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Your data matches 69 different statistics following compositions of up to 3 maps.
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Matching statistic: St000292
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00269: Binary words —flag zeros to zeros⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00269: Binary words —flag zeros to zeros⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1]
=> 10 => 00 => 0
[1,0,1,0,1,0]
=> [2,1]
=> 1010 => 0000 => 0
[1,0,1,1,0,0]
=> [1,1]
=> 110 => 001 => 1
[1,1,0,0,1,0]
=> [2]
=> 100 => 010 => 1
[1,1,0,1,0,0]
=> [1]
=> 10 => 00 => 0
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 101010 => 000000 => 0
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 11010 => 00001 => 1
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 100110 => 001010 => 2
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 10110 => 00100 => 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1110 => 0011 => 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> 10100 => 01000 => 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> 1100 => 0101 => 2
[1,1,0,1,0,0,1,0]
=> [3,1]
=> 10010 => 00010 => 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> 1010 => 0000 => 0
[1,1,0,1,1,0,0,0]
=> [1,1]
=> 110 => 001 => 1
[1,1,1,0,0,0,1,0]
=> [3]
=> 1000 => 0110 => 1
[1,1,1,0,0,1,0,0]
=> [2]
=> 100 => 010 => 1
[1,1,1,0,1,0,0,0]
=> [1]
=> 10 => 00 => 0
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 10101010 => 00000000 => 0
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 1101010 => 0000001 => 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 10011010 => 00001010 => 2
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 1011010 => 0000100 => 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 111010 => 000011 => 1
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 10100110 => 00101000 => 2
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 1100110 => 0010101 => 3
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 10010110 => 00100010 => 2
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 1010110 => 0010000 => 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 110110 => 001001 => 2
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 10001110 => 00110110 => 2
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 1001110 => 0011010 => 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 101110 => 001100 => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 11110 => 00111 => 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 1010100 => 0100000 => 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 110100 => 010001 => 2
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 1001100 => 0101010 => 3
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> 101100 => 010100 => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 11100 => 01011 => 2
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> 1010010 => 0001000 => 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> 110010 => 000101 => 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 1001010 => 0000010 => 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 101010 => 000000 => 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 11010 => 00001 => 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> 1000110 => 0010110 => 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> 100110 => 001010 => 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> 10110 => 00100 => 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 1110 => 0011 => 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 101000 => 011000 => 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 11000 => 01101 => 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> 100100 => 010010 => 2
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 10100 => 01000 => 1
Description
The number of ascents of a binary word.
Matching statistic: St000390
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00269: Binary words —flag zeros to zeros⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00269: Binary words —flag zeros to zeros⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1]
=> 10 => 00 => 0
[1,0,1,0,1,0]
=> [2,1]
=> 1010 => 0000 => 0
[1,0,1,1,0,0]
=> [1,1]
=> 110 => 001 => 1
[1,1,0,0,1,0]
=> [2]
=> 100 => 010 => 1
[1,1,0,1,0,0]
=> [1]
=> 10 => 00 => 0
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 101010 => 000000 => 0
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 11010 => 00001 => 1
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 100110 => 001010 => 2
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 10110 => 00100 => 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1110 => 0011 => 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> 10100 => 01000 => 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> 1100 => 0101 => 2
[1,1,0,1,0,0,1,0]
=> [3,1]
=> 10010 => 00010 => 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> 1010 => 0000 => 0
[1,1,0,1,1,0,0,0]
=> [1,1]
=> 110 => 001 => 1
[1,1,1,0,0,0,1,0]
=> [3]
=> 1000 => 0110 => 1
[1,1,1,0,0,1,0,0]
=> [2]
=> 100 => 010 => 1
[1,1,1,0,1,0,0,0]
=> [1]
=> 10 => 00 => 0
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 10101010 => 00000000 => 0
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 1101010 => 0000001 => 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 10011010 => 00001010 => 2
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 1011010 => 0000100 => 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 111010 => 000011 => 1
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 10100110 => 00101000 => 2
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 1100110 => 0010101 => 3
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 10010110 => 00100010 => 2
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 1010110 => 0010000 => 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 110110 => 001001 => 2
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 10001110 => 00110110 => 2
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 1001110 => 0011010 => 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 101110 => 001100 => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 11110 => 00111 => 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 1010100 => 0100000 => 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 110100 => 010001 => 2
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 1001100 => 0101010 => 3
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> 101100 => 010100 => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 11100 => 01011 => 2
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> 1010010 => 0001000 => 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> 110010 => 000101 => 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 1001010 => 0000010 => 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 101010 => 000000 => 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 11010 => 00001 => 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> 1000110 => 0010110 => 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> 100110 => 001010 => 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> 10110 => 00100 => 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 1110 => 0011 => 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 101000 => 011000 => 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 11000 => 01101 => 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> 100100 => 010010 => 2
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 10100 => 01000 => 1
Description
The number of runs of ones in a binary word.
Matching statistic: St001036
(load all 236 compositions to match this statistic)
(load all 236 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001036: Dyck paths ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001036: Dyck paths ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 0
[1,0,1,1,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,0,1,0]
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 0
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 2
[1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [2,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 2
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> ? = 2
[1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> ? = 2
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 1
[1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,3,3,3,3,1]
=> [1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 2
[1,1,1,0,1,0,0,0,1,1,1,1,0,0,0,0]
=> [4,4,4,4,1]
=> [1,1,1,0,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 2
[1,1,1,1,0,1,0,0,0,0,1,1,1,0,0,0]
=> [5,5,5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,0,1,0]
=> ? = 2
[1,1,1,1,0,1,0,0,1,0,0,0,1,1,0,0]
=> [6,6,3,1]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0,1,0]
=> ? = 3
[1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [7,5,3,1]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 3
[1,1,1,1,1,0,1,0,0,0,0,0,1,1,0,0]
=> [6,6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 2
[1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,1,0,0]
=> ? = 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
=> [7,7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2
[1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2
[1,1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 2
[1,1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> ? = 2
[1,1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> ? = 2
[1,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 1
[1,1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> [3,3,3,3,3,1]
=> [1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 2
[1,1,1,1,0,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> [4,4,4,4,1]
=> [1,1,1,0,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 2
[1,1,1,1,1,0,1,0,0,0,0,1,1,1,0,0,0,0]
=> [5,5,5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,0,1,0]
=> ? = 2
[1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0]
=> [7,5,3,1]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 3
[1,1,1,1,1,1,0,1,0,0,0,0,0,1,1,0,0,0]
=> [6,6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 2
[1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0,0]
=> [7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,1,0,0]
=> ? = 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0,0]
=> [7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 1
[1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,3,3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2
[1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [4,4,4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 2
[1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0]
=> [5,5,5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 2
[1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0]
=> [6,6,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 2
[1,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,0]
=> [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0,0]
=> [8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 1
[1,1,1,1,0,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [5,5,5,5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 2
[1,1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0,0]
=> [5,5,5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 2
[1,1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0,0]
=> [6,6,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 2
[1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0,0,0]
=> [7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,1,0,0]
=> ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0,0]
=> [7,7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2
[1,1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> [4,4,4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 2
[1,1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,3,3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2
[1,1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0,0,0]
=> [7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 1
[1,1,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 1
[1,1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0,0]
=> [7,5,3,1]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 3
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0,0,0]
=> [8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 1
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0,0,0,0]
=> [7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 1
Description
The number of inner corners of the parallelogram polyomino associated with the Dyck path.
Matching statistic: St001499
(load all 66 compositions to match this statistic)
(load all 66 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St001499: Dyck paths ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St001499: Dyck paths ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,0,0]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4 = 3 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> ? = 1 + 1
[1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> ? = 2 + 1
[1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [2,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,1,0,0,0]
=> ? = 2 + 1
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,1,0,0,0]
=> ? = 2 + 1
[1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,1,0,0,0]
=> ? = 2 + 1
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
=> ? = 1 + 1
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 1 + 1
[1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,5,5,4,3,2]
=> [1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> ? = 2 + 1
[1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,3,3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> ? = 2 + 1
[1,1,0,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,3,3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 3 + 1
[1,1,0,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [3,3,3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 3 + 1
[1,1,0,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [3,3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,1,0,0,1,0]
=> ? = 3 + 1
[1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 2 + 1
[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 2 + 1
[1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,3,3,3,3,1]
=> [1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> ? = 2 + 1
[1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> [6,6,5,4,3]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 2 + 1
[1,1,1,0,0,0,1,1,1,0,1,1,0,0,0,0]
=> [4,4,3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,0,0,1,0]
=> ? = 3 + 1
[1,1,1,0,0,0,1,1,1,1,0,1,0,0,0,0]
=> [4,3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> ? = 2 + 1
[1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> ? = 2 + 1
[1,1,1,0,0,1,0,0,1,1,1,1,0,0,0,0]
=> [4,4,4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[1,1,1,0,1,0,0,0,1,1,1,1,0,0,0,0]
=> [4,4,4,4,1]
=> [1,1,1,0,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 2 + 1
[1,1,1,1,0,0,0,0,1,1,1,0,1,0,0,0]
=> [5,4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,0,1,0]
=> ? = 2 + 1
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [4,4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,0,1,0]
=> ? = 2 + 1
[1,1,1,1,0,0,1,0,0,0,1,1,1,0,0,0]
=> [5,5,5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0,1,0,1,0]
=> ? = 3 + 1
[1,1,1,1,0,0,1,0,1,0,0,1,0,0,1,0]
=> [7,5,3,2]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 3 + 1
[1,1,1,1,0,1,0,0,0,0,1,1,1,0,0,0]
=> [5,5,5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> ? = 2 + 1
[1,1,1,1,0,1,0,0,1,0,0,0,1,1,0,0]
=> [6,6,3,1]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 + 1
[1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [7,5,3,1]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 3 + 1
[1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> [6,6,5]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 2 + 1
[1,1,1,1,1,0,0,0,0,0,1,1,0,1,0,0]
=> [6,5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0,1,0]
=> ? = 2 + 1
[1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [5,5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0,1,0]
=> ? = 2 + 1
[1,1,1,1,1,0,0,0,0,1,0,0,1,1,0,0]
=> [6,6,4]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0,1,0]
=> ? = 3 + 1
[1,1,1,1,1,0,0,0,1,0,0,0,1,1,0,0]
=> [6,6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 3 + 1
[1,1,1,1,1,0,0,1,0,0,0,0,1,1,0,0]
=> [6,6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0,1,0]
=> ? = 3 + 1
[1,1,1,1,1,0,1,0,0,0,0,0,1,1,0,0]
=> [6,6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,1,0,0]
=> ? = 2 + 1
[1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0,1,1,1,0,0,0]
=> ? = 1 + 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [7,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> ? = 1 + 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [6,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> ? = 2 + 1
[1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> [7,5]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
=> ? = 2 + 1
[1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [7,4]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0,1,0]
=> ? = 2 + 1
[1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0,1,0]
=> ? = 2 + 1
[1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0,1,0]
=> ? = 2 + 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,1,0,0]
=> ? = 1 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 1 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> ? = 1 + 1
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> ? = 1 + 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,1,0,0]
=> ? = 1 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
=> [7,7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 2 + 1
[1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 2 + 1
[1,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> ? = 1 + 1
Description
The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra.
We use the bijection in the code by Christian Stump to have a bijection to Dyck paths.
Matching statistic: St000340
(load all 59 compositions to match this statistic)
(load all 59 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000340: Dyck paths ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000340: Dyck paths ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,0]
=> [1]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [1]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 3 = 2 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> 4 = 3 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> 3 = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 4 = 3 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2,2,1]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 1 + 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3,1,1]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2,1,1]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [3,3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 3 + 1
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [3,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 + 1
[1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [3,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [3,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 2 + 1
[1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [3,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> ? = 3 + 1
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 1 + 1
[1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 2 + 1
[1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? = 2 + 1
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 1 + 1
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 1
[1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,2]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 2 + 1
[1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 2 + 1
[1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,2]
=> [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 2 + 1
[1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [4,4,3,3,2]
=> [1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 3 + 1
[1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [4,3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 2 + 1
[1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [4,4,4,2,2]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,1,0,0,0,0]
=> ? = 4 + 1
[1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [4,4,3,2,2]
=> [1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 3 + 1
[1,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [4,3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 3 + 1
[1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> [4,4,2,2,2]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,1,1,0,0,0,0]
=> ? = 4 + 1
[1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> [4,3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 2 + 1
[1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [3,3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 3 + 1
[1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 3 + 1
[1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 + 1
[1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,1]
=> [1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,1,0,0,0]
=> ? = 2 + 1
[1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,1]
=> [1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,1,0,1,0,0,0]
=> ? = 2 + 1
[1,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,1]
=> [1,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,1,1,0,0,0,0]
=> ? = 2 + 1
[1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,1]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,1,1,0,0,0]
=> ? = 4 + 1
[1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,1]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,1,0,1,0,0,0]
=> ? = 2 + 1
[1,1,0,1,0,0,1,1,0,1,1,0,0,0]
=> [4,4,3,3,1]
=> [1,1,0,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 3 + 1
[1,1,0,1,0,0,1,1,1,0,1,0,0,0]
=> [4,3,3,3,1]
=> [1,1,0,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,1,0,1,0,0,0,0]
=> ? = 2 + 1
[1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3,1]
=> [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,1]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 1 + 1
[1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,1]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,1,0,0,0]
=> ? = 2 + 1
[1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [5,4,4,2,1]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,1,0,1,0,0,0]
=> ? = 2 + 1
[1,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> [4,4,4,2,1]
=> [1,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,1,1,0,0,0,0]
=> ? = 2 + 1
Description
The number of non-final maximal constant sub-paths of length greater than one.
This is the total number of occurrences of the patterns $110$ and $001$.
Matching statistic: St000249
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000249: Set partitions ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 100%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000249: Set partitions ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,1,0,0]
=> [2,1] => {{1,2}}
=> 2 = 0 + 2
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => {{1,3},{2}}
=> 2 = 0 + 2
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => {{1,2,3}}
=> 3 = 1 + 2
[1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => {{1},{2},{3}}
=> 3 = 1 + 2
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => {{1},{2,3}}
=> 2 = 0 + 2
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => {{1,4},{2,3}}
=> 2 = 0 + 2
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => {{1,4},{2},{3}}
=> 3 = 1 + 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => {{1,2,3,4}}
=> 4 = 2 + 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => {{1,2,4},{3}}
=> 3 = 1 + 2
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => {{1,3,4},{2}}
=> 3 = 1 + 2
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => {{1},{2,3},{4}}
=> 3 = 1 + 2
[1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> 4 = 2 + 2
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => {{1},{2,3,4}}
=> 3 = 1 + 2
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => {{1},{2,4},{3}}
=> 2 = 0 + 2
[1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => {{1},{2},{3,4}}
=> 3 = 1 + 2
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => {{1,2,3},{4}}
=> 3 = 1 + 2
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => {{1,2},{3},{4}}
=> 3 = 1 + 2
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => {{1,2},{3,4}}
=> 2 = 0 + 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => {{1,5},{2,4},{3}}
=> 2 = 0 + 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => {{1,5},{2,3,4}}
=> 3 = 1 + 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [5,2,3,4,1] => {{1,5},{2},{3},{4}}
=> 4 = 2 + 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,2,4,3,1] => {{1,5},{2},{3,4}}
=> 3 = 1 + 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => {{1,5},{2,3},{4}}
=> 3 = 1 + 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => {{1,2,4,5},{3}}
=> 4 = 2 + 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => {{1,2,3,4,5}}
=> 5 = 3 + 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => {{1,2,5},{3},{4}}
=> 4 = 2 + 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => {{1,2,5},{3,4}}
=> 3 = 1 + 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => {{1,2,3,5},{4}}
=> 4 = 2 + 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,2,3,5,1] => {{1,4,5},{2},{3}}
=> 4 = 2 + 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => {{1,3,4,5},{2}}
=> 4 = 2 + 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => {{1,3,5},{2},{4}}
=> 3 = 1 + 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => {{1,4,5},{2,3}}
=> 3 = 1 + 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> 3 = 1 + 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => {{1},{2,3,4},{5}}
=> 4 = 2 + 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 5 = 3 + 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 4 = 2 + 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 4 = 2 + 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => {{1},{2,4,5},{3}}
=> 3 = 1 + 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => {{1},{2,3,4,5}}
=> 4 = 2 + 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => {{1},{2,5},{3},{4}}
=> 3 = 1 + 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => {{1},{2,5},{3,4}}
=> 2 = 0 + 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => {{1},{2,3,5},{4}}
=> 3 = 1 + 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 4 = 2 + 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 4 = 2 + 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> 3 = 1 + 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 3 = 1 + 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => {{1,2,4},{3},{5}}
=> 3 = 1 + 2
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => {{1,2,3,4},{5}}
=> 4 = 2 + 2
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> 4 = 2 + 2
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> 3 = 1 + 2
[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [8,6,5,4,3,2,7,1] => {{1,8},{2,6},{3,5},{4},{7}}
=> ? = 1 + 2
[1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [2,6,5,4,3,8,7,1] => {{1,2,6,8},{3,5},{4},{7}}
=> ? = 2 + 2
[1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,1,0,0,0]
=> [6,2,5,4,3,8,7,1] => ?
=> ? = 2 + 2
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,1,0,0,0]
=> [6,3,5,4,2,8,7,1] => ?
=> ? = 2 + 2
[1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,1,0,0,0]
=> [6,5,3,4,2,8,7,1] => ?
=> ? = 2 + 2
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
=> [6,5,4,3,2,8,7,1] => {{1,6,8},{2,5},{3,4},{7}}
=> ? = 1 + 2
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [7,6,5,4,3,2,8,1] => {{1,7,8},{2,6},{3,5},{4}}
=> ? = 1 + 2
[1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,7,5,4,6,3,2,8] => ?
=> ? = 2 + 2
[1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [1,6,5,4,3,7,2,8] => ?
=> ? = 2 + 2
[1,1,0,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,2,5,4,3,7,6,8] => ?
=> ? = 3 + 2
[1,1,0,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> [1,3,5,4,2,7,6,8] => {{1},{2,3,5},{4},{6,7},{8}}
=> ? = 3 + 2
[1,1,0,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,1,0,0,1,0]
=> [1,5,3,4,2,7,6,8] => ?
=> ? = 3 + 2
[1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,5,4,3,2,7,6,8] => {{1},{2,5},{3,4},{6,7},{8}}
=> ? = 2 + 2
[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,6,5,4,3,2,7,8] => {{1},{2,6},{3,5},{4},{7},{8}}
=> ? = 2 + 2
[1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,6,5,4,3,7,8,2] => ?
=> ? = 2 + 2
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,8,7,4,6,5,3,2] => {{1},{2,8},{3,7},{4},{5,6}}
=> ? = 1 + 2
[1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [1,8,5,4,3,7,6,2] => {{1},{2,8},{3,5},{4},{6,7}}
=> ? = 1 + 2
[1,1,0,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [1,3,5,4,8,7,6,2] => {{1},{2,3,5,8},{4},{6,7}}
=> ? = 2 + 2
[1,1,0,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> [1,5,3,4,8,7,6,2] => ?
=> ? = 2 + 2
[1,1,0,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,5,4,3,8,7,6,2] => {{1},{2,5,8},{3,4},{6,7}}
=> ? = 1 + 2
[1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,6,5,4,3,8,7,2] => {{1},{2,6,8},{3,5},{4},{7}}
=> ? = 1 + 2
[1,1,0,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,2,5,4,3,7,8,6] => {{1},{2},{3,5},{4},{6,7,8}}
=> ? = 3 + 2
[1,1,0,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,2,5,4,8,7,6,3] => {{1},{2},{3,5,8},{4},{6,7}}
=> ? = 2 + 2
[1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,4,8,7,6,5] => {{1},{2},{3},{4},{5,8},{6,7}}
=> ? = 3 + 2
[1,1,0,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,2,4,3,8,7,6,5] => {{1},{2},{3,4},{5,8},{6,7}}
=> ? = 2 + 2
[1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,2,5,4,3,8,7,6] => {{1},{2},{3,5},{4},{6,8},{7}}
=> ? = 2 + 2
[1,1,0,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [1,3,5,4,2,7,8,6] => {{1},{2,3,5},{4},{6,7,8}}
=> ? = 3 + 2
[1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,3,2,4,8,7,6,5] => {{1},{2,3},{4},{5,8},{6,7}}
=> ? = 2 + 2
[1,1,0,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,4,2,8,7,6,5] => {{1},{2,3,4},{5,8},{6,7}}
=> ? = 2 + 2
[1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [1,3,5,4,2,8,7,6] => ?
=> ? = 2 + 2
[1,1,0,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> [1,5,3,4,2,7,8,6] => {{1},{2,5},{3},{4},{6,7,8}}
=> ? = 3 + 2
[1,1,0,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,4,3,2,8,7,6,5] => {{1},{2,4},{3},{5,8},{6,7}}
=> ? = 1 + 2
[1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> [1,5,3,4,2,8,7,6] => {{1},{2,5},{3},{4},{6,8},{7}}
=> ? = 2 + 2
[1,1,0,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,0,1,0,0]
=> [1,5,4,3,2,7,8,6] => {{1},{2,5},{3,4},{6,7,8}}
=> ? = 2 + 2
[1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,5,4,3,2,8,7,6] => {{1},{2,5},{3,4},{6,8},{7}}
=> ? = 1 + 2
[1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,6,5,4,3,2,8,7] => {{1},{2,6},{3,5},{4},{7,8}}
=> ? = 1 + 2
[1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [2,7,6,4,5,3,1,8] => ?
=> ? = 2 + 2
[1,1,1,0,0,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,0,0,1,0]
=> [2,5,3,4,7,6,1,8] => ?
=> ? = 3 + 2
[1,1,1,0,0,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> [2,5,4,3,7,6,1,8] => ?
=> ? = 2 + 2
[1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [2,6,5,4,3,7,1,8] => ?
=> ? = 2 + 2
[1,1,1,0,0,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,1,5,4,3,6,7,8] => {{1,2},{3,5},{4},{6},{7},{8}}
=> ? = 3 + 2
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,1,7,6,5,4,3,8] => {{1,2},{3,7},{4,6},{5},{8}}
=> ? = 1 + 2
[1,1,1,0,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [2,1,7,4,6,5,3,8] => ?
=> ? = 2 + 2
[1,1,1,0,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,1,4,7,6,5,3,8] => ?
=> ? = 2 + 2
[1,1,1,0,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [2,1,5,4,7,6,3,8] => ?
=> ? = 2 + 2
[1,1,1,0,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,7,6,5,8] => ?
=> ? = 3 + 2
[1,1,1,0,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,4,3,7,6,5,8] => ?
=> ? = 2 + 2
[1,1,1,0,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,5,4,3,7,6,8] => {{1,2},{3,5},{4},{6,7},{8}}
=> ? = 2 + 2
[1,1,1,0,0,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,5,4,1,6,7,8] => {{1,2,3,5},{4},{6},{7},{8}}
=> ? = 4 + 2
[1,1,1,0,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [2,3,1,4,7,6,5,8] => ?
=> ? = 3 + 2
Description
The number of singletons ([[St000247]]) plus the number of antisingletons ([[St000248]]) of a set partition.
Matching statistic: St000312
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
Mp00046: Ordered trees —to graph⟶ Graphs
St000312: Graphs ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 100%
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
Mp00046: Ordered trees —to graph⟶ Graphs
St000312: Graphs ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,1,0,0]
=> [[[]]]
=> ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [[[[]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [[[],[]]]
=> ([(0,3),(1,3),(2,3)],4)
=> 3 = 1 + 2
[1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> 3 = 1 + 2
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [[[[[]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3 = 1 + 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 2 + 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [[[],[[]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3 = 1 + 2
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [[[[]],[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3 = 1 + 2
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3 = 1 + 2
[1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 2 + 2
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [[],[[],[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3 = 1 + 2
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[],[[[]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3 = 1 + 2
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [[[],[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3 = 1 + 2
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3 = 1 + 2
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> [[[]],[[]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[[[[[]]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 0 + 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[[[[],[]]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3 = 1 + 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[[[],[],[]]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 4 = 2 + 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [[[[],[[]]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3 = 1 + 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[[[[]],[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3 = 1 + 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [[[],[[]],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 4 = 2 + 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[[],[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 5 = 3 + 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [[[],[[],[]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 4 = 2 + 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [[[],[[[]]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3 = 1 + 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [[[],[],[[]]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 4 = 2 + 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[[[],[]],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 4 = 2 + 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[[[]],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 4 = 2 + 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[[[]],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3 = 1 + 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[[[[]]],[]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3 = 1 + 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[],[[[]]],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3 = 1 + 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [[],[[],[]],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 4 = 2 + 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 5 = 3 + 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[],[],[[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 4 = 2 + 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[],[[]],[],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 4 = 2 + 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[],[[[]],[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3 = 1 + 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[],[[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 4 = 2 + 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[],[[[],[]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3 = 1 + 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[],[[[[]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 0 + 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[],[[],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3 = 1 + 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[],[],[],[[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 4 = 2 + 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[],[],[[],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 4 = 2 + 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[],[],[[[]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3 = 1 + 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3 = 1 + 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[[],[[]]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3 = 1 + 2
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[[],[],[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 4 = 2 + 2
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[[]],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 4 = 2 + 2
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[[]],[[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3 = 1 + 2
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [[[[],[[]],[[]]]]]
=> ([(0,7),(1,6),(2,5),(3,4),(4,7),(5,7),(6,7)],8)
=> ? = 2 + 2
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [[[[[],[]],[[]]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,7),(5,7),(6,7)],8)
=> ? = 2 + 2
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [[[[[[]]],[[]]]]]
=> ([(0,6),(1,5),(2,4),(3,4),(3,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [[[[[[[]]]],[]]]]
=> ([(0,6),(1,5),(2,7),(3,4),(3,5),(4,7),(6,7)],8)
=> ? = 1 + 2
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [[[],[[[]],[[]]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,7),(5,7),(6,7)],8)
=> ? = 2 + 2
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [[[],[[]],[[[]]]]]
=> ([(0,7),(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> ? = 2 + 2
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [[[],[[[]]],[[]]]]
=> ([(0,7),(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> ? = 2 + 2
[1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [[[[],[[]]],[],[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
=> ? = 3 + 2
[1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [[[[]],[],[[[]]]]]
=> ([(0,7),(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> ? = 2 + 2
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> [[[[],[]],[[[]]]]]
=> ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
=> ? = 2 + 2
[1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> [[[[],[[]]],[[]]]]
=> ([(0,7),(1,6),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ? = 2 + 2
[1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,1,0,0,0]
=> [[[[[],[]]],[[]]]]
=> ([(0,7),(1,6),(2,6),(3,4),(4,7),(5,6),(5,7)],8)
=> ? = 2 + 2
[1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [[[[[[]]]],[],[]]]
=> ([(0,7),(1,7),(2,7),(3,4),(4,6),(5,6),(5,7)],8)
=> ? = 2 + 2
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [[[[[[]]]],[[]]]]
=> ([(0,6),(1,5),(2,7),(3,4),(3,5),(4,7),(6,7)],8)
=> ? = 1 + 2
[1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [[],[[[[],[]]]],[]]
=> ([(0,6),(1,6),(2,7),(3,7),(4,5),(4,7),(5,6)],8)
=> ? = 2 + 2
[1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [[],[[[],[[]]]],[]]
=> ([(0,7),(1,6),(2,6),(3,4),(4,7),(5,6),(5,7)],8)
=> ? = 2 + 2
[1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [[],[[[[]],[]]],[]]
=> ([(0,7),(1,6),(2,6),(3,4),(4,7),(5,6),(5,7)],8)
=> ? = 2 + 2
[1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [[],[[],[],[[]]],[]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
=> ? = 3 + 2
[1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [[],[[[]],[[]]],[]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,7),(5,7),(6,7)],8)
=> ? = 2 + 2
[1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [[],[[[[]]],[]],[]]
=> ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
=> ? = 2 + 2
[1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [[],[],[[]],[],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ? = 4 + 2
[1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [[],[],[[],[[]]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
=> ? = 3 + 2
[1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [[],[],[[]],[[]],[]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ? = 3 + 2
[1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [[],[[],[]],[],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ? = 4 + 2
[1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [[],[[]],[[[]]],[]]
=> ([(0,7),(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> ? = 2 + 2
[1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [[],[[],[]],[[]],[]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
=> ? = 3 + 2
[1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [[],[[[]]],[[]],[]]
=> ([(0,7),(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> ? = 2 + 2
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [[],[[[[]]]],[],[]]
=> ([(0,7),(1,7),(2,7),(3,4),(4,6),(5,6),(5,7)],8)
=> ? = 2 + 2
[1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [[],[[[[],[]]],[]]]
=> ([(0,7),(1,6),(2,6),(3,4),(4,7),(5,6),(5,7)],8)
=> ? = 2 + 2
[1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [[],[[[],[[]]],[]]]
=> ([(0,7),(1,6),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ? = 2 + 2
[1,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [[],[[[[]],[]],[]]]
=> ([(0,7),(1,6),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ? = 2 + 2
[1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [[],[[],[],[],[],[]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ? = 4 + 2
[1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [[],[[],[[[]]],[]]]
=> ([(0,7),(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> ? = 2 + 2
[1,1,0,1,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [[],[[],[],[[]],[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ? = 3 + 2
[1,1,0,1,0,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [[],[[[]],[[]],[]]]
=> ([(0,7),(1,6),(2,5),(3,4),(4,7),(5,7),(6,7)],8)
=> ? = 2 + 2
[1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [[],[[[[]]],[],[]]]
=> ([(0,7),(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> ? = 2 + 2
[1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [[],[[[[],[]],[]]]]
=> ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
=> ? = 2 + 2
[1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [[],[[[],[[]],[]]]]
=> ([(0,7),(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> ? = 2 + 2
[1,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [[],[[[[]],[],[]]]]
=> ([(0,7),(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> ? = 2 + 2
[1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [[],[[[[],[],[]]]]]
=> ([(0,7),(1,7),(2,7),(3,4),(4,6),(5,6),(5,7)],8)
=> ? = 2 + 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[],[[[[[[]]]]]]]
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ? = 0 + 2
[1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [[],[[[[],[[]]]]]]
=> ([(0,6),(1,5),(2,7),(3,4),(3,5),(4,7),(6,7)],8)
=> ? = 1 + 2
[1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [[],[[[],[[],[]]]]]
=> ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
=> ? = 2 + 2
[1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [[],[[[[]],[[]]]]]
=> ([(0,6),(1,5),(2,4),(3,4),(3,7),(5,7),(6,7)],8)
=> ? = 1 + 2
[1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [[],[[],[[[]],[]]]]
=> ([(0,7),(1,6),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ? = 2 + 2
[1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [[],[[],[],[[],[]]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
=> ? = 3 + 2
[1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [[],[[],[[[],[]]]]]
=> ([(0,7),(1,6),(2,6),(3,4),(4,7),(5,6),(5,7)],8)
=> ? = 2 + 2
[1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [[],[[],[[[[]]]]]]
=> ([(0,6),(1,5),(2,7),(3,4),(3,5),(4,7),(6,7)],8)
=> ? = 1 + 2
[1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [[],[[],[],[[[]]]]]
=> ([(0,7),(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> ? = 2 + 2
[1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [[],[[[]],[],[[]]]]
=> ([(0,7),(1,6),(2,5),(3,4),(4,7),(5,7),(6,7)],8)
=> ? = 2 + 2
Description
The number of leaves in a graph.
That is, the number of vertices of a graph that have degree 1.
Matching statistic: St000619
(load all 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
St000619: Permutations ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 100%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
St000619: Permutations ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,1,0,0]
=> [1,2] => [2,1] => 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => [2,3,1] => 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,1,3] => [3,2,1] => 2 = 1 + 1
[1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [3,2,1] => [1,3,2] => 2 = 1 + 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => [1,2,3] => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [2,3,4,1] => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [3,2,4,1] => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [4,3,2,1] => 3 = 2 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [4,2,3,1] => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [3,4,2,1] => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,3,4,2] => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,4,3,2] => 3 = 2 + 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [1,3,2,4] => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => 1 = 0 + 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,4,2,3] => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [4,3,1,2] => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [4,1,3,2] => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,1,2,3] => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [2,3,4,5,1] => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => [3,2,4,5,1] => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => [4,3,2,5,1] => 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => [4,2,3,5,1] => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => [3,4,2,5,1] => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [5,3,4,2,1] => 3 = 2 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [5,4,3,2,1] => 4 = 3 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [5,3,2,4,1] => 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [5,2,3,4,1] => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [5,4,2,3,1] => 3 = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => [4,3,5,2,1] => 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => [4,5,3,2,1] => 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => [4,5,2,3,1] => 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => [3,4,5,2,1] => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,3,4,5,2] => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,4,3,5,2] => 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,5,4,3,2] => 4 = 3 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,5,3,4,2] => 3 = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,4,5,3,2] => 3 = 2 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [1,3,4,2,5] => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [1,4,3,2,5] => 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [1,3,2,4,5] => 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [1,4,2,3,5] => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,5,4,2,3] => 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [1,5,3,2,4] => 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,5,2,3,4] => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,4,5,2,3] => 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [5,3,4,1,2] => 2 = 1 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [5,4,3,1,2] => 3 = 2 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [5,1,4,3,2] => 3 = 2 + 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [5,1,3,4,2] => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7] => [3,2,4,5,6,7,1] => ? = 1 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [4,1,2,3,5,6,7] => [3,4,5,2,6,7,1] => ? = 1 + 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [4,5,2,3,1,6,7] => [6,4,5,2,3,7,1] => ? = 2 + 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [4,5,2,1,3,6,7] => [5,4,6,2,3,7,1] => ? = 2 + 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [4,5,1,2,3,6,7] => [4,5,6,2,3,7,1] => ? = 1 + 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [5,1,2,3,4,6,7] => [3,4,5,6,2,7,1] => ? = 1 + 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [6,5,2,3,4,1,7] => [7,4,5,6,3,2,1] => ? = 3 + 1
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [4,5,2,3,6,1,7] => [7,4,5,2,3,6,1] => ? = 2 + 1
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,2,1,7] => [7,6,5,2,3,4,1] => ? = 3 + 1
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [4,5,6,2,3,1,7] => [7,5,6,2,3,4,1] => ? = 2 + 1
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [5,6,2,3,4,1,7] => [7,4,5,6,2,3,1] => ? = 2 + 1
[1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [6,5,2,3,1,4,7] => [6,4,5,7,3,2,1] => ? = 3 + 1
[1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,1,2,7] => [6,7,5,2,3,4,1] => ? = 2 + 1
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> [4,5,6,2,1,3,7] => [6,5,7,2,3,4,1] => ? = 2 + 1
[1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> [5,6,2,3,1,4,7] => [6,4,5,7,2,3,1] => ? = 2 + 1
[1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [6,5,2,1,3,4,7] => [5,4,6,7,3,2,1] => ? = 3 + 1
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [4,5,6,1,2,3,7] => [5,6,7,2,3,4,1] => ? = 1 + 1
[1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,1,0,0,0]
=> [5,6,2,1,3,4,7] => [5,4,6,7,2,3,1] => ? = 2 + 1
[1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [6,5,1,2,3,4,7] => [4,5,6,7,3,2,1] => ? = 2 + 1
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [5,6,1,2,3,4,7] => [4,5,6,7,2,3,1] => ? = 1 + 1
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [6,1,2,3,4,5,7] => [3,4,5,6,7,2,1] => ? = 1 + 1
[1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [7,3,4,2,5,6,1] => [1,5,3,4,6,7,2] => ? = 2 + 1
[1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [7,4,5,3,2,6,1] => [1,6,5,3,4,7,2] => ? = 3 + 1
[1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [7,4,5,2,3,6,1] => [1,5,6,3,4,7,2] => ? = 2 + 1
[1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,4,2,1] => [1,7,5,6,4,3,2] => ? = 4 + 1
[1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [7,4,5,3,6,2,1] => [1,7,5,3,4,6,2] => ? = 3 + 1
[1,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,3,2,1] => [1,7,6,3,4,5,2] => ? = 3 + 1
[1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [7,5,6,3,4,2,1] => [1,7,5,6,3,4,2] => ? = 3 + 1
[1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,2,4,1] => [1,6,5,7,4,3,2] => ? = 4 + 1
[1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,2,3,1] => [1,6,7,3,4,5,2] => ? = 2 + 1
[1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [7,5,6,3,2,4,1] => [1,6,5,7,3,4,2] => ? = 3 + 1
[1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [7,6,5,2,3,4,1] => [1,5,6,7,4,3,2] => ? = 3 + 1
[1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [7,5,6,2,3,4,1] => [1,5,6,7,3,4,2] => ? = 2 + 1
[1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,7,1] => [1,5,3,4,6,2,7] => ? = 2 + 1
[1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,7,1] => [1,6,5,4,3,2,7] => ? = 4 + 1
[1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [6,3,4,5,2,7,1] => [1,6,3,4,5,2,7] => ? = 2 + 1
[1,1,0,1,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [6,4,5,3,2,7,1] => [1,6,5,3,4,2,7] => ? = 3 + 1
[1,1,0,1,0,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [6,4,5,2,3,7,1] => [1,5,6,3,4,2,7] => ? = 2 + 1
[1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [5,3,4,2,6,7,1] => [1,5,3,4,2,6,7] => ? = 2 + 1
[1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [4,3,5,2,6,7,1] => [1,5,3,2,4,6,7] => ? = 2 + 1
[1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [3,4,5,2,6,7,1] => [1,5,2,3,4,6,7] => ? = 1 + 1
[1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [5,3,4,6,2,7,1] => [1,6,3,4,2,5,7] => ? = 2 + 1
[1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [5,4,6,3,2,7,1] => [1,6,5,3,2,4,7] => ? = 3 + 1
[1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,2,7,1] => [1,6,3,2,4,5,7] => ? = 2 + 1
[1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,2,7,1] => [1,6,2,3,4,5,7] => ? = 1 + 1
[1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,2,7,1] => [1,6,5,2,3,4,7] => ? = 2 + 1
[1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [5,6,4,2,3,7,1] => [1,5,6,4,2,3,7] => ? = 2 + 1
[1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [5,4,6,2,3,7,1] => [1,5,6,3,2,4,7] => ? = 2 + 1
[1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [4,5,6,2,3,7,1] => [1,5,6,2,3,4,7] => ? = 1 + 1
[1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [6,7,5,3,4,2,1] => [1,7,5,6,4,2,3] => ? = 3 + 1
Description
The number of cyclic descents of a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is given by the number of indices $1 \leq i \leq n$ such that $\pi(i) > \pi(i+1)$ where we set $\pi(n+1) = \pi(1)$.
Matching statistic: St000291
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00096: Binary words —Foata bijection⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 100%
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00096: Binary words —Foata bijection⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,1,0,0]
=> 1100 => 0110 => 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 111000 => 001110 => 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 110010 => 101100 => 2 = 1 + 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 101100 => 011010 => 2 = 1 + 1
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 110100 => 011100 => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 11110000 => 00011110 => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => 10011100 => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => 01011010 => 3 = 2 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 01011100 => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 11011000 => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 00110110 => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 10110100 => 3 = 2 + 1
[1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 11011000 => 00111010 => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 11101000 => 00111100 => 1 = 0 + 1
[1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 10111000 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 01110010 => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 01110100 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 11010100 => 01111000 => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1111100000 => 0000111110 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => 1000111100 => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => 0100111010 => 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1111000100 => 0100111100 => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1110001010 => 1100111000 => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => 0010110110 => 3 = 2 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1100110010 => 1010110100 => 4 = 3 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1110011000 => 0010111010 => 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1111001000 => 0010111100 => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => 1010111000 => 3 = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1100101100 => 0110110010 => 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1100110100 => 0110110100 => 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1110010100 => 0110111000 => 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1100101010 => 1110110000 => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => 0001101110 => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1011100010 => 1001101100 => 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1011001100 => 0101101010 => 4 = 3 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => 0101101100 => 3 = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1011001010 => 1101101000 => 3 = 2 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1101110000 => 0001110110 => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => 1001110100 => 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1110110000 => 0001111010 => 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1111010000 => 0001111100 => 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => 1001111000 => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1101001100 => 0101110010 => 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1101100100 => 0101110100 => 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1110100100 => 0101111000 => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1101001010 => 1101110000 => 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => 0011100110 => 2 = 1 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1010110010 => 1011100100 => 3 = 2 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1011011000 => 0011101010 => 3 = 2 + 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => 0011101100 => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 11111100000010 => 10000011111100 => ? = 1 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> 11110000101010 => 11100011110000 => ? = 1 + 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> 11110001001010 => 11010011110000 => ? = 2 + 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> 11110001010010 => 10110011110000 => ? = 2 + 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> 11110001010100 => 01110011110000 => ? = 1 + 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> 11100010101010 => 11110011100000 => ? = 1 + 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> 11001100101010 => 11101011010000 => ? = 3 + 1
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> 11110010001010 => 11001011110000 => ? = 2 + 1
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> 11110010010010 => 10101011110000 => ? = 3 + 1
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> 11110010010100 => 01101011110000 => ? = 2 + 1
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> 11100100101010 => 11101011100000 => ? = 2 + 1
[1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> 11001101001010 => 11011011010000 => ? = 3 + 1
[1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> 11110010100010 => 10011011110000 => ? = 2 + 1
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0]
=> 11110010100100 => 01011011110000 => ? = 2 + 1
[1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> 11100101001010 => 11011011100000 => ? = 2 + 1
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> 11110010101000 => 00111011110000 => ? = 1 + 1
[1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> 11001101010100 => 01111011010000 => ? = 2 + 1
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> 11100101010100 => 01111011100000 => ? = 1 + 1
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> 11001010101010 => 11111011000000 => ? = 1 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 10111111000000 => 00000110111110 => ? = 1 + 1
[1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> 10111110000010 => 10000110111100 => ? = 2 + 1
[1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> 10111110000100 => 01000110111100 => ? = 2 + 1
[1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> 10111100001010 => 11000110111000 => ? = 2 + 1
[1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> 10111100010010 => 10100110111000 => ? = 3 + 1
[1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> 10111100010100 => 01100110111000 => ? = 2 + 1
[1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> 10111000101010 => 11100110110000 => ? = 2 + 1
[1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> 10110011001010 => 11010110101000 => ? = 4 + 1
[1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> 10111100100010 => 10010110111000 => ? = 3 + 1
[1,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> 10111100100100 => 01010110111000 => ? = 3 + 1
[1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> 10111001001010 => 11010110110000 => ? = 3 + 1
[1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> 10110011010010 => 10110110101000 => ? = 4 + 1
[1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> 10111100101000 => 00110110111000 => ? = 2 + 1
[1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> 10111001010010 => 10110110110000 => ? = 3 + 1
[1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> 10110011010100 => 01110110101000 => ? = 3 + 1
[1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> 10111001010100 => 01110110110000 => ? = 2 + 1
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> 10110010101010 => 11110110100000 => ? = 2 + 1
[1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> 11011110000010 => 10000111011100 => ? = 2 + 1
[1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> 11011110000100 => 01000111011100 => ? = 2 + 1
[1,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> 11011100001010 => 11000111011000 => ? = 2 + 1
[1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> 11011000110010 => 10100111010100 => ? = 4 + 1
[1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> 11011110001000 => 00100111011100 => ? = 2 + 1
[1,1,0,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> 11011100010010 => 10100111011000 => ? = 3 + 1
[1,1,0,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> 11011100010100 => 01100111011000 => ? = 2 + 1
[1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> 11011000101010 => 11100111010000 => ? = 2 + 1
[1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> 11101111000000 => 00000111101110 => ? = 1 + 1
[1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> 11101110000010 => 10000111101100 => ? = 2 + 1
[1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> 11101110000100 => 01000111101100 => ? = 2 + 1
[1,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0,1,0]
=> 11101100001010 => 11000111101000 => ? = 2 + 1
[1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> 11110110000010 => 10000111110100 => ? = 2 + 1
[1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> 11111011000000 => 00000111111010 => ? = 1 + 1
Description
The number of descents of a binary word.
Matching statistic: St000925
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000925: Set partitions ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 100%
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000925: Set partitions ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 2 = 0 + 2
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 2 = 0 + 2
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 3 = 1 + 2
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 3 = 1 + 2
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 2 = 0 + 2
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 2 = 0 + 2
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> 3 = 1 + 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> 4 = 2 + 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 3 = 1 + 2
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 3 = 1 + 2
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> 3 = 1 + 2
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> {{1,3},{2},{4},{5}}
=> 4 = 2 + 2
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> {{1,2},{3,5},{4}}
=> 3 = 1 + 2
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> 2 = 0 + 2
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> {{1,3},{2},{4,5}}
=> 3 = 1 + 2
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> {{1,4},{2,3},{5}}
=> 3 = 1 + 2
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> 3 = 1 + 2
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 2 = 0 + 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> {{1},{2,3,4,5,6}}
=> 2 = 0 + 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> {{1},{2,3,5,6},{4}}
=> 3 = 1 + 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> {{1},{2,4,6},{3},{5}}
=> 4 = 2 + 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> {{1},{2,4,5,6},{3}}
=> 3 = 1 + 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> {{1},{2,5,6},{3,4}}
=> 3 = 1 + 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> {{1},{2},{3,6},{4,5}}
=> 4 = 2 + 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3},{4,6},{5}}
=> 5 = 3 + 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> {{1},{2},{3,4,6},{5}}
=> 4 = 2 + 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> {{1},{2},{3,4,5,6}}
=> 3 = 1 + 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3},{4,5,6}}
=> 4 = 2 + 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> {{1},{2,3},{4},{5,6}}
=> 4 = 2 + 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> {{1},{2,3},{4,6},{5}}
=> 4 = 2 + 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> {{1},{2,3},{4,5,6}}
=> 3 = 1 + 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> {{1},{2,3,4},{5,6}}
=> 3 = 1 + 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> {{1,2},{3,4,5},{6}}
=> 3 = 1 + 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> {{1,2},{3},{4,5},{6}}
=> 4 = 2 + 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> {{1,3},{2},{4},{5},{6}}
=> 5 = 3 + 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> {{1,3},{2},{4,5},{6}}
=> 4 = 2 + 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> {{1,3,4},{2},{5},{6}}
=> 4 = 2 + 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> {{1,2},{3,6},{4,5}}
=> 3 = 1 + 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> {{1,2},{3},{4,6},{5}}
=> 4 = 2 + 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> {{1,2},{3,4,6},{5}}
=> 3 = 1 + 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> {{1,2},{3,4,5,6}}
=> 2 = 0 + 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> {{1,2},{3},{4,5,6}}
=> 3 = 1 + 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> {{1,3},{2},{4},{5,6}}
=> 4 = 2 + 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> {{1,3},{2},{4,6},{5}}
=> 4 = 2 + 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> {{1,3},{2},{4,5,6}}
=> 3 = 1 + 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> {{1,3,4},{2},{5,6}}
=> 3 = 1 + 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> {{1,4,5},{2,3},{6}}
=> 3 = 1 + 2
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> {{1,5},{2,4},{3},{6}}
=> 4 = 2 + 2
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> {{1,2,3},{4},{5},{6}}
=> 4 = 2 + 2
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> {{1,2,3},{4,5},{6}}
=> 3 = 1 + 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> {{1},{2,3,7,8},{4,5,6}}
=> ? = 1 + 2
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> {{1},{2,6,7,8},{3},{4,5}}
=> ? = 2 + 2
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> {{1},{2,6,7,8},{3,5},{4}}
=> ? = 2 + 2
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> {{1},{2,6,7,8},{3,4,5}}
=> ? = 1 + 2
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> {{1},{2,7,8},{3,4,5,6}}
=> ? = 1 + 2
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> {{1},{2},{3,4,5},{6,8},{7}}
=> ? = 3 + 2
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> {{1},{2},{3,6,7,8},{4,5}}
=> ? = 2 + 2
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> {{1},{2},{3},{4},{5,6,7,8}}
=> ? = 3 + 2
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> {{1},{2},{3,4},{5,6,7,8}}
=> ? = 2 + 2
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5},{6,7,8}}
=> ? = 2 + 2
[1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> {{1},{2,4,5},{3},{6,8},{7}}
=> ? = 3 + 2
[1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3},{4},{5,6,7,8}}
=> ? = 2 + 2
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> {{1},{2,4},{3},{5,6,7,8}}
=> ? = 2 + 2
[1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> {{1},{2,4,5},{3},{6,7,8}}
=> ? = 2 + 2
[1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> {{1},{2,3,5},{4},{6,8},{7}}
=> ? = 3 + 2
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4},{5,6,7,8}}
=> ? = 1 + 2
[1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> {{1},{2,3,5},{4},{6,7,8}}
=> ? = 2 + 2
[1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,0,1,0,0]
=> {{1},{2,3,4,5},{6,8},{7}}
=> ? = 2 + 2
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> {{1},{2,3,4,5},{6,7,8}}
=> ? = 1 + 2
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> {{1},{2,3,4,5,6},{7,8}}
=> ? = 1 + 2
[1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> {{1,2},{3,4,5,6,7},{8}}
=> ? = 1 + 2
[1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> {{1,2},{3,4,6,7},{5},{8}}
=> ? = 2 + 2
[1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> {{1,2},{3,5,6,7},{4},{8}}
=> ? = 2 + 2
[1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> {{1,2},{3,6,7},{4,5},{8}}
=> ? = 2 + 2
[1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> {{1,2},{3},{4},{5,6,7},{8}}
=> ? = 3 + 2
[1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> {{1,2},{3,4},{5,6,7},{8}}
=> ? = 2 + 2
[1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4,5},{6,7},{8}}
=> ? = 2 + 2
[1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> {{1,4,5},{2},{3},{6},{7},{8}}
=> ? = 4 + 2
[1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> {{1,3},{2},{4},{5,6,7},{8}}
=> ? = 3 + 2
[1,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,1,0,0,0,1,0]
=> {{1,4},{2},{3},{5,6,7},{8}}
=> ? = 3 + 2
[1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> {{1,4,5},{2},{3},{6,7},{8}}
=> ? = 3 + 2
[1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> {{1,3,5},{2},{4},{6},{7},{8}}
=> ? = 4 + 2
[1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> {{1,3,4},{2},{5,6,7},{8}}
=> ? = 2 + 2
[1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,1,0,0,1,0]
=> {{1,3,5},{2},{4},{6,7},{8}}
=> ? = 3 + 2
[1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> {{1,3,4,5},{2},{6},{7},{8}}
=> ? = 3 + 2
[1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> {{1,3,4,5},{2},{6,7},{8}}
=> ? = 2 + 2
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> {{1,3,4,5,6},{2},{7},{8}}
=> ? = 2 + 2
[1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> {{1,2},{3,8},{4,6,7},{5}}
=> ? = 2 + 2
[1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> {{1,2},{3,8},{4},{5,6,7}}
=> ? = 2 + 2
[1,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> {{1,2},{3,8},{4,5},{6,7}}
=> ? = 2 + 2
[1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> {{1,2},{3},{4},{5,8},{6},{7}}
=> ? = 4 + 2
[1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> {{1,2},{3},{4,8},{5,6,7}}
=> ? = 2 + 2
[1,1,0,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> {{1,2},{3},{4},{5,8},{6,7}}
=> ? = 3 + 2
[1,1,0,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> {{1,2},{3,4},{5,8},{6,7}}
=> ? = 2 + 2
[1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> {{1,2},{3,4,5},{6,8},{7}}
=> ? = 2 + 2
[1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> {{1,2},{3,4,8},{5,6,7}}
=> ? = 1 + 2
[1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> {{1,2},{3,4,8},{5},{6,7}}
=> ? = 2 + 2
[1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> {{1,2},{3,5,8},{4},{6,7}}
=> ? = 2 + 2
[1,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> {{1,2},{3,6,8},{4,5},{7}}
=> ? = 2 + 2
[1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> {{1,2},{3,4,6,8},{5},{7}}
=> ? = 2 + 2
Description
The number of topologically connected components of a set partition.
For example, the set partition $\{\{1,5\},\{2,3\},\{4,6\}\}$ has the two connected components $\{1,4,5,6\}$ and $\{2,3\}$.
The number of set partitions with only one block is [[oeis:A099947]].
The following 59 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001488The number of corners of a skew partition. St000871The number of very big ascents of a permutation. St000053The number of valleys of the Dyck path. St000236The number of cyclical small weak excedances. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St000024The number of double up and double down steps of a Dyck path. St000105The number of blocks in the set partition. St000636The hull number of a graph. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001883The mutual visibility number of a graph. St000996The number of exclusive left-to-right maxima of a permutation. St000245The number of ascents of a permutation. St001083The number of boxed occurrences of 132 in a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000167The number of leaves of an ordered tree. St000159The number of distinct parts of the integer partition. St000824The sum of the number of descents and the number of recoils of a permutation. St000164The number of short pairs. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000157The number of descents of a standard tableau. St000318The number of addable cells of the Ferrers diagram of an integer partition. St001902The number of potential covers of a poset. St000052The number of valleys of a Dyck path not on the x-axis. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St000015The number of peaks of a Dyck path. St001960The number of descents of a permutation minus one if its first entry is not one. St000021The number of descents of a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St000155The number of exceedances (also excedences) of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000354The number of recoils of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St001489The maximum of the number of descents and the number of inverse descents. St000062The length of the longest increasing subsequence of the permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000239The number of small weak excedances. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000443The number of long tunnels of a Dyck path. St000470The number of runs in a permutation. St000542The number of left-to-right-minima of a permutation. St000665The number of rafts of a permutation. 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. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001180Number of indecomposable injective modules with projective dimension at most 1. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001964The interval resolution global dimension of a poset. St000075The orbit size of a standard tableau under promotion. St000455The second largest eigenvalue of a graph if it is integral. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St000035The number of left outer peaks of a permutation.
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