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Your data matches 207 different statistics following compositions of up to 3 maps.
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Matching statistic: St001715
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(load all 4 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St001715: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St001715: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [3,1,2] => [2,3,1] => 0
[2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => [1,3,4,2] => 0
[1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => [4,2,1,3] => 0
[3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [1,2,4,5,3] => 0
[2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => [2,3,4,1] => 0
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [3,1,5,2,4] => 0
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => [1,2,3,5,6,4] => 0
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [3,4,2,5,1] => 0
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [5,1,3,2,4] => 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [4,5,2,3,1] => 0
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [3,1,4,6,2,5] => 0
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => [1,2,3,4,6,7,5] => 0
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => [4,2,5,3,6,1] => 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [1,3,4,5,2] => 0
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [4,1,5,2,3] => 0
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [5,2,3,1,4] => 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [5,3,6,2,4,1] => 0
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => [4,5,1,3,6,2] => 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [5,2,1,6,3,4] => 0
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [6,1,2,4,3,5] => 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [2,3,4,5,1] => 0
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [1,5,6,4,2,3] => 0
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [4,1,2,6,3,5] => 0
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [6,4,2,3,1,5] => 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [1,2,4,5,6,3] => 0
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [3,4,5,2,6,1] => 0
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [3,1,5,6,2,4] => 0
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [6,3,2,4,1,5] => 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [5,6,1,3,4,2] => 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [4,5,6,2,3,1] => 0
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [4,6,2,1,3,5] => 0
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [3,4,2,5,6,1] => 0
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [5,1,2,6,3,4] => 0
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [4,5,2,3,6,1] => 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [6,1,3,4,2,5] => 2
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [1,6,4,2,3,5] => 1
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [5,6,2,3,4,1] => 1
[5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,4,7,1,5,6] => [1,2,3,5,6,7,4] => 0
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [1,3,4,5,6,2] => 0
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [4,1,5,6,2,3] => 0
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [5,6,2,1,3,4] => 0
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [6,2,3,4,1,5] => 2
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [2,3,4,5,6,1] => 0
[5,4,3]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,7,1,4,5,6] => [1,2,4,5,6,7,3] => 0
[5,4,3,2]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,7,1,3,4,5,6] => [1,3,4,5,6,7,2] => 0
[]
=> []
=> [1] => [1] => 0
Description
The number of non-records in a permutation.
A record in a permutation $\pi$ is a value $\pi(j)$ which is a left-to-right minimum, a left-to-right maximum, a right-to-left minimum, or a right-to-left maximum.
For example, in the permutation $\pi = [1, 4, 3, 2, 5]$, the values $1$ is a left-to-right minimum, $1, 4, 5$ are left-to-right maxima, $5, 2, 1$ are right-to-left minima and $5$ is a right-to-left maximum. Hence, $3$ is the unique non-record.
Permutations without non-records are called square [1].
Matching statistic: St000392
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 67%
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 67%
Values
[1]
=> [1,0,1,0]
=> 1010 => 0110 => 2 = 0 + 2
[2]
=> [1,1,0,0,1,0]
=> 110010 => 010110 => 2 = 0 + 2
[1,1]
=> [1,0,1,1,0,0]
=> 101100 => 110010 => 2 = 0 + 2
[3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => 01010110 => 2 = 0 + 2
[2,1]
=> [1,0,1,0,1,0]
=> 101010 => 100110 => 2 = 0 + 2
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 11010010 => 2 = 0 + 2
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? => ? = 0 + 2
[3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 10010110 => 2 = 0 + 2
[2,2]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => 00111010 => 3 = 1 + 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 10110010 => 2 = 0 + 2
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? => ? = 0 + 2
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 111110000010 => ? => ? = 0 + 2
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? => ? = 1 + 2
[3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 00110110 => 2 = 0 + 2
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 10100110 => 2 = 0 + 2
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 01110010 => 3 = 1 + 2
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? => ? = 0 + 2
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? => ? = 1 + 2
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? => ? = 0 + 2
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => ? => ? = 1 + 2
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 01100110 => 2 = 0 + 2
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? => ? = 0 + 2
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => ? => ? = 0 + 2
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1011011000 => ? => ? = 1 + 2
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1110001010 => ? => ? = 0 + 2
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1101010010 => ? => ? = 0 + 2
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1011100010 => ? => ? = 0 + 2
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1101001100 => ? => ? = 1 + 2
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1100110100 => ? => ? = 1 + 2
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => ? => ? = 0 + 2
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? => ? = 0 + 2
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1101001010 => ? => ? = 0 + 2
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1100110010 => ? => ? = 0 + 2
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1011010010 => ? => ? = 1 + 2
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1100101100 => ? => ? = 2 + 2
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1011001100 => ? => ? = 1 + 2
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? => ? = 1 + 2
[5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 111100001010 => ? => ? = 0 + 2
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1100101010 => ? => ? = 0 + 2
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1011001010 => ? => ? = 0 + 2
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1010110010 => ? => ? = 0 + 2
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? => ? = 2 + 2
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => ? => ? = 0 + 2
[5,4,3]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> 111000101010 => ? => ? = 0 + 2
[5,4,3,2]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 110010101010 => ? => ? = 0 + 2
[]
=> []
=> => ? => ? = 0 + 2
Description
The length of the longest run of ones in a binary word.
Matching statistic: St000982
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
St000982: Binary words ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 67%
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
St000982: Binary words ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 67%
Values
[1]
=> [1,0,1,0]
=> 1010 => 0110 => 2 = 0 + 2
[2]
=> [1,1,0,0,1,0]
=> 110010 => 010110 => 2 = 0 + 2
[1,1]
=> [1,0,1,1,0,0]
=> 101100 => 110010 => 2 = 0 + 2
[3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => 01010110 => 2 = 0 + 2
[2,1]
=> [1,0,1,0,1,0]
=> 101010 => 100110 => 2 = 0 + 2
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 11010010 => 2 = 0 + 2
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? => ? = 0 + 2
[3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 10010110 => 2 = 0 + 2
[2,2]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => 00111010 => 3 = 1 + 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 10110010 => 2 = 0 + 2
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? => ? = 0 + 2
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 111110000010 => ? => ? = 0 + 2
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? => ? = 1 + 2
[3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 00110110 => 2 = 0 + 2
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 10100110 => 2 = 0 + 2
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 01110010 => 3 = 1 + 2
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? => ? = 0 + 2
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? => ? = 1 + 2
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? => ? = 0 + 2
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => ? => ? = 1 + 2
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 01100110 => 2 = 0 + 2
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? => ? = 0 + 2
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => ? => ? = 0 + 2
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1011011000 => ? => ? = 1 + 2
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1110001010 => ? => ? = 0 + 2
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1101010010 => ? => ? = 0 + 2
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1011100010 => ? => ? = 0 + 2
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1101001100 => ? => ? = 1 + 2
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1100110100 => ? => ? = 1 + 2
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => ? => ? = 0 + 2
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? => ? = 0 + 2
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1101001010 => ? => ? = 0 + 2
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1100110010 => ? => ? = 0 + 2
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1011010010 => ? => ? = 1 + 2
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1100101100 => ? => ? = 2 + 2
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1011001100 => ? => ? = 1 + 2
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? => ? = 1 + 2
[5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 111100001010 => ? => ? = 0 + 2
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1100101010 => ? => ? = 0 + 2
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1011001010 => ? => ? = 0 + 2
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1010110010 => ? => ? = 0 + 2
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? => ? = 2 + 2
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => ? => ? = 0 + 2
[5,4,3]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> 111000101010 => ? => ? = 0 + 2
[5,4,3,2]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 110010101010 => ? => ? = 0 + 2
[]
=> []
=> => ? => ? = 0 + 2
Description
The length of the longest constant subword.
Matching statistic: St001001
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001001: Dyck paths ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 33%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001001: Dyck paths ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 33%
Values
[1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 0
[2]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[1,1]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 0
[3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 0
[2,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0
[1,1,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[4]
=> [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]
=> ? = 0
[3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
[2,2]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 1
[2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0
[1,1,1,1]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0
[5]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 0
[4,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[3,2]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[3,1,1]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
[2,2,1]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1
[2,1,1,1]
=> [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]
=> ? = 0
[4,2]
=> [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
[4,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 0
[3,3]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 1
[3,2,1]
=> [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]
=> ? = 0
[3,1,1,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[2,2,2]
=> [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]
=> ? = 0
[2,2,1,1]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 1
[4,3]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 0
[4,2,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[4,1,1,1]
=> [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]
=> ? = 0
[3,3,1]
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
=> ? = 1
[3,2,2]
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 1
[3,2,1,1]
=> [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]
=> ? = 0
[2,2,2,1]
=> [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]
=> ? = 0
[4,3,1]
=> [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]
=> ? = 0
[4,2,2]
=> [4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> ? = 0
[4,2,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 1
[3,3,2]
=> [6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,1,0,0]
=> ? = 2
[3,3,1,1]
=> [6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> ? = 1
[3,2,2,1]
=> [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]
=> ? = 1
[5,4]
=> [5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 0
[4,3,2]
=> [3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 0
[4,3,1,1]
=> [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]
=> ? = 0
[4,2,2,1]
=> [4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> ? = 0
[3,3,2,1]
=> [6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 2
[4,3,2,1]
=> [3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> ? = 0
[5,4,3]
=> [5,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> ? = 0
[5,4,3,2]
=> [5,3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0,0]
=> ? = 0
[]
=> ?
=> ?
=> ?
=> ? = 0
Description
The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001371
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001371: Binary words ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 33%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001371: Binary words ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 33%
Values
[1]
=> [1]
=> [1,0,1,0]
=> 1010 => 0
[2]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 0
[1,1]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => 0
[3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => 0
[2,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0
[1,1,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 0
[4]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? = 0
[3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 0
[2,2]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? = 1
[2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
[1,1,1,1]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => 0
[5]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 111110000010 => ? = 0
[4,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 101111100000 => ? = 1
[3,2]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[3,1,1]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
[2,2,1]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 1
[2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
[4,2]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 10111111000000 => ? = 1
[4,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 101111010000 => ? = 0
[3,3]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 11111100000010 => ? = 1
[3,2,1]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 0
[3,1,1,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
[2,2,2]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 0
[2,2,1,1]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? = 1
[4,3]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> 101111001000 => ? = 0
[4,2,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 1011111110000000 => ? = 0
[4,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> 10111110100000 => ? = 0
[3,3,1]
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> 11111010000010 => ? = 1
[3,2,2]
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1110001010 => ? = 1
[3,2,1,1]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => ? = 0
[2,2,2,1]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1011100010 => ? = 0
[4,3,1]
=> [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> 10111110010000 => ? = 0
[4,2,2]
=> [4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> 101111000100 => ? = 0
[4,2,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 1011111101000000 => ? = 1
[3,3,2]
=> [6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> 11110110000010 => ? = 2
[3,3,1,1]
=> [6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> 11111001000010 => ? = 1
[3,2,2,1]
=> [4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1101001010 => ? = 1
[5,4]
=> [5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 101111000010 => ? = 0
[4,3,2]
=> [3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> 1011111100100000 => ? = 0
[4,3,1,1]
=> [3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> 10111101010000 => ? = 0
[4,2,2,1]
=> [4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> 10111110001000 => ? = 0
[3,3,2,1]
=> [6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> 11101110000010 => ? = 2
[4,3,2,1]
=> [3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> 101111111001000000 => ? = 0
[5,4,3]
=> [5,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> 10111100100100 => ? = 0
[5,4,3,2]
=> [5,3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> 101111110010010000 => ? = 0
[]
=> ?
=> ?
=> ? => ? = 0
Description
The length of the longest Yamanouchi prefix of a binary word.
This is the largest index $i$ such that in each of the prefixes $w_1$, $w_1w_2$, $w_1w_2\dots w_i$ the number of zeros is greater than or equal to the number of ones.
Matching statistic: St001730
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001730: Binary words ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 33%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001730: Binary words ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 33%
Values
[1]
=> [1]
=> [1,0,1,0]
=> 1010 => 0
[2]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 0
[1,1]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => 0
[3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => 0
[2,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0
[1,1,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 0
[4]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? = 0
[3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 0
[2,2]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? = 1
[2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
[1,1,1,1]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => 0
[5]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 111110000010 => ? = 0
[4,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 101111100000 => ? = 1
[3,2]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[3,1,1]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
[2,2,1]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 1
[2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
[4,2]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 10111111000000 => ? = 1
[4,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 101111010000 => ? = 0
[3,3]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 11111100000010 => ? = 1
[3,2,1]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 0
[3,1,1,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0
[2,2,2]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 0
[2,2,1,1]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? = 1
[4,3]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> 101111001000 => ? = 0
[4,2,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 1011111110000000 => ? = 0
[4,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> 10111110100000 => ? = 0
[3,3,1]
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> 11111010000010 => ? = 1
[3,2,2]
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1110001010 => ? = 1
[3,2,1,1]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => ? = 0
[2,2,2,1]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1011100010 => ? = 0
[4,3,1]
=> [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> 10111110010000 => ? = 0
[4,2,2]
=> [4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> 101111000100 => ? = 0
[4,2,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 1011111101000000 => ? = 1
[3,3,2]
=> [6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> 11110110000010 => ? = 2
[3,3,1,1]
=> [6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> 11111001000010 => ? = 1
[3,2,2,1]
=> [4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1101001010 => ? = 1
[5,4]
=> [5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 101111000010 => ? = 0
[4,3,2]
=> [3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> 1011111100100000 => ? = 0
[4,3,1,1]
=> [3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> 10111101010000 => ? = 0
[4,2,2,1]
=> [4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> 10111110001000 => ? = 0
[3,3,2,1]
=> [6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> 11101110000010 => ? = 2
[4,3,2,1]
=> [3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> 101111111001000000 => ? = 0
[5,4,3]
=> [5,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> 10111100100100 => ? = 0
[5,4,3,2]
=> [5,3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> 101111110010010000 => ? = 0
[]
=> ?
=> ?
=> ? => ? = 0
Description
The number of times the path corresponding to a binary word crosses the base line.
Interpret each $0$ as a step $(1,-1)$ and $1$ as a step $(1,1)$. Then this statistic counts the number of times the path crosses the $x$-axis.
Matching statistic: St001803
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St001803: Standard tableaux ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 33%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St001803: Standard tableaux ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 33%
Values
[1]
=> [1]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> 0
[2]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0
[1,1]
=> [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 0
[3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> 0
[2,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 0
[1,1,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
[4]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? = 0
[3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 0
[2,2]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 1
[2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 0
[1,1,1,1]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> 0
[5]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,11],[6,7,8,9,10,12]]
=> ? = 0
[4,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,3,4,5,6,7],[2,8,9,10,11,12]]
=> ? = 1
[3,2]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
[3,1,1]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 0
[2,2,1]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 1
[2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0
[4,2]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[1,3,4,5,6,7,8],[2,9,10,11,12,13,14]]
=> ? = 1
[4,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [[1,3,4,5,6,8],[2,7,9,10,11,12]]
=> ? = 0
[3,3]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,6,13],[7,8,9,10,11,12,14]]
=> ? = 1
[3,2,1]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? = 0
[3,1,1,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0
[2,2,2]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 0
[2,2,1,1]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? = 1
[4,3]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [[1,3,4,5,6,9],[2,7,8,10,11,12]]
=> ? = 0
[4,2,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,3,4,5,6,7,8,9],[2,10,11,12,13,14,15,16]]
=> ? = 0
[4,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [[1,3,4,5,6,7,9],[2,8,10,11,12,13,14]]
=> ? = 0
[3,3,1]
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,7,13],[6,8,9,10,11,12,14]]
=> ? = 1
[3,2,2]
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[1,2,3,7,9],[4,5,6,8,10]]
=> ? = 1
[3,2,1,1]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[1,3,4,6,8],[2,5,7,9,10]]
=> ? = 0
[2,2,2,1]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[1,3,4,5,9],[2,6,7,8,10]]
=> ? = 0
[4,3,1]
=> [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [[1,3,4,5,6,7,10],[2,8,9,11,12,13,14]]
=> ? = 0
[4,2,2]
=> [4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [[1,3,4,5,6,10],[2,7,8,9,11,12]]
=> ? = 0
[4,2,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [[1,3,4,5,6,7,8,10],[2,9,11,12,13,14,15,16]]
=> ? = 1
[3,3,2]
=> [6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [[1,2,3,4,6,7,13],[5,8,9,10,11,12,14]]
=> ? = 2
[3,3,1,1]
=> [6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [[1,2,3,4,5,8,13],[6,7,9,10,11,12,14]]
=> ? = 1
[3,2,2,1]
=> [4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[1,2,4,7,9],[3,5,6,8,10]]
=> ? = 1
[5,4]
=> [5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [[1,3,4,5,6,11],[2,7,8,9,10,12]]
=> ? = 0
[4,3,2]
=> [3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [[1,3,4,5,6,7,8,11],[2,9,10,12,13,14,15,16]]
=> ? = 0
[4,3,1,1]
=> [3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [[1,3,4,5,6,8,10],[2,7,9,11,12,13,14]]
=> ? = 0
[4,2,2,1]
=> [4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [[1,3,4,5,6,7,11],[2,8,9,10,12,13,14]]
=> ? = 0
[3,3,2,1]
=> [6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [[1,2,3,5,6,7,13],[4,8,9,10,11,12,14]]
=> ? = 2
[4,3,2,1]
=> [3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [[1,3,4,5,6,7,8,9,12],[2,10,11,13,14,15,16,17,18]]
=> ? = 0
[5,4,3]
=> [5,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [[1,3,4,5,6,9,12],[2,7,8,10,11,13,14]]
=> ? = 0
[5,4,3,2]
=> [5,3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> [[1,3,4,5,6,7,8,11,14],[2,9,10,12,13,15,16,17,18]]
=> ? = 0
[]
=> ?
=> ?
=> ?
=> ? = 0
Description
The maximal overlap of the cylindrical tableau associated with a tableau.
A cylindrical tableau associated with a standard Young tableau $T$ is the skew row-strict tableau obtained by gluing two copies of $T$ such that the inner shape is a rectangle.
The overlap, recorded in this statistic, equals $\max_C\big(2\ell(T) - \ell(C)\big)$, where $\ell$ denotes the number of rows of a tableau and the maximum is taken over all cylindrical tableaux.
In particular, the statistic equals $0$, if and only if the last entry of the first row is larger than or equal to the first entry of the last row. Moreover, the statistic attains its maximal value, the number of rows of the tableau minus 1, if and only if the tableau consists of a single column.
Matching statistic: St001882
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001882: Signed permutations ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 67%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001882: Signed permutations ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 67%
Values
[1]
=> [[1]]
=> [1] => [1] => 0
[2]
=> [[1,2]]
=> [1,2] => [1,2] => 0
[1,1]
=> [[1],[2]]
=> [2,1] => [2,1] => 0
[3]
=> [[1,2,3]]
=> [1,2,3] => [1,2,3] => 0
[2,1]
=> [[1,3],[2]]
=> [2,1,3] => [2,1,3] => 0
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => [3,2,1] => 0
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => 0
[3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => [2,1,3,4] => 0
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => [3,4,1,2] => 1
[2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => 0
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => 0
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0
[4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ? = 1
[3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => [3,4,1,2,5] => ? = 0
[3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4,5] => ? = 0
[2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [4,2,5,1,3] => ? = 1
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [4,3,2,1,5] => ? = 0
[4,2]
=> [[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => ? = 1
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => [3,2,1,4,5,6] => ? = 0
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [4,5,6,1,2,3] => ? = 1
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => [4,2,5,1,3,6] => ? = 0
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => [4,3,2,1,5,6] => ? = 0
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [5,6,3,4,1,2] => ? = 0
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => [5,3,2,6,1,4] => ? = 1
[4,3]
=> [[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => [4,5,6,1,2,3,7] => ? = 0
[4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => [4,2,5,1,3,6,7] => ? = 0
[4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => [4,3,2,1,5,6,7] => ? = 0
[3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [5,2,6,7,1,3,4] => [5,2,6,7,1,3,4] => ? = 1
[3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => [5,6,3,4,1,2,7] => ? = 1
[3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => [5,3,2,6,1,4,7] => ? = 0
[2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => [6,4,7,2,5,1,3] => ? = 0
[4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8] => [5,2,6,7,1,3,4,8] => ? = 0
[4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> [5,6,3,4,1,2,7,8] => [5,6,3,4,1,2,7,8] => ? = 0
[4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7,8] => [5,3,2,6,1,4,7,8] => ? = 1
[3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> [6,7,3,4,8,1,2,5] => [6,7,3,4,8,1,2,5] => ? = 2
[3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> [6,3,2,7,8,1,4,5] => [6,3,2,7,8,1,4,5] => ? = 1
[3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3,8] => [6,4,7,2,5,1,3,8] => ? = 1
[5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4,9] => [5,6,7,8,1,2,3,4,9] => ? = 0
[4,3,2]
=> [[1,2,5,9],[3,4,8],[6,7]]
=> [6,7,3,4,8,1,2,5,9] => [6,7,3,4,8,1,2,5,9] => ? = 0
[4,3,1,1]
=> [[1,4,5,9],[2,7,8],[3],[6]]
=> [6,3,2,7,8,1,4,5,9] => [6,3,2,7,8,1,4,5,9] => ? = 0
[4,2,2,1]
=> [[1,3,8,9],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3,8,9] => [6,4,7,2,5,1,3,8,9] => ? = 0
[3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8],[7]]
=> [7,4,8,2,5,9,1,3,6] => [7,4,8,2,5,9,1,3,6] => ? = 2
[4,3,2,1]
=> [[1,3,6,10],[2,5,9],[4,8],[7]]
=> [7,4,8,2,5,9,1,3,6,10] => [7,4,8,2,5,9,1,3,6,10] => ? = 0
[5,4,3]
=> [[1,2,3,7,12],[4,5,6,11],[8,9,10]]
=> [8,9,10,4,5,6,11,1,2,3,7,12] => [8,9,10,4,5,6,11,1,2,3,7,12] => ? = 0
[5,4,3,2]
=> [[1,2,5,9,14],[3,4,8,13],[6,7,12],[10,11]]
=> [10,11,6,7,12,3,4,8,13,1,2,5,9,14] => [10,11,6,7,12,3,4,8,13,1,2,5,9,14] => ? = 0
[]
=> []
=> [] => [] => ? = 0
Description
The number of occurrences of a type-B 231 pattern in a signed permutation.
For a signed permutation $\pi\in\mathfrak H_n$, a triple $-n \leq i < j < k\leq n$ is an occurrence of the type-B $231$ pattern, if $1 \leq j < k$, $\pi(i) < \pi(j)$ and $\pi(i)$ is one larger than $\pi(k)$, i.e., $\pi(i) = \pi(k) + 1$ if $\pi(k) \neq -1$ and $\pi(i) = 1$ otherwise.
Matching statistic: St001195
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001195: Dyck paths ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 33%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001195: Dyck paths ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 33%
Values
[1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[2]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,1]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 0 + 1
[2,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[4]
=> [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]
=> ? = 0 + 1
[3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1 = 0 + 1
[2,2]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 1 + 1
[2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,1]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1 = 0 + 1
[5]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 0 + 1
[4,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[3,2]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[3,1,1]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[2,2,1]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1 + 1
[2,1,1,1]
=> [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]
=> ? = 0 + 1
[4,2]
=> [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
[4,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 0 + 1
[3,3]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 1 + 1
[3,2,1]
=> [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]
=> ? = 0 + 1
[3,1,1,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[2,2,2]
=> [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]
=> ? = 0 + 1
[2,2,1,1]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 1 + 1
[4,3]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 0 + 1
[4,2,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[4,1,1,1]
=> [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]
=> ? = 0 + 1
[3,3,1]
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
=> ? = 1 + 1
[3,2,2]
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 1 + 1
[3,2,1,1]
=> [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]
=> ? = 0 + 1
[2,2,2,1]
=> [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]
=> ? = 0 + 1
[4,3,1]
=> [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]
=> ? = 0 + 1
[4,2,2]
=> [4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> ? = 0 + 1
[4,2,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 1 + 1
[3,3,2]
=> [6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,1,0,0]
=> ? = 2 + 1
[3,3,1,1]
=> [6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> ? = 1 + 1
[3,2,2,1]
=> [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]
=> ? = 1 + 1
[5,4]
=> [5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 0 + 1
[4,3,2]
=> [3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 0 + 1
[4,3,1,1]
=> [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]
=> ? = 0 + 1
[4,2,2,1]
=> [4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> ? = 0 + 1
[3,3,2,1]
=> [6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 2 + 1
[4,3,2,1]
=> [3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> ? = 0 + 1
[5,4,3]
=> [5,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> ? = 0 + 1
[5,4,3,2]
=> [5,3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0,0]
=> ? = 0 + 1
[]
=> ?
=> ?
=> ?
=> ? = 0 + 1
Description
The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$.
Matching statistic: St001208
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001208: Permutations ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 33%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001208: Permutations ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 33%
Values
[1]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 1 = 0 + 1
[2]
=> [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 1 = 0 + 1
[1,1]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 1 = 0 + 1
[3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 1 = 0 + 1
[2,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1 = 0 + 1
[1,1,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1 = 0 + 1
[4]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ? = 0 + 1
[3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 1 = 0 + 1
[2,2]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? = 1 + 1
[2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1 = 0 + 1
[1,1,1,1]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 1 = 0 + 1
[5]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? = 0 + 1
[4,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 1 + 1
[3,2]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
[3,1,1]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
[2,2,1]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 1 + 1
[2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 0 + 1
[4,2]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,2] => ? = 1 + 1
[4,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? = 0 + 1
[3,3]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => ? = 1 + 1
[3,2,1]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ? = 0 + 1
[3,1,1,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1 = 0 + 1
[2,2,2]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => ? = 0 + 1
[2,2,1,1]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => ? = 1 + 1
[4,3]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => ? = 0 + 1
[4,2,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,2] => ? = 0 + 1
[4,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [8,1,4,5,6,7,2,3] => ? = 0 + 1
[3,3,1]
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [8,3,4,5,6,1,2,7] => ? = 1 + 1
[3,2,2]
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => ? = 1 + 1
[3,2,1,1]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ? = 0 + 1
[2,2,2,1]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => ? = 0 + 1
[4,3,1]
=> [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [3,1,8,5,6,7,2,4] => ? = 0 + 1
[4,2,2]
=> [4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => ? = 0 + 1
[4,2,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [9,1,4,5,6,7,8,2,3] => ? = 1 + 1
[3,3,2]
=> [6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [6,3,4,5,1,8,2,7] => ? = 2 + 1
[3,3,1,1]
=> [6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [2,8,4,5,6,1,3,7] => ? = 1 + 1
[3,2,2,1]
=> [4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => ? = 1 + 1
[5,4]
=> [5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 0 + 1
[4,3,2]
=> [3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [3,1,9,5,6,7,8,2,4] => ? = 0 + 1
[4,3,1,1]
=> [3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [8,1,7,5,6,2,3,4] => ? = 0 + 1
[4,2,2,1]
=> [4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [3,1,4,8,6,7,2,5] => ? = 0 + 1
[3,3,2,1]
=> [6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [5,3,4,1,6,8,2,7] => ? = 2 + 1
[4,3,2,1]
=> [3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [3,1,10,5,6,7,8,9,2,4] => ? = 0 + 1
[5,4,3]
=> [5,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [3,1,8,5,7,2,4,6] => ? = 0 + 1
[5,4,3,2]
=> [5,3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> [3,1,10,5,9,7,8,2,4,6] => ? = 0 + 1
[]
=> ?
=> ?
=> ? => ? = 0 + 1
Description
The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$.
The following 197 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001845The number of join irreducibles minus the rank of a lattice. St000058The order of a permutation. St001130The number of two successive successions in a permutation. St001857The number of edges in the reduced word graph of a signed permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001490The number of connected components of a skew partition. St001722The number of minimal chains with small intervals between a binary word and the top element. St000879The number of long braid edges in the graph of braid moves of a permutation. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000295The length of the border of a binary word. St000943The number of spots the most unlucky car had to go further in a parking function. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001866The nesting alignments of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St000068The number of minimal elements in a poset. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001462The number of factors of a standard tableaux under concatenation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001884The number of borders of a binary word. St001889The size of the connectivity set of a signed permutation. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000039The number of crossings of a permutation. St000217The number of occurrences of the pattern 312 in a permutation. St000219The number of occurrences of the pattern 231 in a permutation. St000221The number of strong fixed points of a permutation. St000234The number of global ascents of a permutation. St000247The number of singleton blocks of a set partition. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000317The cycle descent number of a permutation. St000338The number of pixed points of a permutation. St000355The number of occurrences of the pattern 21-3. St000358The number of occurrences of the pattern 31-2. St000360The number of occurrences of the pattern 32-1. St000365The number of double ascents of a permutation. St000367The number of simsun double descents of a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000406The number of occurrences of the pattern 3241 in a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000462The major index minus the number of excedences of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000488The number of cycles of a permutation of length at most 2. St000496The rcs statistic of a set partition. St000500Eigenvalues of the random-to-random operator acting on the regular representation. St000516The number of stretching pairs of a permutation. St000557The number of occurrences of the pattern {{1},{2},{3}} in a set partition. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000563The number of overlapping pairs of blocks of a set partition. St000573The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton and 2 a maximal element. St000575The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element and 2 a singleton. St000576The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal and 2 a minimal element. St000578The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton. St000580The number of occurrences of the pattern {{1},{2},{3}} such that 2 is minimal, 3 is maximal. St000583The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1, 2 are maximal. St000584The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal, 3 is maximal. St000587The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. St000590The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is maximal, (2,3) are consecutive in a block. St000591The number of occurrences of the pattern {{1},{2},{3}} such that 2 is maximal. St000592The number of occurrences of the pattern {{1},{2},{3}} such that 1 is maximal. St000593The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal. St000596The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is maximal. St000603The number of occurrences of the pattern {{1},{2},{3}} such that 2,3 are minimal. St000604The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 is maximal. St000608The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal, 3 is maximal. St000615The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St000649The number of 3-excedences of a permutation. St000664The number of right ropes of a permutation. St000666The number of right tethers of a permutation. St000732The number of double deficiencies of a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000779The tier of a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000850The number of 1/2-balanced pairs in a poset. St000872The number of very big descents of a permutation. St000961The shifted major index of a permutation. St000962The 3-shifted major index of a permutation. St000963The 2-shifted major index of a permutation. St000989The number of final rises of a permutation. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001082The number of boxed occurrences of 123 in a permutation. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001301The first Betti number of the order complex associated with the poset. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001381The fertility of a permutation. St001396Number of triples of incomparable elements in a finite poset. St001402The number of separators in a permutation. St001403The number of vertical separators in a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001498The normalised height of a Nakayama algebra with magnitude 1. St001513The number of nested exceedences of a permutation. St001537The number of cyclic crossings of a permutation. St001549The number of restricted non-inversions between exceedances. St001550The number of inversions between exceedances where the greater exceedance is linked. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001705The number of occurrences of the pattern 2413 in a permutation. St001728The number of invisible descents of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001781The interlacing number of a set partition. St001810The number of fixed points of a permutation smaller than its largest moved point. St001847The number of occurrences of the pattern 1432 in a permutation. St001850The number of Hecke atoms of a permutation. St000021The number of descents of a permutation. St000023The number of inner peaks of a permutation. St000056The decomposition (or block) number of a permutation. St000154The sum of the descent bottoms of a permutation. St000210Minimum over maximum difference of elements in cycles. St000253The crossing number of a set partition. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000353The number of inner valleys 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. St000570The Edelman-Greene number of a permutation. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000601The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, (2,3) are consecutive in a block. St000633The size of the automorphism group of a poset. St000654The first descent of a permutation. St000694The number of affine bounded permutations that project to a given permutation. St000729The minimal arc length of a set partition. St000782The indicator function of whether a given perfect matching is an L & P matching. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St000864The number of circled entries of the shifted recording tableau of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000886The number of permutations with the same antidiagonal sums. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St000925The number of topologically connected components of a set partition. St000990The first ascent of a permutation. St001162The minimum jump of a permutation. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001344The neighbouring number of a permutation. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001399The distinguishing number of a poset. St001461The number of topologically connected components of the chord diagram of a permutation. St001472The permanent of the Coxeter matrix of the poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001665The number of pure excedances of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St001806The upper middle entry of a permutation. St001839The number of excedances of a set partition. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001928The number of non-overlapping descents in a permutation. St000084The number of subtrees. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000105The number of blocks in the set partition. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000251The number of nonsingleton blocks of a set partition. St000325The width of the tree associated to a permutation. St000328The maximum number of child nodes in a tree. St000470The number of runs in a permutation. St000485The length of the longest cycle of a permutation. St000487The length of the shortest cycle of a permutation. St000504The cardinality of the first block of a set partition. St000542The number of left-to-right-minima of a permutation. St000574The number of occurrences of the pattern {{1},{2}} such that 1 is a minimal and 2 a maximal element. St000619The number of cyclic descents of a permutation. St000679The pruning number of an ordered tree. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St000836The number of descents of distance 2 of a permutation. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001062The maximal size of a block of a set partition. St001075The minimal size of a block of a set partition. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001624The breadth of a lattice. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St001735The number of permutations with the same set of runs. St001741The largest integer such that all patterns of this size are contained in the permutation. St000495The number of inversions of distance at most 2 of a permutation. St000638The number of up-down runs of a permutation. St000831The number of indices that are either descents or recoils.
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