Your data matches 18 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001430
(load all 8 compositions to match this statistic)
Mapping
St001430: Signed permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1
[-1] => 0
[1,2] => 2
[1,-2] => 1
[-1,2] => 1
[-1,-2] => 0
[2,1] => 2
[2,-1] => 1
[-2,1] => 1
[-2,-1] => 0
[1,2,3] => 3
[1,2,-3] => 2
[1,-2,3] => 2
[1,-2,-3] => 1
[-1,2,3] => 2
[-1,2,-3] => 1
[-1,-2,3] => 1
[-1,-2,-3] => 0
[1,3,2] => 3
[1,3,-2] => 2
[1,-3,2] => 2
[1,-3,-2] => 1
[-1,3,2] => 2
[-1,3,-2] => 1
[-1,-3,2] => 1
[-1,-3,-2] => 0
[2,1,3] => 3
[2,1,-3] => 2
[2,-1,3] => 2
[2,-1,-3] => 1
[-2,1,3] => 2
[-2,1,-3] => 1
[-2,-1,3] => 1
[-2,-1,-3] => 0
[2,3,1] => 3
[2,3,-1] => 2
[2,-3,1] => 2
[2,-3,-1] => 1
[-2,3,1] => 2
[-2,3,-1] => 1
[-2,-3,1] => 1
[-2,-3,-1] => 0
[3,1,2] => 3
[3,1,-2] => 2
[3,-1,2] => 2
[3,-1,-2] => 1
[-3,1,2] => 2
[-3,1,-2] => 1
[-3,-1,2] => 1
[-3,-1,-2] => 0
Description
The number of positive entries in a signed permutation.
Matching statistic: St000288
(load all 15 compositions to match this statistic)
Mapping
Mp00267: Signed permutations —signs⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 100%
Values
[1] => 0 => 1 => 1
[-1] => 1 => 0 => 0
[1,2] => 00 => 11 => 2
[1,-2] => 01 => 10 => 1
[-1,2] => 10 => 01 => 1
[-1,-2] => 11 => 00 => 0
[2,1] => 00 => 11 => 2
[2,-1] => 01 => 10 => 1
[-2,1] => 10 => 01 => 1
[-2,-1] => 11 => 00 => 0
[1,2,3] => 000 => 111 => 3
[1,2,-3] => 001 => 110 => 2
[1,-2,3] => 010 => 101 => 2
[1,-2,-3] => 011 => 100 => 1
[-1,2,3] => 100 => 011 => 2
[-1,2,-3] => 101 => 010 => 1
[-1,-2,3] => 110 => 001 => 1
[-1,-2,-3] => 111 => 000 => 0
[1,3,2] => 000 => 111 => 3
[1,3,-2] => 001 => 110 => 2
[1,-3,2] => 010 => 101 => 2
[1,-3,-2] => 011 => 100 => 1
[-1,3,2] => 100 => 011 => 2
[-1,3,-2] => 101 => 010 => 1
[-1,-3,2] => 110 => 001 => 1
[-1,-3,-2] => 111 => 000 => 0
[2,1,3] => 000 => 111 => 3
[2,1,-3] => 001 => 110 => 2
[2,-1,3] => 010 => 101 => 2
[2,-1,-3] => 011 => 100 => 1
[-2,1,3] => 100 => 011 => 2
[-2,1,-3] => 101 => 010 => 1
[-2,-1,3] => 110 => 001 => 1
[-2,-1,-3] => 111 => 000 => 0
[2,3,1] => 000 => 111 => 3
[2,3,-1] => 001 => 110 => 2
[2,-3,1] => 010 => 101 => 2
[2,-3,-1] => 011 => 100 => 1
[-2,3,1] => 100 => 011 => 2
[-2,3,-1] => 101 => 010 => 1
[-2,-3,1] => 110 => 001 => 1
[-2,-3,-1] => 111 => 000 => 0
[3,1,2] => 000 => 111 => 3
[3,1,-2] => 001 => 110 => 2
[3,-1,2] => 010 => 101 => 2
[3,-1,-2] => 011 => 100 => 1
[-3,1,2] => 100 => 011 => 2
[-3,1,-2] => 101 => 010 => 1
[-3,-1,2] => 110 => 001 => 1
[-3,-1,-2] => 111 => 000 => 0
[1,2,3,4,6,5] => ? => ? => ? = 6
[1,2,3,5,4,6] => ? => ? => ? = 6
[1,2,3,5,6,4] => ? => ? => ? = 6
[1,2,3,6,5,4] => ? => ? => ? = 6
[1,2,4,3,5,6] => ? => ? => ? = 6
[1,2,4,3,6,5] => ? => ? => ? = 6
[1,2,4,5,3,6] => ? => ? => ? = 6
[1,2,4,5,6,3] => ? => ? => ? = 6
[1,2,4,6,3,5] => ? => ? => ? = 6
[1,2,4,6,5,3] => ? => ? => ? = 6
[1,2,5,3,6,4] => ? => ? => ? = 6
[1,2,5,4,3,6] => ? => ? => ? = 6
[1,2,5,4,6,3] => ? => ? => ? = 6
[1,2,5,6,4,3] => ? => ? => ? = 6
[1,2,6,3,4,5] => ? => ? => ? = 6
[1,2,6,4,5,3] => ? => ? => ? = 6
[1,2,6,5,4,3] => ? => ? => ? = 6
[1,3,2,4,5,6] => ? => ? => ? = 6
[1,3,2,4,6,5] => ? => ? => ? = 6
[1,3,2,5,4,6] => ? => ? => ? = 6
[1,3,2,5,6,4] => ? => ? => ? = 6
[1,3,2,6,4,5] => ? => ? => ? = 6
[1,3,2,6,5,4] => ? => ? => ? = 6
[1,3,4,2,5,6] => ? => ? => ? = 6
[1,3,4,2,6,5] => ? => ? => ? = 6
[1,3,4,5,2,6] => ? => ? => ? = 6
[1,3,4,5,6,2] => ? => ? => ? = 6
[1,3,4,6,2,5] => ? => ? => ? = 6
[1,3,4,6,5,2] => ? => ? => ? = 6
[1,3,5,2,4,6] => ? => ? => ? = 6
[1,3,5,2,6,4] => ? => ? => ? = 6
[1,3,5,4,2,6] => ? => ? => ? = 6
[1,3,5,4,6,2] => ? => ? => ? = 6
[1,3,5,6,2,4] => ? => ? => ? = 6
[1,3,5,6,4,2] => ? => ? => ? = 6
[1,3,6,2,4,5] => ? => ? => ? = 6
[1,3,6,4,5,2] => ? => ? => ? = 6
[1,3,6,5,4,2] => ? => ? => ? = 6
[1,4,2,3,6,5] => ? => ? => ? = 6
[1,4,2,5,3,6] => ? => ? => ? = 6
[1,4,2,5,6,3] => ? => ? => ? = 6
[1,4,3,2,5,6] => ? => ? => ? = 6
[1,4,3,2,6,5] => ? => ? => ? = 6
[1,4,3,5,2,6] => ? => ? => ? = 6
[1,4,3,5,6,2] => ? => ? => ? = 6
[1,4,3,6,5,2] => ? => ? => ? = 6
[1,4,5,2,6,3] => ? => ? => ? = 6
[1,4,5,3,2,6] => ? => ? => ? = 6
[1,4,5,3,6,2] => ? => ? => ? = 6
[1,4,5,6,3,2] => ? => ? => ? = 6
Description
The number of ones in a binary word. This is also known as the Hamming weight of the word.
