3] => [3,1] => [[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3
[1,3,2,4] => [2,2] => [[3,2],[1]]
 => ([(0,3),(1,2),(1,3)],4)
 => 5
[1,3,4,2] => [3,1] => [[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3
[1,4,2,3] => [2,2] => [[3,2],[1]]
 => ([(0,3),(1,2),(1,3)],4)
 => 5
[1,4,3,2] => [2,1,1] => [[2,2,2],[1,1]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3
[2,1,3,4] => [1,3] => [[3,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => 3
[2,1,4,3] => [1,2,1] => [[2,2,1],[1]]
 => ([(0,3),(1,2),(1,3)],4)
 => 5
[2,3,1,4] => [2,2] => [[3,2],[1]]
 => ([(0,3),(1,2),(1,3)],4)
 => 5
[2,3,4,1] => [3,1] => [[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3
[2,4,1,3] => [2,2] => [[3,2],[1]]
 => ([(0,3),(1,2),(1,3)],4)
 => 5
[2,4,3,1] => [2,1,1] => [[2,2,2],[1,1]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3
[3,1,2,4] => [1,3] => [[3,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => 3
[3,1,4,2] => [1,2,1] => [[2,2,1],[1]]
 => ([(0,3),(1,2),(1,3)],4)
 => 5
[3,2,1,4] => [1,1,2] => [[2,1,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => 3
[3,2,4,1] => [1,2,1] => [[2,2,1],[1]]
 => ([(0,3),(1,2),(1,3)],4)
 => 5
[3,4,1,2] => [2,2] => [[3,2],[1]]
 => ([(0,3),(1,2),(1,3)],4)
 => 5
[3,4,2,1] => [2,1,1] => [[2,2,2],[1,1]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3
[4,1,2,3] => [1,3] => [[3,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => 3
[4,1,3,2] => [1,2,1] => [[2,2,1],[1]]
 => ([(0,3),(1,2),(1,3)],4)
 => 5
[4,2,1,3] => [1,1,2] => [[2,1,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => 3
[4,2,3,1] => [1,2,1] => [[2,2,1],[1]]
 => ([(0,3),(1,2),(1,3)],4)
 => 5
[4,3,1,2] => [1,1,2] => [[2,1,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => 3
[4,3,2,1] => [1,1,1,1] => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[1,2,3,4,5] => [5] => [[5],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,2,3,5,4] => [4,1] => [[4,4],[3]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 4
[1,2,4,3,5] => [3,2] => [[4,3],[2]]
 => ([(0,3),(1,2),(1,4),(3,4)],5)
 => 9
[1,2,4,5,3] => [4,1] => [[4,4],[3]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 4
[1,2,5,3,4] => [3,2] => [[4,3],[2]]
 => ([(0,3),(1,2),(1,4),(3,4)],5)
 => 9
[1,2,5,4,3] => [3,1,1] => [[3,3,3],[2,2]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 6
[1,3,2,4,5] => [2,3] => [[4,2],[1]]
 => ([(0,4),(1,2),(1,4),(2,3)],5)
 => 9
[1,3,2,5,4] => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 16
[1,3,4,2,5] => [3,2] => [[4,3],[2]]
 => ([(0,3),(1,2),(1,4),(3,4)],5)
 => 9
[1,3,4,5,2] => [4,1] => [[4,4],[3]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 4
[1,3,5,2,4] => [3,2] => [[4,3],[2]]
 => ([(0,3),(1,2),(1,4),(3,4)],5)
 => 9
[1,3,5,4,2] => [3,1,1] => [[3,3,3],[2,2]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 6
[1,4,2,3,5] => [2,3] => [[4,2],[1]]
 => ([(0,4),(1,2),(1,4),(2,3)],5)
 => 9
[1,4,2,5,3] => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 16
[1,4,3,2,5] => [2,1,2] => [[3,2,2],[1,1]]
 => ([(0,4),(1,2),(1,3),(3,4)],5)
 => 11
[1,4,3,5,2] => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 16
[1,4,5,2,3] => [3,2] => [[4,3],[2]]
 => ([(0,3),(1,2),(1,4),(3,4)],5)
 => 9
[1,4,5,3,2] => [3,1,1] => [[3,3,3],[2,2]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 6
[7,8,6,5,4,3,2,1] => [2,1,1,1,1,1,1] => [[2,2,2,2,2,2,2],[1,1,1,1,1,1]]
 => ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
 => ? = 7
[8,6,7,5,4,3,2,1] => [1,2,1,1,1,1,1] => [[2,2,2,2,2,2,1],[1,1,1,1,1]]
 => ([(0,6),(1,3),(1,7),(2,7),(4,5),(5,2),(6,4)],8)
 => ? = 27
[7,6,8,5,4,3,2,1] => [1,2,1,1,1,1,1] => [[2,2,2,2,2,2,1],[1,1,1,1,1]]
 => ([(0,6),(1,3),(1,7),(2,7),(4,5),(5,2),(6,4)],8)
 => ? = 27
[6,7,8,5,4,3,2,1] => [3,1,1,1,1,1] => [[3,3,3,3,3,3],[2,2,2,2,2]]
 => ([(0,6),(1,3),(2,7),(3,7),(4,5),(5,2),(6,4)],8)
 => ? = 21
[8,7,5,6,4,3,2,1] => [1,1,2,1,1,1,1] => [[2,2,2,2,2,1,1],[1,1,1,1]]
 => ([(0,5),(1,6),(1,7),(3,7),(4,3),(5,4),(6,2)],8)
 => ? = 55
[7,8,5,6,4,3,2,1] => [2,2,1,1,1,1] => [[3,3,3,3,3,2],[2,2,2,2,1]]
 => ([(0,3),(1,6),(2,6),(2,7),(3,5),(4,7),(5,4)],8)
 => ? = 105
[8,6,5,7,4,3,2,1] => [1,1,2,1,1,1,1] => [[2,2,2,2,2,1,1],[1,1,1,1]]
 => ([(0,5),(1,6),(1,7),(3,7),(4,3),(5,4),(6,2)],8)
 => ? = 55
[8,5,6,7,4,3,2,1] => [1,3,1,1,1,1] => [[3,3,3,3,3,1],[2,2,2,2]]
 => ?
 => ? = 85
[7,6,5,8,4,3,2,1] => [1,1,2,1,1,1,1] => [[2,2,2,2,2,1,1],[1,1,1,1]]
 => ([(0,5),(1,6),(1,7),(3,7),(4,3),(5,4),(6,2)],8)
 => ? = 55
[6,7,5,8,4,3,2,1] => [2,2,1,1,1,1] => [[3,3,3,3,3,2],[2,2,2,2,1]]
 => ([(0,3),(1,6),(2,6),(2,7),(3,5),(4,7),(5,4)],8)
 => ? = 105
[7,5,6,8,4,3,2,1] => [1,3,1,1,1,1] => [[3,3,3,3,3,1],[2,2,2,2]]
 => ?
 => ? = 85
[6,5,7,8,4,3,2,1] => [1,3,1,1,1,1] => [[3,3,3,3,3,1],[2,2,2,2]]
 => ?
 => ? = 85
[5,6,7,8,4,3,2,1] => [4,1,1,1,1] => [[4,4,4,4,4],[3,3,3,3]]
 => ([(0,5),(1,6),(2,7),(3,7),(4,3),(5,4),(6,2)],8)
 => ? = 35
[8,7,6,4,5,3,2,1] => [1,1,1,2,1,1,1] => [[2,2,2,2,1,1,1],[1,1,1]]
 => ([(0,5),(1,6),(1,7),(3,7),(4,2),(5,3),(6,4)],8)
 => ? = 69
[7,8,6,4,5,3,2,1] => [2,1,2,1,1,1] => [[3,3,3,3,2,2],[2,2,2,1,1]]
 => ([(0,7),(1,4),(2,3),(2,6),(3,7),(4,5),(5,6)],8)
 => ? = 203
[8,6,7,4,5,3,2,1] => [1,2,2,1,1,1] => [[3,3,3,3,2,1],[2,2,2,1]]
 => ([(0,6),(0,7),(1,4),(2,3),(2,6),(4,5),(5,7)],8)
 => ? = 323
[7,6,8,4,5,3,2,1] => [1,2,2,1,1,1] => [[3,3,3,3,2,1],[2,2,2,1]]
 => ([(0,6),(0,7),(1,4),(2,3),(2,6),(4,5),(5,7)],8)
 => ? = 323
[6,7,8,4,5,3,2,1] => [3,2,1,1,1] => [[4,4,4,4,3],[3,3,3,2]]
 => ([(0,6),(0,7),(1,3),(2,4),(3,7),(4,5),(5,6)],8)
 => ? = 189
[8,7,5,4,6,3,2,1] => [1,1,1,2,1,1,1] => [[2,2,2,2,1,1,1],[1,1,1]]
 => ([(0,5),(1,6),(1,7),(3,7),(4,2),(5,3),(6,4)],8)
 => ? = 69
[7,8,5,4,6,3,2,1] => [2,1,2,1,1,1] => [[3,3,3,3,2,2],[2,2,2,1,1]]
 => ([(0,7),(1,4),(2,3),(2,6),(3,7),(4,5),(5,6)],8)
 => ? = 203
[8,7,4,5,6,3,2,1] => [1,1,3,1,1,1] => [[3,3,3,3,1,1],[2,2,2]]
 => ?
 => ? = 155
[7,8,4,5,6,3,2,1] => [2,3,1,1,1] => [[4,4,4,4,2],[3,3,3,1]]
 => ([(0,6),(1,4),(2,3),(2,6),(3,7),(4,5),(5,7)],8)
 => ? = 245
[8,6,5,4,7,3,2,1] => [1,1,1,2,1,1,1] => [[2,2,2,2,1,1,1],[1,1,1]]
 => ([(0,5),(1,6),(1,7),(3,7),(4,2),(5,3),(6,4)],8)
 => ? = 69
[8,5,6,4,7,3,2,1] => [1,2,2,1,1,1] => [[3,3,3,3,2,1],[2,2,2,1]]
 => ([(0,6),(0,7),(1,4),(2,3),(2,6),(4,5),(5,7)],8)
 => ? = 323
[8,6,4,5,7,3,2,1] => [1,1,3,1,1,1] => [[3,3,3,3,1,1],[2,2,2]]
 => ?
 => ? = 155
[8,5,4,6,7,3,2,1] => [1,1,3,1,1,1] => [[3,3,3,3,1,1],[2,2,2]]
 => ?
 => ? = 155
[8,4,5,6,7,3,2,1] => [1,4,1,1,1] => [[4,4,4,4,1],[3,3,3]]
 => ([(0,6),(1,2),(1,5),(3,7),(4,7),(5,4),(6,3)],8)
 => ? = 125
[7,6,5,4,8,3,2,1] => [1,1,1,2,1,1,1] => [[2,2,2,2,1,1,1],[1,1,1]]
 => ([(0,5),(1,6),(1,7),(3,7),(4,2),(5,3),(6,4)],8)
 => ? = 69
[6,7,5,4,8,3,2,1] => [2,1,2,1,1,1] => [[3,3,3,3,2,2],[2,2,2,1,1]]
 => ([(0,7),(1,4),(2,3),(2,6),(3,7),(4,5),(5,6)],8)
 => ? = 203
[7,5,6,4,8,3,2,1] => [1,2,2,1,1,1] => [[3,3,3,3,2,1],[2,2,2,1]]
 => ([(0,6),(0,7),(1,4),(2,3),(2,6),(4,5),(5,7)],8)
 => ? = 323
[6,5,7,4,8,3,2,1] => [1,2,2,1,1,1] => [[3,3,3,3,2,1],[2,2,2,1]]
 => ([(0,6),(0,7),(1,4),(2,3),(2,6),(4,5),(5,7)],8)
 => ? = 323
[5,6,7,4,8,3,2,1] => [3,2,1,1,1] => [[4,4,4,4,3],[3,3,3,2]]
 => ([(0,6),(0,7),(1,3),(2,4),(3,7),(4,5),(5,6)],8)
 => ? = 189
[7,6,4,5,8,3,2,1] => [1,1,3,1,1,1] => [[3,3,3,3,1,1],[2,2,2]]
 => ?
 => ? = 155
[6,7,4,5,8,3,2,1] => [2,3,1,1,1] => [[4,4,4,4,2],[3,3,3,1]]
 => ([(0,6),(1,4),(2,3),(2,6),(3,7),(4,5),(5,7)],8)
 => ? = 245
[7,5,4,6,8,3,2,1] => [1,1,3,1,1,1] => [[3,3,3,3,1,1],[2,2,2]]
 => ?
 => ? = 155
[7,4,5,6,8,3,2,1] => [1,4,1,1,1] => [[4,4,4,4,1],[3,3,3]]
 => ([(0,6),(1,2),(1,5),(3,7),(4,7),(5,4),(6,3)],8)
 => ? = 125
[6,5,4,7,8,3,2,1] => [1,1,3,1,1,1] => [[3,3,3,3,1,1],[2,2,2]]
 => ?
 => ? = 155
[5,6,4,7,8,3,2,1] => [2,3,1,1,1] => [[4,4,4,4,2],[3,3,3,1]]
 => ([(0,6),(1,4),(2,3),(2,6),(3,7),(4,5),(5,7)],8)
 => ? = 245
[6,4,5,7,8,3,2,1] => [1,4,1,1,1] => [[4,4,4,4,1],[3,3,3]]
 => ([(0,6),(1,2),(1,5),(3,7),(4,7),(5,4),(6,3)],8)
 => ? = 125
[5,4,6,7,8,3,2,1] => [1,4,1,1,1] => [[4,4,4,4,1],[3,3,3]]
 => ([(0,6),(1,2),(1,5),(3,7),(4,7),(5,4),(6,3)],8)
 => ? = 125
[4,5,6,7,8,3,2,1] => [5,1,1,1] => [[5,5,5,5],[4,4,4]]
 => ([(0,5),(1,6),(2,7),(3,7),(4,3),(5,4),(6,2)],8)
 => ? = 35
[8,7,6,5,3,4,2,1] => [1,1,1,1,2,1,1] => [[2,2,2,1,1,1,1],[1,1]]
 => ([(0,3),(1,6),(1,7),(3,7),(4,5),(5,2),(6,4)],8)
 => ? = 55
[7,8,6,5,3,4,2,1] => [2,1,1,2,1,1] => [[3,3,3,2,2,2],[2,2,1,1,1]]
 => ([(0,6),(1,3),(2,4),(2,7),(3,7),(4,5),(5,6)],8)
 => ? = 217
[8,6,7,5,3,4,2,1] => [1,2,1,2,1,1] => [[3,3,3,2,2,1],[2,2,1,1]]
 => ([(0,5),(1,4),(1,7),(2,3),(2,6),(4,6),(5,7)],8)
 => ? = 477
[7,6,8,5,3,4,2,1] => [1,2,1,2,1,1] => [[3,3,3,2,2,1],[2,2,1,1]]
 => ([(0,5),(1,4),(1,7),(2,3),(2,6),(4,6),(5,7)],8)
 => ? = 477
[6,7,8,5,3,4,2,1] => [3,1,2,1,1] => [[4,4,4,3,3],[3,3,2,2]]
 => ([(0,3),(1,5),(2,4),(2,6),(3,7),(4,7),(5,6)],8)
 => ? = 315
[8,7,5,6,3,4,2,1] => [1,1,2,2,1,1] => [[3,3,3,2,1,1],[2,2,1]]
 => ([(0,6),(0,7),(1,3),(2,4),(2,6),(3,7),(4,5)],8)
 => ? = 449
[7,8,5,6,3,4,2,1] => [2,2,2,1,1] => [[4,4,4,3,2],[3,3,2,1]]
 => ([(0,5),(1,6),(1,7),(2,5),(2,6),(3,4),(4,7)],8)
 => ? = 791
[8,6,5,7,3,4,2,1] => [1,1,2,2,1,1] => [[3,3,3,2,1,1],[2,2,1]]
 => ([(0,6),(0,7),(1,3),(2,4),(2,6),(3,7),(4,5)],8)
 => ? = 449
Description
The number of linear extensions of a poset.
Matching statistic: St001595
(load all 4 compositions to match this statistic)
Mapping
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
St001595: Skew partitions ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 15%
Values
[1] => [1] => [[1],[]]
 => 1
[1,2] => [2] => [[2],[]]
 => 1
[2,1] => [1,1] => [[1,1],[]]
 => 1
[1,2,3] => [3] => [[3],[]]
 => 1
[1,3,2] => [2,1] => [[2,2],[1]]
 => 2
[2,1,3] => [1,2] => [[2,1],[]]
 => 2
[2,3,1] => [2,1] => [[2,2],[1]]
 => 2
[3,1,2] => [1,2] => [[2,1],[]]
 => 2
[3,2,1] => [1,1,1] => [[1,1,1],[]]
 => 1
[1,2,3,4] => [4] => [[4],[]]
 => 1
[1,2,4,3] => [3,1] => [[3,3],[2]]
 => 3
[1,3,2,4] => [2,2] => [[3,2],[1]]
 => 5
[1,3,4,2] => [3,1] => [[3,3],[2]]
 => 3
[1,4,2,3] => [2,2] => [[3,2],[1]]
 => 5
[1,4,3,2] => [2,1,1] => [[2,2,2],[1,1]]
 => 3
[2,1,3,4] => [1,3] => [[3,1],[]]
 => 3
[2,1,4,3] => [1,2,1] => [[2,2,1],[1]]
 => 5
[2,3,1,4] => [2,2] => [[3,2],[1]]
 => 5
[2,3,4,1] => [3,1] => [[3,3],[2]]
 => 3
[2,4,1,3] => [2,2] => [[3,2],[1]]
 => 5
[2,4,3,1] => [2,1,1] => [[2,2,2],[1,1]]
 => 3
[3,1,2,4] => [1,3] => [[3,1],[]]
 => 3
[3,1,4,2] => [1,2,1] => [[2,2,1],[1]]
 => 5
[3,2,1,4] => [1,1,2] => [[2,1,1],[]]
 => 3
[3,2,4,1] => [1,2,1] => [[2,2,1],[1]]
 => 5
[3,4,1,2] => [2,2] => [[3,2],[1]]
 => 5
[3,4,2,1] => [2,1,1] => [[2,2,2],[1,1]]
 => 3
[4,1,2,3] => [1,3] => [[3,1],[]]
 => 3
[4,1,3,2] => [1,2,1] => [[2,2,1],[1]]
 => 5
[4,2,1,3] => [1,1,2] => [[2,1,1],[]]
 => 3
[4,2,3,1] => [1,2,1] => [[2,2,1],[1]]
 => 5
[4,3,1,2] => [1,1,2] => [[2,1,1],[]]
 => 3
[4,3,2,1] => [1,1,1,1] => [[1,1,1,1],[]]
 => 1
[1,2,3,4,5] => [5] => [[5],[]]
 => 1
[1,2,3,5,4] => [4,1] => [[4,4],[3]]
 => 4
[1,2,4,3,5] => [3,2] => [[4,3],[2]]
 => 9
[1,2,4,5,3] => [4,1] => [[4,4],[3]]
 => 4
[1,2,5,3,4] => [3,2] => [[4,3],[2]]
 => 9
[1,2,5,4,3] => [3,1,1] => [[3,3,3],[2,2]]
 => 6
[1,3,2,4,5] => [2,3] => [[4,2],[1]]
 => 9
[1,3,2,5,4] => [2,2,1] => [[3,3,2],[2,1]]
 => 16
[1,3,4,2,5] => [3,2] => [[4,3],[2]]
 => 9
[1,3,4,5,2] => [4,1] => [[4,4],[3]]
 => 4
[1,3,5,2,4] => [3,2] => [[4,3],[2]]
 => 9
[1,3,5,4,2] => [3,1,1] => [[3,3,3],[2,2]]
 => 6
[1,4,2,3,5] => [2,3] => [[4,2],[1]]
 => 9
[1,4,2,5,3] => [2,2,1] => [[3,3,2],[2,1]]
 => 16
[1,4,3,2,5] => [2,1,2] => [[3,2,2],[1,1]]
 => 11
[1,4,3,5,2] => [2,2,1] => [[3,3,2],[2,1]]
 => 16
[1,4,5,2,3] => [3,2] => [[4,3],[2]]
 => 9
[8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1,1],[]]
 => ? = 1
[7,8,6,5,4,3,2,1] => [2,1,1,1,1,1,1] => [[2,2,2,2,2,2,2],[1,1,1,1,1,1]]
 => ? = 7
[8,6,7,5,4,3,2,1] => [1,2,1,1,1,1,1] => [[2,2,2,2,2,2,1],[1,1,1,1,1]]
 => ? = 27
[7,6,8,5,4,3,2,1] => [1,2,1,1,1,1,1] => [[2,2,2,2,2,2,1],[1,1,1,1,1]]
 => ? = 27
[6,7,8,5,4,3,2,1] => [3,1,1,1,1,1] => [[3,3,3,3,3,3],[2,2,2,2,2]]
 => ? = 21
[8,7,5,6,4,3,2,1] => [1,1,2,1,1,1,1] => [[2,2,2,2,2,1,1],[1,1,1,1]]
 => ? = 55
[7,8,5,6,4,3,2,1] => [2,2,1,1,1,1] => [[3,3,3,3,3,2],[2,2,2,2,1]]
 => ? = 105
[8,6,5,7,4,3,2,1] => [1,1,2,1,1,1,1] => [[2,2,2,2,2,1,1],[1,1,1,1]]
 => ? = 55
[8,5,6,7,4,3,2,1] => [1,3,1,1,1,1] => [[3,3,3,3,3,1],[2,2,2,2]]
 => ? = 85
[7,6,5,8,4,3,2,1] => [1,1,2,1,1,1,1] => [[2,2,2,2,2,1,1],[1,1,1,1]]
 => ? = 55
[6,7,5,8,4,3,2,1] => [2,2,1,1,1,1] => [[3,3,3,3,3,2],[2,2,2,2,1]]
 => ? = 105
[7,5,6,8,4,3,2,1] => [1,3,1,1,1,1] => [[3,3,3,3,3,1],[2,2,2,2]]
 => ? = 85
[6,5,7,8,4,3,2,1] => [1,3,1,1,1,1] => [[3,3,3,3,3,1],[2,2,2,2]]
 => ? = 85
[5,6,7,8,4,3,2,1] => [4,1,1,1,1] => [[4,4,4,4,4],[3,3,3,3]]
 => ? = 35
[8,7,6,4,5,3,2,1] => [1,1,1,2,1,1,1] => [[2,2,2,2,1,1,1],[1,1,1]]
 => ? = 69
[7,8,6,4,5,3,2,1] => [2,1,2,1,1,1] => [[3,3,3,3,2,2],[2,2,2,1,1]]
 => ? = 203
[8,6,7,4,5,3,2,1] => [1,2,2,1,1,1] => [[3,3,3,3,2,1],[2,2,2,1]]
 => ? = 323
[7,6,8,4,5,3,2,1] => [1,2,2,1,1,1] => [[3,3,3,3,2,1],[2,2,2,1]]
 => ? = 323
[6,7,8,4,5,3,2,1] => [3,2,1,1,1] => [[4,4,4,4,3],[3,3,3,2]]
 => ? = 189
[8,7,5,4,6,3,2,1] => [1,1,1,2,1,1,1] => [[2,2,2,2,1,1,1],[1,1,1]]
 => ? = 69
[7,8,5,4,6,3,2,1] => [2,1,2,1,1,1] => [[3,3,3,3,2,2],[2,2,2,1,1]]
 => ? = 203
[8,7,4,5,6,3,2,1] => [1,1,3,1,1,1] => [[3,3,3,3,1,1],[2,2,2]]
 => ? = 155
[7,8,4,5,6,3,2,1] => [2,3,1,1,1] => [[4,4,4,4,2],[3,3,3,1]]
 => ? = 245
[8,6,5,4,7,3,2,1] => [1,1,1,2,1,1,1] => [[2,2,2,2,1,1,1],[1,1,1]]
 => ? = 69
[8,5,6,4,7,3,2,1] => [1,2,2,1,1,1] => [[3,3,3,3,2,1],[2,2,2,1]]
 => ? = 323
[8,6,4,5,7,3,2,1] => [1,1,3,1,1,1] => [[3,3,3,3,1,1],[2,2,2]]
 => ? = 155
[8,5,4,6,7,3,2,1] => [1,1,3,1,1,1] => [[3,3,3,3,1,1],[2,2,2]]
 => ? = 155
[8,4,5,6,7,3,2,1] => [1,4,1,1,1] => [[4,4,4,4,1],[3,3,3]]
 => ? = 125
[7,6,5,4,8,3,2,1] => [1,1,1,2,1,1,1] => [[2,2,2,2,1,1,1],[1,1,1]]
 => ? = 69
[6,7,5,4,8,3,2,1] => [2,1,2,1,1,1] => [[3,3,3,3,2,2],[2,2,2,1,1]]
 => ? = 203
[7,5,6,4,8,3,2,1] => [1,2,2,1,1,1] => [[3,3,3,3,2,1],[2,2,2,1]]
 => ? = 323
[6,5,7,4,8,3,2,1] => [1,2,2,1,1,1] => [[3,3,3,3,2,1],[2,2,2,1]]
 => ? = 323
[5,6,7,4,8,3,2,1] => [3,2,1,1,1] => [[4,4,4,4,3],[3,3,3,2]]
 => ? = 189
[7,6,4,5,8,3,2,1] => [1,1,3,1,1,1] => [[3,3,3,3,1,1],[2,2,2]]
 => ? = 155
[6,7,4,5,8,3,2,1] => [2,3,1,1,1] => [[4,4,4,4,2],[3,3,3,1]]
 => ? = 245
[7,5,4,6,8,3,2,1] => [1,1,3,1,1,1] => [[3,3,3,3,1,1],[2,2,2]]
 => ? = 155
[7,4,5,6,8,3,2,1] => [1,4,1,1,1] => [[4,4,4,4,1],[3,3,3]]
 => ? = 125
[6,5,4,7,8,3,2,1] => [1,1,3,1,1,1] => [[3,3,3,3,1,1],[2,2,2]]
 => ? = 155
[5,6,4,7,8,3,2,1] => [2,3,1,1,1] => [[4,4,4,4,2],[3,3,3,1]]
 => ? = 245
[6,4,5,7,8,3,2,1] => [1,4,1,1,1] => [[4,4,4,4,1],[3,3,3]]
 => ? = 125
[5,4,6,7,8,3,2,1] => [1,4,1,1,1] => [[4,4,4,4,1],[3,3,3]]
 => ? = 125
[4,5,6,7,8,3,2,1] => [5,1,1,1] => [[5,5,5,5],[4,4,4]]
 => ? = 35
[8,7,6,5,3,4,2,1] => [1,1,1,1,2,1,1] => [[2,2,2,1,1,1,1],[1,1]]
 => ? = 55
[7,8,6,5,3,4,2,1] => [2,1,1,2,1,1] => [[3,3,3,2,2,2],[2,2,1,1,1]]
 => ? = 217
[8,6,7,5,3,4,2,1] => [1,2,1,2,1,1] => [[3,3,3,2,2,1],[2,2,1,1]]
 => ? = 477
[7,6,8,5,3,4,2,1] => [1,2,1,2,1,1] => [[3,3,3,2,2,1],[2,2,1,1]]
 => ? = 477
[6,7,8,5,3,4,2,1] => [3,1,2,1,1] => [[4,4,4,3,3],[3,3,2,2]]
 => ? = 315
[8,7,5,6,3,4,2,1] => [1,1,2,2,1,1] => [[3,3,3,2,1,1],[2,2,1]]
 => ? = 449
[7,8,5,6,3,4,2,1] => [2,2,2,1,1] => [[4,4,4,3,2],[3,3,2,1]]
 => ? = 791
[8,6,5,7,3,4,2,1] => [1,1,2,2,1,1] => [[3,3,3,2,1,1],[2,2,1]]
 => ? = 449
Description
The number of standard Young tableaux of the skew partition.
Matching statistic: St000530
(load all 43 compositions to match this statistic)
Mapping
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000530: Permutations ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 14%
Values
[1] => [1] => [1,0]
 => [1] => ? = 1
[1,2] => [2] => [1,1,0,0]
 => [2,1] => 1
[2,1] => [1,1] => [1,0,1,0]
 => [1,2] => 1
[1,2,3] => [3] => [1,1,1,0,0,0]
 => [3,2,1] => 1
[1,3,2] => [2,1] => [1,1,0,0,1,0]
 => [2,1,3] => 2
[2,1,3] => [1,2] => [1,0,1,1,0,0]
 => [1,3,2] => 2
[2,3,1] => [2,1] => [1,1,0,0,1,0]
 => [2,1,3] => 2
[3,1,2] => [1,2] => [1,0,1,1,0,0]
 => [1,3,2] => 2
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
 => [1,2,3] => 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 1
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 3
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 5
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 3
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 5
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 3
