- FindStat

Your data matches 40 different statistics following compositions of up to 3 maps.
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Matching statistic: St000288
(load all 58 compositions to match this statistic)

Mapping

Mp00267: Signed permutations —signs⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => 0 => 0
[-1] => 1 => 1
[1,2] => 00 => 0
[1,-2] => 01 => 1
[-1,2] => 10 => 1
[-1,-2] => 11 => 2
[2,1] => 00 => 0
[2,-1] => 01 => 1
[-2,1] => 10 => 1
[-2,-1] => 11 => 2
[1,2,3] => 000 => 0
[1,2,-3] => 001 => 1
[1,-2,3] => 010 => 1
[1,-2,-3] => 011 => 2
[-1,2,3] => 100 => 1
[-1,2,-3] => 101 => 2
[-1,-2,3] => 110 => 2
[-1,-2,-3] => 111 => 3
[1,3,2] => 000 => 0
[1,3,-2] => 001 => 1
[1,-3,2] => 010 => 1
[1,-3,-2] => 011 => 2
[-1,3,2] => 100 => 1
[-1,3,-2] => 101 => 2
[-1,-3,2] => 110 => 2
[-1,-3,-2] => 111 => 3
[2,1,3] => 000 => 0
[2,1,-3] => 001 => 1
[2,-1,3] => 010 => 1
[2,-1,-3] => 011 => 2
[-2,1,3] => 100 => 1
[-2,1,-3] => 101 => 2
[-2,-1,3] => 110 => 2
[-2,-1,-3] => 111 => 3
[2,3,1] => 000 => 0
[2,3,-1] => 001 => 1
[2,-3,1] => 010 => 1
[2,-3,-1] => 011 => 2
[-2,3,1] => 100 => 1
[-2,3,-1] => 101 => 2
[-2,-3,1] => 110 => 2
[-2,-3,-1] => 111 => 3
[3,1,2] => 000 => 0
[3,1,-2] => 001 => 1
[3,-1,2] => 010 => 1
[3,-1,-2] => 011 => 2
[-3,1,2] => 100 => 1
[-3,1,-2] => 101 => 2
[-3,-1,2] => 110 => 2
[-3,-1,-2] => 111 => 3

Description

The number of ones in a binary word. This is also known as the Hamming weight of the word.

Matching statistic: St000010

Mapping

Mp00267: Signed permutations —signs⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => 0 => [2] => [2]
 => 1 = 0 + 1
[-1] => 1 => [1,1] => [1,1]
 => 2 = 1 + 1
[1,2] => 00 => [3] => [3]
 => 1 = 0 + 1
[1,-2] => 01 => [2,1] => [2,1]
 => 2 = 1 + 1
[-1,2] => 10 => [1,2] => [2,1]
 => 2 = 1 + 1
[-1,-2] => 11 => [1,1,1] => [1,1,1]
 => 3 = 2 + 1
[2,1] => 00 => [3] => [3]
 => 1 = 0 + 1
[2,-1] => 01 => [2,1] => [2,1]
 => 2 = 1 + 1
[-2,1] => 10 => [1,2] => [2,1]
 => 2 = 1 + 1
[-2,-1] => 11 => [1,1,1] => [1,1,1]
 => 3 = 2 + 1
[1,2,3] => 000 => [4] => [4]
 => 1 = 0 + 1
[1,2,-3] => 001 => [3,1] => [3,1]
 => 2 = 1 + 1
[1,-2,3] => 010 => [2,2] => [2,2]
 => 2 = 1 + 1
[1,-2,-3] => 011 => [2,1,1] => [2,1,1]
 => 3 = 2 + 1
[-1,2,3] => 100 => [1,3] => [3,1]
 => 2 = 1 + 1
[-1,2,-3] => 101 => [1,2,1] => [2,1,1]
 => 3 = 2 + 1
[-1,-2,3] => 110 => [1,1,2] => [2,1,1]
 => 3 = 2 + 1
[-1,-2,-3] => 111 => [1,1,1,1] => [1,1,1,1]
 => 4 = 3 + 1
[1,3,2] => 000 => [4] => [4]
 => 1 = 0 + 1
[1,3,-2] => 001 => [3,1] => [3,1]
 => 2 = 1 + 1
[1,-3,2] => 010 => [2,2] => [2,2]
 => 2 = 1 + 1
[1,-3,-2] => 011 => [2,1,1] => [2,1,1]
 => 3 = 2 + 1
[-1,3,2] => 100 => [1,3] => [3,1]
 => 2 = 1 + 1
[-1,3,-2] => 101 => [1,2,1] => [2,1,1]
 => 3 = 2 + 1
[-1,-3,2] => 110 => [1,1,2] => [2,1,1]
 => 3 = 2 + 1
[-1,-3,-2] => 111 => [1,1,1,1] => [1,1,1,1]
 => 4 = 3 + 1
[2,1,3] => 000 => [4] => [4]
 => 1 = 0 + 1
[2,1,-3] => 001 => [3,1] => [3,1]
 => 2 = 1 + 1
[2,-1,3] => 010 => [2,2] => [2,2]
 => 2 = 1 + 1
[2,-1,-3] => 011 => [2,1,1] => [2,1,1]
 => 3 = 2 + 1
[-2,1,3] => 100 => [1,3] => [3,1]
 => 2 = 1 + 1
[-2,1,-3] => 101 => [1,2,1] => [2,1,1]
 => 3 = 2 + 1
[-2,-1,3] => 110 => [1,1,2] => [2,1,1]
 => 3 = 2 + 1
[-2,-1,-3] => 111 => [1,1,1,1] => [1,1,1,1]
 => 4 = 3 + 1
[2,3,1] => 000 => [4] => [4]
 => 1 = 0 + 1
[2,3,-1] => 001 => [3,1] => [3,1]
 => 2 = 1 + 1
[2,-3,1] => 010 => [2,2] => [2,2]
 => 2 = 1 + 1
[2,-3,-1] => 011 => [2,1,1] => [2,1,1]
 => 3 = 2 + 1
[-2,3,1] => 100 => [1,3] => [3,1]
 => 2 = 1 + 1
[-2,3,-1] => 101 => [1,2,1] => [2,1,1]
 => 3 = 2 + 1
[-2,-3,1] => 110 => [1,1,2] => [2,1,1]
 => 3 = 2 + 1
[-2,-3,-1] => 111 => [1,1,1,1] => [1,1,1,1]
 => 4 = 3 + 1
[3,1,2] => 000 => [4] => [4]
 => 1 = 0 + 1
[3,1,-2] => 001 => [3,1] => [3,1]
 => 2 = 1 + 1
[3,-1,2] => 010 => [2,2] => [2,2]
 => 2 = 1 + 1
[3,-1,-2] => 011 => [2,1,1] => [2,1,1]
 => 3 = 2 + 1
[-3,1,2] => 100 => [1,3] => [3,1]
 => 2 = 1 + 1
[-3,1,-2] => 101 => [1,2,1] => [2,1,1]
 => 3 = 2 + 1
[-3,-1,2] => 110 => [1,1,2] => [2,1,1]
 => 3 = 2 + 1
[-3,-1,-2] => 111 => [1,1,1,1] => [1,1,1,1]
 => 4 = 3 + 1

Description

The length of the partition.

Matching statistic: St000011

Mapping

Mp00267: Signed permutations —signs⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%

Values

[1] => 0 => [2] => [1,1,0,0]
 => 1 = 0 + 1
[-1] => 1 => [1,1] => [1,0,1,0]
 => 2 = 1 + 1
[1,2] => 00 => [3] => [1,1,1,0,0,0]
 => 1 = 0 + 1
[1,-2] => 01 => [2,1] => [1,1,0,0,1,0]
 => 2 = 1 + 1
[-1,2] => 10 => [1,2] => [1,0,1,1,0,0]
 => 2 = 1 + 1
[-1,-2] => 11 => [1,1,1] => [1,0,1,0,1,0]
 => 3 = 2 + 1
[2,1] => 00 => [3] => [1,1,1,0,0,0]
 => 1 = 0 + 1
[2,-1] => 01 => [2,1] => [1,1,0,0,1,0]
 => 2 = 1 + 1
[-2,1] => 10 => [1,2] => [1,0,1,1,0,0]
 => 2 = 1 + 1
[-2,-1] => 11 => [1,1,1] => [1,0,1,0,1,0]
 => 3 = 2 + 1
[1,2,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,2,-3] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[1,-2,3] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[1,-2,-3] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
[-1,2,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[-1,2,-3] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 3 = 2 + 1
[-1,-2,3] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[-1,-2,-3] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[1,3,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,3,-2] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[1,-3,2] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[1,-3,-2] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
[-1,3,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[-1,3,-2] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 3 = 2 + 1
[-1,-3,2] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[-1,-3,-2] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[2,1,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[2,1,-3] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[2,-1,3] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[2,-1,-3] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
[-2,1,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[-2,1,-3] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 3 = 2 + 1
[-2,-1,3] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[-2,-1,-3] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[2,3,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[2,3,-1] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[2,-3,1] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[2,-3,-1] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
[-2,3,1] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[-2,3,-1] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 3 = 2 + 1
[-2,-3,1] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[-2,-3,-1] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[3,1,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[3,1,-2] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[3,-1,2] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[3,-1,-2] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
[-3,1,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[-3,1,-2] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 3 = 2 + 1
[-3,-1,2] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[-3,-1,-2] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[-8,-6,-4,-2,1,3,5,7] => 11110000 => [1,1,1,1,5] => [1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 4 + 1
[6,8,-4,-3,1,2,5,7] => 00110000 => [3,1,5] => [1,1,1,0,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[-3,2,-8,-6,-4,1,5,7] => 10111000 => [1,2,1,1,4] => [1,0,1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 4 + 1
[3,2,8,-6,-5,1,4,7] => 00011000 => [4,1,4] => [1,1,1,1,0,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 2 + 1
[-5,-3,2,4,-8,-6,1,7] => 11001100 => [1,1,3,1,3] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 4 + 1
[5,-3,2,4,8,-7,1,6] => 01000100 => [2,4,3] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => ? = 2 + 1
[-7,-5,-3,2,4,6,-8,1] => 11100010 => [1,1,1,4,2] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => ? = 4 + 1
[5,7,-4,2,3,6,-8,1] => 00100010 => [3,4,2] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => ? = 2 + 1
[-4,5,2,3,8,-7,1,6] => 10000100 => [1,5,3] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 2 + 1
[4,5,2,3,-8,-6,1,7] => 00001100 => [5,1,3] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 2 + 1
[2,-8,-4,-6,-5,1,3,7] => 01111000 => [2,1,1,1,4] => [1,1,0,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 4 + 1
[-8,-6,1,-5,-4,2,3,7] => 11011000 => [1,1,2,1,4] => [1,0,1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 4 + 1
[-3,6,8,-5,1,2,4,7] => 10010000 => [1,3,5] => [1,0,1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[2,8,-6,-5,1,3,4,7] => 00110000 => [3,1,5] => [1,1,1,0,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[-5,4,-8,-6,-2,1,3,7] => 10111000 => [1,2,1,1,4] => [1,0,1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 4 + 1
[4,-5,-8,-6,-3,1,2,7] => 01111000 => [2,1,1,1,4] => [1,1,0,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 4 + 1
[3,-8,-5,-6,-4,1,2,7] => 01111000 => [2,1,1,1,4] => [1,1,0,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 4 + 1
[-6,4,3,8,-5,1,2,7] => 10001000 => [1,4,4] => [1,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 2 + 1
[-4,3,-5,2,-8,-6,1,7] => 10101100 => [1,2,2,1,3] => [1,0,1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => ? = 4 + 1
[4,3,-5,2,8,-7,1,6] => 00100100 => [3,3,3] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 2 + 1
[-4,3,-7,-5,2,6,-8,1] => 10110010 => [1,2,1,3,2] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 4 + 1
[4,3,7,-6,2,5,-8,1] => 00010010 => [4,3,2] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 2 + 1
[3,-4,-8,-6,2,-7,1,5] => 01110100 => [2,1,1,2,3] => [1,1,0,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 4 + 1
[8,4,-7,-6,3,1,2,5] => 00110000 => [3,1,5] => [1,1,1,0,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[5,8,-4,3,-7,1,2,6] => 00101000 => [3,2,4] => [1,1,1,0,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 2 + 1
[5,-3,4,8,-7,1,2,6] => 01001000 => [2,3,4] => [1,1,0,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 2 + 1
[-5,2,4,8,-7,1,3,6] => 10001000 => [1,4,4] => [1,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 2 + 1
[-7,-5,4,6,-8,-2,1,3] => 11001100 => [1,1,3,1,3] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 4 + 1
[4,-7,-5,6,-8,-3,1,2] => 01101100 => [2,1,2,1,3] => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => ? = 4 + 1
[-7,3,-5,6,-8,-4,1,2] => 10101100 => [1,2,2,1,3] => [1,0,1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => ? = 4 + 1
[-4,3,6,-7,-8,-5,1,2] => 10011100 => [1,3,1,1,3] => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 4 + 1
[-4,3,5,8,7,-6,1,2] => 10000100 => [1,5,3] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 2 + 1
[4,-8,6,-7,3,5,1,2] => 01010000 => [2,2,5] => [1,1,0,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[5,6,3,4,-7,2,-8,1] => 00001010 => [5,2,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 2 + 1
[3,-8,5,7,4,-6,1,2] => 01000100 => [2,4,3] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => ? = 2 + 1
[-7,-6,3,-8,4,-5,1,2] => 11010100 => [1,1,2,2,3] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 4 + 1
[-7,-6,5,3,-8,-4,1,2] => 11001100 => [1,1,3,1,3] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 4 + 1
[-7,4,-6,5,-8,-3,1,2] => 10101100 => [1,2,2,1,3] => [1,0,1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => ? = 4 + 1
[6,7,4,5,-8,-2,1,3] => 00001100 => [5,1,3] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 2 + 1
[8,-6,2,4,-7,1,3,5] => 01001000 => [2,3,4] => [1,1,0,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 2 + 1
[4,-8,-6,-3,-7,1,2,5] => 01111000 => [2,1,1,1,4] => [1,1,0,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 4 + 1
[-4,3,-8,-6,-7,1,2,5] => 10111000 => [1,2,1,1,4] => [1,0,1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 4 + 1
[-7,6,8,1,3,-5,2,4] => 10000100 => [1,5,3] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 2 + 1
[5,-6,3,8,2,-7,1,4] => 01000100 => [2,4,3] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => ? = 2 + 1
[3,-7,-5,-6,2,4,-8,1] => 01110010 => [2,1,1,3,2] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 4 + 1
[-7,-6,2,-5,3,4,-8,1] => 11010010 => [1,1,2,3,2] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 4 + 1
[-5,2,6,8,3,-7,1,4] => 10000100 => [1,5,3] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 2 + 1
[-5,7,8,-6,2,1,3,4] => 10010000 => [1,3,5] => [1,0,1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[-4,-8,-7,1,-6,2,3,5] => 11101000 => [1,1,1,2,4] => [1,0,1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 4 + 1
[8,-7,-6,3,1,2,4,5] => 01100000 => [2,1,6] => [1,1,0,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2 + 1

Description

The number of touch points (or returns) of a Dyck path. This is the number of points, excluding the origin, where the Dyck path has height 0.

Matching statistic: St000097

Mapping

Mp00267: Signed permutations —signs⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 88% ●values known / values provided: 91%●distinct values known / distinct values provided: 88%

Values

[1] => 0 => [2] => ([],2)
 => 1 = 0 + 1
[-1] => 1 => [1,1] => ([(0,1)],2)
 => 2 = 1 + 1
[1,2] => 00 => [3] => ([],3)
 => 1 = 0 + 1
[1,-2] => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[-1,2] => 10 => [1,2] => ([(1,2)],3)
 => 2 = 1 + 1
[-1,-2] => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[2,1] => 00 => [3] => ([],3)
 => 1 = 0 + 1
[2,-1] => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[-2,1] => 10 => [1,2] => ([(1,2)],3)
 => 2 = 1 + 1
[-2,-1] => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[1,2,3] => 000 => [4] => ([],4)
 => 1 = 0 + 1
[1,2,-3] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,-2,3] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,-2,-3] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[-1,2,3] => 100 => [1,3] => ([(2,3)],4)
 => 2 = 1 + 1
[-1,2,-3] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[-1,-2,3] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[-1,-2,-3] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[1,3,2] => 000 => [4] => ([],4)
 => 1 = 0 + 1
[1,3,-2] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,-3,2] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,-3,-2] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[-1,3,2] => 100 => [1,3] => ([(2,3)],4)
 => 2 = 1 + 1
[-1,3,-2] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[-1,-3,2] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[-1,-3,-2] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[2,1,3] => 000 => [4] => ([],4)
 => 1 = 0 + 1
[2,1,-3] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[2,-1,3] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[2,-1,-3] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[-2,1,3] => 100 => [1,3] => ([(2,3)],4)
 => 2 = 1 + 1
[-2,1,-3] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[-2,-1,3] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[-2,-1,-3] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[2,3,1] => 000 => [4] => ([],4)
 => 1 = 0 + 1
[2,3,-1] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[2,-3,1] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[2,-3,-1] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[-2,3,1] => 100 => [1,3] => ([(2,3)],4)
 => 2 = 1 + 1
[-2,3,-1] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[-2,-3,1] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[-2,-3,-1] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[3,1,2] => 000 => [4] => ([],4)
 => 1 = 0 + 1
[3,1,-2] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[3,-1,2] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[3,-1,-2] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[-3,1,2] => 100 => [1,3] => ([(2,3)],4)
 => 2 = 1 + 1
[-3,1,-2] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[-3,-1,2] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[-3,-1,-2] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[5,6,7,8,3,4,1,2] => 00000000 => [9] => ([],9)
 => ? = 0 + 1
[5,6,7,8,3,1,2,4] => 00000000 => [9] => ([],9)
 => ? = 0 + 1
[5,6,7,8,4,1,2,3] => 00000000 => [9] => ([],9)
 => ? = 0 + 1
[5,6,7,8,1,4,2,3] => 00000000 => [9] => ([],9)
 => ? = 0 + 1
[7,8,5,6,3,4,1,2] => 00000000 => [9] => ([],9)
 => ? = 0 + 1
[-8,-6,-4,-2,1,3,5,7] => 11110000 => [1,1,1,1,5] => ([(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 4 + 1
[6,8,-4,-3,1,2,5,7] => 00110000 => [3,1,5] => ([(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[-3,2,-8,-6,-4,1,5,7] => 10111000 => [1,2,1,1,4] => ([(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 4 + 1
[3,2,8,-6,-5,1,4,7] => 00011000 => [4,1,4] => ([(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[-5,-3,2,4,-8,-6,1,7] => 11001100 => [1,1,3,1,3] => ([(2,7),(2,8),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 4 + 1
[5,-3,2,4,8,-7,1,6] => 01000100 => [2,4,3] => ([(2,8),(3,8),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[-7,-5,-3,2,4,6,-8,1] => 11100010 => [1,1,1,4,2] => ([(1,8),(2,8),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 4 + 1
[5,7,-4,2,3,6,-8,1] => 00100010 => [3,4,2] => ([(1,8),(2,8),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[-4,5,2,3,8,-7,1,6] => 10000100 => [1,5,3] => ([(2,8),(3,8),(4,8),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[4,5,2,3,-8,-6,1,7] => 00001100 => [5,1,3] => ([(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[2,-8,-4,-6,-5,1,3,7] => 01111000 => [2,1,1,1,4] => ([(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 4 + 1
[-8,-6,1,-5,-4,2,3,7] => 11011000 => [1,1,2,1,4] => ([(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 4 + 1
[-3,6,8,-5,1,2,4,7] => 10010000 => [1,3,5] => ([(4,8),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[2,8,-6,-5,1,3,4,7] => 00110000 => [3,1,5] => ([(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[-5,4,-8,-6,-2,1,3,7] => 10111000 => [1,2,1,1,4] => ([(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 4 + 1
[4,-5,-8,-6,-3,1,2,7] => 01111000 => [2,1,1,1,4] => ([(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 4 + 1
[3,-8,-5,-6,-4,1,2,7] => 01111000 => [2,1,1,1,4] => ([(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 4 + 1
[-6,4,3,8,-5,1,2,7] => 10001000 => [1,4,4] => ([(3,8),(4,8),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[-4,3,-5,2,-8,-6,1,7] => 10101100 => [1,2,2,1,3] => ([(2,7),(2,8),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 4 + 1
[4,3,-5,2,8,-7,1,6] => 00100100 => [3,3,3] => ([(2,8),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[-4,3,-7,-5,2,6,-8,1] => 10110010 => [1,2,1,3,2] => ([(1,8),(2,8),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 4 + 1
[4,3,7,-6,2,5,-8,1] => 00010010 => [4,3,2] => ([(1,8),(2,8),(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[3,-4,-8,-6,2,-7,1,5] => 01110100 => [2,1,1,2,3] => ([(2,8),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 4 + 1
[8,4,-7,-6,3,1,2,5] => 00110000 => [3,1,5] => ([(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[-5,4,-7,3,8,1,2,6] => 10100000 => [1,2,6] => ([(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[5,8,-4,3,-7,1,2,6] => 00101000 => [3,2,4] => ([(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[5,-3,4,8,-7,1,2,6] => 01001000 => [2,3,4] => ([(3,8),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[-5,2,4,8,-7,1,3,6] => 10001000 => [1,4,4] => ([(3,8),(4,8),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[-7,-5,4,6,-8,-2,1,3] => 11001100 => [1,1,3,1,3] => ([(2,7),(2,8),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 4 + 1
[4,-7,-5,6,-8,-3,1,2] => 01101100 => [2,1,2,1,3] => ([(2,7),(2,8),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 4 + 1
[-7,3,-5,6,-8,-4,1,2] => 10101100 => [1,2,2,1,3] => ([(2,7),(2,8),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 4 + 1
[-4,3,6,-7,-8,-5,1,2] => 10011100 => [1,3,1,1,3] => ([(2,6),(2,7),(2,8),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 4 + 1
[-4,3,5,8,7,-6,1,2] => 10000100 => [1,5,3] => ([(2,8),(3,8),(4,8),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[4,-8,6,-7,3,5,1,2] => 01010000 => [2,2,5] => ([(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[-6,-4,3,5,-7,2,-8,1] => 11001010 => [1,1,3,2,2] => ([(1,8),(2,7),(2,8),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 4 + 1
[5,6,3,4,-7,2,-8,1] => 00001010 => [5,2,2] => ([(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[3,-8,5,7,4,-6,1,2] => 01000100 => [2,4,3] => ([(2,8),(3,8),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[-7,-6,3,-8,4,-5,1,2] => 11010100 => [1,1,2,2,3] => ([(2,8),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 4 + 1
[-7,-6,5,3,-8,-4,1,2] => 11001100 => [1,1,3,1,3] => ([(2,7),(2,8),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 4 + 1
[-7,4,-6,5,-8,-3,1,2] => 10101100 => [1,2,2,1,3] => ([(2,7),(2,8),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 4 + 1
[6,7,4,5,-8,-2,1,3] => 00001100 => [5,1,3] => ([(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[8,-6,2,4,-7,1,3,5] => 01001000 => [2,3,4] => ([(3,8),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[4,-8,-6,-3,-7,1,2,5] => 01111000 => [2,1,1,1,4] => ([(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 4 + 1
[-4,3,-8,-6,-7,1,2,5] => 10111000 => [1,2,1,1,4] => ([(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 4 + 1
[-7,6,8,1,3,-5,2,4] => 10000100 => [1,5,3] => ([(2,8),(3,8),(4,8),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2 + 1

Description

The order of the largest clique of the graph. A clique in a graph

$G$ is a subset

$U \subseteq V(G)$ such that any pair of vertices in

$U$ are adjacent. I.e. the subgraph induced by

$U$ is a complete graph.

Matching statistic: St001581

Mapping

Mp00267: Signed permutations —signs⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001581: Graphs ⟶ ℤResult quality: 75% ●values known / values provided: 91%●distinct values known / distinct values provided: 75%

Values

[1] => 0 => [2] => ([],2)
 => 1 = 0 + 1
[-1] => 1 => [1,1] => ([(0,1)],2)
 => 2 = 1 + 1
[1,2] => 00 => [3] => ([],3)
 => 1 = 0 + 1
[1,-2] => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[-1,2] => 10 => [1,2] => ([(1,2)],3)
 => 2 = 1 + 1
[-1,-2] => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[2,1] => 00 => [3] => ([],3)
 => 1 = 0 + 1
[2,-1] => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[-2,1] => 10 => [1,2] => ([(1,2)],3)
 => 2 = 1 + 1
[-2,-1] => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[1,2,3] => 000 => [4] => ([],4)
 => 1 = 0 + 1
[1,2,-3] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,-2,3] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,-2,-3] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[-1,2,3] => 100 => [1,3] => ([(2,3)],4)
 => 2 = 1 + 1
[-1,2,-3] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[-1,-2,3] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[-1,-2,-3] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[1,3,2] => 000 => [4] => ([],4)
 => 1 = 0 + 1
[1,3,-2] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,-3,2] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,-3,-2] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[-1,3,2] => 100 => [1,3] => ([(2,3)],4)
 => 2 = 1 + 1
[-1,3,-2] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[-1,-3,2] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[-1,-3,-2] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[2,1,3] => 000 => [4] => ([],4)
 => 1 = 0 + 1
[2,1,-3] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[2,-1,3] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[2,-1,-3] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[-2,1,3] => 100 => [1,3] => ([(2,3)],4)
 => 2 = 1 + 1
[-2,1,-3] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[-2,-1,3] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[-2,-1,-3] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[2,3,1] => 000 => [4] => ([],4)
 => 1 = 0 + 1
[2,3,-1] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[2,-3,1] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[2,-3,-1] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[-2,3,1] => 100 => [1,3] => ([(2,3)],4)
 => 2 = 1 + 1
[-2,3,-1] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[-2,-3,1] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[-2,-3,-1] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[3,1,2] => 000 => [4] => ([],4)
 => 1 = 0 + 1
[3,1,-2] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[3,-1,2] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[3,-1,-2] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[-3,1,2] => 100 => [1,3] => ([(2,3)],4)
 => 2 = 1 + 1
[-3,1,-2] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[-3,-1,2] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[-3,-1,-2] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[5,6,7,8,3,4,1,2] => 00000000 => [9] => ([],9)
 => ? = 0 + 1
[5,6,7,8,3,1,2,4] => 00000000 => [9] => ([],9)
 => ? = 0 + 1
[5,6,7,8,4,1,2,3] => 00000000 => [9] => ([],9)
 => ? = 0 + 1
[5,6,7,8,1,4,2,3] => 00000000 => [9] => ([],9)
 => ? = 0 + 1
[7,8,5,6,3,4,1,2] => 00000000 => [9] => ([],9)
 => ? = 0 + 1
[-8,-6,-4,-2,1,3,5,7] => 11110000 => [1,1,1,1,5] => ([(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 4 + 1
[6,8,-4,-3,1,2,5,7] => 00110000 => [3,1,5] => ([(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[-3,2,-8,-6,-4,1,5,7] => 10111000 => [1,2,1,1,4] => ([(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 4 + 1
[3,2,8,-6,-5,1,4,7] => 00011000 => [4,1,4] => ([(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[-5,-3,2,4,-8,-6,1,7] => 11001100 => [1,1,3,1,3] => ([(2,7),(2,8),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 4 + 1
[5,-3,2,4,8,-7,1,6] => 01000100 => [2,4,3] => ([(2,8),(3,8),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[-7,-5,-3,2,4,6,-8,1] => 11100010 => [1,1,1,4,2] => ([(1,8),(2,8),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 4 + 1
[5,7,-4,2,3,6,-8,1] => 00100010 => [3,4,2] => ([(1,8),(2,8),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[-4,5,2,3,8,-7,1,6] => 10000100 => [1,5,3] => ([(2,8),(3,8),(4,8),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[4,5,2,3,-8,-6,1,7] => 00001100 => [5,1,3] => ([(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[2,-8,-4,-6,-5,1,3,7] => 01111000 => [2,1,1,1,4] => ([(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 4 + 1
[-8,-6,1,-5,-4,2,3,7] => 11011000 => [1,1,2,1,4] => ([(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 4 + 1
[-3,6,8,-5,1,2,4,7] => 10010000 => [1,3,5] => ([(4,8),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[2,8,-6,-5,1,3,4,7] => 00110000 => [3,1,5] => ([(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[-5,4,-8,-6,-2,1,3,7] => 10111000 => [1,2,1,1,4] => ([(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 4 + 1
[4,-5,-8,-6,-3,1,2,7] => 01111000 => [2,1,1,1,4] => ([(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 4 + 1
[3,-8,-5,-6,-4,1,2,7] => 01111000 => [2,1,1,1,4] => ([(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 4 + 1
[-6,4,3,8,-5,1,2,7] => 10001000 => [1,4,4] => ([(3,8),(4,8),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[-4,3,-5,2,-8,-6,1,7] => 10101100 => [1,2,2,1,3] => ([(2,7),(2,8),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 4 + 1
[4,3,-5,2,8,-7,1,6] => 00100100 => [3,3,3] => ([(2,8),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[-4,3,-7,-5,2,6,-8,1] => 10110010 => [1,2,1,3,2] => ([(1,8),(2,8),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 4 + 1
[4,3,7,-6,2,5,-8,1] => 00010010 => [4,3,2] => ([(1,8),(2,8),(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[3,-4,-8,-6,2,-7,1,5] => 01110100 => [2,1,1,2,3] => ([(2,8),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 4 + 1
[8,4,-7,-6,3,1,2,5] => 00110000 => [3,1,5] => ([(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[-5,4,-7,3,8,1,2,6] => 10100000 => [1,2,6] => ([(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[5,8,-4,3,-7,1,2,6] => 00101000 => [3,2,4] => ([(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[5,-3,4,8,-7,1,2,6] => 01001000 => [2,3,4] => ([(3,8),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[-5,2,4,8,-7,1,3,6] => 10001000 => [1,4,4] => ([(3,8),(4,8),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[-7,-5,4,6,-8,-2,1,3] => 11001100 => [1,1,3,1,3] => ([(2,7),(2,8),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 4 + 1
[4,-7,-5,6,-8,-3,1,2] => 01101100 => [2,1,2,1,3] => ([(2,7),(2,8),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 4 + 1
[-7,3,-5,6,-8,-4,1,2] => 10101100 => [1,2,2,1,3] => ([(2,7),(2,8),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 4 + 1
[-4,3,6,-7,-8,-5,1,2] => 10011100 => [1,3,1,1,3] => ([(2,6),(2,7),(2,8),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 4 + 1
[-4,3,5,8,7,-6,1,2] => 10000100 => [1,5,3] => ([(2,8),(3,8),(4,8),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[4,-8,6,-7,3,5,1,2] => 01010000 => [2,2,5] => ([(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[-6,-4,3,5,-7,2,-8,1] => 11001010 => [1,1,3,2,2] => ([(1,8),(2,7),(2,8),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 4 + 1
[5,6,3,4,-7,2,-8,1] => 00001010 => [5,2,2] => ([(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[3,-8,5,7,4,-6,1,2] => 01000100 => [2,4,3] => ([(2,8),(3,8),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[-7,-6,3,-8,4,-5,1,2] => 11010100 => [1,1,2,2,3] => ([(2,8),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 4 + 1
[-7,-6,5,3,-8,-4,1,2] => 11001100 => [1,1,3,1,3] => ([(2,7),(2,8),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 4 + 1
[-7,4,-6,5,-8,-3,1,2] => 10101100 => [1,2,2,1,3] => ([(2,7),(2,8),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 4 + 1
[6,7,4,5,-8,-2,1,3] => 00001100 => [5,1,3] => ([(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[8,-6,2,4,-7,1,3,5] => 01001000 => [2,3,4] => ([(3,8),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[4,-8,-6,-3,-7,1,2,5] => 01111000 => [2,1,1,1,4] => ([(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 4 + 1
[-4,3,-8,-6,-7,1,2,5] => 10111000 => [1,2,1,1,4] => ([(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 4 + 1
[-7,6,8,1,3,-5,2,4] => 10000100 => [1,5,3] => ([(2,8),(3,8),(4,8),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2 + 1

Description

The achromatic number of a graph. This is the maximal number of colours of a proper colouring, such that for any pair of colours there are two adjacent vertices with these colours.

Matching statistic: St000306

Mapping

Mp00267: Signed permutations —signs⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000306: Dyck paths ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%

Values

[1] => 0 => [2] => [1,1,0,0]
 => 0
[-1] => 1 => [1,1] => [1,0,1,0]
 => 1
[1,2] => 00 => [3] => [1,1,1,0,0,0]
 => 0
[1,-2] => 01 => [2,1] => [1,1,0,0,1,0]
 => 1
[-1,2] => 10 => [1,2] => [1,0,1,1,0,0]
 => 1
[-1,-2] => 11 => [1,1,1] => [1,0,1,0,1,0]
 => 2
[2,1] => 00 => [3] => [1,1,1,0,0,0]
 => 0
[2,-1] => 01 => [2,1] => [1,1,0,0,1,0]
 => 1
[-2,1] => 10 => [1,2] => [1,0,1,1,0,0]
 => 1
[-2,-1] => 11 => [1,1,1] => [1,0,1,0,1,0]
 => 2
[1,2,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[1,2,-3] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[1,-2,3] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[1,-2,-3] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[-1,2,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[-1,2,-3] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[-1,-2,3] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[-1,-2,-3] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3
[1,3,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[1,3,-2] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[1,-3,2] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[1,-3,-2] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[-1,3,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[-1,3,-2] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[-1,-3,2] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[-1,-3,-2] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3
[2,1,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[2,1,-3] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[2,-1,3] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[2,-1,-3] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[-2,1,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[-2,1,-3] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[-2,-1,3] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[-2,-1,-3] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3
[2,3,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[2,3,-1] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[2,-3,1] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[2,-3,-1] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[-2,3,1] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[-2,3,-1] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[-2,-3,1] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[-2,-3,-1] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3
[3,1,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[3,1,-2] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[3,-1,2] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[3,-1,-2] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[-3,1,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[-3,1,-2] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[-3,-1,2] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[-3,-1,-2] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3
[-6,-4,-2,1,3,5] => 111000 => [1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 3
[-3,2,-6,-4,1,5] => 101100 => [1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => ? = 3
[-6,-5,1,-4,2,3] => 110100 => [1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 3
[-5,4,-6,-2,1,3] => 101100 => [1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => ? = 3
[-4,3,-5,2,-6,1] => 101010 => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 3
[-8,-6,-4,-2,1,3,5,7] => 11110000 => [1,1,1,1,5] => [1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 4
[6,8,-4,-3,1,2,5,7] => 00110000 => [3,1,5] => [1,1,1,0,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 2
[-3,2,-8,-6,-4,1,5,7] => 10111000 => [1,2,1,1,4] => [1,0,1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 4
[3,2,8,-6,-5,1,4,7] => 00011000 => [4,1,4] => [1,1,1,1,0,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 2
[-5,-3,2,4,-8,-6,1,7] => 11001100 => [1,1,3,1,3] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 4
[5,-3,2,4,8,-7,1,6] => 01000100 => [2,4,3] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => ? = 2
[-7,-5,-3,2,4,6,-8,1] => 11100010 => [1,1,1,4,2] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => ? = 4
[5,7,-4,2,3,6,-8,1] => 00100010 => [3,4,2] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => ? = 2
[-4,5,2,3,8,-7,1,6] => 10000100 => [1,5,3] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 2
[4,5,2,3,-8,-6,1,7] => 00001100 => [5,1,3] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 2
[2,-8,-4,-6,-5,1,3,7] => 01111000 => [2,1,1,1,4] => [1,1,0,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 4
[-8,-6,1,-5,-4,2,3,7] => 11011000 => [1,1,2,1,4] => [1,0,1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 4
[-3,6,8,-5,1,2,4,7] => 10010000 => [1,3,5] => [1,0,1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 2
[2,8,-6,-5,1,3,4,7] => 00110000 => [3,1,5] => [1,1,1,0,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 2
[-5,4,-8,-6,-2,1,3,7] => 10111000 => [1,2,1,1,4] => [1,0,1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 4
[4,-5,-8,-6,-3,1,2,7] => 01111000 => [2,1,1,1,4] => [1,1,0,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 4
[3,-8,-5,-6,-4,1,2,7] => 01111000 => [2,1,1,1,4] => [1,1,0,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 4
[-6,4,3,8,-5,1,2,7] => 10001000 => [1,4,4] => [1,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 2
[-4,3,-5,2,-8,-6,1,7] => 10101100 => [1,2,2,1,3] => [1,0,1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => ? = 4
[4,3,-5,2,8,-7,1,6] => 00100100 => [3,3,3] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 2
[-4,3,-7,-5,2,6,-8,1] => 10110010 => [1,2,1,3,2] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 4
[4,3,7,-6,2,5,-8,1] => 00010010 => [4,3,2] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 2
[3,-4,-8,-6,2,-7,1,5] => 01110100 => [2,1,1,2,3] => [1,1,0,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 4
[8,4,-7,-6,3,1,2,5] => 00110000 => [3,1,5] => [1,1,1,0,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 2
[5,8,-4,3,-7,1,2,6] => 00101000 => [3,2,4] => [1,1,1,0,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 2
[5,-3,4,8,-7,1,2,6] => 01001000 => [2,3,4] => [1,1,0,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 2
[-5,2,4,8,-7,1,3,6] => 10001000 => [1,4,4] => [1,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 2
[-7,-5,4,6,-8,-2,1,3] => 11001100 => [1,1,3,1,3] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 4
[4,-7,-5,6,-8,-3,1,2] => 01101100 => [2,1,2,1,3] => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => ? = 4
[-7,3,-5,6,-8,-4,1,2] => 10101100 => [1,2,2,1,3] => [1,0,1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => ? = 4
[-4,3,6,-7,-8,-5,1,2] => 10011100 => [1,3,1,1,3] => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 4
[-4,3,5,8,7,-6,1,2] => 10000100 => [1,5,3] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 2
[4,-8,6,-7,3,5,1,2] => 01010000 => [2,2,5] => [1,1,0,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 2
[-6,-4,3,5,-7,2,-8,1] => 11001010 => [1,1,3,2,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 4
[5,6,3,4,-7,2,-8,1] => 00001010 => [5,2,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 2
[3,-8,5,7,4,-6,1,2] => 01000100 => [2,4,3] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => ? = 2
[-7,-6,3,-8,4,-5,1,2] => 11010100 => [1,1,2,2,3] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 4
[-7,-6,5,3,-8,-4,1,2] => 11001100 => [1,1,3,1,3] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 4
[-7,4,-6,5,-8,-3,1,2] => 10101100 => [1,2,2,1,3] => [1,0,1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => ? = 4
[6,7,4,5,-8,-2,1,3] => 00001100 => [5,1,3] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 2
[8,-6,2,4,-7,1,3,5] => 01001000 => [2,3,4] => [1,1,0,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 2
[4,-8,-6,-3,-7,1,2,5] => 01111000 => [2,1,1,1,4] => [1,1,0,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 4
[-4,3,-8,-6,-7,1,2,5] => 10111000 => [1,2,1,1,4] => [1,0,1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 4
[-7,6,8,1,3,-5,2,4] => 10000100 => [1,5,3] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 2
[5,-6,3,8,2,-7,1,4] => 01000100 => [2,4,3] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => ? = 2

Description

The bounce count of a Dyck path. For a Dyck path

$D$ of length

$2n$ , this is the number of points

$(i,i)$ for

$1 \leq i < n$ that are touching points of the [[Mp00099|bounce path]] of

$D$ .

Matching statistic: St000053

Mapping

Mp00267: Signed permutations —signs⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 85%●distinct values known / distinct values provided: 75%

Values

[1] => 0 => [2] => [1,1,0,0]
 => 0
[-1] => 1 => [1,1] => [1,0,1,0]
 => 1
[1,2] => 00 => [3] => [1,1,1,0,0,0]
 => 0
[1,-2] => 01 => [2,1] => [1,1,0,0,1,0]
 => 1
[-1,2] => 10 => [1,2] => [1,0,1,1,0,0]
 => 1
[-1,-2] => 11 => [1,1,1] => [1,0,1,0,1,0]
 => 2
[2,1] => 00 => [3] => [1,1,1,0,0,0]
 => 0
[2,-1] => 01 => [2,1] => [1,1,0,0,1,0]
 => 1
[-2,1] => 10 => [1,2] => [1,0,1,1,0,0]
 => 1
[-2,-1] => 11 => [1,1,1] => [1,0,1,0,1,0]
 => 2
[1,2,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[1,2,-3] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[1,-2,3] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[1,-2,-3] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[-1,2,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[-1,2,-3] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[-1,-2,3] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[-1,-2,-3] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3
[1,3,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[1,3,-2] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[1,-3,2] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[1,-3,-2] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[-1,3,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[-1,3,-2] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[-1,-3,2] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[-1,-3,-2] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3
[2,1,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[2,1,-3] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[2,-1,3] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[2,-1,-3] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[-2,1,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[-2,1,-3] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[-2,-1,3] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[-2,-1,-3] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3
[2,3,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[2,3,-1] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[2,-3,1] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[2,-3,-1] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[-2,3,1] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[-2,3,-1] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[-2,-3,1] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[-2,-3,-1] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3
[3,1,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[3,1,-2] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[3,-1,2] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[3,-1,-2] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[-3,1,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[-3,1,-2] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[-3,-1,2] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[-3,-1,-2] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3
[1,2,3,4,7,5,6] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[1,2,5,3,7,4,6] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[1,2,7,3,4,5,6] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[1,2,7,3,6,4,5] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[1,4,7,2,3,5,6] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[1,5,7,2,3,4,6] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[1,6,7,2,3,4,5] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[1,6,7,2,5,3,4] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[1,6,7,4,2,3,5] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[3,1,7,2,4,5,6] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[3,1,7,5,2,4,6] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[3,4,7,1,2,5,6] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[3,4,7,1,6,2,5] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[3,4,7,5,1,2,6] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[3,4,7,5,6,1,2] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[3,5,7,1,2,4,6] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[3,6,1,2,7,4,5] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[3,7,4,1,2,5,6] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[3,7,4,5,1,2,6] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[3,7,4,5,6,1,2] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[3,7,4,6,1,2,5] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[3,7,5,1,2,4,6] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[3,7,5,6,1,2,4] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[3,7,6,1,2,4,5] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[4,7,5,6,1,2,3] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[5,3,4,7,1,2,6] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[5,3,7,1,2,4,6] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[5,4,7,1,2,3,6] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[7,1,2,5,3,4,6] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[7,1,3,2,4,5,6] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[7,1,4,2,3,5,6] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[7,1,4,2,6,3,5] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[7,1,4,5,2,3,6] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[7,1,4,6,2,3,5] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[7,1,5,2,3,4,6] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[7,1,5,6,2,3,4] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[7,1,6,2,3,4,5] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[7,1,6,4,5,2,3] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[7,2,4,1,3,5,6] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[7,2,5,1,3,4,6] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[7,2,6,1,3,4,5] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[7,3,1,2,4,5,6] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[7,3,5,1,2,4,6] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[7,3,5,6,1,2,4] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[7,3,6,1,2,4,5] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[7,5,1,4,2,3,6] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[5,6,7,8,3,4,1,2] => 00000000 => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ? = 0
[5,6,7,8,3,1,2,4] => 00000000 => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ? = 0
[5,6,7,8,4,1,2,3] => 00000000 => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ? = 0
[5,6,7,8,1,4,2,3] => 00000000 => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ? = 0

Description

The number of valleys of the Dyck path.

Matching statistic: St000272

Mapping

Mp00267: Signed permutations —signs⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000272: Graphs ⟶ ℤResult quality: 75% ●values known / values provided: 85%●distinct values known / distinct values provided: 75%

Values

[1] => 0 => [2] => ([],2)
 => 0
[-1] => 1 => [1,1] => ([(0,1)],2)
 => 1
[1,2] => 00 => [3] => ([],3)
 => 0
[1,-2] => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 1
[-1,2] => 10 => [1,2] => ([(1,2)],3)
 => 1
[-1,-2] => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[2,1] => 00 => [3] => ([],3)
 => 0
[2,-1] => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 1
[-2,1] => 10 => [1,2] => ([(1,2)],3)
 => 1
[-2,-1] => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,2,3] => 000 => [4] => ([],4)
 => 0
[1,2,-3] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[1,-2,3] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 1
[1,-2,-3] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[-1,2,3] => 100 => [1,3] => ([(2,3)],4)
 => 1
[-1,2,-3] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[-1,-2,3] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[-1,-2,-3] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,3,2] => 000 => [4] => ([],4)
 => 0
[1,3,-2] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[1,-3,2] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 1
[1,-3,-2] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[-1,3,2] => 100 => [1,3] => ([(2,3)],4)
 => 1
[-1,3,-2] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[-1,-3,2] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[-1,-3,-2] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[2,1,3] => 000 => [4] => ([],4)
 => 0
[2,1,-3] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[2,-1,3] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 1
[2,-1,-3] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[-2,1,3] => 100 => [1,3] => ([(2,3)],4)
 => 1
[-2,1,-3] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[-2,-1,3] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[-2,-1,-3] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[2,3,1] => 000 => [4] => ([],4)
 => 0
[2,3,-1] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[2,-3,1] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 1
[2,-3,-1] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[-2,3,1] => 100 => [1,3] => ([(2,3)],4)
 => 1
[-2,3,-1] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[-2,-3,1] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[-2,-3,-1] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[3,1,2] => 000 => [4] => ([],4)
 => 0
[3,1,-2] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[3,-1,2] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 1
[3,-1,-2] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[-3,1,2] => 100 => [1,3] => ([(2,3)],4)
 => 1
[-3,1,-2] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[-3,-1,2] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[-3,-1,-2] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,2,3,4,7,5,6] => 0000000 => [8] => ([],8)
 => ? = 0
[1,2,5,3,7,4,6] => 0000000 => [8] => ([],8)
 => ? = 0
[1,2,7,3,4,5,6] => 0000000 => [8] => ([],8)
 => ? = 0
[1,2,7,3,6,4,5] => 0000000 => [8] => ([],8)
 => ? = 0
[1,4,7,2,3,5,6] => 0000000 => [8] => ([],8)
 => ? = 0
[1,5,7,2,3,4,6] => 0000000 => [8] => ([],8)
 => ? = 0
[1,6,7,2,3,4,5] => 0000000 => [8] => ([],8)
 => ? = 0
[1,6,7,2,5,3,4] => 0000000 => [8] => ([],8)
 => ? = 0
[1,6,7,4,2,3,5] => 0000000 => [8] => ([],8)
 => ? = 0
[3,1,7,2,4,5,6] => 0000000 => [8] => ([],8)
 => ? = 0
[3,1,7,5,2,4,6] => 0000000 => [8] => ([],8)
 => ? = 0
[3,4,7,1,2,5,6] => 0000000 => [8] => ([],8)
 => ? = 0
[3,4,7,1,6,2,5] => 0000000 => [8] => ([],8)
 => ? = 0
[3,4,7,5,1,2,6] => 0000000 => [8] => ([],8)
 => ? = 0
[3,4,7,5,6,1,2] => 0000000 => [8] => ([],8)
 => ? = 0
[3,5,7,1,2,4,6] => 0000000 => [8] => ([],8)
 => ? = 0
[3,6,1,2,7,4,5] => 0000000 => [8] => ([],8)
 => ? = 0
[3,7,4,1,2,5,6] => 0000000 => [8] => ([],8)
 => ? = 0
[3,7,4,5,1,2,6] => 0000000 => [8] => ([],8)
 => ? = 0
[3,7,4,5,6,1,2] => 0000000 => [8] => ([],8)
 => ? = 0
[3,7,4,6,1,2,5] => 0000000 => [8] => ([],8)
 => ? = 0
[3,7,5,1,2,4,6] => 0000000 => [8] => ([],8)
 => ? = 0
[3,7,5,6,1,2,4] => 0000000 => [8] => ([],8)
 => ? = 0
[3,7,6,1,2,4,5] => 0000000 => [8] => ([],8)
 => ? = 0
[4,7,5,6,1,2,3] => 0000000 => [8] => ([],8)
 => ? = 0
[5,3,4,7,1,2,6] => 0000000 => [8] => ([],8)
 => ? = 0
[5,3,7,1,2,4,6] => 0000000 => [8] => ([],8)
 => ? = 0
[5,4,7,1,2,3,6] => 0000000 => [8] => ([],8)
 => ? = 0
[7,1,2,5,3,4,6] => 0000000 => [8] => ([],8)
 => ? = 0
[7,1,3,2,4,5,6] => 0000000 => [8] => ([],8)
 => ? = 0
[7,1,4,2,3,5,6] => 0000000 => [8] => ([],8)
 => ? = 0
[7,1,4,2,6,3,5] => 0000000 => [8] => ([],8)
 => ? = 0
[7,1,4,5,2,3,6] => 0000000 => [8] => ([],8)
 => ? = 0
[7,1,4,6,2,3,5] => 0000000 => [8] => ([],8)
 => ? = 0
[7,1,5,2,3,4,6] => 0000000 => [8] => ([],8)
 => ? = 0
[7,1,5,6,2,3,4] => 0000000 => [8] => ([],8)
 => ? = 0
[7,1,6,2,3,4,5] => 0000000 => [8] => ([],8)
 => ? = 0
[7,1,6,4,5,2,3] => 0000000 => [8] => ([],8)
 => ? = 0
[7,2,4,1,3,5,6] => 0000000 => [8] => ([],8)
 => ? = 0
[7,2,5,1,3,4,6] => 0000000 => [8] => ([],8)
 => ? = 0
[7,2,6,1,3,4,5] => 0000000 => [8] => ([],8)
 => ? = 0
[7,3,1,2,4,5,6] => 0000000 => [8] => ([],8)
 => ? = 0
[7,3,5,1,2,4,6] => 0000000 => [8] => ([],8)
 => ? = 0
[7,3,5,6,1,2,4] => 0000000 => [8] => ([],8)
 => ? = 0
[7,3,6,1,2,4,5] => 0000000 => [8] => ([],8)
 => ? = 0
[7,5,1,4,2,3,6] => 0000000 => [8] => ([],8)
 => ? = 0
[5,6,7,8,3,4,1,2] => 00000000 => [9] => ([],9)
 => ? = 0
[5,6,7,8,3,1,2,4] => 00000000 => [9] => ([],9)
 => ? = 0
[5,6,7,8,4,1,2,3] => 00000000 => [9] => ([],9)
 => ? = 0
[5,6,7,8,1,4,2,3] => 00000000 => [9] => ([],9)
 => ? = 0

Description

The treewidth of a graph. A graph has treewidth zero if and only if it has no edges. A connected graph has treewidth at most one if and only if it is a tree. A connected graph has treewidth at most two if and only if it is a series-parallel graph.

Matching statistic: St000362

Mapping

Mp00267: Signed permutations —signs⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000362: Graphs ⟶ ℤResult quality: 75% ●values known / values provided: 85%●distinct values known / distinct values provided: 75%

Values

[1] => 0 => [2] => ([],2)
 => 0
[-1] => 1 => [1,1] => ([(0,1)],2)
 => 1
[1,2] => 00 => [3] => ([],3)
 => 0
[1,-2] => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 1
[-1,2] => 10 => [1,2] => ([(1,2)],3)
 => 1
[-1,-2] => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[2,1] => 00 => [3] => ([],3)
 => 0
[2,-1] => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 1
[-2,1] => 10 => [1,2] => ([(1,2)],3)
 => 1
[-2,-1] => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,2,3] => 000 => [4] => ([],4)
 => 0
[1,2,-3] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[1,-2,3] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 1
[1,-2,-3] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[-1,2,3] => 100 => [1,3] => ([(2,3)],4)
 => 1
[-1,2,-3] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[-1,-2,3] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[-1,-2,-3] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,3,2] => 000 => [4] => ([],4)
 => 0
[1,3,-2] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[1,-3,2] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 1
[1,-3,-2] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[-1,3,2] => 100 => [1,3] => ([(2,3)],4)
 => 1
[-1,3,-2] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[-1,-3,2] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[-1,-3,-2] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[2,1,3] => 000 => [4] => ([],4)
 => 0
[2,1,-3] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[2,-1,3] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 1
[2,-1,-3] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[-2,1,3] => 100 => [1,3] => ([(2,3)],4)
 => 1
[-2,1,-3] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[-2,-1,3] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[-2,-1,-3] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[2,3,1] => 000 => [4] => ([],4)
 => 0
[2,3,-1] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[2,-3,1] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 1
[2,-3,-1] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[-2,3,1] => 100 => [1,3] => ([(2,3)],4)
 => 1
[-2,3,-1] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[-2,-3,1] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[-2,-3,-1] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[3,1,2] => 000 => [4] => ([],4)
 => 0
[3,1,-2] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[3,-1,2] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 1
[3,-1,-2] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[-3,1,2] => 100 => [1,3] => ([(2,3)],4)
 => 1
[-3,1,-2] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[-3,-1,2] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[-3,-1,-2] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,2,3,4,7,5,6] => 0000000 => [8] => ([],8)
 => ? = 0
[1,2,5,3,7,4,6] => 0000000 => [8] => ([],8)
 => ? = 0
[1,2,7,3,4,5,6] => 0000000 => [8] => ([],8)
 => ? = 0
[1,2,7,3,6,4,5] => 0000000 => [8] => ([],8)
 => ? = 0
[1,4,7,2,3,5,6] => 0000000 => [8] => ([],8)
 => ? = 0
[1,5,7,2,3,4,6] => 0000000 => [8] => ([],8)
 => ? = 0
[1,6,7,2,3,4,5] => 0000000 => [8] => ([],8)
 => ? = 0
[1,6,7,2,5,3,4] => 0000000 => [8] => ([],8)
 => ? = 0
[1,6,7,4,2,3,5] => 0000000 => [8] => ([],8)
 => ? = 0
[3,1,7,2,4,5,6] => 0000000 => [8] => ([],8)
 => ? = 0
[3,1,7,5,2,4,6] => 0000000 => [8] => ([],8)
 => ? = 0
[3,4,7,1,2,5,6] => 0000000 => [8] => ([],8)
 => ? = 0
[3,4,7,1,6,2,5] => 0000000 => [8] => ([],8)
 => ? = 0
[3,4,7,5,1,2,6] => 0000000 => [8] => ([],8)
 => ? = 0
[3,4,7,5,6,1,2] => 0000000 => [8] => ([],8)
 => ? = 0
[3,5,7,1,2,4,6] => 0000000 => [8] => ([],8)
 => ? = 0
[3,6,1,2,7,4,5] => 0000000 => [8] => ([],8)
 => ? = 0
[3,7,4,1,2,5,6] => 0000000 => [8] => ([],8)
 => ? = 0
[3,7,4,5,1,2,6] => 0000000 => [8] => ([],8)
 => ? = 0
[3,7,4,5,6,1,2] => 0000000 => [8] => ([],8)
 => ? = 0
[3,7,4,6,1,2,5] => 0000000 => [8] => ([],8)
 => ? = 0
[3,7,5,1,2,4,6] => 0000000 => [8] => ([],8)
 => ? = 0
[3,7,5,6,1,2,4] => 0000000 => [8] => ([],8)
 => ? = 0
[3,7,6,1,2,4,5] => 0000000 => [8] => ([],8)
 => ? = 0
[4,7,5,6,1,2,3] => 0000000 => [8] => ([],8)
 => ? = 0
[5,3,4,7,1,2,6] => 0000000 => [8] => ([],8)
 => ? = 0
[5,3,7,1,2,4,6] => 0000000 => [8] => ([],8)
 => ? = 0
[5,4,7,1,2,3,6] => 0000000 => [8] => ([],8)
 => ? = 0
[7,1,2,5,3,4,6] => 0000000 => [8] => ([],8)
 => ? = 0
[7,1,3,2,4,5,6] => 0000000 => [8] => ([],8)
 => ? = 0
[7,1,4,2,3,5,6] => 0000000 => [8] => ([],8)
 => ? = 0
[7,1,4,2,6,3,5] => 0000000 => [8] => ([],8)
 => ? = 0
[7,1,4,5,2,3,6] => 0000000 => [8] => ([],8)
 => ? = 0
[7,1,4,6,2,3,5] => 0000000 => [8] => ([],8)
 => ? = 0
[7,1,5,2,3,4,6] => 0000000 => [8] => ([],8)
 => ? = 0
[7,1,5,6,2,3,4] => 0000000 => [8] => ([],8)
 => ? = 0
[7,1,6,2,3,4,5] => 0000000 => [8] => ([],8)
 => ? = 0
[7,1,6,4,5,2,3] => 0000000 => [8] => ([],8)
 => ? = 0
[7,2,4,1,3,5,6] => 0000000 => [8] => ([],8)
 => ? = 0
[7,2,5,1,3,4,6] => 0000000 => [8] => ([],8)
 => ? = 0
[7,2,6,1,3,4,5] => 0000000 => [8] => ([],8)
 => ? = 0
[7,3,1,2,4,5,6] => 0000000 => [8] => ([],8)
 => ? = 0
[7,3,5,1,2,4,6] => 0000000 => [8] => ([],8)
 => ? = 0
[7,3,5,6,1,2,4] => 0000000 => [8] => ([],8)
 => ? = 0
[7,3,6,1,2,4,5] => 0000000 => [8] => ([],8)
 => ? = 0
[7,5,1,4,2,3,6] => 0000000 => [8] => ([],8)
 => ? = 0
[5,6,7,8,3,4,1,2] => 00000000 => [9] => ([],9)
 => ? = 0
[5,6,7,8,3,1,2,4] => 00000000 => [9] => ([],9)
 => ? = 0
[5,6,7,8,4,1,2,3] => 00000000 => [9] => ([],9)
 => ? = 0
[5,6,7,8,1,4,2,3] => 00000000 => [9] => ([],9)
 => ? = 0

Description

The size of a minimal vertex cover of a graph. A '''vertex cover''' of a graph

$G$ is a subset

$S$ of the vertices of

$G$ such that each edge of

$G$ contains at least one vertex of

$S$ . Finding a minimal vertex cover is an NP-hard optimization problem.

Matching statistic: St000536

Mapping

Mp00267: Signed permutations —signs⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000536: Graphs ⟶ ℤResult quality: 75% ●values known / values provided: 85%●distinct values known / distinct values provided: 75%

Values

[1] => 0 => [2] => ([],2)
 => 0
[-1] => 1 => [1,1] => ([(0,1)],2)
 => 1
[1,2] => 00 => [3] => ([],3)
 => 0
[1,-2] => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 1
[-1,2] => 10 => [1,2] => ([(1,2)],3)
 => 1
[-1,-2] => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[2,1] => 00 => [3] => ([],3)
 => 0
[2,-1] => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 1
[-2,1] => 10 => [1,2] => ([(1,2)],3)
 => 1
[-2,-1] => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,2,3] => 000 => [4] => ([],4)
 => 0
[1,2,-3] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[1,-2,3] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 1
[1,-2,-3] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[-1,2,3] => 100 => [1,3] => ([(2,3)],4)
 => 1
[-1,2,-3] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[-1,-2,3] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[-1,-2,-3] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,3,2] => 000 => [4] => ([],4)
 => 0
[1,3,-2] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[1,-3,2] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 1
[1,-3,-2] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[-1,3,2] => 100 => [1,3] => ([(2,3)],4)
 => 1
[-1,3,-2] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[-1,-3,2] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[-1,-3,-2] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[2,1,3] => 000 => [4] => ([],4)
 => 0
[2,1,-3] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[2,-1,3] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 1
[2,-1,-3] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[-2,1,3] => 100 => [1,3] => ([(2,3)],4)
 => 1
[-2,1,-3] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[-2,-1,3] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[-2,-1,-3] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[2,3,1] => 000 => [4] => ([],4)
 => 0
[2,3,-1] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[2,-3,1] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 1
[2,-3,-1] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[-2,3,1] => 100 => [1,3] => ([(2,3)],4)
 => 1
[-2,3,-1] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[-2,-3,1] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[-2,-3,-1] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[3,1,2] => 000 => [4] => ([],4)
 => 0
[3,1,-2] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[3,-1,2] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 1
[3,-1,-2] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[-3,1,2] => 100 => [1,3] => ([(2,3)],4)
 => 1
[-3,1,-2] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[-3,-1,2] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[-3,-1,-2] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,2,3,4,7,5,6] => 0000000 => [8] => ([],8)
 => ? = 0
[1,2,5,3,7,4,6] => 0000000 => [8] => ([],8)
 => ? = 0
[1,2,7,3,4,5,6] => 0000000 => [8] => ([],8)
 => ? = 0
[1,2,7,3,6,4,5] => 0000000 => [8] => ([],8)
 => ? = 0
[1,4,7,2,3,5,6] => 0000000 => [8] => ([],8)
 => ? = 0
[1,5,7,2,3,4,6] => 0000000 => [8] => ([],8)
 => ? = 0
[1,6,7,2,3,4,5] => 0000000 => [8] => ([],8)
 => ? = 0
[1,6,7,2,5,3,4] => 0000000 => [8] => ([],8)
 => ? = 0
[1,6,7,4,2,3,5] => 0000000 => [8] => ([],8)
 => ? = 0
[3,1,7,2,4,5,6] => 0000000 => [8] => ([],8)
 => ? = 0
[3,1,7,5,2,4,6] => 0000000 => [8] => ([],8)
 => ? = 0
[3,4,7,1,2,5,6] => 0000000 => [8] => ([],8)
 => ? = 0
[3,4,7,1,6,2,5] => 0000000 => [8] => ([],8)
 => ? = 0
[3,4,7,5,1,2,6] => 0000000 => [8] => ([],8)
 => ? = 0
[3,4,7,5,6,1,2] => 0000000 => [8] => ([],8)
 => ? = 0
[3,5,7,1,2,4,6] => 0000000 => [8] => ([],8)
 => ? = 0
[3,6,1,2,7,4,5] => 0000000 => [8] => ([],8)
 => ? = 0
[3,7,4,1,2,5,6] => 0000000 => [8] => ([],8)
 => ? = 0
[3,7,4,5,1,2,6] => 0000000 => [8] => ([],8)
 => ? = 0
[3,7,4,5,6,1,2] => 0000000 => [8] => ([],8)
 => ? = 0
[3,7,4,6,1,2,5] => 0000000 => [8] => ([],8)
 => ? = 0
[3,7,5,1,2,4,6] => 0000000 => [8] => ([],8)
 => ? = 0
[3,7,5,6,1,2,4] => 0000000 => [8] => ([],8)
 => ? = 0
[3,7,6,1,2,4,5] => 0000000 => [8] => ([],8)
 => ? = 0
[4,7,5,6,1,2,3] => 0000000 => [8] => ([],8)
 => ? = 0
[5,3,4,7,1,2,6] => 0000000 => [8] => ([],8)
 => ? = 0
[5,3,7,1,2,4,6] => 0000000 => [8] => ([],8)
 => ? = 0
[5,4,7,1,2,3,6] => 0000000 => [8] => ([],8)
 => ? = 0
[7,1,2,5,3,4,6] => 0000000 => [8] => ([],8)
 => ? = 0
[7,1,3,2,4,5,6] => 0000000 => [8] => ([],8)
 => ? = 0
[7,1,4,2,3,5,6] => 0000000 => [8] => ([],8)
 => ? = 0
[7,1,4,2,6,3,5] => 0000000 => [8] => ([],8)
 => ? = 0
[7,1,4,5,2,3,6] => 0000000 => [8] => ([],8)
 => ? = 0
[7,1,4,6,2,3,5] => 0000000 => [8] => ([],8)
 => ? = 0
[7,1,5,2,3,4,6] => 0000000 => [8] => ([],8)
 => ? = 0
[7,1,5,6,2,3,4] => 0000000 => [8] => ([],8)
 => ? = 0
[7,1,6,2,3,4,5] => 0000000 => [8] => ([],8)
 => ? = 0
[7,1,6,4,5,2,3] => 0000000 => [8] => ([],8)
 => ? = 0
[7,2,4,1,3,5,6] => 0000000 => [8] => ([],8)
 => ? = 0
[7,2,5,1,3,4,6] => 0000000 => [8] => ([],8)
 => ? = 0
[7,2,6,1,3,4,5] => 0000000 => [8] => ([],8)
 => ? = 0
[7,3,1,2,4,5,6] => 0000000 => [8] => ([],8)
 => ? = 0
[7,3,5,1,2,4,6] => 0000000 => [8] => ([],8)
 => ? = 0
[7,3,5,6,1,2,4] => 0000000 => [8] => ([],8)
 => ? = 0
[7,3,6,1,2,4,5] => 0000000 => [8] => ([],8)
 => ? = 0
[7,5,1,4,2,3,6] => 0000000 => [8] => ([],8)
 => ? = 0
[5,6,7,8,3,4,1,2] => 00000000 => [9] => ([],9)
 => ? = 0
[5,6,7,8,3,1,2,4] => 00000000 => [9] => ([],9)
 => ? = 0
[5,6,7,8,4,1,2,3] => 00000000 => [9] => ([],9)
 => ? = 0
[5,6,7,8,1,4,2,3] => 00000000 => [9] => ([],9)
 => ? = 0

Description

The pathwidth of a graph.

The following 30 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001197The global dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St000098The chromatic number of a graph. St000172The Grundy number of a graph. St001029The size of the core of a graph. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n-1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a Dyck path as follows: St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001670The connected partition number of a graph. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001963The tree-depth of a graph. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001205The number of non-simple indecomposable projective-injective modules of the algebra

$eAe$ in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by

$\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St000015The number of peaks of a Dyck path. St000822The Hadwiger number of the graph. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n−1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a special CNakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001812The biclique partition number of a graph. St001330The hat guessing number of a graph. St001870The number of positive entries followed by a negative entry in a signed permutation. St001430The number of positive entries in a signed permutation. St001429The number of negative entries in a signed permutation.