Matching statistic: St000394
Mapping
Mp00267: Signed permutations —signs⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 100%
Values
[1] => 0 => [2] => [1,1,0,0]
 => 1
[-1] => 1 => [1,1] => [1,0,1,0]
 => 0
[1,2] => 00 => [3] => [1,1,1,0,0,0]
 => 2
[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]
 => 0
[2,1] => 00 => [3] => [1,1,1,0,0,0]
 => 2
[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]
 => 0
[1,2,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 3
[1,2,-3] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2
[1,-2,3] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[1,-2,-3] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1
[-1,2,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 2
[-1,2,-3] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[-1,-2,3] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
[-1,-2,-3] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 0
[1,3,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 3
[1,3,-2] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2
[1,-3,2] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[1,-3,-2] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1
[-1,3,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 2
[-1,3,-2] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[-1,-3,2] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
[-1,-3,-2] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 0
[2,1,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 3
[2,1,-3] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2
[2,-1,3] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[2,-1,-3] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1
[-2,1,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 2
[-2,1,-3] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[-2,-1,3] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
[-2,-1,-3] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 0
[2,3,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 3
[2,3,-1] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2
[2,-3,1] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[2,-3,-1] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1
[-2,3,1] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 2
[-2,3,-1] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[-2,-3,1] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
[-2,-3,-1] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 0
[3,1,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 3
[3,1,-2] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2
[3,-1,2] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[3,-1,-2] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1
[-3,1,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 2
[-3,1,-2] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[-3,-1,2] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
[-3,-1,-2] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 0
[1,2,3,4,6,5] => ? => ? => ?
 => ? = 6
[1,2,3,5,4,6] => ? => ? => ?
 => ? = 6
[1,2,3,5,6,4] => ? => ? => ?
 => ? = 6
[1,2,3,6,5,4] => ? => ? => ?
 => ? = 6
[1,2,4,3,5,6] => ? => ? => ?
 => ? = 6
[1,2,4,3,6,5] => ? => ? => ?
 => ? = 6
[1,2,4,5,3,6] => ? => ? => ?
 => ? = 6
[1,2,4,5,6,3] => ? => ? => ?
 => ? = 6
[1,2,4,6,3,5] => ? => ? => ?
 => ? = 6
[1,2,4,6,5,3] => ? => ? => ?
 => ? = 6
[1,2,5,3,6,4] => ? => ? => ?
 => ? = 6
[1,2,5,4,3,6] => ? => ? => ?
 => ? = 6
[1,2,5,4,6,3] => ? => ? => ?
 => ? = 6
[1,2,5,6,4,3] => ? => ? => ?
 => ? = 6
[1,2,6,3,4,5] => ? => ? => ?
 => ? = 6
[1,2,6,4,5,3] => ? => ? => ?
 => ? = 6
[1,2,6,5,4,3] => ? => ? => ?
 => ? = 6
[1,3,2,4,5,6] => ? => ? => ?
 => ? = 6
[1,3,2,4,6,5] => ? => ? => ?
 => ? = 6
[1,3,2,5,4,6] => ? => ? => ?
 => ? = 6
[1,3,2,5,6,4] => ? => ? => ?
 => ? = 6
[1,3,2,6,4,5] => ? => ? => ?
 => ? = 6
[1,3,2,6,5,4] => ? => ? => ?
 => ? = 6
[1,3,4,2,5,6] => ? => ? => ?
 => ? = 6
[1,3,4,2,6,5] => ? => ? => ?
 => ? = 6
[1,3,4,5,2,6] => ? => ? => ?
 => ? = 6
[1,3,4,5,6,2] => ? => ? => ?
 => ? = 6
[1,3,4,6,2,5] => ? => ? => ?
 => ? = 6
[1,3,4,6,5,2] => ? => ? => ?
 => ? = 6
[1,3,5,2,4,6] => ? => ? => ?
 => ? = 6
[1,3,5,2,6,4] => ? => ? => ?
 => ? = 6
[1,3,5,4,2,6] => ? => ? => ?
 => ? = 6
[1,3,5,4,6,2] => ? => ? => ?
 => ? = 6
[1,3,5,6,2,4] => ? => ? => ?
 => ? = 6
[1,3,5,6,4,2] => ? => ? => ?
 => ? = 6
[1,3,6,2,4,5] => ? => ? => ?
 => ? = 6
[1,3,6,4,5,2] => ? => ? => ?
 => ? = 6
[1,3,6,5,4,2] => ? => ? => ?
 => ? = 6
[1,4,2,3,6,5] => ? => ? => ?
 => ? = 6
[1,4,2,5,3,6] => ? => ? => ?
 => ? = 6
[1,4,2,5,6,3] => ? => ? => ?
 => ? = 6
[1,4,3,2,5,6] => ? => ? => ?
 => ? = 6
[1,4,3,2,6,5] => ? => ? => ?
 => ? = 6
[1,4,3,5,2,6] => ? => ? => ?
 => ? = 6
[1,4,3,5,6,2] => ? => ? => ?
 => ? = 6
[1,4,3,6,5,2] => ? => ? => ?
 => ? = 6
[1,4,5,2,6,3] => ? => ? => ?
 => ? = 6
[1,4,5,3,2,6] => ? => ? => ?
 => ? = 6
[1,4,5,3,6,2] => ? => ? => ?
 => ? = 6
[1,4,5,6,3,2] => ? => ? => ?
 => ? = 6
Description
The sum of the heights of the peaks of a Dyck path minus the number of peaks.
Matching statistic: St000024
Mapping
Mp00267: Signed permutations —signs⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 78%
Values
[1] => 0 => [2] => [1,1,0,0]
 => 1
[-1] => 1 => [1,1] => [1,0,1,0]
 => 0
[1,2] => 00 => [3] => [1,1,1,0,0,0]
 => 2
[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]
 => 0
[2,1] => 00 => [3] => [1,1,1,0,0,0]
 => 2
[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]
 => 0
[1,2,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 3
[1,2,-3] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2
[1,-2,3] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[1,-2,-3] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1
[-1,2,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 2
[-1,2,-3] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[-1,-2,3] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
[-1,-2,-3] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 0
[1,3,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 3
[1,3,-2] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2
[1,-3,2] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[1,-3,-2] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1
[-1,3,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 2
[-1,3,-2] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[-1,-3,2] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
[-1,-3,-2] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 0
[2,1,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 3
[2,1,-3] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2
[2,-1,3] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[2,-1,-3] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1
[-2,1,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 2
[-2,1,-3] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[-2,-1,3] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
[-2,-1,-3] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 0
[2,3,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 3
[2,3,-1] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2
[2,-3,1] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[2,-3,-1] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1
[-2,3,1] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 2
[-2,3,-1] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[-2,-3,1] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
[-2,-3,-1] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 0
[3,1,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 3
[3,1,-2] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2
[3,-1,2] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[3,-1,-2] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1
[-3,1,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 2
[-3,1,-2] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[-3,-1,2] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
[-3,-1,-2] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 0
[1,2,3,4,6,5] => ? => ? => ?
 => ? = 6
[1,2,3,5,4,6] => ? => ? => ?
 => ? = 6
[1,2,3,5,6,4] => ? => ? => ?
 => ? = 6
[1,2,3,6,5,4] => ? => ? => ?
 => ? = 6
[1,2,4,3,5,6] => ? => ? => ?
 => ? = 6
[1,2,4,3,6,5] => ? => ? => ?
 => ? = 6
[1,2,4,5,3,6] => ? => ? => ?
 => ? = 6
[1,2,4,5,6,3] => ? => ? => ?
 => ? = 6
[1,2,4,6,3,5] => ? => ? => ?
 => ? = 6
[1,2,4,6,5,3] => ? => ? => ?
 => ? = 6
[1,2,5,3,6,4] => ? => ? => ?
 => ? = 6
[1,2,5,4,3,6] => ? => ? => ?
 => ? = 6
[1,2,5,4,6,3] => ? => ? => ?
 => ? = 6
[1,2,5,6,4,3] => ? => ? => ?
 => ? = 6
[1,2,6,3,4,5] => ? => ? => ?
 => ? = 6
[1,2,6,4,5,3] => ? => ? => ?
 => ? = 6
[1,2,6,5,4,3] => ? => ? => ?
 => ? = 6
[1,3,2,4,5,6] => ? => ? => ?
 => ? = 6
[1,3,2,4,6,5] => ? => ? => ?
 => ? = 6
[1,3,2,5,4,6] => ? => ? => ?
 => ? = 6
[1,3,2,5,6,4] => ? => ? => ?
 => ? = 6
[1,3,2,6,4,5] => ? => ? => ?
 => ? = 6
[1,3,2,6,5,4] => ? => ? => ?
 => ? = 6
[1,3,4,2,5,6] => ? => ? => ?
 => ? = 6
[1,3,4,2,6,5] => ? => ? => ?
 => ? = 6
[1,3,4,5,2,6] => ? => ? => ?
 => ? = 6
[1,3,4,5,6,2] => ? => ? => ?
 => ? = 6
[1,3,4,6,2,5] => ? => ? => ?
 => ? = 6
[1,3,4,6,5,2] => ? => ? => ?
 => ? = 6
[1,3,5,2,4,6] => ? => ? => ?
 => ? = 6
[1,3,5,2,6,4] => ? => ? => ?
 => ? = 6
[1,3,5,4,2,6] => ? => ? => ?
 => ? = 6
[1,3,5,4,6,2] => ? => ? => ?
 => ? = 6
[1,3,5,6,2,4] => ? => ? => ?
 => ? = 6
[1,3,5,6,4,2] => ? => ? => ?
 => ? = 6
[1,3,6,2,4,5] => ? => ? => ?
 => ? = 6
[1,3,6,4,5,2] => ? => ? => ?
 => ? = 6
[1,3,6,5,4,2] => ? => ? => ?
 => ? = 6
[1,4,2,3,6,5] => ? => ? => ?
 => ? = 6
[1,4,2,5,3,6] => ? => ? => ?
 => ? = 6
[1,4,2,5,6,3] => ? => ? => ?
 => ? = 6
[1,4,3,2,5,6] => ? => ? => ?
 => ? = 6
[1,4,3,2,6,5] => ? => ? => ?
 => ? = 6
[1,4,3,5,2,6] => ? => ? => ?
 => ? = 6
[1,4,3,5,6,2] => ? => ? => ?
 => ? = 6
[1,4,3,6,5,2] => ? => ? => ?
 => ? = 6
[1,4,5,2,6,3] => ? => ? => ?
 => ? = 6
[1,4,5,3,2,6] => ? => ? => ?
 => ? = 6
[1,4,5,3,6,2] => ? => ? => ?
 => ? = 6
[1,4,5,6,3,2] => ? => ? => ?
 => ? = 6
Description
The number of double up and double down steps of a Dyck path. In other words, this is the number of double rises (and, equivalently, the number of double falls) of a Dyck path.
Matching statistic: St001189
Mapping
Mp00267: Signed permutations —signs⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001189: Dyck paths ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 78%
Values
[1] => 0 => [2] => [1,1,0,0]
 => 1
[-1] => 1 => [1,1] => [1,0,1,0]
 => 0
[1,2] => 00 => [3] => [1,1,1,0,0,0]
 => 2
[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]
 => 0
[2,1] => 00 => [3] => [1,1,1,0,0,0]
 => 2
[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]
 => 0
[1,2,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 3
[1,2,-3] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2
[1,-2,3] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[1,-2,-3] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1
[-1,2,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 2
[-1,2,-3] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[-1,-2,3] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
[-1,-2,-3] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 0
[1,3,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 3
[1,3,-2] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2
[1,-3,2] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[1,-3,-2] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1
[-1,3,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 2
[-1,3,-2] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[-1,-3,2] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
[-1,-3,-2] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 0
[2,1,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 3
[2,1,-3] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2
[2,-1,3] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[2,-1,-3] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1
[-2,1,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 2
[-2,1,-3] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[-2,-1,3] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
[-2,-1,-3] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 0
[2,3,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 3
[2,3,-1] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2
[2,-3,1] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[2,-3,-1] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1
[-2,3,1] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 2
[-2,3,-1] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[-2,-3,1] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
[-2,-3,-1] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 0
[3,1,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 3
[3,1,-2] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2
[3,-1,2] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[3,-1,-2] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1
[-3,1,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 2
[-3,1,-2] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[-3,-1,2] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
[-3,-1,-2] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 0
[1,2,3,4,6,5] => ? => ? => ?
 => ? = 6
[1,2,3,5,4,6] => ? => ? => ?
 => ? = 6
[1,2,3,5,6,4] => ? => ? => ?
 => ? = 6
[1,2,3,6,5,4] => ? => ? => ?
 => ? = 6
[1,2,4,3,5,6] => ? => ? => ?
 => ? = 6
[1,2,4,3,6,5] => ? => ? => ?
 => ? = 6
[1,2,4,5,3,6] => ? => ? => ?
 => ? = 6
[1,2,4,5,6,3] => ? => ? => ?
 => ? = 6
[1,2,4,6,3,5] => ? => ? => ?
 => ? = 6
[1,2,4,6,5,3] => ? => ? => ?
 => ? = 6
[1,2,5,3,6,4] => ? => ? => ?
 => ? = 6
[1,2,5,4,3,6] => ? => ? => ?
 => ? = 6
[1,2,5,4,6,3] => ? => ? => ?
 => ? = 6
[1,2,5,6,4,3] => ? => ? => ?
 => ? = 6
[1,2,6,3,4,5] => ? => ? => ?
 => ? = 6
[1,2,6,4,5,3] => ? => ? => ?
 => ? = 6
[1,2,6,5,4,3] => ? => ? => ?
 => ? = 6
[1,3,2,4,5,6] => ? => ? => ?
 => ? = 6
[1,3,2,4,6,5] => ? => ? => ?
 => ? = 6
[1,3,2,5,4,6] => ? => ? => ?
 => ? = 6
[1,3,2,5,6,4] => ? => ? => ?
 => ? = 6
[1,3,2,6,4,5] => ? => ? => ?
 => ? = 6
[1,3,2,6,5,4] => ? => ? => ?
 => ? = 6
[1,3,4,2,5,6] => ? => ? => ?
 => ? = 6
[1,3,4,2,6,5] => ? => ? => ?
 => ? = 6
[1,3,4,5,2,6] => ? => ? => ?
 => ? = 6
[1,3,4,5,6,2] => ? => ? => ?
 => ? = 6
[1,3,4,6,2,5] => ? => ? => ?
 => ? = 6
[1,3,4,6,5,2] => ? => ? => ?
 => ? = 6
[1,3,5,2,4,6] => ? => ? => ?
 => ? = 6
[1,3,5,2,6,4] => ? => ? => ?
 => ? = 6
[1,3,5,4,2,6] => ? => ? => ?
 => ? = 6
[1,3,5,4,6,2] => ? => ? => ?
 => ? = 6
[1,3,5,6,2,4] => ? => ? => ?
 => ? = 6
[1,3,5,6,4,2] => ? => ? => ?
 => ? = 6
[1,3,6,2,4,5] => ? => ? => ?
 => ? = 6
[1,3,6,4,5,2] => ? => ? => ?
 => ? = 6
[1,3,6,5,4,2] => ? => ? => ?
 => ? = 6
[1,4,2,3,6,5] => ? => ? => ?
 => ? = 6
[1,4,2,5,3,6] => ? => ? => ?
 => ? = 6
[1,4,2,5,6,3] => ? => ? => ?
 => ? = 6
[1,4,3,2,5,6] => ? => ? => ?
 => ? = 6
[1,4,3,2,6,5] => ? => ? => ?
 => ? = 6
[1,4,3,5,2,6] => ? => ? => ?
 => ? = 6
[1,4,3,5,6,2] => ? => ? => ?
 => ? = 6
[1,4,3,6,5,2] => ? => ? => ?
 => ? = 6
[1,4,5,2,6,3] => ? => ? => ?
 => ? = 6
[1,4,5,3,2,6] => ? => ? => ?
 => ? = 6
[1,4,5,3,6,2] => ? => ? => ?
 => ? = 6
[1,4,5,6,3,2] => ? => ? => ?
 => ? = 6
Description
The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000093
Mapping
Mp00267: Signed permutations —signs⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000093: Graphs ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 78%
Values
[1] => 0 => [2] => ([],2)
 => 2 = 1 + 1
[-1] => 1 => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
[1,2] => 00 => [3] => ([],3)
 => 3 = 2 + 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)
 => 1 = 0 + 1
[2,1] => 00 => [3] => ([],3)
 => 3 = 2 + 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)
 => 1 = 0 + 1
[1,2,3] => 000 => [4] => ([],4)
 => 4 = 3 + 1
[1,2,-3] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,-2,3] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,-2,-3] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[-1,2,3] => 100 => [1,3] => ([(2,3)],4)
 => 3 = 2 + 1
[-1,2,-3] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[-1,-2,3] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[-1,-2,-3] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[1,3,2] => 000 => [4] => ([],4)
 => 4 = 3 + 1
[1,3,-2] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,-3,2] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,-3,-2] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[-1,3,2] => 100 => [1,3] => ([(2,3)],4)
 => 3 = 2 + 1
[-1,3,-2] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[-1,-3,2] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[-1,-3,-2] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[2,1,3] => 000 => [4] => ([],4)
 => 4 = 3 + 1
[2,1,-3] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3 = 2 + 1
[2,-1,3] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 3 = 2 + 1
[2,-1,-3] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[-2,1,3] => 100 => [1,3] => ([(2,3)],4)
 => 3 = 2 + 1
[-2,1,-3] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[-2,-1,3] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[-2,-1,-3] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[2,3,1] => 000 => [4] => ([],4)
 => 4 = 3 + 1
[2,3,-1] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3 = 2 + 1
[2,-3,1] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 3 = 2 + 1
[2,-3,-1] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[-2,3,1] => 100 => [1,3] => ([(2,3)],4)
 => 3 = 2 + 1
[-2,3,-1] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[-2,-3,1] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[-2,-3,-1] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[3,1,2] => 000 => [4] => ([],4)
 => 4 = 3 + 1
[3,1,-2] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,-1,2] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,-1,-2] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[-3,1,2] => 100 => [1,3] => ([(2,3)],4)
 => 3 = 2 + 1
[-3,1,-2] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[-3,-1,2] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[-3,-1,-2] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[1,2,3,4,6,5] => ? => ? => ?
 => ? = 6 + 1
[1,2,3,5,4,6] => ? => ? => ?
 => ? = 6 + 1
[1,2,3,5,6,4] => ? => ? => ?
 => ? = 6 + 1
[1,2,3,6,5,4] => ? => ? => ?
 => ? = 6 + 1
[1,2,4,3,5,6] => ? => ? => ?
 => ? = 6 + 1
[1,2,4,3,6,5] => ? => ? => ?
 => ? = 6 + 1
[1,2,4,5,3,6] => ? => ? => ?
 => ? = 6 + 1
[1,2,4,5,6,3] => ? => ? => ?
 => ? = 6 + 1
[1,2,4,6,3,5] => ? => ? => ?
 => ? = 6 + 1
[1,2,4,6,5,3] => ? => ? => ?
 => ? = 6 + 1
[1,2,5,3,6,4] => ? => ? => ?
 => ? = 6 + 1
[1,2,5,4,3,6] => ? => ? => ?
 => ? = 6 + 1
[1,2,5,4,6,3] => ? => ? => ?
 => ? = 6 + 1
[1,2,5,6,4,3] => ? => ? => ?
 => ? = 6 + 1
[1,2,6,3,4,5] => ? => ? => ?
 => ? = 6 + 1
[1,2,6,4,5,3] => ? => ? => ?
 => ? = 6 + 1
[1,2,6,5,4,3] => ? => ? => ?
 => ? = 6 + 1
[1,3,2,4,5,6] => ? => ? => ?
 => ? = 6 + 1
[1,3,2,4,6,5] => ? => ? => ?
 => ? = 6 + 1
[1,3,2,5,4,6] => ? => ? => ?
 => ? = 6 + 1
[1,3,2,5,6,4] => ? => ? => ?
 => ? = 6 + 1
[1,3,2,6,4,5] => ? => ? => ?
 => ? = 6 + 1
[1,3,2,6,5,4] => ? => ? => ?
 => ? = 6 + 1
[1,3,4,2,5,6] => ? => ? => ?
 => ? = 6 + 1
[1,3,4,2,6,5] => ? => ? => ?
 => ? = 6 + 1
[1,3,4,5,2,6] => ? => ? => ?
 => ? = 6 + 1
[1,3,4,5,6,2] => ? => ? => ?
 => ? = 6 + 1
[1,3,4,6,2,5] => ? => ? => ?
 => ? = 6 + 1
[1,3,4,6,5,2] => ? => ? => ?
 => ? = 6 + 1
[1,3,5,2,4,6] => ? => ? => ?
 => ? = 6 + 1
[1,3,5,2,6,4] => ? => ? => ?
 => ? = 6 + 1
[1,3,5,4,2,6] => ? => ? => ?
 => ? = 6 + 1
[1,3,5,4,6,2] => ? => ? => ?
 => ? = 6 + 1
[1,3,5,6,2,4] => ? => ? => ?
 => ? = 6 + 1
[1,3,5,6,4,2] => ? => ? => ?
 => ? = 6 + 1
[1,3,6,2,4,5] => ? => ? => ?
 => ? = 6 + 1
[1,3,6,4,5,2] => ? => ? => ?
 => ? = 6 + 1
[1,3,6,5,4,2] => ? => ? => ?
 => ? = 6 + 1
[1,4,2,3,6,5] => ? => ? => ?
 => ? = 6 + 1
[1,4,2,5,3,6] => ? => ? => ?
 => ? = 6 + 1
[1,4,2,5,6,3] => ? => ? => ?
 => ? = 6 + 1
[1,4,3,2,5,6] => ? => ? => ?
 => ? = 6 + 1
[1,4,3,2,6,5] => ? => ? => ?
 => ? = 6 + 1
[1,4,3,5,2,6] => ? => ? => ?
 => ? = 6 + 1
[1,4,3,5,6,2] => ? => ? => ?
 => ? = 6 + 1
[1,4,3,6,5,2] => ? => ? => ?
 => ? = 6 + 1
[1,4,5,2,6,3] => ? => ? => ?
 => ? = 6 + 1
[1,4,5,3,2,6] => ? => ? => ?
 => ? = 6 + 1
[1,4,5,3,6,2] => ? => ? => ?
 => ? = 6 + 1
[1,4,5,6,3,2] => ? => ? => ?
 => ? = 6 + 1
Description
The cardinality of a maximal independent set of vertices of a graph. An independent set of a graph is a set of pairwise non-adjacent vertices. A maximum independent set is an independent set of maximum cardinality. This statistic is also called the independence number or stability number $\alpha(G)$ of $G$.
Matching statistic: St000786
Mapping
Mp00267: Signed permutations —signs⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000786: Graphs ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 78%
Values
[1] => 0 => [2] => ([],2)
 => 2 = 1 + 1
[-1] => 1 => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
[1,2] => 00 => [3] => ([],3)
 => 3 = 2 + 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)
 => 1 = 0 + 1
[2,1] => 00 => [3] => ([],3)
 => 3 = 2 + 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)
 => 1 = 0 + 1
[1,2,3] => 000 => [4] => ([],4)
 => 4 = 3 + 1
[1,2,-3] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,-2,3] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,-2,-3] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[-1,2,3] => 100 => [1,3] => ([(2,3)],4)
 => 3 = 2 + 1
[-1,2,-3] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[-1,-2,3] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[-1,-2,-3] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[1,3,2] => 000 => [4] => ([],4)
 => 4 = 3 + 1
[1,3,-2] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,-3,2] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,-3,-2] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[-1,3,2] => 100 => [1,3] => ([(2,3)],4)
 => 3 = 2 + 1
[-1,3,-2] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[-1,-3,2] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[-1,-3,-2] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[2,1,3] => 000 => [4] => ([],4)
 => 4 = 3 + 1
[2,1,-3] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3 = 2 + 1
[2,-1,3] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 3 = 2 + 1
[2,-1,-3] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[-2,1,3] => 100 => [1,3] => ([(2,3)],4)
 => 3 = 2 + 1
[-2,1,-3] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[-2,-1,3] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[-2,-1,-3] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[2,3,1] => 000 => [4] => ([],4)
 => 4 = 3 + 1
[2,3,-1] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3 = 2 + 1
[2,-3,1] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 3 = 2 + 1
[2,-3,-1] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[-2,3,1] => 100 => [1,3] => ([(2,3)],4)
 => 3 = 2 + 1
[-2,3,-1] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[-2,-3,1] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[-2,-3,-1] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[3,1,2] => 000 => [4] => ([],4)
 => 4 = 3 + 1
[3,1,-2] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,-1,2] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,-1,-2] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[-3,1,2] => 100 => [1,3] => ([(2,3)],4)
 => 3 = 2 + 1
[-3,1,-2] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[-3,-1,2] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[-3,-1,-2] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[1,2,3,4,6,5] => ? => ? => ?
 => ? = 6 + 1
[1,2,3,5,4,6] => ? => ? => ?
 => ? = 6 + 1
[1,2,3,5,6,4] => ? => ? => ?
 => ? = 6 + 1
[1,2,3,6,5,4] => ? => ? => ?
 => ? = 6 + 1
[1,2,4,3,5,6] => ? => ? => ?
 => ? = 6 + 1
[1,2,4,3,6,5] => ? => ? => ?
 => ? = 6 + 1
[1,2,4,5,3,6] => ? => ? => ?
 => ? = 6 + 1
[1,2,4,5,6,3] => ? => ? => ?
 => ? = 6 + 1
[1,2,4,6,3,5] => ? => ? => ?
 => ? = 6 + 1
[1,2,4,6,5,3] => ? => ? => ?
 => ? = 6 + 1
[1,2,5,3,6,4] => ? => ? => ?
 => ? = 6 + 1
[1,2,5,4,3,6] => ? => ? => ?
 => ? = 6 + 1
[1,2,5,4,6,3] => ? => ? => ?
 => ? = 6 + 1
[1,2,5,6,4,3] => ? => ? => ?
 => ? = 6 + 1
[1,2,6,3,4,5] => ? => ? => ?
 => ? = 6 + 1
[1,2,6,4,5,3] => ? => ? => ?
 => ? = 6 + 1
[1,2,6,5,4,3] => ? => ? => ?
 => ? = 6 + 1
[1,3,2,4,5,6] => ? => ? => ?
 => ? = 6 + 1
[1,3,2,4,6,5] => ? => ? => ?
 => ? = 6 + 1
[1,3,2,5,4,6] => ? => ? => ?
 => ? = 6 + 1
[1,3,2,5,6,4] => ? => ? => ?
 => ? = 6 + 1
[1,3,2,6,4,5] => ? => ? => ?
 => ? = 6 + 1
[1,3,2,6,5,4] => ? => ? => ?
 => ? = 6 + 1
[1,3,4,2,5,6] => ? => ? => ?
 => ? = 6 + 1
[1,3,4,2,6,5] => ? => ? => ?
 => ? = 6 + 1
[1,3,4,5,2,6] => ? => ? => ?
 => ? = 6 + 1
[1,3,4,5,6,2] => ? => ? => ?
 => ? = 6 + 1
[1,3,4,6,2,5] => ? => ? => ?
 => ? = 6 + 1
[1,3,4,6,5,2] => ? => ? => ?
 => ? = 6 + 1
[1,3,5,2,4,6] => ? => ? => ?
 => ? = 6 + 1
[1,3,5,2,6,4] => ? => ? => ?
 => ? = 6 + 1
[1,3,5,4,2,6] => ? => ? => ?
 => ? = 6 + 1
[1,3,5,4,6,2] => ? => ? => ?
 => ? = 6 + 1
[1,3,5,6,2,4] => ? => ? => ?
 => ? = 6 + 1
[1,3,5,6,4,2] => ? => ? => ?
 => ? = 6 + 1
[1,3,6,2,4,5] => ? => ? => ?
 => ? = 6 + 1
[1,3,6,4,5,2] => ? => ? => ?
 => ? = 6 + 1
[1,3,6,5,4,2] => ? => ? => ?
 => ? = 6 + 1
[1,4,2,3,6,5] => ? => ? => ?
 => ? = 6 + 1
[1,4,2,5,3,6] => ? => ? => ?
 => ? = 6 + 1
[1,4,2,5,6,3] => ? => ? => ?
 => ? = 6 + 1
[1,4,3,2,5,6] => ? => ? => ?
 => ? = 6 + 1
[1,4,3,2,6,5] => ? => ? => ?
 => ? = 6 + 1
[1,4,3,5,2,6] => ? => ? => ?
 => ? = 6 + 1
[1,4,3,5,6,2] => ? => ? => ?
 => ? = 6 + 1
[1,4,3,6,5,2] => ? => ? => ?
 => ? = 6 + 1
[1,4,5,2,6,3] => ? => ? => ?
 => ? = 6 + 1
[1,4,5,3,2,6] => ? => ? => ?
 => ? = 6 + 1
[1,4,5,3,6,2] => ? => ? => ?
 => ? = 6 + 1
[1,4,5,6,3,2] => ? => ? => ?
 => ? = 6 + 1
Description
The maximal number of occurrences of a colour in a proper colouring of a graph. To any proper colouring with the minimal number of colours possible we associate the integer partition recording how often each colour is used. This statistic records the largest part occurring in any of these partitions. For example, the graph on six vertices consisting of a square together with two attached triangles - ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6) in the list of values - is three-colourable and admits two colouring schemes, $[2,2,2]$ and $[3,2,1]$. Therefore, the statistic on this graph is $3$.
Matching statistic: St001007
Mapping
Mp00267: Signed permutations —signs⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001007: Dyck paths ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 78%
Values
[1] => 0 => [2] => [1,1,0,0]
 => 2 = 1 + 1
[-1] => 1 => [1,1] => [1,0,1,0]
 => 1 = 0 + 1
[1,2] => 00 => [3] => [1,1,1,0,0,0]
 => 3 = 2 + 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]
 => 1 = 0 + 1
[2,1] => 00 => [3] => [1,1,1,0,0,0]
 => 3 = 2 + 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]
 => 1 = 0 + 1
[1,2,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,2,-3] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[1,-2,3] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[1,-2,-3] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[-1,2,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[-1,2,-3] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[-1,-2,3] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[-1,-2,-3] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[1,3,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,3,-2] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[1,-3,2] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[1,-3,-2] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[-1,3,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[-1,3,-2] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[-1,-3,2] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[-1,-3,-2] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[2,1,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[2,1,-3] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[2,-1,3] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[2,-1,-3] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[-2,1,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[-2,1,-3] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[-2,-1,3] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[-2,-1,-3] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[2,3,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[2,3,-1] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[2,-3,1] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[2,-3,-1] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[-2,3,1] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[-2,3,-1] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[-2,-3,1] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[-2,-3,-1] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[3,1,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[3,1,-2] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[3,-1,2] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[3,-1,-2] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[-3,1,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[-3,1,-2] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[-3,-1,2] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[-3,-1,-2] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[1,2,3,4,6,5] => ? => ? => ?
 => ? = 6 + 1
[1,2,3,5,4,6] => ? => ? => ?
 => ? = 6 + 1
[1,2,3,5,6,4] => ? => ? => ?
 => ? = 6 + 1
[1,2,3,6,5,4] => ? => ? => ?
 => ? = 6 + 1
[1,2,4,3,5,6] => ? => ? => ?
 => ? = 6 + 1
[1,2,4,3,6,5] => ? => ? => ?
 => ? = 6 + 1
[1,2,4,5,3,6] => ? => ? => ?
 => ? = 6 + 1
[1,2,4,5,6,3] => ? => ? => ?
 => ? = 6 + 1
[1,2,4,6,3,5] => ? => ? => ?
 => ? = 6 + 1
[1,2,4,6,5,3] => ? => ? => ?
 => ? = 6 + 1
[1,2,5,3,6,4] => ? => ? => ?
 => ? = 6 + 1
[1,2,5,4,3,6] => ? => ? => ?
 => ? = 6 + 1
[1,2,5,4,6,3] => ? => ? => ?
 => ? = 6 + 1
[1,2,5,6,4,3] => ? => ? => ?
 => ? = 6 + 1
[1,2,6,3,4,5] => ? => ? => ?
 => ? = 6 + 1
[1,2,6,4,5,3] => ? => ? => ?
 => ? = 6 + 1
[1,2,6,5,4,3] => ? => ? => ?
 => ? = 6 + 1
[1,3,2,4,5,6] => ? => ? => ?
 => ? = 6 + 1
[1,3,2,4,6,5] => ? => ? => ?
 => ? = 6 + 1
[1,3,2,5,4,6] => ? => ? => ?
 => ? = 6 + 1
[1,3,2,5,6,4] => ? => ? => ?
 => ? = 6 + 1
[1,3,2,6,4,5] => ? => ? => ?
 => ? = 6 + 1
[1,3,2,6,5,4] => ? => ? => ?
 => ? = 6 + 1
[1,3,4,2,5,6] => ? => ? => ?
 => ? = 6 + 1
[1,3,4,2,6,5] => ? => ? => ?
 => ? = 6 + 1
[1,3,4,5,2,6] => ? => ? => ?
 => ? = 6 + 1
[1,3,4,5,6,2] => ? => ? => ?
 => ? = 6 + 1
[1,3,4,6,2,5] => ? => ? => ?
 => ? = 6 + 1
[1,3,4,6,5,2] => ? => ? => ?
 => ? = 6 + 1
[1,3,5,2,4,6] => ? => ? => ?
 => ? = 6 + 1
[1,3,5,2,6,4] => ? => ? => ?
 => ? = 6 + 1
[1,3,5,4,2,6] => ? => ? => ?
 => ? = 6 + 1
[1,3,5,4,6,2] => ? => ? => ?
 => ? = 6 + 1
[1,3,5,6,2,4] => ? => ? => ?
 => ? = 6 + 1
[1,3,5,6,4,2] => ? => ? => ?
 => ? = 6 + 1
[1,3,6,2,4,5] => ? => ? => ?
 => ? = 6 + 1
[1,3,6,4,5,2] => ? => ? => ?
 => ? = 6 + 1
[1,3,6,5,4,2] => ? => ? => ?
 => ? = 6 + 1
[1,4,2,3,6,5] => ? => ? => ?
 => ? = 6 + 1
[1,4,2,5,3,6] => ? => ? => ?
 => ? = 6 + 1
[1,4,2,5,6,3] => ? => ? => ?
 => ? = 6 + 1
[1,4,3,2,5,6] => ? => ? => ?
 => ? = 6 + 1
[1,4,3,2,6,5] => ? => ? => ?
 => ? = 6 + 1
[1,4,3,5,2,6] => ? => ? => ?
 => ? = 6 + 1
[1,4,3,5,6,2] => ? => ? => ?
 => ? = 6 + 1
[1,4,3,6,5,2] => ? => ? => ?
 => ? = 6 + 1
[1,4,5,2,6,3] => ? => ? => ?
 => ? = 6 + 1
[1,4,5,3,2,6] => ? => ? => ?
 => ? = 6 + 1
[1,4,5,3,6,2] => ? => ? => ?
 => ? = 6 + 1
[1,4,5,6,3,2] => ? => ? => ?
 => ? = 6 + 1
Description
Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001088
Mapping
Mp00267: Signed permutations —signs⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001088: Dyck paths ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 78%
Values
[1] => 0 => [2] => [1,1,0,0]
 => 2 = 1 + 1
[-1] => 1 => [1,1] => [1,0,1,0]
 => 1 = 0 + 1
[1,2] => 00 => [3] => [1,1,1,0,0,0]
 => 3 = 2 + 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]
 => 1 = 0 + 1
[2,1] => 00 => [3] => [1,1,1,0,0,0]
 => 3 = 2 + 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]
 => 1 = 0 + 1
[1,2,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,2,-3] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[1,-2,3] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[1,-2,-3] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[-1,2,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[-1,2,-3] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[-1,-2,3] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[-1,-2,-3] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[1,3,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,3,-2] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[1,-3,2] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[1,-3,-2] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[-1,3,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[-1,3,-2] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[-1,-3,2] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[-1,-3,-2] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[2,1,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[2,1,-3] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[2,-1,3] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[2,-1,-3] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[-2,1,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[-2,1,-3] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[-2,-1,3] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[-2,-1,-3] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[2,3,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[2,3,-1] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[2,-3,1] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[2,-3,-1] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[-2,3,1] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[-2,3,-1] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[-2,-3,1] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[-2,-3,-1] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[3,1,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[3,1,-2] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[3,-1,2] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[3,-1,-2] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[-3,1,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[-3,1,-2] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[-3,-1,2] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[-3,-1,-2] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[1,2,3,4,6,5] => ? => ? => ?
 => ? = 6 + 1
[1,2,3,5,4,6] => ? => ? => ?
 => ? = 6 + 1
[1,2,3,5,6,4] => ? => ? => ?
 => ? = 6 + 1
[1,2,3,6,5,4] => ? => ? => ?
 => ? = 6 + 1
[1,2,4,3,5,6] => ? => ? => ?
 => ? = 6 + 1
[1,2,4,3,6,5] => ? => ? => ?
 => ? = 6 + 1
[1,2,4,5,3,6] => ? => ? => ?
 => ? = 6 + 1
[1,2,4,5,6,3] => ? => ? => ?
 => ? = 6 + 1
[1,2,4,6,3,5] => ? => ? => ?
 => ? = 6 + 1
[1,2,4,6,5,3] => ? => ? => ?
 => ? = 6 + 1
[1,2,5,3,6,4] => ? => ? => ?
 => ? = 6 + 1
[1,2,5,4,3,6] => ? => ? => ?
 => ? = 6 + 1
[1,2,5,4,6,3] => ? => ? => ?
 => ? = 6 + 1
[1,2,5,6,4,3] => ? => ? => ?
 => ? = 6 + 1
[1,2,6,3,4,5] => ? => ? => ?
 => ? = 6 + 1
[1,2,6,4,5,3] => ? => ? => ?
 => ? = 6 + 1
[1,2,6,5,4,3] => ? => ? => ?
 => ? = 6 + 1
[1,3,2,4,5,6] => ? => ? => ?
 => ? = 6 + 1
[1,3,2,4,6,5] => ? => ? => ?
 => ? = 6 + 1
[1,3,2,5,4,6] => ? => ? => ?
 => ? = 6 + 1
[1,3,2,5,6,4] => ? => ? => ?
 => ? = 6 + 1
[1,3,2,6,4,5] => ? => ? => ?
 => ? = 6 + 1
[1,3,2,6,5,4] => ? => ? => ?
 => ? = 6 + 1
[1,3,4,2,5,6] => ? => ? => ?
 => ? = 6 + 1
[1,3,4,2,6,5] => ? => ? => ?
 => ? = 6 + 1
[1,3,4,5,2,6] => ? => ? => ?
 => ? = 6 + 1
[1,3,4,5,6,2] => ? => ? => ?
 => ? = 6 + 1
[1,3,4,6,2,5] => ? => ? => ?
 => ? = 6 + 1
[1,3,4,6,5,2] => ? => ? => ?
 => ? = 6 + 1
[1,3,5,2,4,6] => ? => ? => ?
 => ? = 6 + 1
[1,3,5,2,6,4] => ? => ? => ?
 => ? = 6 + 1
[1,3,5,4,2,6] => ? => ? => ?
 => ? = 6 + 1
[1,3,5,4,6,2] => ? => ? => ?
 => ? = 6 + 1
[1,3,5,6,2,4] => ? => ? => ?
 => ? = 6 + 1
[1,3,5,6,4,2] => ? => ? => ?
 => ? = 6 + 1
[1,3,6,2,4,5] => ? => ? => ?
 => ? = 6 + 1
[1,3,6,4,5,2] => ? => ? => ?
 => ? = 6 + 1
[1,3,6,5,4,2] => ? => ? => ?
 => ? = 6 + 1
[1,4,2,3,6,5] => ? => ? => ?
 => ? = 6 + 1
[1,4,2,5,3,6] => ? => ? => ?
 => ? = 6 + 1
[1,4,2,5,6,3] => ? => ? => ?
 => ? = 6 + 1
[1,4,3,2,5,6] => ? => ? => ?
 => ? = 6 + 1
[1,4,3,2,6,5] => ? => ? => ?
 => ? = 6 + 1
[1,4,3,5,2,6] => ? => ? => ?
 => ? = 6 + 1
[1,4,3,5,6,2] => ? => ? => ?
 => ? = 6 + 1
[1,4,3,6,5,2] => ? => ? => ?
 => ? = 6 + 1
[1,4,5,2,6,3] => ? => ? => ?
 => ? = 6 + 1
[1,4,5,3,2,6] => ? => ? => ?
 => ? = 6 + 1
[1,4,5,3,6,2] => ? => ? => ?
 => ? = 6 + 1
[1,4,5,6,3,2] => ? => ? => ?
 => ? = 6 + 1
Description
Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra.
Matching statistic: St001337
Mapping
Mp00267: Signed permutations —signs⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001337: Graphs ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 78%
Values
[1] => 0 => [2] => ([],2)
 => 2 = 1 + 1
[-1] => 1 => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
[1,2] => 00 => [3] => ([],3)
 => 3 = 2 + 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)
 => 1 = 0 + 1
[2,1] => 00 => [3] => ([],3)
 => 3 = 2 + 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)
 => 1 = 0 + 1
[1,2,3] => 000 => [4] => ([],4)
 => 4 = 3 + 1
[1,2,-3] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,-2,3] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,-2,-3] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[-1,2,3] => 100 => [1,3] => ([(2,3)],4)
 => 3 = 2 + 1
[-1,2,-3] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[-1,-2,3] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[-1,-2,-3] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[1,3,2] => 000 => [4] => ([],4)
 => 4 = 3 + 1
[1,3,-2] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,-3,2] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,-3,-2] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[-1,3,2] => 100 => [1,3] => ([(2,3)],4)
 => 3 = 2 + 1
[-1,3,-2] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[-1,-3,2] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[-1,-3,-2] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[2,1,3] => 000 => [4] => ([],4)
 => 4 = 3 + 1
[2,1,-3] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3 = 2 + 1
[2,-1,3] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 3 = 2 + 1
[2,-1,-3] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[-2,1,3] => 100 => [1,3] => ([(2,3)],4)
 => 3 = 2 + 1
[-2,1,-3] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[-2,-1,3] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[-2,-1,-3] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[2,3,1] => 000 => [4] => ([],4)
 => 4 = 3 + 1
[2,3,-1] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3 = 2 + 1
[2,-3,1] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 3 = 2 + 1
[2,-3,-1] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[-2,3,1] => 100 => [1,3] => ([(2,3)],4)
 => 3 = 2 + 1
[-2,3,-1] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[-2,-3,1] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[-2,-3,-1] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[3,1,2] => 000 => [4] => ([],4)
 => 4 = 3 + 1
[3,1,-2] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,-1,2] => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,-1,-2] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[-3,1,2] => 100 => [1,3] => ([(2,3)],4)
 => 3 = 2 + 1
[-3,1,-2] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[-3,-1,2] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[-3,-1,-2] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[1,2,3,4,6,5] => ? => ? => ?
 => ? = 6 + 1
[1,2,3,5,4,6] => ? => ? => ?
 => ? = 6 + 1
[1,2,3,5,6,4] => ? => ? => ?
 => ? = 6 + 1
[1,2,3,6,5,4] => ? => ? => ?
 => ? = 6 + 1
[1,2,4,3,5,6] => ? => ? => ?
 => ? = 6 + 1
[1,2,4,3,6,5] => ? => ? => ?
 => ? = 6 + 1
[1,2,4,5,3,6] => ? => ? => ?
 => ? = 6 + 1
[1,2,4,5,6,3] => ? => ? => ?
 => ? = 6 + 1
[1,2,4,6,3,5] => ? => ? => ?
 => ? = 6 + 1
[1,2,4,6,5,3] => ? => ? => ?
 => ? = 6 + 1
[1,2,5,3,6,4] => ? => ? => ?
 => ? = 6 + 1
[1,2,5,4,3,6] => ? => ? => ?
 => ? = 6 + 1
[1,2,5,4,6,3] => ? => ? => ?
 => ? = 6 + 1
[1,2,5,6,4,3] => ? => ? => ?
 => ? = 6 + 1
[1,2,6,3,4,5] => ? => ? => ?
 => ? = 6 + 1
[1,2,6,4,5,3] => ? => ? => ?
 => ? = 6 + 1
[1,2,6,5,4,3] => ? => ? => ?
 => ? = 6 + 1
[1,3,2,4,5,6] => ? => ? => ?
 => ? = 6 + 1
[1,3,2,4,6,5] => ? => ? => ?
 => ? = 6 + 1
[1,3,2,5,4,6] => ? => ? => ?
 => ? = 6 + 1
[1,3,2,5,6,4] => ? => ? => ?
 => ? = 6 + 1
[1,3,2,6,4,5] => ? => ? => ?
 => ? = 6 + 1
[1,3,2,6,5,4] => ? => ? => ?
 => ? = 6 + 1
[1,3,4,2,5,6] => ? => ? => ?
 => ? = 6 + 1
[1,3,4,2,6,5] => ? => ? => ?
 => ? = 6 + 1
[1,3,4,5,2,6] => ? => ? => ?
 => ? = 6 + 1
[1,3,4,5,6,2] => ? => ? => ?
 => ? = 6 + 1
[1,3,4,6,2,5] => ? => ? => ?
 => ? = 6 + 1
[1,3,4,6,5,2] => ? => ? => ?
 => ? = 6 + 1
[1,3,5,2,4,6] => ? => ? => ?
 => ? = 6 + 1
[1,3,5,2,6,4] => ? => ? => ?
 => ? = 6 + 1
[1,3,5,4,2,6] => ? => ? => ?
 => ? = 6 + 1
[1,3,5,4,6,2] => ? => ? => ?
 => ? = 6 + 1
[1,3,5,6,2,4] => ? => ? => ?
 => ? = 6 + 1
[1,3,5,6,4,2] => ? => ? => ?
 => ? = 6 + 1
[1,3,6,2,4,5] => ? => ? => ?
 => ? = 6 + 1
[1,3,6,4,5,2] => ? => ? => ?
 => ? = 6 + 1
[1,3,6,5,4,2] => ? => ? => ?
 => ? = 6 + 1
[1,4,2,3,6,5] => ? => ? => ?
 => ? = 6 + 1
[1,4,2,5,3,6] => ? => ? => ?
 => ? = 6 + 1
[1,4,2,5,6,3] => ? => ? => ?
 => ? = 6 + 1
[1,4,3,2,5,6] => ? => ? => ?
 => ? = 6 + 1
[1,4,3,2,6,5] => ? => ? => ?
 => ? = 6 + 1
[1,4,3,5,2,6] => ? => ? => ?
 => ? = 6 + 1
[1,4,3,5,6,2] => ? => ? => ?
 => ? = 6 + 1
[1,4,3,6,5,2] => ? => ? => ?
 => ? = 6 + 1
[1,4,5,2,6,3] => ? => ? => ?
 => ? = 6 + 1
[1,4,5,3,2,6] => ? => ? => ?
 => ? = 6 + 1
[1,4,5,3,6,2] => ? => ? => ?
 => ? = 6 + 1
[1,4,5,6,3,2] => ? => ? => ?
 => ? = 6 + 1
Description
The upper domination number of a graph. This is the maximum cardinality of a minimal dominating set of $G$. The smallest graph with different upper irredundance number and upper domination number has eight vertices. It is obtained from the disjoint union of two copies of $K_4$ by joining three of the four vertices of the first with three of the four vertices of the second. For bipartite graphs the two parameters always coincide [1].
The following 8 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001338The upper irredundance number of a graph. St001480The number of simple summands of the module J^2/J^3. St000443The number of long tunnels of a Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001429The number of negative entries in a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation.