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Your data matches 10 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000391
Mapping
Mp00131: Permutations —descent bottoms⟶ Binary words
St000391: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 0 => 0
[2,1] => 1 => 1
[1,2,3] => 00 => 0
[1,3,2] => 01 => 2
[2,1,3] => 10 => 1
[2,3,1] => 10 => 1
[3,1,2] => 10 => 1
[3,2,1] => 11 => 3
[1,2,3,4] => 000 => 0
[1,2,4,3] => 001 => 3
[1,3,2,4] => 010 => 2
[1,3,4,2] => 010 => 2
[1,4,2,3] => 010 => 2
[1,4,3,2] => 011 => 5
[2,1,3,4] => 100 => 1
[2,1,4,3] => 101 => 4
[2,3,1,4] => 100 => 1
[2,3,4,1] => 100 => 1
[2,4,1,3] => 100 => 1
[2,4,3,1] => 101 => 4
[3,1,2,4] => 100 => 1
[3,1,4,2] => 110 => 3
[3,2,1,4] => 110 => 3
[3,2,4,1] => 110 => 3
[3,4,1,2] => 100 => 1
[3,4,2,1] => 110 => 3
[4,1,2,3] => 100 => 1
[4,1,3,2] => 110 => 3
[4,2,1,3] => 110 => 3
[4,2,3,1] => 110 => 3
[4,3,1,2] => 101 => 4
[4,3,2,1] => 111 => 6
[1,2,3,4,5] => 0000 => 0
[1,2,3,5,4] => 0001 => 4
[1,2,4,3,5] => 0010 => 3
[1,2,4,5,3] => 0010 => 3
[1,2,5,3,4] => 0010 => 3
[1,2,5,4,3] => 0011 => 7
[1,3,2,4,5] => 0100 => 2
[1,3,2,5,4] => 0101 => 6
[1,3,4,2,5] => 0100 => 2
[1,3,4,5,2] => 0100 => 2
[1,3,5,2,4] => 0100 => 2
[1,3,5,4,2] => 0101 => 6
[1,4,2,3,5] => 0100 => 2
[1,4,2,5,3] => 0110 => 5
[1,4,3,2,5] => 0110 => 5
[1,4,3,5,2] => 0110 => 5
[1,4,5,2,3] => 0100 => 2
[1,4,5,3,2] => 0110 => 5
Description
The sum of the positions of the ones in a binary word.
Matching statistic: St000008
Mapping
Mp00131: Permutations —descent bottoms⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St000008: Integer compositions ⟶ ℤResult quality: 80% ●values known / values provided: 87%●distinct values known / distinct values provided: 80%
Values
[1,2] => 0 => [2] => 0
[2,1] => 1 => [1,1] => 1
[1,2,3] => 00 => [3] => 0
[1,3,2] => 01 => [2,1] => 2
[2,1,3] => 10 => [1,2] => 1
[2,3,1] => 10 => [1,2] => 1
[3,1,2] => 10 => [1,2] => 1
[3,2,1] => 11 => [1,1,1] => 3
[1,2,3,4] => 000 => [4] => 0
[1,2,4,3] => 001 => [3,1] => 3
[1,3,2,4] => 010 => [2,2] => 2
[1,3,4,2] => 010 => [2,2] => 2
[1,4,2,3] => 010 => [2,2] => 2
[1,4,3,2] => 011 => [2,1,1] => 5
[2,1,3,4] => 100 => [1,3] => 1
[2,1,4,3] => 101 => [1,2,1] => 4
[2,3,1,4] => 100 => [1,3] => 1
[2,3,4,1] => 100 => [1,3] => 1
[2,4,1,3] => 100 => [1,3] => 1
[2,4,3,1] => 101 => [1,2,1] => 4
[3,1,2,4] => 100 => [1,3] => 1
[3,1,4,2] => 110 => [1,1,2] => 3
[3,2,1,4] => 110 => [1,1,2] => 3
[3,2,4,1] => 110 => [1,1,2] => 3
[3,4,1,2] => 100 => [1,3] => 1
[3,4,2,1] => 110 => [1,1,2] => 3
[4,1,2,3] => 100 => [1,3] => 1
[4,1,3,2] => 110 => [1,1,2] => 3
[4,2,1,3] => 110 => [1,1,2] => 3
[4,2,3,1] => 110 => [1,1,2] => 3
[4,3,1,2] => 101 => [1,2,1] => 4
[4,3,2,1] => 111 => [1,1,1,1] => 6
[1,2,3,4,5] => 0000 => [5] => 0
[1,2,3,5,4] => 0001 => [4,1] => 4
[1,2,4,3,5] => 0010 => [3,2] => 3
[1,2,4,5,3] => 0010 => [3,2] => 3
[1,2,5,3,4] => 0010 => [3,2] => 3
[1,2,5,4,3] => 0011 => [3,1,1] => 7
[1,3,2,4,5] => 0100 => [2,3] => 2
[1,3,2,5,4] => 0101 => [2,2,1] => 6
[1,3,4,2,5] => 0100 => [2,3] => 2
[1,3,4,5,2] => 0100 => [2,3] => 2
[1,3,5,2,4] => 0100 => [2,3] => 2
[1,3,5,4,2] => 0101 => [2,2,1] => 6
[1,4,2,3,5] => 0100 => [2,3] => 2
[1,4,2,5,3] => 0110 => [2,1,2] => 5
[1,4,3,2,5] => 0110 => [2,1,2] => 5
[1,4,3,5,2] => 0110 => [2,1,2] => 5
[1,4,5,2,3] => 0100 => [2,3] => 2
[1,4,5,3,2] => 0110 => [2,1,2] => 5
[2,1,4,3,6,5,8,7,10,9] => 101010101 => [1,2,2,2,2,1] => ? = 25
[3,4,1,2,6,5,8,7,10,9] => 100010101 => [1,4,2,2,1] => ? = 22
[4,3,2,1,6,5,8,7,10,9] => 111010101 => [1,1,1,2,2,2,1] => ? = 27
[6,3,2,5,4,1,8,7,10,9] => 111100101 => [1,1,1,1,3,2,1] => ? = 26
[8,3,2,5,4,7,6,1,10,9] => 111101001 => [1,1,1,1,2,3,1] => ? = 25
[10,3,2,5,4,7,6,9,8,1] => 111101010 => [1,1,1,1,2,2,2] => ? = 24
[3,5,1,6,2,4,8,7,10,9] => 110000101 => [1,1,5,2,1] => ? = 19
[2,1,5,6,3,4,8,7,10,9] => 101000101 => [1,2,4,2,1] => ? = 20
[2,1,6,5,4,3,8,7,10,9] => 101110101 => [1,2,1,1,2,2,1] => ? = 29
[6,5,4,3,2,1,8,7,10,9] => 111110101 => [1,1,1,1,1,2,2,1] => ? = 31
[8,5,4,3,2,7,6,1,10,9] => 111111001 => [1,1,1,1,1,1,3,1] => ? = 30
[2,1,8,5,4,7,6,3,10,9] => 101111001 => [1,2,1,1,1,3,1] => ? = 28
[8,7,4,3,6,5,2,1,10,9] => 111110101 => [1,1,1,1,1,2,2,1] => ? = 31
[10,7,4,3,6,5,2,9,8,1] => 111110110 => [1,1,1,1,1,2,1,2] => ? = 30
[2,1,10,5,4,7,6,9,8,3] => 101111010 => [1,2,1,1,1,2,2] => ? = 27
[10,9,4,3,6,5,8,7,2,1] => 111110101 => [1,1,1,1,1,2,2,1] => ? = 31
[8,9,5,6,3,4,10,1,2,7] => 101010000 => [1,2,2,5] => ? = 9
[2,1,5,7,3,8,4,6,10,9] => 101100001 => [1,2,1,5,1] => ? = 17
[3,5,1,7,2,8,4,6,10,9] => 110100001 => [1,1,2,5,1] => ? = 16
[2,1,4,3,7,8,5,6,10,9] => 101010001 => [1,2,2,4,1] => ? = 18
[2,1,4,3,8,7,6,5,10,9] => 101011101 => [1,2,2,1,1,2,1] => ? = 31
[4,3,2,1,8,7,6,5,10,9] => 111011101 => [1,1,1,2,1,1,2,1] => ? = 33
[8,3,2,7,6,5,4,1,10,9] => 111111001 => [1,1,1,1,1,1,3,1] => ? = 30
[2,1,8,7,6,5,4,3,10,9] => 101111101 => [1,2,1,1,1,1,2,1] => ? = 35
[8,7,6,5,4,3,2,1,10,9] => 111111101 => [1,1,1,1,1,1,1,2,1] => ? = 37
[9,7,6,5,4,3,2,10,1,8] => 111111100 => [1,1,1,1,1,1,1,3] => ? = 28
[10,7,6,5,4,3,2,9,8,1] => 111111110 => [1,1,1,1,1,1,1,1,2] => ? = 36
[2,1,10,7,6,5,4,9,8,3] => 101111110 => [1,2,1,1,1,1,1,2] => ? = 34
[10,9,6,5,4,3,8,7,2,1] => 111111101 => [1,1,1,1,1,1,1,2,1] => ? = 37
[2,1,4,3,10,7,6,9,8,5] => 101011110 => [1,2,2,1,1,1,2] => ? = 30
[4,3,2,1,10,7,6,9,8,5] => 111011110 => [1,1,1,2,1,1,1,2] => ? = 32
[10,3,2,9,6,5,8,7,4,1] => 111111100 => [1,1,1,1,1,1,1,3] => ? = 28
[2,1,10,9,6,5,8,7,4,3] => 101111101 => [1,2,1,1,1,1,2,1] => ? = 35
[10,9,8,5,4,7,6,3,2,1] => 111111011 => [1,1,1,1,1,1,2,1,1] => ? = 38
[2,1,7,8,9,10,3,4,5,6] => 101000000 => [1,2,7] => ? = 4
[3,7,1,8,9,10,2,4,5,6] => 110000000 => [1,1,8] => ? = 3
[4,7,8,1,9,10,2,3,5,6] => 110000000 => [1,1,8] => ? = 3
[5,7,8,9,1,10,2,3,4,6] => 110000000 => [1,1,8] => ? = 3
[5,6,8,9,1,2,10,3,4,7] => 101000000 => [1,2,7] => ? = 4
[4,6,8,1,9,2,10,3,5,7] => 111000000 => [1,1,1,7] => ? = 6
[3,6,1,8,9,2,10,4,5,7] => 110100000 => [1,1,2,6] => ? = 7
[2,1,6,8,9,3,10,4,5,7] => 101100000 => [1,2,1,6] => ? = 8
[2,1,5,8,3,9,10,4,6,7] => 101100000 => [1,2,1,6] => ? = 8
[3,5,1,8,2,9,10,4,6,7] => 110100000 => [1,1,2,6] => ? = 7
[4,5,8,1,2,9,10,3,6,7] => 101000000 => [1,2,7] => ? = 4
[7,3,2,8,9,10,1,4,5,6] => 111000000 => [1,1,1,7] => ? = 6
[3,4,1,2,8,9,10,5,6,7] => 100010000 => [1,4,5] => ? = 6
[2,1,4,3,8,9,10,5,6,7] => 101010000 => [1,2,2,5] => ? = 9
[2,1,4,3,7,9,5,10,6,8] => 101011000 => [1,2,2,1,4] => ? = 15
[3,4,1,2,7,9,5,10,6,8] => 100011000 => [1,4,1,4] => ? = 12
Description
The major index of the composition. The descents of a composition $[c_1,c_2,\dots,c_k]$ are the partial sums $c_1, c_1+c_2,\dots, c_1+\dots+c_{k-1}$, excluding the sum of all parts. The major index of a composition is the sum of its descents. For details about the major index see [[Permutations/Descents-Major]].
Matching statistic: St000947
Mapping
Mp00131: Permutations —descent bottoms⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000947: Dyck paths ⟶ ℤResult quality: 63% ●values known / values provided: 84%●distinct values known / distinct values provided: 63%
Values
[1,2] => 0 => [2] => [1,1,0,0]
 => 0
[2,1] => 1 => [1,1] => [1,0,1,0]
 => 1
[1,2,3] => 00 => [3] => [1,1,1,0,0,0]
 => 0
[1,3,2] => 01 => [2,1] => [1,1,0,0,1,0]
 => 2
[2,1,3] => 10 => [1,2] => [1,0,1,1,0,0]
 => 1
[2,3,1] => 10 => [1,2] => [1,0,1,1,0,0]
 => 1
[3,1,2] => 10 => [1,2] => [1,0,1,1,0,0]
 => 1
[3,2,1] => 11 => [1,1,1] => [1,0,1,0,1,0]
 => 3
[1,2,3,4] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[1,2,4,3] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 3
[1,3,2,4] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[1,3,4,2] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[1,4,2,3] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[1,4,3,2] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 5
[2,1,3,4] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[2,1,4,3] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 4
[2,3,1,4] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[2,3,4,1] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[2,4,1,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[2,4,3,1] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 4
[3,1,2,4] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[3,1,4,2] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 3
[3,2,1,4] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 3
[3,2,4,1] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 3
[3,4,1,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[3,4,2,1] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 3
[4,1,2,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[4,1,3,2] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 3
[4,2,1,3] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 3
[4,2,3,1] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 3
[4,3,1,2] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 4
[4,3,2,1] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 6
[1,2,3,4,5] => 0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,2,3,5,4] => 0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 4
[1,2,4,3,5] => 0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3
[1,2,4,5,3] => 0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3
[1,2,5,3,4] => 0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3
[1,2,5,4,3] => 0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 7
[1,3,2,4,5] => 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
[1,3,2,5,4] => 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 6
[1,3,4,2,5] => 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
[1,3,4,5,2] => 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
[1,3,5,2,4] => 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
[1,3,5,4,2] => 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 6
[1,4,2,3,5] => 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
[1,4,2,5,3] => 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 5
[1,4,3,2,5] => 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 5
[1,4,3,5,2] => 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 5
[1,4,5,2,3] => 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
[1,4,5,3,2] => 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 5
[1,2,3,4,5,6,7,8] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[1,2,8,7,6,5,4,3] => 0011111 => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 25
[1,2,7,8,6,5,4,3] => 0011110 => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 18
[1,2,7,6,8,5,4,3] => 0011110 => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 18
[1,2,6,7,8,5,4,3] => 0011100 => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 12
[1,2,5,8,7,6,4,3] => 0011011 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 20
[1,2,6,5,8,7,4,3] => 0011101 => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 19
[1,2,5,6,8,7,4,3] => 0011001 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 14
[1,2,7,6,5,8,4,3] => 0011110 => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 18
[1,2,5,7,6,8,4,3] => 0011010 => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 13
[1,2,6,5,7,8,4,3] => 0011100 => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 12
[1,2,5,6,7,8,4,3] => 0011000 => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 7
[1,2,4,8,7,6,5,3] => 0010111 => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 21
[1,2,4,7,8,6,5,3] => 0010110 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 14
[1,2,4,6,8,7,5,3] => 0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 15
[1,2,4,7,6,8,5,3] => 0010110 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 14
[1,2,4,6,7,8,5,3] => 0010100 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 8
[1,2,5,4,8,7,6,3] => 0011011 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 20
[1,2,5,4,7,8,6,3] => 0011010 => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 13
[1,2,4,5,8,7,6,3] => 0010011 => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 16
[1,2,4,5,7,8,6,3] => 0010010 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 9
[1,2,6,5,4,8,7,3] => 0011101 => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 19
[1,2,4,6,5,8,7,3] => 0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 15
[1,2,5,4,6,8,7,3] => 0011001 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 14
[1,2,4,5,6,8,7,3] => 0010001 => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 10
[1,2,7,6,5,4,8,3] => 0011110 => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 18
[1,2,5,7,6,4,8,3] => 0011010 => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 13
[1,2,5,6,7,4,8,3] => 0011000 => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 7
[1,2,4,7,6,5,8,3] => 0010110 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 14
[1,2,5,4,7,6,8,3] => 0011010 => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 13
[1,2,6,5,4,7,8,3] => 0011100 => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 12
[1,2,5,6,4,7,8,3] => 0011000 => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 7
[1,2,4,6,5,7,8,3] => 0010100 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 8
[1,2,5,4,6,7,8,3] => 0011000 => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 7
[1,2,4,5,6,7,8,3] => 0010000 => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 3
[1,2,3,8,7,6,5,4] => 0001111 => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 22
[1,2,3,7,8,6,5,4] => 0001110 => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => ? = 15
[1,2,3,6,8,7,5,4] => 0001101 => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => ? = 16
[1,2,3,7,6,8,5,4] => 0001110 => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => ? = 15
[1,2,3,6,7,8,5,4] => 0001100 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 9
[1,2,3,5,8,7,6,4] => 0001011 => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
 => ? = 17
[1,2,3,5,7,8,6,4] => 0001010 => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 10
[1,2,3,6,5,8,7,4] => 0001101 => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => ? = 16
[1,2,3,5,6,8,7,4] => 0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 11
[1,2,3,7,6,5,8,4] => 0001110 => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => ? = 15
[1,2,3,6,7,5,8,4] => 0001100 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 9
[1,2,3,5,7,6,8,4] => 0001010 => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 10
[1,2,3,6,5,7,8,4] => 0001100 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 9
[1,2,3,5,6,7,8,4] => 0001000 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 4
[1,2,4,3,8,7,6,5] => 0010111 => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 21
Description
The major index east count of a Dyck path. The descent set $\operatorname{des}(D)$ of a Dyck path $D = D_1 \cdots D_{2n}$ with $D_i \in \{N,E\}$ is given by all indices $i$ such that $D_i = E$ and $D_{i+1} = N$. This is, the positions of the valleys of $D$. The '''major index''' of a Dyck path is then the sum of the positions of the valleys, $\sum_{i \in \operatorname{des}(D)} i$, see [[St000027]]. The '''major index east count''' is given by $\sum_{i \in \operatorname{des}(D)} \#\{ j \leq i \mid D_j = E\}$.
Matching statistic: St001161
Mapping
Mp00131: Permutations —descent bottoms⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001161: Dyck paths ⟶ ℤResult quality: 63% ●values known / values provided: 84%●distinct values known / distinct values provided: 63%
Values
[1,2] => 0 => [2] => [1,1,0,0]
 => 0
[2,1] => 1 => [1,1] => [1,0,1,0]
 => 1
[1,2,3] => 00 => [3] => [1,1,1,0,0,0]
 => 0
[1,3,2] => 01 => [2,1] => [1,1,0,0,1,0]
 => 2
[2,1,3] => 10 => [1,2] => [1,0,1,1,0,0]
 => 1
[2,3,1] => 10 => [1,2] => [1,0,1,1,0,0]
 => 1
[3,1,2] => 10 => [1,2] => [1,0,1,1,0,0]
 => 1
[3,2,1] => 11 => [1,1,1] => [1,0,1,0,1,0]
 => 3
[1,2,3,4] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[1,2,4,3] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 3
[1,3,2,4] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[1,3,4,2] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[1,4,2,3] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[1,4,3,2] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 5
[2,1,3,4] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[2,1,4,3] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 4
[2,3,1,4] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[2,3,4,1] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[2,4,1,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[2,4,3,1] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 4
[3,1,2,4] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[3,1,4,2] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 3
[3,2,1,4] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 3
[3,2,4,1] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 3
[3,4,1,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[3,4,2,1] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 3
[4,1,2,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[4,1,3,2] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 3
[4,2,1,3] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 3
[4,2,3,1] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 3
[4,3,1,2] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 4
[4,3,2,1] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 6
[1,2,3,4,5] => 0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,2,3,5,4] => 0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 4
[1,2,4,3,5] => 0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3
[1,2,4,5,3] => 0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3
[1,2,5,3,4] => 0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3
[1,2,5,4,3] => 0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 7
[1,3,2,4,5] => 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
[1,3,2,5,4] => 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 6
[1,3,4,2,5] => 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
[1,3,4,5,2] => 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
[1,3,5,2,4] => 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
[1,3,5,4,2] => 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 6
[1,4,2,3,5] => 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
[1,4,2,5,3] => 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 5
[1,4,3,2,5] => 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 5
[1,4,3,5,2] => 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 5
[1,4,5,2,3] => 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
[1,4,5,3,2] => 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 5
[1,2,3,4,5,6,7,8] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[1,2,8,7,6,5,4,3] => 0011111 => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 25
[1,2,7,8,6,5,4,3] => 0011110 => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 18
[1,2,7,6,8,5,4,3] => 0011110 => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 18
[1,2,6,7,8,5,4,3] => 0011100 => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 12
[1,2,5,8,7,6,4,3] => 0011011 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 20
[1,2,6,5,8,7,4,3] => 0011101 => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 19
[1,2,5,6,8,7,4,3] => 0011001 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 14
[1,2,7,6,5,8,4,3] => 0011110 => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 18
[1,2,5,7,6,8,4,3] => 0011010 => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 13
[1,2,6,5,7,8,4,3] => 0011100 => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 12
[1,2,5,6,7,8,4,3] => 0011000 => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 7
[1,2,4,8,7,6,5,3] => 0010111 => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 21
[1,2,4,7,8,6,5,3] => 0010110 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 14
[1,2,4,6,8,7,5,3] => 0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 15
[1,2,4,7,6,8,5,3] => 0010110 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 14
[1,2,4,6,7,8,5,3] => 0010100 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 8
[1,2,5,4,8,7,6,3] => 0011011 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 20
[1,2,5,4,7,8,6,3] => 0011010 => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 13
[1,2,4,5,8,7,6,3] => 0010011 => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 16
[1,2,4,5,7,8,6,3] => 0010010 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 9
[1,2,6,5,4,8,7,3] => 0011101 => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 19
[1,2,4,6,5,8,7,3] => 0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 15
[1,2,5,4,6,8,7,3] => 0011001 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 14
[1,2,4,5,6,8,7,3] => 0010001 => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 10
[1,2,7,6,5,4,8,3] => 0011110 => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 18
[1,2,5,7,6,4,8,3] => 0011010 => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 13
[1,2,5,6,7,4,8,3] => 0011000 => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 7
[1,2,4,7,6,5,8,3] => 0010110 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 14
[1,2,5,4,7,6,8,3] => 0011010 => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 13
[1,2,6,5,4,7,8,3] => 0011100 => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 12
[1,2,5,6,4,7,8,3] => 0011000 => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 7
[1,2,4,6,5,7,8,3] => 0010100 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 8
[1,2,5,4,6,7,8,3] => 0011000 => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 7
[1,2,4,5,6,7,8,3] => 0010000 => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 3
[1,2,3,8,7,6,5,4] => 0001111 => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 22
[1,2,3,7,8,6,5,4] => 0001110 => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => ? = 15
[1,2,3,6,8,7,5,4] => 0001101 => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => ? = 16
[1,2,3,7,6,8,5,4] => 0001110 => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => ? = 15
[1,2,3,6,7,8,5,4] => 0001100 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 9
[1,2,3,5,8,7,6,4] => 0001011 => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
 => ? = 17
[1,2,3,5,7,8,6,4] => 0001010 => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 10
[1,2,3,6,5,8,7,4] => 0001101 => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => ? = 16
[1,2,3,5,6,8,7,4] => 0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 11
[1,2,3,7,6,5,8,4] => 0001110 => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => ? = 15
[1,2,3,6,7,5,8,4] => 0001100 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 9
[1,2,3,5,7,6,8,4] => 0001010 => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 10
[1,2,3,6,5,7,8,4] => 0001100 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 9
[1,2,3,5,6,7,8,4] => 0001000 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 4
[1,2,4,3,8,7,6,5] => 0010111 => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 21
Description
The major index north count of a Dyck path. The descent set $\operatorname{des}(D)$ of a Dyck path $D = D_1 \cdots D_{2n}$ with $D_i \in \{N,E\}$ is given by all indices $i$ such that $D_i = E$ and $D_{i+1} = N$. This is, the positions of the valleys of $D$. The '''major index''' of a Dyck path is then the sum of the positions of the valleys, $\sum_{i \in \operatorname{des}(D)} i$, see [[St000027]]. The '''major index north count''' is given by $\sum_{i \in \operatorname{des}(D)} \#\{ j \leq i \mid D_j = N\}$.
Matching statistic: St000081
Mapping
Mp00131: Permutations —descent bottoms⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000081: Graphs ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 70%
Values
[1,2] => 0 => [2] => ([],2)
 => 0
[2,1] => 1 => [1,1] => ([(0,1)],2)
 => 1
[1,2,3] => 00 => [3] => ([],3)
 => 0
[1,3,2] => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[2,1,3] => 10 => [1,2] => ([(1,2)],3)
 => 1
[2,3,1] => 10 => [1,2] => ([(1,2)],3)
 => 1
[3,1,2] => 10 => [1,2] => ([(1,2)],3)
 => 1
[3,2,1] => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[1,2,3,4] => 000 => [4] => ([],4)
 => 0
[1,2,4,3] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[1,3,2,4] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 2
[1,3,4,2] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 2
[1,4,2,3] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 2
[1,4,3,2] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 5
[2,1,3,4] => 100 => [1,3] => ([(2,3)],4)
 => 1
[2,1,4,3] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[2,3,1,4] => 100 => [1,3] => ([(2,3)],4)
 => 1
[2,3,4,1] => 100 => [1,3] => ([(2,3)],4)
 => 1
[2,4,1,3] => 100 => [1,3] => ([(2,3)],4)
 => 1
[2,4,3,1] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[3,1,2,4] => 100 => [1,3] => ([(2,3)],4)
 => 1
[3,1,4,2] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3
[3,2,1,4] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3
[3,2,4,1] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3
[3,4,1,2] => 100 => [1,3] => ([(2,3)],4)
 => 1
[3,4,2,1] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3
[4,1,2,3] => 100 => [1,3] => ([(2,3)],4)
 => 1
[4,1,3,2] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3
[4,2,1,3] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3
[4,2,3,1] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3
[4,3,1,2] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,3,2,1] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 6
[1,2,3,4,5] => 0000 => [5] => ([],5)
 => 0
[1,2,3,5,4] => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[1,2,4,3,5] => 0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[1,2,4,5,3] => 0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[1,2,5,3,4] => 0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[1,2,5,4,3] => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 7
[1,3,2,4,5] => 0100 => [2,3] => ([(2,4),(3,4)],5)
 => 2
[1,3,2,5,4] => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 6
[1,3,4,2,5] => 0100 => [2,3] => ([(2,4),(3,4)],5)
 => 2
[1,3,4,5,2] => 0100 => [2,3] => ([(2,4),(3,4)],5)
 => 2
[1,3,5,2,4] => 0100 => [2,3] => ([(2,4),(3,4)],5)
 => 2
[1,3,5,4,2] => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 6
[1,4,2,3,5] => 0100 => [2,3] => ([(2,4),(3,4)],5)
 => 2
[1,4,2,5,3] => 0110 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,4,3,2,5] => 0110 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,4,3,5,2] => 0110 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,4,5,2,3] => 0100 => [2,3] => ([(2,4),(3,4)],5)
 => 2
[1,4,5,3,2] => 0110 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[7,8,6,5,4,3,2,1] => 1111110 => [1,1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 21
[8,6,7,5,4,3,2,1] => 1111110 => [1,1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 21
[7,6,8,5,4,3,2,1] => 1111110 => [1,1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 21
[6,7,8,5,4,3,2,1] => 1111100 => [1,1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 15
[7,8,5,6,4,3,2,1] => 1111100 => [1,1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 15
[8,6,5,7,4,3,2,1] => 1111110 => [1,1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 21
[8,5,6,7,4,3,2,1] => 1111100 => [1,1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 15
[7,6,5,8,4,3,2,1] => 1111110 => [1,1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 21
[6,7,5,8,4,3,2,1] => 1111100 => [1,1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 15
[7,5,6,8,4,3,2,1] => 1111100 => [1,1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 15
[6,5,7,8,4,3,2,1] => 1111100 => [1,1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 15
[5,6,7,8,4,3,2,1] => 1111000 => [1,1,1,1,4] => ([(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 10
[8,7,6,4,5,3,2,1] => 1111011 => [1,1,1,1,2,1,1] => ([(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 23
[7,8,6,4,5,3,2,1] => 1111010 => [1,1,1,1,2,2] => ([(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 16
[8,6,7,4,5,3,2,1] => 1111010 => [1,1,1,1,2,2] => ([(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 16
[7,6,8,4,5,3,2,1] => 1111010 => [1,1,1,1,2,2] => ([(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 16
[6,7,8,4,5,3,2,1] => 1111000 => [1,1,1,1,4] => ([(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 10
[8,7,4,5,6,3,2,1] => 1111001 => [1,1,1,1,3,1] => ([(0,7),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 17
[8,6,5,4,7,3,2,1] => 1111110 => [1,1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 21
[8,5,6,4,7,3,2,1] => 1111100 => [1,1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 15
[8,6,4,5,7,3,2,1] => 1111010 => [1,1,1,1,2,2] => ([(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 16
[8,5,4,6,7,3,2,1] => 1111100 => [1,1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 15
[8,4,5,6,7,3,2,1] => 1111000 => [1,1,1,1,4] => ([(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 10
[7,6,5,4,8,3,2,1] => 1111110 => [1,1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 21
[6,7,5,4,8,3,2,1] => 1111100 => [1,1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 15
[7,5,6,4,8,3,2,1] => 1111100 => [1,1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 15
[6,5,7,4,8,3,2,1] => 1111100 => [1,1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 15
[5,6,7,4,8,3,2,1] => 1111000 => [1,1,1,1,4] => ([(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 10
[7,6,4,5,8,3,2,1] => 1111010 => [1,1,1,1,2,2] => ([(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 16
[6,7,4,5,8,3,2,1] => 1111000 => [1,1,1,1,4] => ([(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 10
[7,5,4,6,8,3,2,1] => 1111100 => [1,1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 15
[7,4,5,6,8,3,2,1] => 1111000 => [1,1,1,1,4] => ([(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 10
[6,5,4,7,8,3,2,1] => 1111100 => [1,1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 15
[5,6,4,7,8,3,2,1] => 1111000 => [1,1,1,1,4] => ([(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 10
[6,4,5,7,8,3,2,1] => 1111000 => [1,1,1,1,4] => ([(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 10
[5,4,6,7,8,3,2,1] => 1111000 => [1,1,1,1,4] => ([(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 10
[4,5,6,7,8,3,2,1] => 1110000 => [1,1,1,5] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6
[8,7,6,5,3,4,2,1] => 1110111 => [1,1,1,2,1,1,1] => ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 24
[7,8,6,5,3,4,2,1] => 1110110 => [1,1,1,2,1,2] => ([(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 17
[8,6,7,5,3,4,2,1] => 1110110 => [1,1,1,2,1,2] => ([(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 17
[7,6,8,5,3,4,2,1] => 1110110 => [1,1,1,2,1,2] => ([(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 17
[8,7,5,6,3,4,2,1] => 1110101 => [1,1,1,2,2,1] => ([(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 18
[7,8,5,6,3,4,2,1] => 1110100 => [1,1,1,2,3] => ([(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 11
[8,5,6,7,3,4,2,1] => 1110100 => [1,1,1,2,3] => ([(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 11
[7,6,5,8,3,4,2,1] => 1110110 => [1,1,1,2,1,2] => ([(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 17
[6,7,5,8,3,4,2,1] => 1110100 => [1,1,1,2,3] => ([(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 11
[6,5,7,8,3,4,2,1] => 1110100 => [1,1,1,2,3] => ([(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 11
[5,6,7,8,3,4,2,1] => 1110000 => [1,1,1,5] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6
[8,7,6,4,3,5,2,1] => 1111011 => [1,1,1,1,2,1,1] => ([(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 23
[7,8,6,4,3,5,2,1] => 1111010 => [1,1,1,1,2,2] => ([(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 16
Description
The number of edges of a graph.
Matching statistic: St000492
(load all 3 compositions to match this statistic)
Mapping
Mp00237: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000492: Set partitions ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 48%
Values
[1,2] => [1,2] => [2,1] => {{1,2}}
 => 0
[2,1] => [2,1] => [1,2] => {{1},{2}}
 => 1
[1,2,3] => [1,2,3] => [2,3,1] => {{1,2,3}}
 => 0
[1,3,2] => [1,3,2] => [2,1,3] => {{1,2},{3}}
 => 2
[2,1,3] => [2,1,3] => [3,2,1] => {{1,3},{2}}
 => 1
[2,3,1] => [3,2,1] => [1,3,2] => {{1},{2,3}}
 => 1
[3,1,2] => [3,1,2] => [3,1,2] => {{1,3},{2}}
 => 1
[3,2,1] => [2,3,1] => [1,2,3] => {{1},{2},{3}}
 => 3
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => {{1,2,3,4}}
 => 0
[1,2,4,3] => [1,2,4,3] => [2,3,1,4] => {{1,2,3},{4}}
 => 3
[1,3,2,4] => [1,3,2,4] => [2,4,3,1] => {{1,2,4},{3}}
 => 2
[1,3,4,2] => [1,4,3,2] => [2,1,4,3] => {{1,2},{3,4}}
 => 2
[1,4,2,3] => [1,4,2,3] => [2,4,1,3] => {{1,2,4},{3}}
 => 2
[1,4,3,2] => [1,3,4,2] => [2,1,3,4] => {{1,2},{3},{4}}
 => 5
[2,1,3,4] => [2,1,3,4] => [3,2,4,1] => {{1,3,4},{2}}
 => 1
[2,1,4,3] => [2,1,4,3] => [3,2,1,4] => {{1,3},{2},{4}}
 => 4
[2,3,1,4] => [3,2,1,4] => [4,3,2,1] => {{1,4},{2,3}}
 => 1
[2,3,4,1] => [4,2,3,1] => [1,3,4,2] => {{1},{2,3,4}}
 => 1
[2,4,1,3] => [4,2,1,3] => [4,3,1,2] => {{1,4},{2,3}}
 => 1
[2,4,3,1] => [3,2,4,1] => [1,3,2,4] => {{1},{2,3},{4}}
 => 4
[3,1,2,4] => [3,1,2,4] => [3,4,2,1] => {{1,3},{2,4}}
 => 1
[3,1,4,2] => [3,4,1,2] => [4,1,2,3] => {{1,4},{2},{3}}
 => 3
[3,2,1,4] => [2,3,1,4] => [4,2,3,1] => {{1,4},{2},{3}}
 => 3
[3,2,4,1] => [4,3,2,1] => [1,4,3,2] => {{1},{2,4},{3}}
 => 3
[3,4,1,2] => [4,1,3,2] => [3,1,4,2] => {{1,3,4},{2}}
 => 1
[3,4,2,1] => [2,4,3,1] => [1,2,4,3] => {{1},{2},{3,4}}
 => 3
[4,1,2,3] => [4,1,2,3] => [3,4,1,2] => {{1,3},{2,4}}
 => 1
[4,1,3,2] => [4,3,1,2] => [4,1,3,2] => {{1,4},{2},{3}}
 => 3
[4,2,1,3] => [2,4,1,3] => [4,2,1,3] => {{1,4},{2},{3}}
 => 3
[4,2,3,1] => [3,4,2,1] => [1,4,2,3] => {{1},{2,4},{3}}
 => 3
[4,3,1,2] => [3,1,4,2] => [3,1,2,4] => {{1,3},{2},{4}}
 => 4
[4,3,2,1] => [2,3,4,1] => [1,2,3,4] => {{1},{2},{3},{4}}
 => 6
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => {{1,2,3,4,5}}
 => 0
[1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,1,5] => {{1,2,3,4},{5}}
 => 4
[1,2,4,3,5] => [1,2,4,3,5] => [2,3,5,4,1] => {{1,2,3,5},{4}}
 => 3
[1,2,4,5,3] => [1,2,5,4,3] => [2,3,1,5,4] => {{1,2,3},{4,5}}
 => 3
[1,2,5,3,4] => [1,2,5,3,4] => [2,3,5,1,4] => {{1,2,3,5},{4}}
 => 3
[1,2,5,4,3] => [1,2,4,5,3] => [2,3,1,4,5] => {{1,2,3},{4},{5}}
 => 7
[1,3,2,4,5] => [1,3,2,4,5] => [2,4,3,5,1] => {{1,2,4,5},{3}}
 => 2
[1,3,2,5,4] => [1,3,2,5,4] => [2,4,3,1,5] => {{1,2,4},{3},{5}}
 => 6
[1,3,4,2,5] => [1,4,3,2,5] => [2,5,4,3,1] => {{1,2,5},{3,4}}
 => 2
[1,3,4,5,2] => [1,5,3,4,2] => [2,1,4,5,3] => {{1,2},{3,4,5}}
 => 2
[1,3,5,2,4] => [1,5,3,2,4] => [2,5,4,1,3] => {{1,2,5},{3,4}}
 => 2
[1,3,5,4,2] => [1,4,3,5,2] => [2,1,4,3,5] => {{1,2},{3,4},{5}}
 => 6
[1,4,2,3,5] => [1,4,2,3,5] => [2,4,5,3,1] => {{1,2,4},{3,5}}
 => 2
[1,4,2,5,3] => [1,4,5,2,3] => [2,5,1,3,4] => {{1,2,5},{3},{4}}
 => 5
[1,4,3,2,5] => [1,3,4,2,5] => [2,5,3,4,1] => {{1,2,5},{3},{4}}
 => 5
[1,4,3,5,2] => [1,5,4,3,2] => [2,1,5,4,3] => {{1,2},{3,5},{4}}
 => 5
[1,4,5,2,3] => [1,5,2,4,3] => [2,4,1,5,3] => {{1,2,4,5},{3}}
 => 2
[1,4,5,3,2] => [1,3,5,4,2] => [2,1,3,5,4] => {{1,2},{3},{4,5}}
 => 5
[8,7,6,5,4,3,2,1] => [2,3,4,5,6,7,8,1] => [1,2,3,4,5,6,7,8] => {{1},{2},{3},{4},{5},{6},{7},{8}}
 => ? = 28
[7,8,6,5,4,3,2,1] => [2,3,4,5,6,8,7,1] => [1,2,3,4,5,6,8,7] => {{1},{2},{3},{4},{5},{6},{7,8}}
 => ? = 21
[8,6,7,5,4,3,2,1] => [2,3,4,5,7,8,6,1] => [1,2,3,4,5,8,6,7] => {{1},{2},{3},{4},{5},{6,8},{7}}
 => ? = 21
[7,6,8,5,4,3,2,1] => [2,3,4,5,8,7,6,1] => [1,2,3,4,5,8,7,6] => {{1},{2},{3},{4},{5},{6,8},{7}}
 => ? = 21
[6,7,8,5,4,3,2,1] => [2,3,4,5,8,6,7,1] => [1,2,3,4,5,7,8,6] => {{1},{2},{3},{4},{5},{6,7,8}}
 => ? = 15
[8,7,5,6,4,3,2,1] => [2,3,4,6,7,5,8,1] => [1,2,3,4,7,5,6,8] => ?
 => ? = 22
[7,8,5,6,4,3,2,1] => [2,3,4,6,8,5,7,1] => [1,2,3,4,7,5,8,6] => {{1},{2},{3},{4},{5,7,8},{6}}
 => ? = 15
[8,6,5,7,4,3,2,1] => [2,3,4,7,6,8,5,1] => [1,2,3,4,8,6,5,7] => ?
 => ? = 21
[8,5,6,7,4,3,2,1] => [2,3,4,7,8,5,6,1] => [1,2,3,4,7,8,5,6] => {{1},{2},{3},{4},{5,7},{6,8}}
 => ? = 15
[7,6,5,8,4,3,2,1] => [2,3,4,8,6,7,5,1] => [1,2,3,4,8,6,7,5] => ?
 => ? = 21
[6,7,5,8,4,3,2,1] => [2,3,4,8,7,6,5,1] => [1,2,3,4,8,7,6,5] => {{1},{2},{3},{4},{5,8},{6,7}}
 => ? = 15
[7,5,6,8,4,3,2,1] => [2,3,4,8,7,5,6,1] => [1,2,3,4,7,8,6,5] => {{1},{2},{3},{4},{5,7},{6,8}}
 => ? = 15
[6,5,7,8,4,3,2,1] => [2,3,4,8,6,5,7,1] => [1,2,3,4,7,6,8,5] => {{1},{2},{3},{4},{5,7,8},{6}}
 => ? = 15
[5,6,7,8,4,3,2,1] => [2,3,4,8,5,6,7,1] => [1,2,3,4,6,7,8,5] => {{1},{2},{3},{4},{5,6,7,8}}
 => ? = 10
[8,7,6,4,5,3,2,1] => [2,3,5,6,4,7,8,1] => [1,2,3,6,4,5,7,8] => {{1},{2},{3},{4,6},{5},{7},{8}}
 => ? = 23
[7,8,6,4,5,3,2,1] => [2,3,5,6,4,8,7,1] => [1,2,3,6,4,5,8,7] => ?
 => ? = 16
[8,6,7,4,5,3,2,1] => [2,3,5,7,4,8,6,1] => [1,2,3,6,4,8,5,7] => {{1},{2},{3},{4,6,8},{5},{7}}
 => ? = 16
[7,6,8,4,5,3,2,1] => [2,3,5,8,4,7,6,1] => [1,2,3,6,4,8,7,5] => ?
 => ? = 16
[6,7,8,4,5,3,2,1] => [2,3,5,8,4,6,7,1] => [1,2,3,6,4,7,8,5] => {{1},{2},{3},{4,6,7,8},{5}}
 => ? = 10
[8,7,5,4,6,3,2,1] => [2,3,6,5,7,4,8,1] => ? => ?
 => ? = 22
[8,7,4,5,6,3,2,1] => [2,3,6,7,4,5,8,1] => [1,2,3,6,7,4,5,8] => ?
 => ? = 17
[8,6,5,4,7,3,2,1] => [2,3,7,5,6,8,4,1] => [1,2,3,8,5,6,4,7] => ?
 => ? = 21
[8,5,6,4,7,3,2,1] => [2,3,7,6,8,5,4,1] => [1,2,3,8,7,5,4,6] => ?
 => ? = 15
[8,6,4,5,7,3,2,1] => [2,3,7,6,4,8,5,1] => [1,2,3,6,8,5,4,7] => ?
 => ? = 16
[8,5,4,6,7,3,2,1] => [2,3,7,5,8,4,6,1] => [1,2,3,7,5,8,4,6] => ?
 => ? = 15
[8,4,5,6,7,3,2,1] => [2,3,7,8,4,5,6,1] => [1,2,3,6,7,8,4,5] => ?
 => ? = 10
[7,6,5,4,8,3,2,1] => [2,3,8,5,6,7,4,1] => [1,2,3,8,5,6,7,4] => {{1},{2},{3},{4,8},{5},{6},{7}}
 => ? = 21
[6,7,5,4,8,3,2,1] => [2,3,8,5,7,6,4,1] => [1,2,3,8,5,7,6,4] => {{1},{2},{3},{4,8},{5},{6,7}}
 => ? = 15
[7,5,6,4,8,3,2,1] => [2,3,8,6,7,5,4,1] => [1,2,3,8,7,5,6,4] => {{1},{2},{3},{4,8},{5,7},{6}}
 => ? = 15
[6,5,7,4,8,3,2,1] => [2,3,8,7,6,5,4,1] => [1,2,3,8,7,6,5,4] => {{1},{2},{3},{4,8},{5,7},{6}}
 => ? = 15
[5,6,7,4,8,3,2,1] => [2,3,8,7,5,6,4,1] => [1,2,3,8,6,7,5,4] => {{1},{2},{3},{4,8},{5,6,7}}
 => ? = 10
[7,6,4,5,8,3,2,1] => [2,3,8,6,4,7,5,1] => [1,2,3,6,8,5,7,4] => {{1},{2},{3},{4,6},{5,8},{7}}
 => ? = 16
[6,7,4,5,8,3,2,1] => [2,3,8,7,4,6,5,1] => ? => ?
 => ? = 10
[7,5,4,6,8,3,2,1] => [2,3,8,5,7,4,6,1] => [1,2,3,7,5,8,6,4] => {{1},{2},{3},{4,7},{5},{6,8}}
 => ? = 15
[7,4,5,6,8,3,2,1] => [2,3,8,7,4,5,6,1] => [1,2,3,6,7,8,5,4] => {{1},{2},{3},{4,6,8},{5,7}}
 => ? = 10
[6,5,4,7,8,3,2,1] => [2,3,8,5,6,4,7,1] => [1,2,3,7,5,6,8,4] => {{1},{2},{3},{4,7,8},{5},{6}}
 => ? = 15
[5,6,4,7,8,3,2,1] => [2,3,8,6,5,4,7,1] => [1,2,3,7,6,5,8,4] => {{1},{2},{3},{4,7,8},{5,6}}
 => ? = 10
[6,4,5,7,8,3,2,1] => [2,3,8,6,4,5,7,1] => [1,2,3,6,7,5,8,4] => {{1},{2},{3},{4,6},{5,7,8}}
 => ? = 10
[5,4,6,7,8,3,2,1] => [2,3,8,5,4,6,7,1] => [1,2,3,6,5,7,8,4] => {{1},{2},{3},{4,6,7,8},{5}}
 => ? = 10
[4,5,6,7,8,3,2,1] => [2,3,8,4,5,6,7,1] => [1,2,3,5,6,7,8,4] => {{1},{2},{3},{4,5,6,7,8}}
 => ? = 6
[8,7,6,5,3,4,2,1] => [2,4,5,3,6,7,8,1] => [1,2,5,3,4,6,7,8] => ?
 => ? = 24
[7,8,6,5,3,4,2,1] => [2,4,5,3,6,8,7,1] => [1,2,5,3,4,6,8,7] => ?
 => ? = 17
[8,6,7,5,3,4,2,1] => [2,4,5,3,7,8,6,1] => [1,2,5,3,4,8,6,7] => ?
 => ? = 17
[7,6,8,5,3,4,2,1] => [2,4,5,3,8,7,6,1] => [1,2,5,3,4,8,7,6] => ?
 => ? = 17
[8,7,5,6,3,4,2,1] => [2,4,6,3,7,5,8,1] => [1,2,5,3,7,4,6,8] => ?
 => ? = 18
[7,8,5,6,3,4,2,1] => [2,4,6,3,8,5,7,1] => [1,2,5,3,7,4,8,6] => ?
 => ? = 11
[8,5,6,7,3,4,2,1] => [2,4,7,3,8,5,6,1] => [1,2,5,3,7,8,4,6] => ?
 => ? = 11
[7,6,5,8,3,4,2,1] => [2,4,8,3,6,7,5,1] => ? => ?
 => ? = 17
[6,7,5,8,3,4,2,1] => [2,4,8,3,7,6,5,1] => [1,2,5,3,8,7,6,4] => {{1},{2},{3,5,8},{4},{6,7}}
 => ? = 11
[6,5,7,8,3,4,2,1] => [2,4,8,3,6,5,7,1] => [1,2,5,3,7,6,8,4] => {{1},{2},{3,5,7,8},{4},{6}}
 => ? = 11
Description
The rob statistic of a set partition. Let $S = B_1,\ldots,B_k$ be a set partition with ordered blocks $B_i$ and with $\operatorname{min} B_a < \operatorname{min} B_b$ for $a < b$. According to [1, Definition 3], a '''rob''' (right-opener-bigger) of $S$ is given by a pair $i < j$ such that $j = \operatorname{min} B_b$ and $i \in B_a$ for $a < b$. This is also the number of occurrences of the pattern {{1}, {2}}, such that 2 is the minimal element of a block.
Matching statistic: St000472
(load all 8 compositions to match this statistic)
Mapping
Mp00064: Permutations —reverse⟶ Permutations
St000472: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 46%
Values
[1,2] => [2,1] => 0
[2,1] => [1,2] => 1
[1,2,3] => [3,2,1] => 0
[1,3,2] => [2,3,1] => 2
[2,1,3] => [3,1,2] => 1
[2,3,1] => [1,3,2] => 1
[3,1,2] => [2,1,3] => 1
[3,2,1] => [1,2,3] => 3
[1,2,3,4] => [4,3,2,1] => 0
[1,2,4,3] => [3,4,2,1] => 3
[1,3,2,4] => [4,2,3,1] => 2
[1,3,4,2] => [2,4,3,1] => 2
[1,4,2,3] => [3,2,4,1] => 2
[1,4,3,2] => [2,3,4,1] => 5
[2,1,3,4] => [4,3,1,2] => 1
[2,1,4,3] => [3,4,1,2] => 4
[2,3,1,4] => [4,1,3,2] => 1
[2,3,4,1] => [1,4,3,2] => 1
[2,4,1,3] => [3,1,4,2] => 1
[2,4,3,1] => [1,3,4,2] => 4
[3,1,2,4] => [4,2,1,3] => 1
[3,1,4,2] => [2,4,1,3] => 3
[3,2,1,4] => [4,1,2,3] => 3
[3,2,4,1] => [1,4,2,3] => 3
[3,4,1,2] => [2,1,4,3] => 1
[3,4,2,1] => [1,2,4,3] => 3
[4,1,2,3] => [3,2,1,4] => 1
[4,1,3,2] => [2,3,1,4] => 3
[4,2,1,3] => [3,1,2,4] => 3
[4,2,3,1] => [1,3,2,4] => 3
[4,3,1,2] => [2,1,3,4] => 4
[4,3,2,1] => [1,2,3,4] => 6
[1,2,3,4,5] => [5,4,3,2,1] => 0
[1,2,3,5,4] => [4,5,3,2,1] => 4
[1,2,4,3,5] => [5,3,4,2,1] => 3
[1,2,4,5,3] => [3,5,4,2,1] => 3
[1,2,5,3,4] => [4,3,5,2,1] => 3
[1,2,5,4,3] => [3,4,5,2,1] => 7
[1,3,2,4,5] => [5,4,2,3,1] => 2
[1,3,2,5,4] => [4,5,2,3,1] => 6
[1,3,4,2,5] => [5,2,4,3,1] => 2
[1,3,4,5,2] => [2,5,4,3,1] => 2
[1,3,5,2,4] => [4,2,5,3,1] => 2
[1,3,5,4,2] => [2,4,5,3,1] => 6
[1,4,2,3,5] => [5,3,2,4,1] => 2
[1,4,2,5,3] => [3,5,2,4,1] => 5
[1,4,3,2,5] => [5,2,3,4,1] => 5
[1,4,3,5,2] => [2,5,3,4,1] => 5
[1,4,5,2,3] => [3,2,5,4,1] => 2
[1,4,5,3,2] => [2,3,5,4,1] => 5
[1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 0
[1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => ? = 6
[1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => ? = 5
[1,2,3,4,6,7,5] => [5,7,6,4,3,2,1] => ? = 5
[1,2,3,4,7,5,6] => [6,5,7,4,3,2,1] => ? = 5
[1,2,3,4,7,6,5] => [5,6,7,4,3,2,1] => ? = 11
[1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => ? = 4
[1,2,3,5,4,7,6] => [6,7,4,5,3,2,1] => ? = 10
[1,2,3,5,6,4,7] => [7,4,6,5,3,2,1] => ? = 4
[1,2,3,5,6,7,4] => [4,7,6,5,3,2,1] => ? = 4
[1,2,3,5,7,4,6] => [6,4,7,5,3,2,1] => ? = 4
[1,2,3,5,7,6,4] => [4,6,7,5,3,2,1] => ? = 10
[1,2,3,6,4,5,7] => [7,5,4,6,3,2,1] => ? = 4
[1,2,3,6,4,7,5] => [5,7,4,6,3,2,1] => ? = 9
[1,2,3,6,5,4,7] => [7,4,5,6,3,2,1] => ? = 9
[1,2,3,6,5,7,4] => [4,7,5,6,3,2,1] => ? = 9
[1,2,3,6,7,4,5] => [5,4,7,6,3,2,1] => ? = 4
[1,2,3,6,7,5,4] => [4,5,7,6,3,2,1] => ? = 9
[1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => ? = 4
[1,2,3,7,4,6,5] => [5,6,4,7,3,2,1] => ? = 9
[1,2,3,7,5,4,6] => [6,4,5,7,3,2,1] => ? = 9
[1,2,3,7,5,6,4] => [4,6,5,7,3,2,1] => ? = 9
[1,2,3,7,6,4,5] => [5,4,6,7,3,2,1] => ? = 10
[1,2,3,7,6,5,4] => [4,5,6,7,3,2,1] => ? = 15
[1,2,4,3,5,6,7] => [7,6,5,3,4,2,1] => ? = 3
[1,2,4,3,5,7,6] => [6,7,5,3,4,2,1] => ? = 9
[1,2,4,3,6,5,7] => [7,5,6,3,4,2,1] => ? = 8
[1,2,4,3,6,7,5] => [5,7,6,3,4,2,1] => ? = 8
[1,2,4,3,7,5,6] => [6,5,7,3,4,2,1] => ? = 8
[1,2,4,3,7,6,5] => [5,6,7,3,4,2,1] => ? = 14
[1,2,4,5,3,6,7] => [7,6,3,5,4,2,1] => ? = 3
[1,2,4,5,3,7,6] => [6,7,3,5,4,2,1] => ? = 9
[1,2,4,5,6,3,7] => [7,3,6,5,4,2,1] => ? = 3
[1,2,4,5,6,7,3] => [3,7,6,5,4,2,1] => ? = 3
[1,2,4,5,7,3,6] => [6,3,7,5,4,2,1] => ? = 3
[1,2,4,5,7,6,3] => [3,6,7,5,4,2,1] => ? = 9
[1,2,4,6,3,5,7] => [7,5,3,6,4,2,1] => ? = 3
[1,2,4,6,3,7,5] => [5,7,3,6,4,2,1] => ? = 8
[1,2,4,6,5,3,7] => [7,3,5,6,4,2,1] => ? = 8
[1,2,4,6,5,7,3] => [3,7,5,6,4,2,1] => ? = 8
[1,2,4,6,7,3,5] => [5,3,7,6,4,2,1] => ? = 3
[1,2,4,6,7,5,3] => [3,5,7,6,4,2,1] => ? = 8
[1,2,4,7,3,5,6] => [6,5,3,7,4,2,1] => ? = 3
[1,2,4,7,3,6,5] => [5,6,3,7,4,2,1] => ? = 8
[1,2,4,7,5,3,6] => [6,3,5,7,4,2,1] => ? = 8
[1,2,4,7,5,6,3] => [3,6,5,7,4,2,1] => ? = 8
[1,2,4,7,6,3,5] => [5,3,6,7,4,2,1] => ? = 9
[1,2,4,7,6,5,3] => [3,5,6,7,4,2,1] => ? = 14
[1,2,5,3,4,6,7] => [7,6,4,3,5,2,1] => ? = 3
[1,2,5,3,4,7,6] => [6,7,4,3,5,2,1] => ? = 9
Description
The sum of the ascent bottoms of a permutation.
Matching statistic: St000154
(load all 13 compositions to match this statistic)
Mapping
St000154: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 35%
Values
[1,2] => 0
[2,1] => 1
[1,2,3] => 0
[1,3,2] => 2
[2,1,3] => 1
[2,3,1] => 1
[3,1,2] => 1
[3,2,1] => 3
[1,2,3,4] => 0
[1,2,4,3] => 3
[1,3,2,4] => 2
[1,3,4,2] => 2
[1,4,2,3] => 2
[1,4,3,2] => 5
[2,1,3,4] => 1
[2,1,4,3] => 4
[2,3,1,4] => 1
[2,3,4,1] => 1
[2,4,1,3] => 1
[2,4,3,1] => 4
[3,1,2,4] => 1
[3,1,4,2] => 3
[3,2,1,4] => 3
[3,2,4,1] => 3
[3,4,1,2] => 1
[3,4,2,1] => 3
[4,1,2,3] => 1
[4,1,3,2] => 3
[4,2,1,3] => 3
[4,2,3,1] => 3
[4,3,1,2] => 4
[4,3,2,1] => 6
[1,2,3,4,5] => 0
[1,2,3,5,4] => 4
[1,2,4,3,5] => 3
[1,2,4,5,3] => 3
[1,2,5,3,4] => 3
[1,2,5,4,3] => 7
[1,3,2,4,5] => 2
[1,3,2,5,4] => 6
[1,3,4,2,5] => 2
[1,3,4,5,2] => 2
[1,3,5,2,4] => 2
[1,3,5,4,2] => 6
[1,4,2,3,5] => 2
[1,4,2,5,3] => 5
[1,4,3,2,5] => 5
[1,4,3,5,2] => 5
[1,4,5,2,3] => 2
[1,4,5,3,2] => 5
[1,2,3,4,5,6,7] => ? = 0
[1,2,3,4,5,7,6] => ? = 6
[1,2,3,4,6,5,7] => ? = 5
[1,2,3,4,6,7,5] => ? = 5
[1,2,3,4,7,5,6] => ? = 5
[1,2,3,4,7,6,5] => ? = 11
[1,2,3,5,4,6,7] => ? = 4
[1,2,3,5,4,7,6] => ? = 10
[1,2,3,5,6,4,7] => ? = 4
[1,2,3,5,6,7,4] => ? = 4
[1,2,3,5,7,4,6] => ? = 4
[1,2,3,5,7,6,4] => ? = 10
[1,2,3,6,4,5,7] => ? = 4
[1,2,3,6,4,7,5] => ? = 9
[1,2,3,6,5,4,7] => ? = 9
[1,2,3,6,5,7,4] => ? = 9
[1,2,3,6,7,4,5] => ? = 4
[1,2,3,6,7,5,4] => ? = 9
[1,2,3,7,4,5,6] => ? = 4
[1,2,3,7,4,6,5] => ? = 9
[1,2,3,7,5,4,6] => ? = 9
[1,2,3,7,5,6,4] => ? = 9
[1,2,3,7,6,4,5] => ? = 10
[1,2,3,7,6,5,4] => ? = 15
[1,2,4,3,5,6,7] => ? = 3
[1,2,4,3,5,7,6] => ? = 9
[1,2,4,3,6,5,7] => ? = 8
[1,2,4,3,6,7,5] => ? = 8
[1,2,4,3,7,5,6] => ? = 8
[1,2,4,3,7,6,5] => ? = 14
[1,2,4,5,3,6,7] => ? = 3
[1,2,4,5,3,7,6] => ? = 9
[1,2,4,5,6,3,7] => ? = 3
[1,2,4,5,6,7,3] => ? = 3
[1,2,4,5,7,3,6] => ? = 3
[1,2,4,5,7,6,3] => ? = 9
[1,2,4,6,3,5,7] => ? = 3
[1,2,4,6,3,7,5] => ? = 8
[1,2,4,6,5,3,7] => ? = 8
[1,2,4,6,5,7,3] => ? = 8
[1,2,4,6,7,3,5] => ? = 3
[1,2,4,6,7,5,3] => ? = 8
[1,2,4,7,3,5,6] => ? = 3
[1,2,4,7,3,6,5] => ? = 8
[1,2,4,7,5,3,6] => ? = 8
[1,2,4,7,5,6,3] => ? = 8
[1,2,4,7,6,3,5] => ? = 9
[1,2,4,7,6,5,3] => ? = 14
[1,2,5,3,4,6,7] => ? = 3
[1,2,5,3,4,7,6] => ? = 9
Description
The sum of the descent bottoms of a permutation. This statistic is given by $$\pi \mapsto \sum_{i\in\operatorname{Des}(\pi)} \pi_{i+1}.$$ For the descent tops, see [[St000111]].
Matching statistic: St000005
Mapping
Mp00131: Permutations —descent bottoms⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000005: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 35%
Values
[1,2] => 0 => [2] => [1,1,0,0]
 => 0
[2,1] => 1 => [1,1] => [1,0,1,0]
 => 1
[1,2,3] => 00 => [3] => [1,1,1,0,0,0]
 => 0
[1,3,2] => 01 => [2,1] => [1,1,0,0,1,0]
 => 2
[2,1,3] => 10 => [1,2] => [1,0,1,1,0,0]
 => 1
[2,3,1] => 10 => [1,2] => [1,0,1,1,0,0]
 => 1
[3,1,2] => 10 => [1,2] => [1,0,1,1,0,0]
 => 1
[3,2,1] => 11 => [1,1,1] => [1,0,1,0,1,0]
 => 3
[1,2,3,4] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[1,2,4,3] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 3
[1,3,2,4] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[1,3,4,2] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[1,4,2,3] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[1,4,3,2] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 5
[2,1,3,4] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[2,1,4,3] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 4
[2,3,1,4] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[2,3,4,1] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[2,4,1,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[2,4,3,1] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 4
[3,1,2,4] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[3,1,4,2] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 3
[3,2,1,4] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 3
[3,2,4,1] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 3
[3,4,1,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[3,4,2,1] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 3
[4,1,2,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[4,1,3,2] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 3
[4,2,1,3] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 3
[4,2,3,1] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 3
[4,3,1,2] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 4
[4,3,2,1] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 6
[1,2,3,4,5] => 0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,2,3,5,4] => 0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 4
[1,2,4,3,5] => 0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3
[1,2,4,5,3] => 0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3
[1,2,5,3,4] => 0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3
[1,2,5,4,3] => 0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 7
[1,3,2,4,5] => 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
[1,3,2,5,4] => 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 6
[1,3,4,2,5] => 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
[1,3,4,5,2] => 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
[1,3,5,2,4] => 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
[1,3,5,4,2] => 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 6
[1,4,2,3,5] => 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
[1,4,2,5,3] => 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 5
[1,4,3,2,5] => 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 5
[1,4,3,5,2] => 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 5
[1,4,5,2,3] => 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
[1,4,5,3,2] => 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 5
[1,2,3,4,5,6,7] => 000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0
[1,2,3,4,5,7,6] => 000001 => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 6
[1,2,3,4,6,5,7] => 000010 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 5
[1,2,3,4,6,7,5] => 000010 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 5
[1,2,3,4,7,5,6] => 000010 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 5
[1,2,3,4,7,6,5] => 000011 => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => ? = 11
[1,2,3,5,4,6,7] => 000100 => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => ? = 4
[1,2,3,5,4,7,6] => 000101 => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => ? = 10
[1,2,3,5,6,4,7] => 000100 => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => ? = 4
[1,2,3,5,6,7,4] => 000100 => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => ? = 4
[1,2,3,5,7,4,6] => 000100 => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => ? = 4
[1,2,3,5,7,6,4] => 000101 => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => ? = 10
[1,2,3,6,4,5,7] => 000100 => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => ? = 4
[1,2,3,6,4,7,5] => 000110 => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => ? = 9
[1,2,3,6,5,4,7] => 000110 => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => ? = 9
[1,2,3,6,5,7,4] => 000110 => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => ? = 9
[1,2,3,6,7,4,5] => 000100 => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => ? = 4
[1,2,3,6,7,5,4] => 000110 => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => ? = 9
[1,2,3,7,4,5,6] => 000100 => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => ? = 4
[1,2,3,7,4,6,5] => 000110 => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => ? = 9
[1,2,3,7,5,4,6] => 000110 => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => ? = 9
[1,2,3,7,5,6,4] => 000110 => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => ? = 9
[1,2,3,7,6,4,5] => 000101 => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => ? = 10
[1,2,3,7,6,5,4] => 000111 => [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => ? = 15
[1,2,4,3,5,6,7] => 001000 => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 3
[1,2,4,3,5,7,6] => 001001 => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 9
[1,2,4,3,6,5,7] => 001010 => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 8
[1,2,4,3,6,7,5] => 001010 => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 8
[1,2,4,3,7,5,6] => 001010 => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 8
[1,2,4,3,7,6,5] => 001011 => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => ? = 14
[1,2,4,5,3,6,7] => 001000 => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 3
[1,2,4,5,3,7,6] => 001001 => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 9
[1,2,4,5,6,3,7] => 001000 => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 3
[1,2,4,5,6,7,3] => 001000 => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 3
[1,2,4,5,7,3,6] => 001000 => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 3
[1,2,4,5,7,6,3] => 001001 => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 9
[1,2,4,6,3,5,7] => 001000 => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 3
[1,2,4,6,3,7,5] => 001010 => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 8
[1,2,4,6,5,3,7] => 001010 => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 8
[1,2,4,6,5,7,3] => 001010 => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 8
[1,2,4,6,7,3,5] => 001000 => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 3
[1,2,4,6,7,5,3] => 001010 => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 8
[1,2,4,7,3,5,6] => 001000 => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 3
[1,2,4,7,3,6,5] => 001010 => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 8
[1,2,4,7,5,3,6] => 001010 => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 8
[1,2,4,7,5,6,3] => 001010 => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 8
[1,2,4,7,6,3,5] => 001001 => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 9
[1,2,4,7,6,5,3] => 001011 => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => ? = 14
[1,2,5,3,4,6,7] => 001000 => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 3
[1,2,5,3,4,7,6] => 001001 => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 9
Description
The bounce statistic of a Dyck path. The '''bounce path''' $D'$ of a Dyck path $D$ is the Dyck path obtained from $D$ by starting at the end point $(2n,0)$, traveling north-west until hitting $D$, then bouncing back south-west to the $x$-axis, and repeating this procedure until finally reaching the point $(0,0)$. The points where $D'$ touches the $x$-axis are called '''bounce points''', and a bounce path is uniquely determined by its bounce points. This statistic is given by the sum of all $i$ for which the bounce path $D'$ of $D$ touches the $x$-axis at $(2i,0)$. In particular, the bounce statistics of $D$ and $D'$ coincide.
Matching statistic: St001232
Mapping
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 15%
Values
[1,2] => [1,2] => [1,0,1,0]
 => [1,1,0,0]
 => 0
[2,1] => [2,1] => [1,1,0,0]
 => [1,0,1,0]
 => 1
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 2
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 1
[2,3,1] => [3,2,1] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => ? = 1
[3,1,2] => [3,2,1] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => ? = 1
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => ? = 3
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => ? = 2
[1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => ? = 2
[1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => ? = 5
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 1
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => ? = 4
[2,3,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => ? = 1
[2,3,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => ? = 1
[2,4,1,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => ? = 1
[2,4,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => ? = 4
[3,1,2,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => ? = 1
[3,1,4,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => ? = 3
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => ? = 3
[3,2,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => ? = 3
[3,4,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => ? = 1
[3,4,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => ? = 3
[4,1,2,3] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => ? = 1
[4,1,3,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => ? = 3
[4,2,1,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => ? = 3
[4,2,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => ? = 3
[4,3,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => ? = 4
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => ? = 6
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 3
[1,2,4,5,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => ? = 3
[1,2,5,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => ? = 3
[1,2,5,4,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => ? = 7
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => ? = 6
[1,3,4,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => ? = 2
[1,3,4,5,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => ? = 2
[1,3,5,2,4] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => ? = 2
[1,3,5,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => ? = 6
[1,4,2,3,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => ? = 2
[1,4,2,5,3] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => ? = 5
[1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => ? = 5
[1,4,3,5,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => ? = 5
[1,4,5,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => ? = 2
[1,4,5,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => ? = 5
[1,5,2,3,4] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => ? = 2
[1,5,2,4,3] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => ? = 5
[1,5,3,2,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => ? = 5
[1,5,3,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => ? = 5
[1,5,4,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => ? = 6
[1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => ? = 9
[2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[2,1,3,5,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => ? = 5
[2,1,4,3,5] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => ? = 4
[2,1,4,5,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => ? = 4
[2,1,5,3,4] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => ? = 4
[2,1,5,4,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => ? = 8
[2,3,1,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ? = 1
[2,3,1,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => ? = 5
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
[1,2,3,4,6,5] => [1,2,3,4,6,5] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5
[1,2,3,5,4,6] => [1,2,3,5,4,6] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 4
[1,2,4,3,5,6] => [1,2,4,3,5,6] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 3
[1,3,2,4,5,6] => [1,3,2,4,5,6] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 2
[2,1,3,4,5,6] => [2,1,3,4,5,6] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
[1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => 0
[1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => 6
[1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => 5
[1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => 4
[1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => 3
[1,3,2,4,5,6,7] => [1,3,2,4,5,6,7] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => 2
[2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.