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 3
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 5
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 5
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 3
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 5
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 3
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 3
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 5
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 3
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 5
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 5
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 3
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 3
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 5
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 3
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 5
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 3
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 1
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => 1
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => 4
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => 9
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => 4
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => 9
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => 6
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => 9
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 16
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => 9
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => 4
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => 9
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => 6
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => 9
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 16
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => 11
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 16
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => 9
[1,4,5,3,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => 6
[1,2,3,4,5,6,7] => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [7,6,5,4,3,2,1] => ? = 1
[1,2,3,4,5,7,6] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [6,5,4,3,2,1,7] => ? = 6
[1,2,3,4,6,5,7] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [5,4,3,2,1,7,6] => ? = 20
[1,2,3,4,6,7,5] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [6,5,4,3,2,1,7] => ? = 6
[1,2,3,4,7,5,6] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [5,4,3,2,1,7,6] => ? = 20
[1,2,3,4,7,6,5] => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [5,4,3,2,1,6,7] => ? = 15
[1,2,3,5,4,6,7] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [4,3,2,1,7,6,5] => ? = 34
[1,2,3,5,4,7,6] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,3,2,1,6,5,7] => ? = 64
[1,2,3,5,6,4,7] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [5,4,3,2,1,7,6] => ? = 20
[1,2,3,5,6,7,4] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [6,5,4,3,2,1,7] => ? = 6
[1,2,3,5,7,4,6] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [5,4,3,2,1,7,6] => ? = 20
[1,2,3,5,7,6,4] => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [5,4,3,2,1,6,7] => ? = 15
[1,2,3,6,4,5,7] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [4,3,2,1,7,6,5] => ? = 34
[1,2,3,6,4,7,5] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,3,2,1,6,5,7] => ? = 64
[1,2,3,6,5,4,7] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => [4,3,2,1,5,7,6] => ? = 50
[1,2,3,6,5,7,4] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,3,2,1,6,5,7] => ? = 64
[1,2,3,6,7,4,5] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [5,4,3,2,1,7,6] => ? = 20
[1,2,3,6,7,5,4] => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [5,4,3,2,1,6,7] => ? = 15
[1,2,3,7,4,5,6] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [4,3,2,1,7,6,5] => ? = 34
[1,2,3,7,4,6,5] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,3,2,1,6,5,7] => ? = 64
[1,2,3,7,5,4,6] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => [4,3,2,1,5,7,6] => ? = 50
[1,2,3,7,5,6,4] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,3,2,1,6,5,7] => ? = 64
[1,2,3,7,6,4,5] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => [4,3,2,1,5,7,6] => ? = 50
[1,2,3,7,6,5,4] => [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [4,3,2,1,5,6,7] => ? = 20
[1,2,4,3,5,6,7] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [3,2,1,7,6,5,4] => ? = 34
[1,2,4,3,5,7,6] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [3,2,1,6,5,4,7] => ? = 99
[1,2,4,3,6,5,7] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [3,2,1,5,4,7,6] => ? = 155
[1,2,4,3,6,7,5] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [3,2,1,6,5,4,7] => ? = 99
[1,2,4,3,7,5,6] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [3,2,1,5,4,7,6] => ? = 155
[1,2,4,3,7,6,5] => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => [3,2,1,5,4,6,7] => ? = 90
[1,2,4,5,3,6,7] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [4,3,2,1,7,6,5] => ? = 34
[1,2,4,5,3,7,6] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,3,2,1,6,5,7] => ? = 64
[1,2,4,5,6,3,7] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [5,4,3,2,1,7,6] => ? = 20
[1,2,4,5,6,7,3] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [6,5,4,3,2,1,7] => ? = 6
[1,2,4,5,7,3,6] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [5,4,3,2,1,7,6] => ? = 20
[1,2,4,5,7,6,3] => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [5,4,3,2,1,6,7] => ? = 15
[1,2,4,6,3,5,7] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [4,3,2,1,7,6,5] => ? = 34
[1,2,4,6,3,7,5] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,3,2,1,6,5,7] => ? = 64
[1,2,4,6,5,3,7] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => [4,3,2,1,5,7,6] => ? = 50
[1,2,4,6,5,7,3] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,3,2,1,6,5,7] => ? = 64
[1,2,4,6,7,3,5] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [5,4,3,2,1,7,6] => ? = 20
[1,2,4,6,7,5,3] => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [5,4,3,2,1,6,7] => ? = 15
[1,2,4,7,3,5,6] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [4,3,2,1,7,6,5] => ? = 34
[1,2,4,7,3,6,5] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,3,2,1,6,5,7] => ? = 64
[1,2,4,7,5,3,6] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => [4,3,2,1,5,7,6] => ? = 50
[1,2,4,7,5,6,3] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,3,2,1,6,5,7] => ? = 64
[1,2,4,7,6,3,5] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => [4,3,2,1,5,7,6] => ? = 50
[1,2,4,7,6,5,3] => [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [4,3,2,1,5,6,7] => ? = 20
[1,2,5,3,4,6,7] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [3,2,1,7,6,5,4] => ? = 34
Description
The number of permutations with the same descent word as the given permutation. The descent word of a permutation is the binary word given by [[Mp00109]]. For a given permutation, this statistic is the number of permutations with the same descent word, so the number of elements in the fiber of the map [[Mp00109]] containing a given permutation. This statistic appears as ''up-down analysis'' in statistical applications in genetics, see [1] and the references therein.
Matching statistic: St000001
Mapping
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000001: Permutations ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 5%
Values
[1] => [1] => [1,0]
 => [2,1] => 1
[1,2] => [2] => [1,1,0,0]
 => [2,3,1] => 1
[2,1] => [1,1] => [1,0,1,0]
 => [3,1,2] => 1
[1,2,3] => [3] => [1,1,1,0,0,0]
 => [2,3,4,1] => 1
[1,3,2] => [2,1] => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[2,1,3] => [1,2] => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[2,3,1] => [2,1] => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[3,1,2] => [1,2] => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
 => [4,1,2,3] => 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => 3
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => 5
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => 3
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => 5
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 3
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => 3
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => 5
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => 5
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => 3
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => 5
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 3
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => 3
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => 5
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 3
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => 5
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => 5
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 3
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => 3
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => 5
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 3
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => 5
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 3
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => 1
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [2,3,4,6,1,5] => 4
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => 9
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [2,3,4,6,1,5] => 4
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => 9
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => 6
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => 9
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => 16
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => 9
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [2,3,4,6,1,5] => 4
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => 9
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => 6
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => 9
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => 16
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => 11
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => 16
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => 9
[1,2,4,3,5,6] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,3,5,1,6,7,4] => ? = 19
[1,2,4,3,6,5] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,3,5,1,7,4,6] => ? = 35
[1,2,5,3,4,6] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,3,5,1,6,7,4] => ? = 19
[1,2,5,3,6,4] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,3,5,1,7,4,6] => ? = 35
[1,2,5,4,3,6] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [2,3,6,1,4,7,5] => ? = 26
[1,2,5,4,6,3] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,3,5,1,7,4,6] => ? = 35
[1,2,6,3,4,5] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,3,5,1,6,7,4] => ? = 19
[1,2,6,3,5,4] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,3,5,1,7,4,6] => ? = 35
[1,2,6,4,3,5] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [2,3,6,1,4,7,5] => ? = 26
[1,2,6,4,5,3] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,3,5,1,7,4,6] => ? = 35
[1,2,6,5,3,4] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [2,3,6,1,4,7,5] => ? = 26
[1,2,6,5,4,3] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [2,3,7,1,4,5,6] => ? = 10
[1,3,2,4,5,6] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,4,1,5,6,7,3] => ? = 14
[1,3,2,4,6,5] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,4,1,5,7,3,6] => ? = 40
[1,3,2,5,4,6] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,4,1,6,3,7,5] => ? = 61
[1,3,2,5,6,4] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,4,1,5,7,3,6] => ? = 40
[1,3,2,6,4,5] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,4,1,6,3,7,5] => ? = 61
[1,3,2,6,5,4] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,4,1,7,3,5,6] => ? = 35
[1,3,4,2,5,6] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,3,5,1,6,7,4] => ? = 19
[1,3,4,2,6,5] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,3,5,1,7,4,6] => ? = 35
[1,3,5,2,4,6] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,3,5,1,6,7,4] => ? = 19
[1,3,5,2,6,4] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,3,5,1,7,4,6] => ? = 35
[1,3,5,4,2,6] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [2,3,6,1,4,7,5] => ? = 26
[1,3,5,4,6,2] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,3,5,1,7,4,6] => ? = 35
[1,3,6,2,4,5] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,3,5,1,6,7,4] => ? = 19
[1,3,6,2,5,4] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,3,5,1,7,4,6] => ? = 35
[1,3,6,4,2,5] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [2,3,6,1,4,7,5] => ? = 26
[1,3,6,4,5,2] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,3,5,1,7,4,6] => ? = 35
[1,3,6,5,2,4] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [2,3,6,1,4,7,5] => ? = 26
[1,3,6,5,4,2] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [2,3,7,1,4,5,6] => ? = 10
[1,4,2,3,5,6] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,4,1,5,6,7,3] => ? = 14
[1,4,2,3,6,5] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,4,1,5,7,3,6] => ? = 40
[1,4,2,5,3,6] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,4,1,6,3,7,5] => ? = 61
[1,4,2,5,6,3] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,4,1,5,7,3,6] => ? = 40
[1,4,2,6,3,5] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,4,1,6,3,7,5] => ? = 61
[1,4,2,6,5,3] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,4,1,7,3,5,6] => ? = 35
[1,4,3,2,5,6] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [2,5,1,3,6,7,4] => ? = 26
[1,4,3,2,6,5] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [2,5,1,3,7,4,6] => ? = 40
[1,4,3,5,2,6] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,4,1,6,3,7,5] => ? = 61
[1,4,3,5,6,2] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,4,1,5,7,3,6] => ? = 40
[1,4,3,6,2,5] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,4,1,6,3,7,5] => ? = 61
[1,4,3,6,5,2] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,4,1,7,3,5,6] => ? = 35
[1,4,5,2,3,6] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,3,5,1,6,7,4] => ? = 19
[1,4,5,2,6,3] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,3,5,1,7,4,6] => ? = 35
[1,4,5,3,2,6] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [2,3,6,1,4,7,5] => ? = 26
[1,4,5,3,6,2] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,3,5,1,7,4,6] => ? = 35
[1,4,6,2,3,5] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,3,5,1,6,7,4] => ? = 19
[1,4,6,2,5,3] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,3,5,1,7,4,6] => ? = 35
[1,4,6,3,2,5] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [2,3,6,1,4,7,5] => ? = 26
[1,4,6,3,5,2] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,3,5,1,7,4,6] => ? = 35
Description
The number of reduced words for a permutation. This is the number of ways to write a permutation as a minimal length product of simple transpositions. E.g., there are two reduced words for the permutation [3,2,1], which are (1,2)(2,3)(1,2) = (2,3)(1,2)(2,3).
Matching statistic: St000255
Mapping
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000255: Permutations ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 5%
Values
[1] => [1] => [1,0]
 => [2,1] => 1
[1,2] => [2] => [1,1,0,0]
 => [2,3,1] => 1
[2,1] => [1,1] => [1,0,1,0]
 => [3,1,2] => 1
[1,2,3] => [3] => [1,1,1,0,0,0]
 => [2,3,4,1] => 1
[1,3,2] => [2,1] => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[2,1,3] => [1,2] => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[2,3,1] => [2,1] => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[3,1,2] => [1,2] => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
 => [4,1,2,3] => 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => 3
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => 5
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => 3
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => 5
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 3
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => 3
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => 5
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => 5
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => 3
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => 5
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 3
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => 3
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => 5
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 3
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => 5
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => 5
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 3
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => 3
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => 5
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 3
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => 5
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 3
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => 1
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [2,3,4,6,1,5] => 4
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => 9
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [2,3,4,6,1,5] => 4
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => 9
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => 6
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => 9
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => 16
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => 9
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [2,3,4,6,1,5] => 4
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => 9
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => 6
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => 9
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => 16
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => 11
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => 16
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => 9
[1,2,4,3,5,6] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,3,5,1,6,7,4] => ? = 19
[1,2,4,3,6,5] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,3,5,1,7,4,6] => ? = 35
[1,2,5,3,4,6] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,3,5,1,6,7,4] => ? = 19
[1,2,5,3,6,4] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,3,5,1,7,4,6] => ? = 35
[1,2,5,4,3,6] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [2,3,6,1,4,7,5] => ? = 26
[1,2,5,4,6,3] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,3,5,1,7,4,6] => ? = 35
[1,2,6,3,4,5] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,3,5,1,6,7,4] => ? = 19
[1,2,6,3,5,4] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,3,5,1,7,4,6] => ? = 35
[1,2,6,4,3,5] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [2,3,6,1,4,7,5] => ? = 26
[1,2,6,4,5,3] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,3,5,1,7,4,6] => ? = 35
[1,2,6,5,3,4] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [2,3,6,1,4,7,5] => ? = 26
[1,2,6,5,4,3] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [2,3,7,1,4,5,6] => ? = 10
[1,3,2,4,5,6] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,4,1,5,6,7,3] => ? = 14
[1,3,2,4,6,5] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,4,1,5,7,3,6] => ? = 40
[1,3,2,5,4,6] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,4,1,6,3,7,5] => ? = 61
[1,3,2,5,6,4] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,4,1,5,7,3,6] => ? = 40
[1,3,2,6,4,5] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,4,1,6,3,7,5] => ? = 61
[1,3,2,6,5,4] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,4,1,7,3,5,6] => ? = 35
[1,3,4,2,5,6] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,3,5,1,6,7,4] => ? = 19
[1,3,4,2,6,5] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,3,5,1,7,4,6] => ? = 35
[1,3,5,2,4,6] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,3,5,1,6,7,4] => ? = 19
[1,3,5,2,6,4] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,3,5,1,7,4,6] => ? = 35
[1,3,5,4,2,6] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [2,3,6,1,4,7,5] => ? = 26
[1,3,5,4,6,2] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,3,5,1,7,4,6] => ? = 35
[1,3,6,2,4,5] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,3,5,1,6,7,4] => ? = 19
[1,3,6,2,5,4] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,3,5,1,7,4,6] => ? = 35
[1,3,6,4,2,5] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [2,3,6,1,4,7,5] => ? = 26
[1,3,6,4,5,2] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,3,5,1,7,4,6] => ? = 35
[1,3,6,5,2,4] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [2,3,6,1,4,7,5] => ? = 26
[1,3,6,5,4,2] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [2,3,7,1,4,5,6] => ? = 10
[1,4,2,3,5,6] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,4,1,5,6,7,3] => ? = 14
[1,4,2,3,6,5] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,4,1,5,7,3,6] => ? = 40
[1,4,2,5,3,6] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,4,1,6,3,7,5] => ? = 61
[1,4,2,5,6,3] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,4,1,5,7,3,6] => ? = 40
[1,4,2,6,3,5] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,4,1,6,3,7,5] => ? = 61
[1,4,2,6,5,3] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,4,1,7,3,5,6] => ? = 35
[1,4,3,2,5,6] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [2,5,1,3,6,7,4] => ? = 26
[1,4,3,2,6,5] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [2,5,1,3,7,4,6] => ? = 40
[1,4,3,5,2,6] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,4,1,6,3,7,5] => ? = 61
[1,4,3,5,6,2] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,4,1,5,7,3,6] => ? = 40
[1,4,3,6,2,5] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,4,1,6,3,7,5] => ? = 61
[1,4,3,6,5,2] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,4,1,7,3,5,6] => ? = 35
[1,4,5,2,3,6] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,3,5,1,6,7,4] => ? = 19
[1,4,5,2,6,3] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,3,5,1,7,4,6] => ? = 35
[1,4,5,3,2,6] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [2,3,6,1,4,7,5] => ? = 26
[1,4,5,3,6,2] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,3,5,1,7,4,6] => ? = 35
[1,4,6,2,3,5] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,3,5,1,6,7,4] => ? = 19
[1,4,6,2,5,3] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,3,5,1,7,4,6] => ? = 35
[1,4,6,3,2,5] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [2,3,6,1,4,7,5] => ? = 26
[1,4,6,3,5,2] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,3,5,1,7,4,6] => ? = 35
Description
The number of reduced Kogan faces with the permutation as type. This is equivalent to finding the number of ways to represent the permutation \pi \in S_{n+1} as a reduced subword of s_n (s_{n-1} s_n) (s_{n-2} s_{n-1} s_n) \dotsm (s_1 \dotsm s_n), or the number of reduced pipe dreams for \pi.
Matching statistic: St000880
Mapping
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000880: Permutations ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 4%
Values
[1] => [1] => [1,0]
 => [2,1] => 1
[1,2] => [2] => [1,1,0,0]
 => [2,3,1] => 1
[2,1] => [1,1] => [1,0,1,0]
 => [3,1,2] => 1
[1,2,3] => [3] => [1,1,1,0,0,0]
 => [2,3,4,1] => 1
[1,3,2] => [2,1] => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[2,1,3] => [1,2] => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[2,3,1] => [2,1] => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
[3,1,2] => [1,2] => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
 => [4,1,2,3] => 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => 3
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => 5
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => 3
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => 5
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 3
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => 3
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => 5
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => 5
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => 3
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => 5
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 3
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => 3
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => 5
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 3
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => 5
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => 5
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 3
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => 3
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => 5
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 3
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => 5
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 3
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => 1
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [2,3,4,6,1,5] => 4
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => 9
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [2,3,4,6,1,5] => 4
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => 9
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => 6
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => 9
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => 16
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => 9
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [2,3,4,6,1,5] => 4
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => 9
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => 6
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => 9
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => 16
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => 11
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => 16
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => 9
[3,2,1,4,5] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => ? = 6
[3,2,1,5,4] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => ? = 9
[4,2,1,3,5] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => ? = 6
[4,2,1,5,3] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => ? = 9
[4,3,1,2,5] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => ? = 6
[4,3,1,5,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => ? = 9
[4,3,2,1,5] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => ? = 4
[4,3,2,5,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => ? = 9
[5,2,1,3,4] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => ? = 6
[5,2,1,4,3] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => ? = 9
[5,3,1,2,4] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => ? = 6
[5,3,1,4,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => ? = 9
[5,3,2,1,4] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => ? = 4
[5,3,2,4,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => ? = 9
[5,4,1,2,3] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => ? = 6
[5,4,1,3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => ? = 9
[5,4,2,1,3] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => ? = 4
[5,4,2,3,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => ? = 9
[5,4,3,1,2] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => ? = 4
[5,4,3,2,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => ? = 1
[1,2,3,4,5,6] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,7,1] => ? = 1
[1,2,3,4,6,5] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [2,3,4,5,7,1,6] => ? = 5
[1,2,3,5,4,6] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [2,3,4,6,1,7,5] => ? = 14
[1,2,3,5,6,4] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [2,3,4,5,7,1,6] => ? = 5
[1,2,3,6,4,5] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [2,3,4,6,1,7,5] => ? = 14
[1,2,3,6,5,4] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [2,3,4,7,1,5,6] => ? = 10
[1,2,4,3,5,6] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,3,5,1,6,7,4] => ? = 19
[1,2,4,3,6,5] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,3,5,1,7,4,6] => ? = 35
[1,2,4,5,3,6] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [2,3,4,6,1,7,5] => ? = 14
[1,2,4,5,6,3] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [2,3,4,5,7,1,6] => ? = 5
[1,2,4,6,3,5] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [2,3,4,6,1,7,5] => ? = 14
[1,2,4,6,5,3] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [2,3,4,7,1,5,6] => ? = 10
[1,2,5,3,4,6] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,3,5,1,6,7,4] => ? = 19
[1,2,5,3,6,4] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,3,5,1,7,4,6] => ? = 35
[1,2,5,4,3,6] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [2,3,6,1,4,7,5] => ? = 26
[1,2,5,4,6,3] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,3,5,1,7,4,6] => ? = 35
[1,2,5,6,3,4] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [2,3,4,6,1,7,5] => ? = 14
[1,2,5,6,4,3] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [2,3,4,7,1,5,6] => ? = 10
[1,2,6,3,4,5] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,3,5,1,6,7,4] => ? = 19
[1,2,6,3,5,4] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,3,5,1,7,4,6] => ? = 35
[1,2,6,4,3,5] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [2,3,6,1,4,7,5] => ? = 26
[1,2,6,4,5,3] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,3,5,1,7,4,6] => ? = 35
[1,2,6,5,3,4] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [2,3,6,1,4,7,5] => ? = 26
[1,2,6,5,4,3] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [2,3,7,1,4,5,6] => ? = 10
[1,3,2,4,5,6] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,4,1,5,6,7,3] => ? = 14
[1,3,2,4,6,5] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,4,1,5,7,3,6] => ? = 40
[1,3,2,5,4,6] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,4,1,6,3,7,5] => ? = 61
[1,3,2,5,6,4] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,4,1,5,7,3,6] => ? = 40
[1,3,2,6,4,5] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,4,1,6,3,7,5] => ? = 61
[1,3,2,6,5,4] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,4,1,7,3,5,6] => ? = 35
Description
The number of connected components of long braid edges in the graph of braid moves of a permutation. Given a permutation \pi, let \operatorname{Red}(\pi) denote the set of reduced words for \pi in terms of simple transpositions s_i = (i,i+1). We now say that two reduced words are connected by a long braid move if they are obtained from each other by a modification of the form s_i s_{i+1} s_i \leftrightarrow s_{i+1} s_i s_{i+1} as a consecutive subword of a reduced word. For example, the two reduced words s_1s_3s_2s_3 and s_1s_2s_3s_2 for (124) = (12)(34)(23)(34) = (12)(23)(34)(23) share an edge because they are obtained from each other by interchanging s_3s_2s_3 \leftrightarrow s_3s_2s_3. This statistic counts the number connected components of such long braid moves among all reduced words.
Matching statistic: St001633
(load all 8 compositions to match this statistic)
Mapping
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001633: Posets ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 1%
Values
[1] => [1] => ([],1)
 => 0 = 1 - 1
[1,2] => [1,2] => ([(0,1)],2)
 => 0 = 1 - 1
[2,1] => [2,1] => ([(0,1)],2)
 => 0 = 1 - 1
[1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
 => 0 = 1 - 1
[1,3,2] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[2,3,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[3,1,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[3,2,1] => [3,2,1] => ([(0,2),(2,1)],3)
 => 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[1,2,4,3] => [4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 3 - 1
[1,3,2,4] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 5 - 1
[1,3,4,2] => [3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => 2 = 3 - 1
[1,4,2,3] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 5 - 1
[1,4,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 3 - 1
[2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 3 - 1
[2,1,4,3] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
 => ? = 5 - 1
[2,3,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 5 - 1
[2,3,4,1] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 3 - 1
[2,4,1,3] => [4,2,1,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 5 - 1
[2,4,3,1] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 3 - 1
[3,1,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 3 - 1
[3,1,4,2] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 5 - 1
[3,2,1,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 3 - 1
[3,2,4,1] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 5 - 1
[3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
 => ? = 5 - 1
[3,4,2,1] => [3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 3 - 1
[4,1,2,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 3 - 1
[4,1,3,2] => [4,1,3,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 5 - 1
[4,2,1,3] => [2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => 2 = 3 - 1
[4,2,3,1] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 5 - 1
[4,3,1,2] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 3 - 1
[4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[1,2,3,5,4] => [5,1,2,3,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 4 - 1
[1,2,4,3,5] => [4,1,2,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 9 - 1
[1,2,4,5,3] => [4,5,1,2,3] => ([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9)
 => ? = 4 - 1
[1,2,5,3,4] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 9 - 1
[1,2,5,4,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 6 - 1
[1,3,2,4,5] => [3,1,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 9 - 1
[1,3,2,5,4] => [3,5,1,2,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
 => ? = 16 - 1
[1,3,4,2,5] => [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
 => ? = 9 - 1
[1,3,4,5,2] => [3,4,5,1,2] => ([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9)
 => ? = 4 - 1
[1,3,5,2,4] => [5,3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
 => ? = 9 - 1
[1,3,5,4,2] => [5,3,4,1,2] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
 => ? = 6 - 1
[1,4,2,3,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
 => ? = 9 - 1
[1,4,2,5,3] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
 => ? = 16 - 1
[1,4,3,2,5] => [4,3,1,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ? = 11 - 1
[1,4,3,5,2] => [4,3,5,1,2] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ? = 16 - 1
[1,4,5,2,3] => [4,1,5,2,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
 => ? = 9 - 1
[1,4,5,3,2] => [4,5,3,1,2] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
 => ? = 6 - 1
[1,5,2,3,4] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 9 - 1
[1,5,2,4,3] => [5,1,4,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
 => ? = 16 - 1
[1,5,3,2,4] => [5,1,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
 => ? = 11 - 1
[1,5,3,4,2] => [3,5,4,1,2] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ? = 16 - 1
[1,5,4,2,3] => [1,5,4,2,3] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ? = 11 - 1
[1,5,4,3,2] => [5,4,3,1,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 4 - 1
[2,1,3,4,5] => [2,1,3,4,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 4 - 1
[2,1,3,5,4] => [2,5,1,3,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
 => ? = 11 - 1
[2,1,4,3,5] => [2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
 => ? = 16 - 1
[2,1,4,5,3] => [2,4,5,1,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
 => ? = 11 - 1
[2,1,5,3,4] => [5,2,1,3,4] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ? = 16 - 1
[2,1,5,4,3] => [5,2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
 => ? = 9 - 1
[2,3,1,4,5] => [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 9 - 1
[2,3,1,5,4] => [2,3,5,1,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
 => ? = 16 - 1
[2,3,4,1,5] => [2,3,4,1,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 9 - 1
[2,3,4,5,1] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 4 - 1
[2,3,5,1,4] => [5,2,3,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
 => ? = 9 - 1
[2,3,5,4,1] => [5,2,3,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ? = 6 - 1
[2,4,1,3,5] => [4,2,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
 => ? = 9 - 1
[2,4,1,5,3] => [4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
 => ? = 16 - 1
[2,4,3,1,5] => [4,2,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
 => ? = 11 - 1
[5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[6,5,4,3,2,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 0 = 1 - 1
[7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 0 = 1 - 1
Description
The number of simple modules with projective dimension two in the incidence algebra of the poset.
Matching statistic: St001812
Mapping
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St001812: Graphs ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 1%
Values
[1] => [1] => ([],1)
 => ([],1)
 => 0 = 1 - 1
[1,2] => [1,2] => ([(0,1)],2)
 => ([],2)
 => 0 = 1 - 1
[2,1] => [2,1] => ([(0,1)],2)
 => ([],2)
 => 0 = 1 - 1
[1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
 => ([],3)
 => 0 = 1 - 1
[1,3,2] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 1 = 2 - 1
[2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 1 = 2 - 1
[2,3,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 1 = 2 - 1
[3,1,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 1 = 2 - 1
[3,2,1] => [3,2,1] => ([(0,2),(2,1)],3)
 => ([],3)
 => 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0 = 1 - 1
[1,2,4,3] => [4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 2 = 3 - 1
[1,3,2,4] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => ? = 5 - 1
[1,3,4,2] => [3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(2,5),(3,4)],6)
 => 2 = 3 - 1
[1,4,2,3] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => ? = 5 - 1
[1,4,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 2 = 3 - 1
[2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 2 = 3 - 1
[2,1,4,3] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
 => ([(2,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 - 1
[2,3,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => ? = 5 - 1
[2,3,4,1] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 2 = 3 - 1
[2,4,1,3] => [4,2,1,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => ? = 5 - 1
[2,4,3,1] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => ? = 3 - 1
[3,1,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => ? = 3 - 1
[3,1,4,2] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => ? = 5 - 1
[3,2,1,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 2 = 3 - 1
[3,2,4,1] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => ? = 5 - 1
[3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
 => ([(2,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 - 1
[3,4,2,1] => [3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 2 = 3 - 1
[4,1,2,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 2 = 3 - 1
[4,1,3,2] => [4,1,3,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => ? = 5 - 1
[4,2,1,3] => [2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(2,5),(3,4)],6)
 => 2 = 3 - 1
[4,2,3,1] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => ? = 5 - 1
[4,3,1,2] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 2 = 3 - 1
[4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0 = 1 - 1
[1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0 = 1 - 1
[1,2,3,5,4] => [5,1,2,3,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
 => ? = 4 - 1
[1,2,4,3,5] => [4,1,2,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
 => ? = 9 - 1
[1,2,4,5,3] => [4,5,1,2,3] => ([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9)
 => ([(2,6),(3,4),(3,8),(4,7),(5,7),(5,8),(6,8),(7,8)],9)
 => ? = 4 - 1
[1,2,5,3,4] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
 => ? = 9 - 1
[1,2,5,4,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ([(2,5),(2,8),(3,4),(3,8),(4,7),(5,7),(6,7),(6,8),(7,8)],9)
 => ? = 6 - 1
[1,3,2,4,5] => [3,1,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
 => ? = 9 - 1
[1,3,2,5,4] => [3,5,1,2,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
 => ([(2,3),(3,12),(4,7),(4,9),(4,11),(4,12),(5,6),(5,8),(5,10),(5,12),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
 => ? = 16 - 1
[1,3,4,2,5] => [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
 => ([(2,3),(3,10),(4,5),(4,8),(4,9),(5,7),(5,9),(6,7),(6,8),(6,9),(6,10),(7,8),(7,10),(8,10),(9,10)],11)
 => ? = 9 - 1
[1,3,4,5,2] => [3,4,5,1,2] => ([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9)
 => ([(2,6),(3,4),(3,8),(4,7),(5,7),(5,8),(6,8),(7,8)],9)
 => ? = 4 - 1
[1,3,5,2,4] => [5,3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
 => ([(2,3),(2,11),(3,10),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,10),(6,7),(6,8),(6,11),(7,8),(7,9),(7,10),(8,9),(8,11),(9,10),(9,11),(10,11)],12)
 => ? = 9 - 1
[1,3,5,4,2] => [5,3,4,1,2] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
 => ([(2,5),(3,6),(3,9),(4,7),(4,8),(5,9),(6,8),(6,9),(7,8),(7,9)],10)
 => ? = 6 - 1
[1,4,2,3,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
 => ([(2,3),(3,7),(3,11),(4,5),(4,6),(4,10),(4,11),(5,6),(5,9),(5,11),(6,8),(6,11),(7,8),(7,9),(7,10),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
 => ? = 9 - 1
[1,4,2,5,3] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
 => ([(2,3),(3,10),(4,5),(4,8),(4,9),(5,7),(5,9),(6,7),(6,8),(6,9),(6,10),(7,8),(7,10),(8,10),(9,10)],11)
 => ? = 16 - 1
[1,4,3,2,5] => [4,3,1,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ([(2,3),(2,10),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(5,10),(6,7),(6,9),(6,10),(7,8),(7,10),(8,10),(9,10)],11)
 => ? = 11 - 1
[1,4,3,5,2] => [4,3,5,1,2] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ([(2,3),(3,10),(4,6),(4,7),(4,8),(5,6),(5,8),(5,9),(5,10),(6,7),(6,9),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ? = 16 - 1
[1,4,5,2,3] => [4,1,5,2,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
 => ([(2,3),(3,12),(4,7),(4,9),(4,11),(4,12),(5,6),(5,8),(5,10),(5,12),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
 => ? = 9 - 1
[1,4,5,3,2] => [4,5,3,1,2] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
 => ([(2,3),(3,9),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,9),(7,9),(8,9)],10)
 => ? = 6 - 1
[1,5,2,3,4] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
 => ? = 9 - 1
[1,5,2,4,3] => [5,1,4,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
 => ([(2,3),(2,11),(3,10),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,10),(6,7),(6,8),(6,11),(7,8),(7,9),(7,10),(8,9),(8,11),(9,10),(9,11),(10,11)],12)
 => ? = 16 - 1
[1,5,3,2,4] => [5,1,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
 => ([(2,3),(2,12),(3,11),(4,5),(4,6),(4,7),(4,8),(4,12),(5,6),(5,8),(5,10),(5,11),(6,8),(6,9),(6,11),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
 => ? = 11 - 1
[1,5,3,4,2] => [3,5,4,1,2] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ([(2,3),(3,10),(4,6),(4,7),(4,8),(5,6),(5,8),(5,9),(5,10),(6,7),(6,9),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ? = 16 - 1
[1,5,4,2,3] => [1,5,4,2,3] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ([(2,3),(2,10),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(5,10),(6,7),(6,9),(6,10),(7,8),(7,10),(8,10),(9,10)],11)
 => ? = 11 - 1
[1,5,4,3,2] => [5,4,3,1,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
 => ? = 4 - 1
[2,1,3,4,5] => [2,1,3,4,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
 => ? = 4 - 1
[2,1,3,5,4] => [2,5,1,3,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
 => ([(2,3),(3,11),(4,6),(4,9),(4,10),(4,11),(5,7),(5,8),(5,10),(5,11),(5,12),(6,8),(6,9),(6,10),(6,12),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,11),(8,12),(9,10),(9,12),(10,12),(11,12)],13)
 => ? = 11 - 1
[2,1,4,3,5] => [2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
 => ([(2,3),(3,9),(4,9),(4,11),(4,12),(4,13),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,10),(6,12),(6,13),(7,8),(7,10),(7,11),(7,13),(8,10),(8,11),(8,12),(9,11),(9,12),(9,13),(10,11),(10,12),(10,13),(11,12),(11,13),(12,13)],14)
 => ? = 16 - 1
[2,1,4,5,3] => [2,4,5,1,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
 => ([(2,3),(3,12),(4,7),(4,9),(4,11),(4,12),(5,6),(5,8),(5,10),(5,12),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
 => ? = 11 - 1
[2,1,5,3,4] => [5,2,1,3,4] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ([(2,3),(2,10),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(5,10),(6,7),(6,9),(6,10),(7,8),(7,10),(8,10),(9,10)],11)
 => ? = 16 - 1
[2,1,5,4,3] => [5,2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
 => ([(2,3),(3,9),(4,9),(4,11),(4,12),(4,13),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,10),(6,12),(6,13),(7,8),(7,10),(7,11),(7,13),(8,10),(8,11),(8,12),(9,11),(9,12),(9,13),(10,11),(10,12),(10,13),(11,12),(11,13),(12,13)],14)
 => ? = 9 - 1
[2,3,1,4,5] => [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
 => ? = 9 - 1
[2,3,1,5,4] => [2,3,5,1,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
 => ([(2,3),(3,11),(4,6),(4,9),(4,10),(4,11),(5,7),(5,8),(5,10),(5,11),(5,12),(6,8),(6,9),(6,10),(6,12),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,11),(8,12),(9,10),(9,12),(10,12),(11,12)],13)
 => ? = 16 - 1
[2,3,4,1,5] => [2,3,4,1,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
 => ? = 9 - 1
[2,3,4,5,1] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
 => ? = 4 - 1
[2,3,5,1,4] => [5,2,3,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
 => ([(2,3),(2,12),(3,11),(4,5),(4,6),(4,7),(4,8),(4,12),(5,6),(5,8),(5,10),(5,11),(6,8),(6,9),(6,11),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
 => ? = 9 - 1
[2,3,5,4,1] => [5,2,3,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ([(2,3),(2,10),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(5,10),(6,7),(6,9),(6,10),(7,8),(7,10),(8,10),(9,10)],11)
 => ? = 6 - 1
[2,4,1,3,5] => [4,2,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
 => ([(2,3),(2,11),(3,10),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,10),(6,7),(6,8),(6,11),(7,8),(7,9),(7,10),(8,9),(8,11),(9,10),(9,11),(10,11)],12)
 => ? = 9 - 1
[2,4,1,5,3] => [4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
 => ([(2,3),(3,13),(4,5),(4,6),(4,11),(4,12),(4,13),(5,6),(5,10),(5,11),(5,13),(6,9),(6,11),(6,13),(7,8),(7,9),(7,10),(7,12),(7,13),(8,9),(8,10),(8,12),(8,13),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12),(11,13),(12,13)],14)
 => ? = 16 - 1
[2,4,3,1,5] => [4,2,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
 => ([(2,3),(2,12),(3,11),(4,5),(4,6),(4,7),(4,8),(4,12),(5,6),(5,8),(5,10),(5,11),(6,8),(6,9),(6,11),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
 => ? = 11 - 1
[5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0 = 1 - 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => 0 = 1 - 1
[6,5,4,3,2,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => 0 = 1 - 1
Description
The biclique partition number of a graph. The biclique partition number of a graph is the minimum number of pairwise edge disjoint complete bipartite subgraphs so that each edge belongs to exactly one of them. A theorem of Graham and Pollak [1] asserts that the complete graph K_n has biclique partition number n - 1.
The following 2 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset.