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Your data matches 53 different statistics following compositions of up to 3 maps.
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Matching statistic: St000010
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> []
=> 0
[1,2] => ([],2)
=> []
=> 0
[2,1] => ([(0,1)],2)
=> [1]
=> 1
[1,2,3] => ([],3)
=> []
=> 0
[1,3,2] => ([(1,2)],3)
=> [1]
=> 1
[2,1,3] => ([(1,2)],3)
=> [1]
=> 1
[2,3,1] => ([(0,2),(1,2)],3)
=> [2]
=> 1
[3,1,2] => ([(0,2),(1,2)],3)
=> [2]
=> 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[1,2,3,4] => ([],4)
=> []
=> 0
[1,2,4,3] => ([(2,3)],4)
=> [1]
=> 1
[1,3,2,4] => ([(2,3)],4)
=> [1]
=> 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> [2]
=> 1
[1,4,2,3] => ([(1,3),(2,3)],4)
=> [2]
=> 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 1
[2,1,3,4] => ([(2,3)],4)
=> [1]
=> 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> [1,1]
=> 2
[2,3,1,4] => ([(1,3),(2,3)],4)
=> [2]
=> 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [3]
=> 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[3,1,2,4] => ([(1,3),(2,3)],4)
=> [2]
=> 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 1
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> 1
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> [3]
=> 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> 1
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> 1
[1,2,3,4,5] => ([],5)
=> []
=> 0
[1,2,3,5,4] => ([(3,4)],5)
=> [1]
=> 1
[1,2,4,3,5] => ([(3,4)],5)
=> [1]
=> 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> [2]
=> 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> [2]
=> 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> [3]
=> 1
[1,3,2,4,5] => ([(3,4)],5)
=> [1]
=> 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> [1,1]
=> 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> [2]
=> 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> [3]
=> 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [3]
=> 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4]
=> 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> [2]
=> 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> [3]
=> 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> [3]
=> 1
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4]
=> 1
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> 1
Description
The length of the partition.
Matching statistic: St001011
(load all 39 compositions to match this statistic)
(load all 39 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> 0
[1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[2,1,3] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[2,3,1] => [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
Description
Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001280
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> [1]
=> 0
[1,2] => ([],2)
=> [1,1]
=> 0
[2,1] => ([(0,1)],2)
=> [2]
=> 1
[1,2,3] => ([],3)
=> [1,1,1]
=> 0
[1,3,2] => ([(1,2)],3)
=> [2,1]
=> 1
[2,1,3] => ([(1,2)],3)
=> [2,1]
=> 1
[2,3,1] => ([(0,2),(1,2)],3)
=> [3]
=> 1
[3,1,2] => ([(0,2),(1,2)],3)
=> [3]
=> 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[1,2,3,4] => ([],4)
=> [1,1,1,1]
=> 0
[1,2,4,3] => ([(2,3)],4)
=> [2,1,1]
=> 1
[1,3,2,4] => ([(2,3)],4)
=> [2,1,1]
=> 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> 1
[1,4,2,3] => ([(1,3),(2,3)],4)
=> [3,1]
=> 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 1
[2,1,3,4] => ([(2,3)],4)
=> [2,1,1]
=> 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> [2,2]
=> 2
[2,3,1,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[3,1,2,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 1
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> 1
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[1,2,3,4,5] => ([],5)
=> [1,1,1,1,1]
=> 0
[1,2,3,5,4] => ([(3,4)],5)
=> [2,1,1,1]
=> 1
[1,2,4,3,5] => ([(3,4)],5)
=> [2,1,1,1]
=> 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 1
[1,3,2,4,5] => ([(3,4)],5)
=> [2,1,1,1]
=> 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> [2,2,1]
=> 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 1
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> 1
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St000291
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => => [1] => 1 => 0
[1,2] => 1 => [1,1] => 11 => 0
[2,1] => 0 => [2] => 10 => 1
[1,2,3] => 11 => [1,1,1] => 111 => 0
[1,3,2] => 10 => [1,2] => 110 => 1
[2,1,3] => 01 => [2,1] => 101 => 1
[2,3,1] => 00 => [3] => 100 => 1
[3,1,2] => 00 => [3] => 100 => 1
[3,2,1] => 00 => [3] => 100 => 1
[1,2,3,4] => 111 => [1,1,1,1] => 1111 => 0
[1,2,4,3] => 110 => [1,1,2] => 1110 => 1
[1,3,2,4] => 101 => [1,2,1] => 1101 => 1
[1,3,4,2] => 100 => [1,3] => 1100 => 1
[1,4,2,3] => 100 => [1,3] => 1100 => 1
[1,4,3,2] => 100 => [1,3] => 1100 => 1
[2,1,3,4] => 011 => [2,1,1] => 1011 => 1
[2,1,4,3] => 010 => [2,2] => 1010 => 2
[2,3,1,4] => 001 => [3,1] => 1001 => 1
[2,3,4,1] => 000 => [4] => 1000 => 1
[2,4,1,3] => 000 => [4] => 1000 => 1
[2,4,3,1] => 000 => [4] => 1000 => 1
[3,1,2,4] => 001 => [3,1] => 1001 => 1
[3,1,4,2] => 000 => [4] => 1000 => 1
[3,2,1,4] => 001 => [3,1] => 1001 => 1
[3,2,4,1] => 000 => [4] => 1000 => 1
[3,4,1,2] => 000 => [4] => 1000 => 1
[3,4,2,1] => 000 => [4] => 1000 => 1
[4,1,2,3] => 000 => [4] => 1000 => 1
[4,1,3,2] => 000 => [4] => 1000 => 1
[4,2,1,3] => 000 => [4] => 1000 => 1
[4,2,3,1] => 000 => [4] => 1000 => 1
[4,3,1,2] => 000 => [4] => 1000 => 1
[4,3,2,1] => 000 => [4] => 1000 => 1
[1,2,3,4,5] => 1111 => [1,1,1,1,1] => 11111 => 0
[1,2,3,5,4] => 1110 => [1,1,1,2] => 11110 => 1
[1,2,4,3,5] => 1101 => [1,1,2,1] => 11101 => 1
[1,2,4,5,3] => 1100 => [1,1,3] => 11100 => 1
[1,2,5,3,4] => 1100 => [1,1,3] => 11100 => 1
[1,2,5,4,3] => 1100 => [1,1,3] => 11100 => 1
[1,3,2,4,5] => 1011 => [1,2,1,1] => 11011 => 1
[1,3,2,5,4] => 1010 => [1,2,2] => 11010 => 2
[1,3,4,2,5] => 1001 => [1,3,1] => 11001 => 1
[1,3,4,5,2] => 1000 => [1,4] => 11000 => 1
[1,3,5,2,4] => 1000 => [1,4] => 11000 => 1
[1,3,5,4,2] => 1000 => [1,4] => 11000 => 1
[1,4,2,3,5] => 1001 => [1,3,1] => 11001 => 1
[1,4,2,5,3] => 1000 => [1,4] => 11000 => 1
[1,4,3,2,5] => 1001 => [1,3,1] => 11001 => 1
[1,4,3,5,2] => 1000 => [1,4] => 11000 => 1
[1,4,5,2,3] => 1000 => [1,4] => 11000 => 1
Description
The number of descents of a binary word.
Matching statistic: St000473
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000473: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00247: Graphs —de-duplicate⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000473: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> ([],1)
=> [1]
=> 0
[1,2] => ([],2)
=> ([],1)
=> [1]
=> 0
[2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 1
[1,2,3] => ([],3)
=> ([],1)
=> [1]
=> 0
[1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> [2,1]
=> 1
[2,1,3] => ([(1,2)],3)
=> ([(1,2)],3)
=> [2,1]
=> 1
[2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> [2]
=> 1
[3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> [2]
=> 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[1,2,3,4] => ([],4)
=> ([],1)
=> [1]
=> 0
[1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> [2,1]
=> 1
[1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> [2,1]
=> 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> [2,1]
=> 1
[1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> [2,1]
=> 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 1
[2,1,3,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> [2,1]
=> 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> [2,2]
=> 2
[2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> [2,1]
=> 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [2]
=> 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> [2,1]
=> 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [4]
=> 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 1
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> [2]
=> 1
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [2]
=> 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[1,2,3,4,5] => ([],5)
=> ([],1)
=> [1]
=> 0
[1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> [2,1]
=> 1
[1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> [2,1]
=> 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> [2,1]
=> 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> [2,1]
=> 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 1
[1,3,2,4,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> [2,1]
=> 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> [2,2,1]
=> 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> [2,1]
=> 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> [2,1]
=> 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> [2,1]
=> 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 1
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> [2,1]
=> 1
Description
The number of parts of a partition that are strictly bigger than the number of ones.
This is part of the definition of Dyson's crank of a partition, see [[St000474]].
Matching statistic: St001276
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001276: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001276: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> [1,0]
=> 0
[1,2] => [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[2,1] => [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[2,3,1] => [3,2,1] => [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 1
[3,1,2] => [3,2,1] => [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 1
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 1
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[2,3,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[2,3,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[2,4,1,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[2,4,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[3,1,2,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[3,1,4,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[3,2,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[3,4,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[3,4,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[4,1,2,3] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[4,1,3,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[4,2,1,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[4,2,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[4,3,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,2,4,5,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[1,2,5,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[1,2,5,4,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,3,4,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[1,3,4,5,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[1,3,5,2,4] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[1,3,5,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[1,4,2,3,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[1,4,2,5,3] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[1,4,3,5,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[1,4,5,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
Description
The number of 2-regular indecomposable modules in the corresponding Nakayama algebra.
Generalising the notion of k-regular modules from simple to arbitrary indecomposable modules, we call an indecomposable module $M$ over an algebra $A$ k-regular in case it has projective dimension k and $Ext_A^i(M,A)=0$ for $i \neq k$ and $Ext_A^k(M,A)$ is 1-dimensional.
The number of Dyck paths where the statistic returns 0 might be given by [[OEIS:A035929]] .
Matching statistic: St000390
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00105: Binary words —complement⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => => => ? = 0
[1,2] => 1 => 0 => 0
[2,1] => 0 => 1 => 1
[1,2,3] => 11 => 00 => 0
[1,3,2] => 10 => 01 => 1
[2,1,3] => 01 => 10 => 1
[2,3,1] => 00 => 11 => 1
[3,1,2] => 00 => 11 => 1
[3,2,1] => 00 => 11 => 1
[1,2,3,4] => 111 => 000 => 0
[1,2,4,3] => 110 => 001 => 1
[1,3,2,4] => 101 => 010 => 1
[1,3,4,2] => 100 => 011 => 1
[1,4,2,3] => 100 => 011 => 1
[1,4,3,2] => 100 => 011 => 1
[2,1,3,4] => 011 => 100 => 1
[2,1,4,3] => 010 => 101 => 2
[2,3,1,4] => 001 => 110 => 1
[2,3,4,1] => 000 => 111 => 1
[2,4,1,3] => 000 => 111 => 1
[2,4,3,1] => 000 => 111 => 1
[3,1,2,4] => 001 => 110 => 1
[3,1,4,2] => 000 => 111 => 1
[3,2,1,4] => 001 => 110 => 1
[3,2,4,1] => 000 => 111 => 1
[3,4,1,2] => 000 => 111 => 1
[3,4,2,1] => 000 => 111 => 1
[4,1,2,3] => 000 => 111 => 1
[4,1,3,2] => 000 => 111 => 1
[4,2,1,3] => 000 => 111 => 1
[4,2,3,1] => 000 => 111 => 1
[4,3,1,2] => 000 => 111 => 1
[4,3,2,1] => 000 => 111 => 1
[1,2,3,4,5] => 1111 => 0000 => 0
[1,2,3,5,4] => 1110 => 0001 => 1
[1,2,4,3,5] => 1101 => 0010 => 1
[1,2,4,5,3] => 1100 => 0011 => 1
[1,2,5,3,4] => 1100 => 0011 => 1
[1,2,5,4,3] => 1100 => 0011 => 1
[1,3,2,4,5] => 1011 => 0100 => 1
[1,3,2,5,4] => 1010 => 0101 => 2
[1,3,4,2,5] => 1001 => 0110 => 1
[1,3,4,5,2] => 1000 => 0111 => 1
[1,3,5,2,4] => 1000 => 0111 => 1
[1,3,5,4,2] => 1000 => 0111 => 1
[1,4,2,3,5] => 1001 => 0110 => 1
[1,4,2,5,3] => 1000 => 0111 => 1
[1,4,3,2,5] => 1001 => 0110 => 1
[1,4,3,5,2] => 1000 => 0111 => 1
[1,4,5,2,3] => 1000 => 0111 => 1
[1,4,5,3,2] => 1000 => 0111 => 1
Description
The number of runs of ones in a binary word.
Matching statistic: St000659
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000659: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000659: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => => [1] => [1,0]
=> ? = 0
[1,2] => 1 => [1,1] => [1,0,1,0]
=> 0
[2,1] => 0 => [2] => [1,1,0,0]
=> 1
[1,2,3] => 11 => [1,1,1] => [1,0,1,0,1,0]
=> 0
[1,3,2] => 10 => [1,2] => [1,0,1,1,0,0]
=> 1
[2,1,3] => 01 => [2,1] => [1,1,0,0,1,0]
=> 1
[2,3,1] => 00 => [3] => [1,1,1,0,0,0]
=> 1
[3,1,2] => 00 => [3] => [1,1,1,0,0,0]
=> 1
[3,2,1] => 00 => [3] => [1,1,1,0,0,0]
=> 1
[1,2,3,4] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[1,2,4,3] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,3,2,4] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[1,3,4,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,4,2,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,4,3,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[2,1,3,4] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[2,1,4,3] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[2,3,1,4] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[2,3,4,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 1
[2,4,1,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 1
[2,4,3,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 1
[3,1,2,4] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[3,1,4,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 1
[3,2,1,4] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[3,2,4,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 1
[3,4,1,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 1
[3,4,2,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 1
[4,1,2,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 1
[4,1,3,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 1
[4,2,1,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 1
[4,2,3,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 1
[4,3,1,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 1
[4,3,2,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 1
[1,2,3,4,5] => 1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,2,3,5,4] => 1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,2,4,3,5] => 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,2,4,5,3] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,2,5,3,4] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,2,5,4,3] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,3,2,4,5] => 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,3,2,5,4] => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,3,4,2,5] => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,3,4,5,2] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,3,5,2,4] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,3,5,4,2] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,4,2,3,5] => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,4,2,5,3] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,4,3,2,5] => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,4,3,5,2] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,4,5,2,3] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,4,5,3,2] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
Description
The number of rises of length at least 2 of a Dyck path.
Matching statistic: St001333
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001333: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00239: Permutations —Corteel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001333: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => ([],1)
=> 0
[1,2] => [1,2] => [1,2] => ([],2)
=> 0
[2,1] => [2,1] => [2,1] => ([(0,1)],2)
=> 1
[1,2,3] => [1,2,3] => [1,2,3] => ([],3)
=> 0
[1,3,2] => [1,3,2] => [1,3,2] => ([(1,2)],3)
=> 1
[2,1,3] => [2,1,3] => [2,1,3] => ([(1,2)],3)
=> 1
[2,3,1] => [3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 1
[3,1,2] => [3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 1
[3,2,1] => [3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> 1
[1,3,4,2] => [1,4,3,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 1
[1,4,2,3] => [1,4,3,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 1
[1,4,3,2] => [1,4,3,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 1
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2
[2,3,1,4] => [3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> 1
[2,3,4,1] => [4,2,3,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[2,4,1,3] => [3,4,1,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,4,3,1] => [4,3,2,1] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 1
[3,1,2,4] => [3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> 1
[3,1,4,2] => [4,2,3,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[3,2,1,4] => [3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> 1
[3,2,4,1] => [4,2,3,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[3,4,1,2] => [4,3,2,1] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 1
[3,4,2,1] => [4,3,2,1] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 1
[4,1,2,3] => [4,2,3,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[4,1,3,2] => [4,2,3,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[4,2,1,3] => [4,3,2,1] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 1
[4,2,3,1] => [4,3,2,1] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 1
[4,3,1,2] => [4,3,2,1] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 1
[4,3,2,1] => [4,3,2,1] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> 1
[1,2,4,5,3] => [1,2,5,4,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 1
[1,2,5,3,4] => [1,2,5,4,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 1
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 2
[1,3,4,2,5] => [1,4,3,2,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 1
[1,3,4,5,2] => [1,5,3,4,2] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 1
[1,3,5,2,4] => [1,4,5,2,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,3,5,4,2] => [1,5,4,3,2] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 1
[1,4,2,3,5] => [1,4,3,2,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 1
[1,4,2,5,3] => [1,5,3,4,2] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 1
[1,4,3,2,5] => [1,4,3,2,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 1
[1,4,3,5,2] => [1,5,3,4,2] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 1
[1,4,5,2,3] => [1,5,4,3,2] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 1
[1,3,5,7,2,4,6] => [1,5,6,7,2,3,4] => [1,7,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[2,3,5,7,1,4,6] => [5,2,6,7,1,3,4] => [2,7,6,5,4,3,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[2,4,5,7,1,3,6] => [5,6,3,7,1,2,4] => [6,3,7,5,4,2,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[2,4,6,1,3,5,7] => [4,5,6,1,2,3,7] => [6,5,4,3,2,1,7] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[2,4,6,1,3,7,5] => [4,5,7,1,2,6,3] => [6,5,4,3,2,7,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[2,4,6,1,7,3,5] => [4,6,7,1,5,2,3] => [6,5,4,3,7,2,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[2,4,6,7,1,3,5] => [5,6,7,4,1,2,3] => [6,5,4,7,3,2,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[2,4,7,1,3,5,6] => [4,5,7,1,2,6,3] => [6,5,4,3,2,7,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[2,4,7,1,3,6,5] => [4,5,7,1,2,6,3] => [6,5,4,3,2,7,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[2,4,7,1,5,3,6] => [4,6,7,1,5,2,3] => [6,5,4,3,7,2,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[2,4,7,1,6,3,5] => [4,6,7,1,5,2,3] => [6,5,4,3,7,2,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[2,4,7,5,1,3,6] => [5,6,7,4,1,2,3] => [6,5,4,7,3,2,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[2,4,7,6,1,3,5] => [5,6,7,4,1,2,3] => [6,5,4,7,3,2,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[2,5,3,7,1,4,6] => [5,6,3,7,1,2,4] => [6,3,7,5,4,2,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[2,5,4,7,1,3,6] => [5,6,3,7,1,2,4] => [6,3,7,5,4,2,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[2,5,7,1,3,4,6] => [4,6,7,1,5,2,3] => [6,5,4,3,7,2,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[2,5,7,1,4,3,6] => [4,6,7,1,5,2,3] => [6,5,4,3,7,2,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[2,5,7,3,1,4,6] => [5,6,7,4,1,2,3] => [6,5,4,7,3,2,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[2,5,7,4,1,3,6] => [5,6,7,4,1,2,3] => [6,5,4,7,3,2,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[3,1,5,7,2,4,6] => [5,2,6,7,1,3,4] => [2,7,6,5,4,3,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[3,2,5,7,1,4,6] => [5,2,6,7,1,3,4] => [2,7,6,5,4,3,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[3,5,1,7,2,4,6] => [5,6,3,7,1,2,4] => [6,3,7,5,4,2,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[3,5,2,7,1,4,6] => [5,6,3,7,1,2,4] => [6,3,7,5,4,2,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[3,5,7,1,2,4,6] => [5,6,7,4,1,2,3] => [6,5,4,7,3,2,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[3,5,7,2,1,4,6] => [5,6,7,4,1,2,3] => [6,5,4,7,3,2,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
Description
The cardinality of a minimal edge-isolating set of a graph.
Let $\mathcal F$ be a set of graphs. A set of vertices $S$ is $\mathcal F$-isolating, if the subgraph induced by the vertices in the complement of the closed neighbourhood of $S$ does not contain any graph in $\mathcal F$.
This statistic returns the cardinality of the smallest isolating set when $\mathcal F$ contains only the graph with one edge.
Matching statistic: St000147
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> []
=> []
=> 0
[1,2] => ([],2)
=> []
=> []
=> 0
[2,1] => ([(0,1)],2)
=> [1]
=> [1]
=> 1
[1,2,3] => ([],3)
=> []
=> []
=> 0
[1,3,2] => ([(1,2)],3)
=> [1]
=> [1]
=> 1
[2,1,3] => ([(1,2)],3)
=> [1]
=> [1]
=> 1
[2,3,1] => ([(0,2),(1,2)],3)
=> [2]
=> [1,1]
=> 1
[3,1,2] => ([(0,2),(1,2)],3)
=> [2]
=> [1,1]
=> 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 1
[1,2,3,4] => ([],4)
=> []
=> []
=> 0
[1,2,4,3] => ([(2,3)],4)
=> [1]
=> [1]
=> 1
[1,3,2,4] => ([(2,3)],4)
=> [1]
=> [1]
=> 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> [2]
=> [1,1]
=> 1
[1,4,2,3] => ([(1,3),(2,3)],4)
=> [2]
=> [1,1]
=> 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [3]
=> [1,1,1]
=> 1
[2,1,3,4] => ([(2,3)],4)
=> [1]
=> [1]
=> 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> [1,1]
=> [2]
=> 2
[2,3,1,4] => ([(1,3),(2,3)],4)
=> [2]
=> [1,1]
=> 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [3]
=> [1,1,1]
=> 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> [3]
=> [1,1,1]
=> 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 1
[3,1,2,4] => ([(1,3),(2,3)],4)
=> [2]
=> [1,1]
=> 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> [3]
=> [1,1,1]
=> 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [3]
=> [1,1,1]
=> 1
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,1,1,1]
=> 1
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [1,1,1,1,1]
=> 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> [3]
=> [1,1,1]
=> 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 1
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 1
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [1,1,1,1,1]
=> 1
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [1,1,1,1,1]
=> 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> [1,1,1,1,1,1]
=> 1
[1,2,3,4,5] => ([],5)
=> []
=> []
=> 0
[1,2,3,5,4] => ([(3,4)],5)
=> [1]
=> [1]
=> 1
[1,2,4,3,5] => ([(3,4)],5)
=> [1]
=> [1]
=> 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> [2]
=> [1,1]
=> 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> [2]
=> [1,1]
=> 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,1,1]
=> 1
[1,3,2,4,5] => ([(3,4)],5)
=> [1]
=> [1]
=> 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> [1,1]
=> [2]
=> 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> [2]
=> [1,1]
=> 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> [3]
=> [1,1,1]
=> 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [3]
=> [1,1,1]
=> 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> [2]
=> [1,1]
=> 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> [3]
=> [1,1,1]
=> 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,1,1]
=> 1
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 1
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1]
=> 1
[3,7,6,5,4,2,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[4,6,7,5,3,2,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[4,7,5,6,3,2,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[4,7,6,3,5,2,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[4,7,6,5,2,3,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[4,7,6,5,3,1,2] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[5,4,7,6,3,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[5,6,4,7,3,2,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[5,6,7,3,4,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[5,6,7,4,2,3,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[5,6,7,4,3,1,2] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[5,7,3,6,4,2,1] => ([(0,1),(0,4),(0,5),(0,6),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[5,7,4,3,6,2,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[5,7,4,6,2,3,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[5,7,4,6,3,1,2] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[5,7,6,2,4,3,1] => ([(0,1),(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[5,7,6,3,2,4,1] => ([(0,1),(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[5,7,6,3,4,1,2] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[5,7,6,4,1,3,2] => ([(0,1),(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[5,7,6,4,2,1,3] => ([(0,1),(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[6,3,7,5,4,2,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[6,4,5,7,3,2,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[6,4,7,3,5,2,1] => ([(0,1),(0,4),(0,5),(0,6),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[6,4,7,5,2,3,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[6,4,7,5,3,1,2] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[6,5,3,7,4,2,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[6,5,4,3,7,2,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[6,5,4,7,2,3,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[6,5,4,7,3,1,2] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[6,5,7,2,4,3,1] => ([(0,1),(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[6,5,7,3,2,4,1] => ([(0,1),(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[6,5,7,3,4,1,2] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[6,5,7,4,1,3,2] => ([(0,1),(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[6,5,7,4,2,1,3] => ([(0,1),(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[6,7,2,5,4,3,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[6,7,3,4,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[6,7,3,5,2,4,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[6,7,3,5,4,1,2] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[6,7,4,2,5,3,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[6,7,4,3,2,5,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[6,7,4,3,5,1,2] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[6,7,4,5,1,3,2] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[6,7,4,5,2,1,3] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[6,7,5,1,4,3,2] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[6,7,5,2,3,4,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[6,7,5,2,4,1,3] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[6,7,5,3,1,4,2] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[6,7,5,3,2,1,4] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[6,7,5,4,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[7,2,6,5,4,3,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
Description
The largest part of an integer partition.
The following 43 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000374The number of exclusive right-to-left minima of a permutation. St000670The reversal length of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000035The number of left outer peaks of a permutation. St000668The least common multiple of the parts of the partition. St000260The radius of a connected graph. St000379The number of Hamiltonian cycles in a graph. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000456The monochromatic index of a connected graph. St001354The number of series nodes in the modular decomposition of a graph. St001393The induced matching number of a graph. St001261The Castelnuovo-Mumford regularity of a graph. St001665The number of pure excedances of a permutation. St000455The second largest eigenvalue of a graph if it is integral. St001737The number of descents of type 2 in a permutation. St000287The number of connected components of a graph. St000286The number of connected components of the complement of a graph. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001720The minimal length of a chain of small intervals in a lattice. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001597The Frobenius rank of a skew partition. St000264The girth of a graph, which is not a tree. St001624The breadth of a lattice.
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