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Your data matches 63 different statistics following compositions of up to 3 maps.
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Matching statistic: St000977
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
St000977: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
St000977: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 4
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 6
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 4
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 12
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 8
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 8
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 6
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 18
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 16
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 14
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 12
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 16
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 10
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 8
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 24
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 20
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 12
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
Description
MacMahon's equal index of a Dyck path.
This is the sum of the positions of double (up- or down-)steps of a Dyck path, see [1, p. 135].
Matching statistic: St000008
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
St000008: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
St000008: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 10 => [1,1] => [2] => 0
[1,0,1,0]
=> 1010 => [1,1,1,1] => [4] => 0
[1,1,0,0]
=> 1100 => [2,2] => [1,2,1] => 4
[1,0,1,0,1,0]
=> 101010 => [1,1,1,1,1,1] => [6] => 0
[1,0,1,1,0,0]
=> 101100 => [1,1,2,2] => [3,2,1] => 8
[1,1,0,0,1,0]
=> 110010 => [2,2,1,1] => [1,2,3] => 4
[1,1,0,1,0,0]
=> 110100 => [2,1,1,2] => [1,4,1] => 6
[1,1,1,0,0,0]
=> 111000 => [3,3] => [1,1,2,1,1] => 12
[1,0,1,0,1,0,1,0]
=> 10101010 => [1,1,1,1,1,1,1,1] => [8] => 0
[1,0,1,0,1,1,0,0]
=> 10101100 => [1,1,1,1,2,2] => [5,2,1] => 12
[1,0,1,1,0,0,1,0]
=> 10110010 => [1,1,2,2,1,1] => [3,2,3] => 8
[1,0,1,1,0,1,0,0]
=> 10110100 => [1,1,2,1,1,2] => [3,4,1] => 10
[1,0,1,1,1,0,0,0]
=> 10111000 => [1,1,3,3] => [3,1,2,1,1] => 20
[1,1,0,0,1,0,1,0]
=> 11001010 => [2,2,1,1,1,1] => [1,2,5] => 4
[1,1,0,0,1,1,0,0]
=> 11001100 => [2,2,2,2] => [1,2,2,2,1] => 16
[1,1,0,1,0,0,1,0]
=> 11010010 => [2,1,1,2,1,1] => [1,4,3] => 6
[1,1,0,1,0,1,0,0]
=> 11010100 => [2,1,1,1,1,2] => [1,6,1] => 8
[1,1,0,1,1,0,0,0]
=> 11011000 => [2,1,2,3] => [1,3,2,1,1] => 18
[1,1,1,0,0,0,1,0]
=> 11100010 => [3,3,1,1] => [1,1,2,1,3] => 12
[1,1,1,0,0,1,0,0]
=> 11100100 => [3,2,1,2] => [1,1,2,3,1] => 14
[1,1,1,0,1,0,0,0]
=> 11101000 => [3,1,1,3] => [1,1,4,1,1] => 16
[1,1,1,1,0,0,0,0]
=> 11110000 => [4,4] => [1,1,1,2,1,1,1] => 24
Description
The major index of the composition.
The descents of a composition $[c_1,c_2,\dots,c_k]$ are the partial sums $c_1, c_1+c_2,\dots, c_1+\dots+c_{k-1}$, excluding the sum of all parts. The major index of a composition is the sum of its descents.
For details about the major index see [[Permutations/Descents-Major]].
Matching statistic: St000391
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00269: Binary words —flag zeros to zeros⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000391: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00269: Binary words —flag zeros to zeros⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000391: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 10 => 00 => 00 => 0
[1,0,1,0]
=> 1010 => 0000 => 0000 => 0
[1,1,0,0]
=> 1100 => 0101 => 1010 => 4
[1,0,1,0,1,0]
=> 101010 => 000000 => 000000 => 0
[1,0,1,1,0,0]
=> 101100 => 010100 => 001010 => 8
[1,1,0,0,1,0]
=> 110010 => 000101 => 101000 => 4
[1,1,0,1,0,0]
=> 110100 => 010001 => 100010 => 6
[1,1,1,0,0,0]
=> 111000 => 011011 => 110110 => 12
[1,0,1,0,1,0,1,0]
=> 10101010 => 00000000 => 00000000 => 0
[1,0,1,0,1,1,0,0]
=> 10101100 => 01010000 => 00001010 => 12
[1,0,1,1,0,0,1,0]
=> 10110010 => 00010100 => 00101000 => 8
[1,0,1,1,0,1,0,0]
=> 10110100 => 01000100 => 00100010 => 10
[1,0,1,1,1,0,0,0]
=> 10111000 => 01101100 => 00110110 => 20
[1,1,0,0,1,0,1,0]
=> 11001010 => 00000101 => 10100000 => 4
[1,1,0,0,1,1,0,0]
=> 11001100 => 01010101 => 10101010 => 16
[1,1,0,1,0,0,1,0]
=> 11010010 => 00010001 => 10001000 => 6
[1,1,0,1,0,1,0,0]
=> 11010100 => 01000001 => 10000010 => 8
[1,1,0,1,1,0,0,0]
=> 11011000 => 01101001 => 10010110 => 18
[1,1,1,0,0,0,1,0]
=> 11100010 => 00011011 => 11011000 => 12
[1,1,1,0,0,1,0,0]
=> 11100100 => 01001011 => 11010010 => 14
[1,1,1,0,1,0,0,0]
=> 11101000 => 01100011 => 11000110 => 16
[1,1,1,1,0,0,0,0]
=> 11110000 => 01110111 => 11101110 => 24
Description
The sum of the positions of the ones in a binary word.
Matching statistic: St000043
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
Mp00098: Alternating sign matrices —link pattern⟶ Perfect matchings
St000043: Perfect matchings ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 36%
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
Mp00098: Alternating sign matrices —link pattern⟶ Perfect matchings
St000043: Perfect matchings ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 36%
Values
[1,0]
=> [2,1] => [[0,1],[1,0]]
=> [(1,2),(3,4)]
=> 0
[1,0,1,0]
=> [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [(1,2),(3,4),(5,6)]
=> 0
[1,1,0,0]
=> [2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
=> [(1,6),(2,3),(4,5)]
=> 4
[1,0,1,0,1,0]
=> [4,1,2,3] => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [(1,2),(3,8),(4,5),(6,7)]
=> 4
[1,0,1,1,0,0]
=> [3,1,4,2] => [[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8)]
=> ? ∊ {0,12}
[1,1,0,0,1,0]
=> [2,4,1,3] => [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [(1,8),(2,3),(4,5),(6,7)]
=> 6
[1,1,0,1,0,0]
=> [4,3,1,2] => [[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [(1,2),(3,4),(5,8),(6,7)]
=> ? ∊ {0,12}
[1,1,1,0,0,0]
=> [2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [(1,8),(2,3),(4,7),(5,6)]
=> 8
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0]]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> ? ∊ {0,4,6,8,8,10,12,12,14,16,16,18,20,24}
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> ? ∊ {0,4,6,8,8,10,12,12,14,16,16,18,20,24}
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> ? ∊ {0,4,6,8,8,10,12,12,14,16,16,18,20,24}
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0]]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> ? ∊ {0,4,6,8,8,10,12,12,14,16,16,18,20,24}
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [(1,2),(3,4),(5,8),(6,7),(9,10)]
=> ? ∊ {0,4,6,8,8,10,12,12,14,16,16,18,20,24}
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [(1,10),(2,3),(4,9),(5,6),(7,8)]
=> ? ∊ {0,4,6,8,8,10,12,12,14,16,16,18,20,24}
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [(1,10),(2,3),(4,5),(6,7),(8,9)]
=> ? ∊ {0,4,6,8,8,10,12,12,14,16,16,18,20,24}
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0]]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> ? ∊ {0,4,6,8,8,10,12,12,14,16,16,18,20,24}
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0]]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> ? ∊ {0,4,6,8,8,10,12,12,14,16,16,18,20,24}
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> ? ∊ {0,4,6,8,8,10,12,12,14,16,16,18,20,24}
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [(1,10),(2,3),(4,9),(5,6),(7,8)]
=> ? ∊ {0,4,6,8,8,10,12,12,14,16,16,18,20,24}
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0]]
=> [(1,10),(2,3),(4,5),(6,9),(7,8)]
=> ? ∊ {0,4,6,8,8,10,12,12,14,16,16,18,20,24}
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0]]
=> [(1,10),(2,3),(4,5),(6,7),(8,9)]
=> ? ∊ {0,4,6,8,8,10,12,12,14,16,16,18,20,24}
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [(1,10),(2,9),(3,4),(5,8),(6,7)]
=> ? ∊ {0,4,6,8,8,10,12,12,14,16,16,18,20,24}
Description
The number of crossings plus two-nestings of a perfect matching.
This is $C+2N$ where $C$ is the number of crossings ([[St000042]]) and $N$ is the number of nestings ([[St000041]]).
The generating series $\sum_{m} q^{\textrm{cn}(m)}$, where the sum is over the perfect matchings of $2n$ and $\textrm{cn}(m)$ is this statistic is $[2n-1]_q[2n-3]_q\cdots [3]_q[1]_q$ where $[m]_q = 1+q+q^2+\cdots + q^{m-1}$ [1, Equation (5,4)].
Matching statistic: St000422
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 18% ●values known / values provided: 27%●distinct values known / distinct values provided: 18%
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 18% ●values known / values provided: 27%●distinct values known / distinct values provided: 18%
Values
[1,0]
=> [2,1] => [1,2] => ([],2)
=> 0
[1,0,1,0]
=> [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> ? = 4
[1,1,0,0]
=> [2,3,1] => [1,2,3] => ([],3)
=> 0
[1,0,1,0,1,0]
=> [4,1,2,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4
[1,0,1,1,0,0]
=> [3,1,4,2] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {6,8,12}
[1,1,0,0,1,0]
=> [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {6,8,12}
[1,1,0,1,0,0]
=> [4,3,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {6,8,12}
[1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => ([],4)
=> 0
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {6,8,8,10,12,12,14,16,16,18,20,24}
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [3,5,2,1,4] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ? ∊ {6,8,8,10,12,12,14,16,16,18,20,24}
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? ∊ {6,8,8,10,12,12,14,16,16,18,20,24}
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ? ∊ {6,8,8,10,12,12,14,16,16,18,20,24}
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? ∊ {6,8,8,10,12,12,14,16,16,18,20,24}
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {6,8,8,10,12,12,14,16,16,18,20,24}
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? ∊ {6,8,8,10,12,12,14,16,16,18,20,24}
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [4,5,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {6,8,8,10,12,12,14,16,16,18,20,24}
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {6,8,8,10,12,12,14,16,16,18,20,24}
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {6,8,8,10,12,12,14,16,16,18,20,24}
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [5,2,1,4,3] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ? ∊ {6,8,8,10,12,12,14,16,16,18,20,24}
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [5,1,3,4,2] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {6,8,8,10,12,12,14,16,16,18,20,24}
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => ([],5)
=> 0
Description
The energy of a graph, if it is integral.
The energy of a graph is the sum of the absolute values of its eigenvalues. This statistic is only defined for graphs with integral energy. It is known, that the energy is never an odd integer [2]. In fact, it is never the square root of an odd integer [3].
The energy of a graph is the sum of the energies of the connected components of a graph. The energy of the complete graph $K_n$ equals $2n-2$. For this reason, we do not define the energy of the empty graph.
Matching statistic: St001232
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00123: Dyck paths —Barnabei-Castronuovo involution⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 18% ●values known / values provided: 23%●distinct values known / distinct values provided: 18%
Mp00123: Dyck paths —Barnabei-Castronuovo involution⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 18% ●values known / values provided: 23%●distinct values known / distinct values provided: 18%
Values
[1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> 0
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> ? = 4
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ? ∊ {4,6,8,12}
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ? ∊ {4,6,8,12}
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ? ∊ {4,6,8,12}
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ? ∊ {4,6,8,12}
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {4,6,8,10,12,12,14,16,16,18,20,24}
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> ? ∊ {4,6,8,10,12,12,14,16,16,18,20,24}
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> ? ∊ {4,6,8,10,12,12,14,16,16,18,20,24}
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> ? ∊ {4,6,8,10,12,12,14,16,16,18,20,24}
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? ∊ {4,6,8,10,12,12,14,16,16,18,20,24}
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> ? ∊ {4,6,8,10,12,12,14,16,16,18,20,24}
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> ? ∊ {4,6,8,10,12,12,14,16,16,18,20,24}
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> ? ∊ {4,6,8,10,12,12,14,16,16,18,20,24}
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> ? ∊ {4,6,8,10,12,12,14,16,16,18,20,24}
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? ∊ {4,6,8,10,12,12,14,16,16,18,20,24}
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? ∊ {4,6,8,10,12,12,14,16,16,18,20,24}
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? ∊ {4,6,8,10,12,12,14,16,16,18,20,24}
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 8
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000068
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00158: Binary words —alternating inverse⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St000068: Posets ⟶ ℤResult quality: 9% ●values known / values provided: 18%●distinct values known / distinct values provided: 9%
Mp00158: Binary words —alternating inverse⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St000068: Posets ⟶ ℤResult quality: 9% ●values known / values provided: 18%●distinct values known / distinct values provided: 9%
Values
[1,0]
=> 10 => 11 => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,0,1,0]
=> 1010 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,1,0,0]
=> 1100 => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 4 + 1
[1,0,1,0,1,0]
=> 101010 => 111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> 101100 => 111001 => ([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17)
=> ? ∊ {4,6,8,12} + 1
[1,1,0,0,1,0]
=> 110010 => 100111 => ([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17)
=> ? ∊ {4,6,8,12} + 1
[1,1,0,1,0,0]
=> 110100 => 100001 => ([(0,4),(0,5),(1,3),(1,7),(1,8),(2,13),(2,14),(3,2),(3,11),(3,12),(4,9),(4,10),(5,1),(5,9),(5,10),(7,12),(8,11),(9,8),(10,7),(11,13),(12,14),(13,6),(14,6)],15)
=> ? ∊ {4,6,8,12} + 1
[1,1,1,0,0,0]
=> 111000 => 101101 => ([(0,2),(0,3),(1,10),(1,11),(2,13),(2,14),(3,1),(3,13),(3,14),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9),(12,5),(12,6),(13,10),(13,12),(14,11),(14,12)],15)
=> ? ∊ {4,6,8,12} + 1
[1,0,1,0,1,0,1,0]
=> 10101010 => 11111111 => ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> 10101100 => 11111001 => ([(0,6),(0,7),(1,4),(1,17),(2,16),(3,2),(3,23),(4,3),(4,24),(5,9),(5,18),(6,1),(6,15),(6,22),(7,5),(7,15),(7,22),(9,11),(10,12),(11,10),(12,13),(13,8),(14,8),(15,9),(16,14),(17,21),(17,24),(18,11),(18,21),(19,12),(19,20),(20,13),(20,14),(21,10),(21,19),(22,17),(22,18),(23,16),(23,20),(24,19),(24,23)],25)
=> ? ∊ {4,6,8,8,10,12,12,14,16,16,18,20,24} + 1
[1,0,1,1,0,0,1,0]
=> 10110010 => 11100111 => ([(0,4),(0,5),(1,23),(1,24),(2,3),(2,25),(2,26),(3,7),(3,8),(4,1),(4,19),(4,20),(5,2),(5,19),(5,20),(7,12),(8,11),(9,13),(10,14),(11,9),(12,10),(13,6),(14,6),(15,13),(15,14),(16,21),(16,22),(17,11),(17,21),(18,12),(18,22),(19,23),(19,25),(20,24),(20,26),(21,9),(21,15),(22,10),(22,15),(23,16),(23,17),(24,16),(24,18),(25,8),(25,17),(26,7),(26,18)],27)
=> ? ∊ {4,6,8,8,10,12,12,14,16,16,18,20,24} + 1
[1,0,1,1,0,1,0,0]
=> 10110100 => 11100001 => ([(0,6),(0,7),(1,2),(1,24),(2,9),(3,5),(3,11),(3,23),(4,18),(4,25),(5,4),(5,17),(5,26),(6,3),(6,12),(6,19),(7,1),(7,12),(7,19),(9,13),(10,16),(11,17),(12,11),(13,14),(14,15),(15,8),(16,8),(17,18),(18,10),(19,23),(19,24),(20,15),(20,16),(21,14),(21,20),(22,13),(22,21),(23,22),(23,26),(24,9),(24,22),(25,10),(25,20),(26,21),(26,25)],27)
=> ? ∊ {4,6,8,8,10,12,12,14,16,16,18,20,24} + 1
[1,0,1,1,1,0,0,0]
=> 10111000 => 11101101 => ([(0,3),(0,4),(1,12),(2,1),(2,18),(2,22),(3,20),(3,21),(4,2),(4,20),(4,21),(6,8),(7,9),(8,10),(9,11),(10,5),(11,5),(12,7),(13,14),(13,15),(14,9),(14,16),(15,8),(15,16),(16,10),(16,11),(17,13),(17,23),(18,13),(18,19),(19,6),(19,15),(20,17),(20,18),(21,17),(21,22),(22,12),(22,19),(22,23),(23,6),(23,7),(23,14)],24)
=> ? ∊ {4,6,8,8,10,12,12,14,16,16,18,20,24} + 1
[1,1,0,0,1,0,1,0]
=> 11001010 => 10011111 => ([(0,6),(0,7),(1,4),(1,17),(2,16),(3,2),(3,23),(4,3),(4,24),(5,9),(5,18),(6,1),(6,15),(6,22),(7,5),(7,15),(7,22),(9,11),(10,12),(11,10),(12,13),(13,8),(14,8),(15,9),(16,14),(17,21),(17,24),(18,11),(18,21),(19,12),(19,20),(20,13),(20,14),(21,10),(21,19),(22,17),(22,18),(23,16),(23,20),(24,19),(24,23)],25)
=> ? ∊ {4,6,8,8,10,12,12,14,16,16,18,20,24} + 1
[1,1,0,0,1,1,0,0]
=> 11001100 => 10011001 => ([(0,3),(0,4),(1,21),(1,22),(2,23),(2,24),(3,1),(3,16),(3,17),(4,2),(4,16),(4,17),(6,9),(7,8),(8,10),(9,11),(10,5),(11,5),(12,8),(12,14),(13,9),(13,14),(14,10),(14,11),(15,12),(15,13),(16,21),(16,23),(17,22),(17,24),(18,7),(18,12),(19,6),(19,13),(20,6),(20,7),(21,15),(21,18),(22,15),(22,19),(23,18),(23,20),(24,19),(24,20)],25)
=> ? ∊ {4,6,8,8,10,12,12,14,16,16,18,20,24} + 1
[1,1,0,1,0,0,1,0]
=> 11010010 => 10000111 => ([(0,6),(0,7),(1,2),(1,24),(2,9),(3,5),(3,11),(3,23),(4,18),(4,25),(5,4),(5,17),(5,26),(6,3),(6,12),(6,19),(7,1),(7,12),(7,19),(9,13),(10,16),(11,17),(12,11),(13,14),(14,15),(15,8),(16,8),(17,18),(18,10),(19,23),(19,24),(20,15),(20,16),(21,14),(21,20),(22,13),(22,21),(23,22),(23,26),(24,9),(24,22),(25,10),(25,20),(26,21),(26,25)],27)
=> ? ∊ {4,6,8,8,10,12,12,14,16,16,18,20,24} + 1
[1,1,0,1,0,1,0,0]
=> 11010100 => 10000001 => ([(0,6),(0,7),(1,5),(1,9),(1,10),(2,19),(2,20),(3,4),(3,13),(3,14),(4,2),(4,15),(4,16),(5,3),(5,17),(5,18),(6,11),(6,12),(7,1),(7,11),(7,12),(9,18),(10,17),(11,10),(12,9),(13,15),(14,16),(15,19),(16,20),(17,13),(18,14),(19,8),(20,8)],21)
=> ? ∊ {4,6,8,8,10,12,12,14,16,16,18,20,24} + 1
[1,1,0,1,1,0,0,0]
=> 11011000 => 10001101 => ([(0,4),(0,5),(1,23),(1,24),(2,3),(2,16),(2,25),(3,8),(3,19),(4,1),(4,26),(4,27),(5,2),(5,26),(5,27),(7,11),(8,10),(9,7),(10,12),(11,13),(12,14),(13,15),(14,6),(15,6),(16,8),(17,11),(17,20),(18,20),(18,21),(19,10),(19,21),(20,13),(20,22),(21,12),(21,22),(22,14),(22,15),(23,7),(23,17),(24,17),(24,18),(25,18),(25,19),(26,9),(26,24),(26,25),(27,9),(27,16),(27,23)],28)
=> ? ∊ {4,6,8,8,10,12,12,14,16,16,18,20,24} + 1
[1,1,1,0,0,0,1,0]
=> 11100010 => 10110111 => ([(0,3),(0,4),(1,12),(2,1),(2,18),(2,22),(3,20),(3,21),(4,2),(4,20),(4,21),(6,8),(7,9),(8,10),(9,11),(10,5),(11,5),(12,7),(13,14),(13,15),(14,9),(14,16),(15,8),(15,16),(16,10),(16,11),(17,13),(17,23),(18,13),(18,19),(19,6),(19,15),(20,17),(20,18),(21,17),(21,22),(22,12),(22,19),(22,23),(23,6),(23,7),(23,14)],24)
=> ? ∊ {4,6,8,8,10,12,12,14,16,16,18,20,24} + 1
[1,1,1,0,0,1,0,0]
=> 11100100 => 10110001 => ([(0,4),(0,5),(1,23),(1,24),(2,3),(2,16),(2,25),(3,8),(3,19),(4,1),(4,26),(4,27),(5,2),(5,26),(5,27),(7,11),(8,10),(9,7),(10,12),(11,13),(12,14),(13,15),(14,6),(15,6),(16,8),(17,11),(17,20),(18,20),(18,21),(19,10),(19,21),(20,13),(20,22),(21,12),(21,22),(22,14),(22,15),(23,7),(23,17),(24,17),(24,18),(25,18),(25,19),(26,9),(26,24),(26,25),(27,9),(27,16),(27,23)],28)
=> ? ∊ {4,6,8,8,10,12,12,14,16,16,18,20,24} + 1
[1,1,1,0,1,0,0,0]
=> 11101000 => 10111101 => ([(0,4),(0,5),(1,3),(1,17),(1,18),(2,23),(2,24),(3,2),(3,21),(3,22),(4,19),(4,20),(5,1),(5,19),(5,20),(7,12),(8,11),(9,13),(10,14),(11,9),(12,10),(13,6),(14,6),(15,7),(15,8),(16,13),(16,14),(17,8),(17,21),(18,7),(18,22),(19,15),(19,17),(20,15),(20,18),(21,11),(21,23),(22,12),(22,24),(23,9),(23,16),(24,10),(24,16)],25)
=> ? ∊ {4,6,8,8,10,12,12,14,16,16,18,20,24} + 1
[1,1,1,1,0,0,0,0]
=> 11110000 => 10100101 => ([(0,2),(0,3),(1,17),(1,18),(2,22),(2,23),(3,1),(3,22),(3,23),(5,10),(6,9),(7,11),(8,12),(9,7),(10,8),(11,4),(12,4),(13,20),(13,21),(14,11),(14,12),(15,9),(15,20),(16,10),(16,21),(17,13),(17,15),(18,13),(18,16),(19,5),(19,6),(20,7),(20,14),(21,8),(21,14),(22,17),(22,19),(22,24),(23,18),(23,19),(23,24),(24,5),(24,6),(24,15),(24,16)],25)
=> ? ∊ {4,6,8,8,10,12,12,14,16,16,18,20,24} + 1
Description
The number of minimal elements in a poset.
Matching statistic: St000071
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00158: Binary words —alternating inverse⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St000071: Posets ⟶ ℤResult quality: 9% ●values known / values provided: 18%●distinct values known / distinct values provided: 9%
Mp00158: Binary words —alternating inverse⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St000071: Posets ⟶ ℤResult quality: 9% ●values known / values provided: 18%●distinct values known / distinct values provided: 9%
Values
[1,0]
=> 10 => 11 => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,0,1,0]
=> 1010 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,1,0,0]
=> 1100 => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 4 + 1
[1,0,1,0,1,0]
=> 101010 => 111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> 101100 => 111001 => ([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17)
=> ? ∊ {4,6,8,12} + 1
[1,1,0,0,1,0]
=> 110010 => 100111 => ([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17)
=> ? ∊ {4,6,8,12} + 1
[1,1,0,1,0,0]
=> 110100 => 100001 => ([(0,4),(0,5),(1,3),(1,7),(1,8),(2,13),(2,14),(3,2),(3,11),(3,12),(4,9),(4,10),(5,1),(5,9),(5,10),(7,12),(8,11),(9,8),(10,7),(11,13),(12,14),(13,6),(14,6)],15)
=> ? ∊ {4,6,8,12} + 1
[1,1,1,0,0,0]
=> 111000 => 101101 => ([(0,2),(0,3),(1,10),(1,11),(2,13),(2,14),(3,1),(3,13),(3,14),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9),(12,5),(12,6),(13,10),(13,12),(14,11),(14,12)],15)
=> ? ∊ {4,6,8,12} + 1
[1,0,1,0,1,0,1,0]
=> 10101010 => 11111111 => ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> 10101100 => 11111001 => ([(0,6),(0,7),(1,4),(1,17),(2,16),(3,2),(3,23),(4,3),(4,24),(5,9),(5,18),(6,1),(6,15),(6,22),(7,5),(7,15),(7,22),(9,11),(10,12),(11,10),(12,13),(13,8),(14,8),(15,9),(16,14),(17,21),(17,24),(18,11),(18,21),(19,12),(19,20),(20,13),(20,14),(21,10),(21,19),(22,17),(22,18),(23,16),(23,20),(24,19),(24,23)],25)
=> ? ∊ {4,6,8,8,10,12,12,14,16,16,18,20,24} + 1
[1,0,1,1,0,0,1,0]
=> 10110010 => 11100111 => ([(0,4),(0,5),(1,23),(1,24),(2,3),(2,25),(2,26),(3,7),(3,8),(4,1),(4,19),(4,20),(5,2),(5,19),(5,20),(7,12),(8,11),(9,13),(10,14),(11,9),(12,10),(13,6),(14,6),(15,13),(15,14),(16,21),(16,22),(17,11),(17,21),(18,12),(18,22),(19,23),(19,25),(20,24),(20,26),(21,9),(21,15),(22,10),(22,15),(23,16),(23,17),(24,16),(24,18),(25,8),(25,17),(26,7),(26,18)],27)
=> ? ∊ {4,6,8,8,10,12,12,14,16,16,18,20,24} + 1
[1,0,1,1,0,1,0,0]
=> 10110100 => 11100001 => ([(0,6),(0,7),(1,2),(1,24),(2,9),(3,5),(3,11),(3,23),(4,18),(4,25),(5,4),(5,17),(5,26),(6,3),(6,12),(6,19),(7,1),(7,12),(7,19),(9,13),(10,16),(11,17),(12,11),(13,14),(14,15),(15,8),(16,8),(17,18),(18,10),(19,23),(19,24),(20,15),(20,16),(21,14),(21,20),(22,13),(22,21),(23,22),(23,26),(24,9),(24,22),(25,10),(25,20),(26,21),(26,25)],27)
=> ? ∊ {4,6,8,8,10,12,12,14,16,16,18,20,24} + 1
[1,0,1,1,1,0,0,0]
=> 10111000 => 11101101 => ([(0,3),(0,4),(1,12),(2,1),(2,18),(2,22),(3,20),(3,21),(4,2),(4,20),(4,21),(6,8),(7,9),(8,10),(9,11),(10,5),(11,5),(12,7),(13,14),(13,15),(14,9),(14,16),(15,8),(15,16),(16,10),(16,11),(17,13),(17,23),(18,13),(18,19),(19,6),(19,15),(20,17),(20,18),(21,17),(21,22),(22,12),(22,19),(22,23),(23,6),(23,7),(23,14)],24)
=> ? ∊ {4,6,8,8,10,12,12,14,16,16,18,20,24} + 1
[1,1,0,0,1,0,1,0]
=> 11001010 => 10011111 => ([(0,6),(0,7),(1,4),(1,17),(2,16),(3,2),(3,23),(4,3),(4,24),(5,9),(5,18),(6,1),(6,15),(6,22),(7,5),(7,15),(7,22),(9,11),(10,12),(11,10),(12,13),(13,8),(14,8),(15,9),(16,14),(17,21),(17,24),(18,11),(18,21),(19,12),(19,20),(20,13),(20,14),(21,10),(21,19),(22,17),(22,18),(23,16),(23,20),(24,19),(24,23)],25)
=> ? ∊ {4,6,8,8,10,12,12,14,16,16,18,20,24} + 1
[1,1,0,0,1,1,0,0]
=> 11001100 => 10011001 => ([(0,3),(0,4),(1,21),(1,22),(2,23),(2,24),(3,1),(3,16),(3,17),(4,2),(4,16),(4,17),(6,9),(7,8),(8,10),(9,11),(10,5),(11,5),(12,8),(12,14),(13,9),(13,14),(14,10),(14,11),(15,12),(15,13),(16,21),(16,23),(17,22),(17,24),(18,7),(18,12),(19,6),(19,13),(20,6),(20,7),(21,15),(21,18),(22,15),(22,19),(23,18),(23,20),(24,19),(24,20)],25)
=> ? ∊ {4,6,8,8,10,12,12,14,16,16,18,20,24} + 1
[1,1,0,1,0,0,1,0]
=> 11010010 => 10000111 => ([(0,6),(0,7),(1,2),(1,24),(2,9),(3,5),(3,11),(3,23),(4,18),(4,25),(5,4),(5,17),(5,26),(6,3),(6,12),(6,19),(7,1),(7,12),(7,19),(9,13),(10,16),(11,17),(12,11),(13,14),(14,15),(15,8),(16,8),(17,18),(18,10),(19,23),(19,24),(20,15),(20,16),(21,14),(21,20),(22,13),(22,21),(23,22),(23,26),(24,9),(24,22),(25,10),(25,20),(26,21),(26,25)],27)
=> ? ∊ {4,6,8,8,10,12,12,14,16,16,18,20,24} + 1
[1,1,0,1,0,1,0,0]
=> 11010100 => 10000001 => ([(0,6),(0,7),(1,5),(1,9),(1,10),(2,19),(2,20),(3,4),(3,13),(3,14),(4,2),(4,15),(4,16),(5,3),(5,17),(5,18),(6,11),(6,12),(7,1),(7,11),(7,12),(9,18),(10,17),(11,10),(12,9),(13,15),(14,16),(15,19),(16,20),(17,13),(18,14),(19,8),(20,8)],21)
=> ? ∊ {4,6,8,8,10,12,12,14,16,16,18,20,24} + 1
[1,1,0,1,1,0,0,0]
=> 11011000 => 10001101 => ([(0,4),(0,5),(1,23),(1,24),(2,3),(2,16),(2,25),(3,8),(3,19),(4,1),(4,26),(4,27),(5,2),(5,26),(5,27),(7,11),(8,10),(9,7),(10,12),(11,13),(12,14),(13,15),(14,6),(15,6),(16,8),(17,11),(17,20),(18,20),(18,21),(19,10),(19,21),(20,13),(20,22),(21,12),(21,22),(22,14),(22,15),(23,7),(23,17),(24,17),(24,18),(25,18),(25,19),(26,9),(26,24),(26,25),(27,9),(27,16),(27,23)],28)
=> ? ∊ {4,6,8,8,10,12,12,14,16,16,18,20,24} + 1
[1,1,1,0,0,0,1,0]
=> 11100010 => 10110111 => ([(0,3),(0,4),(1,12),(2,1),(2,18),(2,22),(3,20),(3,21),(4,2),(4,20),(4,21),(6,8),(7,9),(8,10),(9,11),(10,5),(11,5),(12,7),(13,14),(13,15),(14,9),(14,16),(15,8),(15,16),(16,10),(16,11),(17,13),(17,23),(18,13),(18,19),(19,6),(19,15),(20,17),(20,18),(21,17),(21,22),(22,12),(22,19),(22,23),(23,6),(23,7),(23,14)],24)
=> ? ∊ {4,6,8,8,10,12,12,14,16,16,18,20,24} + 1
[1,1,1,0,0,1,0,0]
=> 11100100 => 10110001 => ([(0,4),(0,5),(1,23),(1,24),(2,3),(2,16),(2,25),(3,8),(3,19),(4,1),(4,26),(4,27),(5,2),(5,26),(5,27),(7,11),(8,10),(9,7),(10,12),(11,13),(12,14),(13,15),(14,6),(15,6),(16,8),(17,11),(17,20),(18,20),(18,21),(19,10),(19,21),(20,13),(20,22),(21,12),(21,22),(22,14),(22,15),(23,7),(23,17),(24,17),(24,18),(25,18),(25,19),(26,9),(26,24),(26,25),(27,9),(27,16),(27,23)],28)
=> ? ∊ {4,6,8,8,10,12,12,14,16,16,18,20,24} + 1
[1,1,1,0,1,0,0,0]
=> 11101000 => 10111101 => ([(0,4),(0,5),(1,3),(1,17),(1,18),(2,23),(2,24),(3,2),(3,21),(3,22),(4,19),(4,20),(5,1),(5,19),(5,20),(7,12),(8,11),(9,13),(10,14),(11,9),(12,10),(13,6),(14,6),(15,7),(15,8),(16,13),(16,14),(17,8),(17,21),(18,7),(18,22),(19,15),(19,17),(20,15),(20,18),(21,11),(21,23),(22,12),(22,24),(23,9),(23,16),(24,10),(24,16)],25)
=> ? ∊ {4,6,8,8,10,12,12,14,16,16,18,20,24} + 1
[1,1,1,1,0,0,0,0]
=> 11110000 => 10100101 => ([(0,2),(0,3),(1,17),(1,18),(2,22),(2,23),(3,1),(3,22),(3,23),(5,10),(6,9),(7,11),(8,12),(9,7),(10,8),(11,4),(12,4),(13,20),(13,21),(14,11),(14,12),(15,9),(15,20),(16,10),(16,21),(17,13),(17,15),(18,13),(18,16),(19,5),(19,6),(20,7),(20,14),(21,8),(21,14),(22,17),(22,19),(22,24),(23,18),(23,19),(23,24),(24,5),(24,6),(24,15),(24,16)],25)
=> ? ∊ {4,6,8,8,10,12,12,14,16,16,18,20,24} + 1
Description
The number of maximal chains in a poset.
Matching statistic: St000100
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00158: Binary words —alternating inverse⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St000100: Posets ⟶ ℤResult quality: 9% ●values known / values provided: 18%●distinct values known / distinct values provided: 9%
Mp00158: Binary words —alternating inverse⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St000100: Posets ⟶ ℤResult quality: 9% ●values known / values provided: 18%●distinct values known / distinct values provided: 9%
Values
[1,0]
=> 10 => 11 => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,0,1,0]
=> 1010 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,1,0,0]
=> 1100 => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 4 + 1
[1,0,1,0,1,0]
=> 101010 => 111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> 101100 => 111001 => ([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17)
=> ? ∊ {4,6,8,12} + 1
[1,1,0,0,1,0]
=> 110010 => 100111 => ([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17)
=> ? ∊ {4,6,8,12} + 1
[1,1,0,1,0,0]
=> 110100 => 100001 => ([(0,4),(0,5),(1,3),(1,7),(1,8),(2,13),(2,14),(3,2),(3,11),(3,12),(4,9),(4,10),(5,1),(5,9),(5,10),(7,12),(8,11),(9,8),(10,7),(11,13),(12,14),(13,6),(14,6)],15)
=> ? ∊ {4,6,8,12} + 1
[1,1,1,0,0,0]
=> 111000 => 101101 => ([(0,2),(0,3),(1,10),(1,11),(2,13),(2,14),(3,1),(3,13),(3,14),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9),(12,5),(12,6),(13,10),(13,12),(14,11),(14,12)],15)
=> ? ∊ {4,6,8,12} + 1
[1,0,1,0,1,0,1,0]
=> 10101010 => 11111111 => ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> 10101100 => 11111001 => ([(0,6),(0,7),(1,4),(1,17),(2,16),(3,2),(3,23),(4,3),(4,24),(5,9),(5,18),(6,1),(6,15),(6,22),(7,5),(7,15),(7,22),(9,11),(10,12),(11,10),(12,13),(13,8),(14,8),(15,9),(16,14),(17,21),(17,24),(18,11),(18,21),(19,12),(19,20),(20,13),(20,14),(21,10),(21,19),(22,17),(22,18),(23,16),(23,20),(24,19),(24,23)],25)
=> ? ∊ {4,6,8,8,10,12,12,14,16,16,18,20,24} + 1
[1,0,1,1,0,0,1,0]
=> 10110010 => 11100111 => ([(0,4),(0,5),(1,23),(1,24),(2,3),(2,25),(2,26),(3,7),(3,8),(4,1),(4,19),(4,20),(5,2),(5,19),(5,20),(7,12),(8,11),(9,13),(10,14),(11,9),(12,10),(13,6),(14,6),(15,13),(15,14),(16,21),(16,22),(17,11),(17,21),(18,12),(18,22),(19,23),(19,25),(20,24),(20,26),(21,9),(21,15),(22,10),(22,15),(23,16),(23,17),(24,16),(24,18),(25,8),(25,17),(26,7),(26,18)],27)
=> ? ∊ {4,6,8,8,10,12,12,14,16,16,18,20,24} + 1
[1,0,1,1,0,1,0,0]
=> 10110100 => 11100001 => ([(0,6),(0,7),(1,2),(1,24),(2,9),(3,5),(3,11),(3,23),(4,18),(4,25),(5,4),(5,17),(5,26),(6,3),(6,12),(6,19),(7,1),(7,12),(7,19),(9,13),(10,16),(11,17),(12,11),(13,14),(14,15),(15,8),(16,8),(17,18),(18,10),(19,23),(19,24),(20,15),(20,16),(21,14),(21,20),(22,13),(22,21),(23,22),(23,26),(24,9),(24,22),(25,10),(25,20),(26,21),(26,25)],27)
=> ? ∊ {4,6,8,8,10,12,12,14,16,16,18,20,24} + 1
[1,0,1,1,1,0,0,0]
=> 10111000 => 11101101 => ([(0,3),(0,4),(1,12),(2,1),(2,18),(2,22),(3,20),(3,21),(4,2),(4,20),(4,21),(6,8),(7,9),(8,10),(9,11),(10,5),(11,5),(12,7),(13,14),(13,15),(14,9),(14,16),(15,8),(15,16),(16,10),(16,11),(17,13),(17,23),(18,13),(18,19),(19,6),(19,15),(20,17),(20,18),(21,17),(21,22),(22,12),(22,19),(22,23),(23,6),(23,7),(23,14)],24)
=> ? ∊ {4,6,8,8,10,12,12,14,16,16,18,20,24} + 1
[1,1,0,0,1,0,1,0]
=> 11001010 => 10011111 => ([(0,6),(0,7),(1,4),(1,17),(2,16),(3,2),(3,23),(4,3),(4,24),(5,9),(5,18),(6,1),(6,15),(6,22),(7,5),(7,15),(7,22),(9,11),(10,12),(11,10),(12,13),(13,8),(14,8),(15,9),(16,14),(17,21),(17,24),(18,11),(18,21),(19,12),(19,20),(20,13),(20,14),(21,10),(21,19),(22,17),(22,18),(23,16),(23,20),(24,19),(24,23)],25)
=> ? ∊ {4,6,8,8,10,12,12,14,16,16,18,20,24} + 1
[1,1,0,0,1,1,0,0]
=> 11001100 => 10011001 => ([(0,3),(0,4),(1,21),(1,22),(2,23),(2,24),(3,1),(3,16),(3,17),(4,2),(4,16),(4,17),(6,9),(7,8),(8,10),(9,11),(10,5),(11,5),(12,8),(12,14),(13,9),(13,14),(14,10),(14,11),(15,12),(15,13),(16,21),(16,23),(17,22),(17,24),(18,7),(18,12),(19,6),(19,13),(20,6),(20,7),(21,15),(21,18),(22,15),(22,19),(23,18),(23,20),(24,19),(24,20)],25)
=> ? ∊ {4,6,8,8,10,12,12,14,16,16,18,20,24} + 1
[1,1,0,1,0,0,1,0]
=> 11010010 => 10000111 => ([(0,6),(0,7),(1,2),(1,24),(2,9),(3,5),(3,11),(3,23),(4,18),(4,25),(5,4),(5,17),(5,26),(6,3),(6,12),(6,19),(7,1),(7,12),(7,19),(9,13),(10,16),(11,17),(12,11),(13,14),(14,15),(15,8),(16,8),(17,18),(18,10),(19,23),(19,24),(20,15),(20,16),(21,14),(21,20),(22,13),(22,21),(23,22),(23,26),(24,9),(24,22),(25,10),(25,20),(26,21),(26,25)],27)
=> ? ∊ {4,6,8,8,10,12,12,14,16,16,18,20,24} + 1
[1,1,0,1,0,1,0,0]
=> 11010100 => 10000001 => ([(0,6),(0,7),(1,5),(1,9),(1,10),(2,19),(2,20),(3,4),(3,13),(3,14),(4,2),(4,15),(4,16),(5,3),(5,17),(5,18),(6,11),(6,12),(7,1),(7,11),(7,12),(9,18),(10,17),(11,10),(12,9),(13,15),(14,16),(15,19),(16,20),(17,13),(18,14),(19,8),(20,8)],21)
=> ? ∊ {4,6,8,8,10,12,12,14,16,16,18,20,24} + 1
[1,1,0,1,1,0,0,0]
=> 11011000 => 10001101 => ([(0,4),(0,5),(1,23),(1,24),(2,3),(2,16),(2,25),(3,8),(3,19),(4,1),(4,26),(4,27),(5,2),(5,26),(5,27),(7,11),(8,10),(9,7),(10,12),(11,13),(12,14),(13,15),(14,6),(15,6),(16,8),(17,11),(17,20),(18,20),(18,21),(19,10),(19,21),(20,13),(20,22),(21,12),(21,22),(22,14),(22,15),(23,7),(23,17),(24,17),(24,18),(25,18),(25,19),(26,9),(26,24),(26,25),(27,9),(27,16),(27,23)],28)
=> ? ∊ {4,6,8,8,10,12,12,14,16,16,18,20,24} + 1
[1,1,1,0,0,0,1,0]
=> 11100010 => 10110111 => ([(0,3),(0,4),(1,12),(2,1),(2,18),(2,22),(3,20),(3,21),(4,2),(4,20),(4,21),(6,8),(7,9),(8,10),(9,11),(10,5),(11,5),(12,7),(13,14),(13,15),(14,9),(14,16),(15,8),(15,16),(16,10),(16,11),(17,13),(17,23),(18,13),(18,19),(19,6),(19,15),(20,17),(20,18),(21,17),(21,22),(22,12),(22,19),(22,23),(23,6),(23,7),(23,14)],24)
=> ? ∊ {4,6,8,8,10,12,12,14,16,16,18,20,24} + 1
[1,1,1,0,0,1,0,0]
=> 11100100 => 10110001 => ([(0,4),(0,5),(1,23),(1,24),(2,3),(2,16),(2,25),(3,8),(3,19),(4,1),(4,26),(4,27),(5,2),(5,26),(5,27),(7,11),(8,10),(9,7),(10,12),(11,13),(12,14),(13,15),(14,6),(15,6),(16,8),(17,11),(17,20),(18,20),(18,21),(19,10),(19,21),(20,13),(20,22),(21,12),(21,22),(22,14),(22,15),(23,7),(23,17),(24,17),(24,18),(25,18),(25,19),(26,9),(26,24),(26,25),(27,9),(27,16),(27,23)],28)
=> ? ∊ {4,6,8,8,10,12,12,14,16,16,18,20,24} + 1
[1,1,1,0,1,0,0,0]
=> 11101000 => 10111101 => ([(0,4),(0,5),(1,3),(1,17),(1,18),(2,23),(2,24),(3,2),(3,21),(3,22),(4,19),(4,20),(5,1),(5,19),(5,20),(7,12),(8,11),(9,13),(10,14),(11,9),(12,10),(13,6),(14,6),(15,7),(15,8),(16,13),(16,14),(17,8),(17,21),(18,7),(18,22),(19,15),(19,17),(20,15),(20,18),(21,11),(21,23),(22,12),(22,24),(23,9),(23,16),(24,10),(24,16)],25)
=> ? ∊ {4,6,8,8,10,12,12,14,16,16,18,20,24} + 1
[1,1,1,1,0,0,0,0]
=> 11110000 => 10100101 => ([(0,2),(0,3),(1,17),(1,18),(2,22),(2,23),(3,1),(3,22),(3,23),(5,10),(6,9),(7,11),(8,12),(9,7),(10,8),(11,4),(12,4),(13,20),(13,21),(14,11),(14,12),(15,9),(15,20),(16,10),(16,21),(17,13),(17,15),(18,13),(18,16),(19,5),(19,6),(20,7),(20,14),(21,8),(21,14),(22,17),(22,19),(22,24),(23,18),(23,19),(23,24),(24,5),(24,6),(24,15),(24,16)],25)
=> ? ∊ {4,6,8,8,10,12,12,14,16,16,18,20,24} + 1
Description
The number of linear extensions of a poset.
Matching statistic: St000527
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00158: Binary words —alternating inverse⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St000527: Posets ⟶ ℤResult quality: 9% ●values known / values provided: 18%●distinct values known / distinct values provided: 9%
Mp00158: Binary words —alternating inverse⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St000527: Posets ⟶ ℤResult quality: 9% ●values known / values provided: 18%●distinct values known / distinct values provided: 9%
Values
[1,0]
=> 10 => 11 => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,0,1,0]
=> 1010 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,1,0,0]
=> 1100 => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 4 + 1
[1,0,1,0,1,0]
=> 101010 => 111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> 101100 => 111001 => ([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17)
=> ? ∊ {4,6,8,12} + 1
[1,1,0,0,1,0]
=> 110010 => 100111 => ([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17)
=> ? ∊ {4,6,8,12} + 1
[1,1,0,1,0,0]
=> 110100 => 100001 => ([(0,4),(0,5),(1,3),(1,7),(1,8),(2,13),(2,14),(3,2),(3,11),(3,12),(4,9),(4,10),(5,1),(5,9),(5,10),(7,12),(8,11),(9,8),(10,7),(11,13),(12,14),(13,6),(14,6)],15)
=> ? ∊ {4,6,8,12} + 1
[1,1,1,0,0,0]
=> 111000 => 101101 => ([(0,2),(0,3),(1,10),(1,11),(2,13),(2,14),(3,1),(3,13),(3,14),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9),(12,5),(12,6),(13,10),(13,12),(14,11),(14,12)],15)
=> ? ∊ {4,6,8,12} + 1
[1,0,1,0,1,0,1,0]
=> 10101010 => 11111111 => ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> 10101100 => 11111001 => ([(0,6),(0,7),(1,4),(1,17),(2,16),(3,2),(3,23),(4,3),(4,24),(5,9),(5,18),(6,1),(6,15),(6,22),(7,5),(7,15),(7,22),(9,11),(10,12),(11,10),(12,13),(13,8),(14,8),(15,9),(16,14),(17,21),(17,24),(18,11),(18,21),(19,12),(19,20),(20,13),(20,14),(21,10),(21,19),(22,17),(22,18),(23,16),(23,20),(24,19),(24,23)],25)
=> ? ∊ {4,6,8,8,10,12,12,14,16,16,18,20,24} + 1
[1,0,1,1,0,0,1,0]
=> 10110010 => 11100111 => ([(0,4),(0,5),(1,23),(1,24),(2,3),(2,25),(2,26),(3,7),(3,8),(4,1),(4,19),(4,20),(5,2),(5,19),(5,20),(7,12),(8,11),(9,13),(10,14),(11,9),(12,10),(13,6),(14,6),(15,13),(15,14),(16,21),(16,22),(17,11),(17,21),(18,12),(18,22),(19,23),(19,25),(20,24),(20,26),(21,9),(21,15),(22,10),(22,15),(23,16),(23,17),(24,16),(24,18),(25,8),(25,17),(26,7),(26,18)],27)
=> ? ∊ {4,6,8,8,10,12,12,14,16,16,18,20,24} + 1
[1,0,1,1,0,1,0,0]
=> 10110100 => 11100001 => ([(0,6),(0,7),(1,2),(1,24),(2,9),(3,5),(3,11),(3,23),(4,18),(4,25),(5,4),(5,17),(5,26),(6,3),(6,12),(6,19),(7,1),(7,12),(7,19),(9,13),(10,16),(11,17),(12,11),(13,14),(14,15),(15,8),(16,8),(17,18),(18,10),(19,23),(19,24),(20,15),(20,16),(21,14),(21,20),(22,13),(22,21),(23,22),(23,26),(24,9),(24,22),(25,10),(25,20),(26,21),(26,25)],27)
=> ? ∊ {4,6,8,8,10,12,12,14,16,16,18,20,24} + 1
[1,0,1,1,1,0,0,0]
=> 10111000 => 11101101 => ([(0,3),(0,4),(1,12),(2,1),(2,18),(2,22),(3,20),(3,21),(4,2),(4,20),(4,21),(6,8),(7,9),(8,10),(9,11),(10,5),(11,5),(12,7),(13,14),(13,15),(14,9),(14,16),(15,8),(15,16),(16,10),(16,11),(17,13),(17,23),(18,13),(18,19),(19,6),(19,15),(20,17),(20,18),(21,17),(21,22),(22,12),(22,19),(22,23),(23,6),(23,7),(23,14)],24)
=> ? ∊ {4,6,8,8,10,12,12,14,16,16,18,20,24} + 1
[1,1,0,0,1,0,1,0]
=> 11001010 => 10011111 => ([(0,6),(0,7),(1,4),(1,17),(2,16),(3,2),(3,23),(4,3),(4,24),(5,9),(5,18),(6,1),(6,15),(6,22),(7,5),(7,15),(7,22),(9,11),(10,12),(11,10),(12,13),(13,8),(14,8),(15,9),(16,14),(17,21),(17,24),(18,11),(18,21),(19,12),(19,20),(20,13),(20,14),(21,10),(21,19),(22,17),(22,18),(23,16),(23,20),(24,19),(24,23)],25)
=> ? ∊ {4,6,8,8,10,12,12,14,16,16,18,20,24} + 1
[1,1,0,0,1,1,0,0]
=> 11001100 => 10011001 => ([(0,3),(0,4),(1,21),(1,22),(2,23),(2,24),(3,1),(3,16),(3,17),(4,2),(4,16),(4,17),(6,9),(7,8),(8,10),(9,11),(10,5),(11,5),(12,8),(12,14),(13,9),(13,14),(14,10),(14,11),(15,12),(15,13),(16,21),(16,23),(17,22),(17,24),(18,7),(18,12),(19,6),(19,13),(20,6),(20,7),(21,15),(21,18),(22,15),(22,19),(23,18),(23,20),(24,19),(24,20)],25)
=> ? ∊ {4,6,8,8,10,12,12,14,16,16,18,20,24} + 1
[1,1,0,1,0,0,1,0]
=> 11010010 => 10000111 => ([(0,6),(0,7),(1,2),(1,24),(2,9),(3,5),(3,11),(3,23),(4,18),(4,25),(5,4),(5,17),(5,26),(6,3),(6,12),(6,19),(7,1),(7,12),(7,19),(9,13),(10,16),(11,17),(12,11),(13,14),(14,15),(15,8),(16,8),(17,18),(18,10),(19,23),(19,24),(20,15),(20,16),(21,14),(21,20),(22,13),(22,21),(23,22),(23,26),(24,9),(24,22),(25,10),(25,20),(26,21),(26,25)],27)
=> ? ∊ {4,6,8,8,10,12,12,14,16,16,18,20,24} + 1
[1,1,0,1,0,1,0,0]
=> 11010100 => 10000001 => ([(0,6),(0,7),(1,5),(1,9),(1,10),(2,19),(2,20),(3,4),(3,13),(3,14),(4,2),(4,15),(4,16),(5,3),(5,17),(5,18),(6,11),(6,12),(7,1),(7,11),(7,12),(9,18),(10,17),(11,10),(12,9),(13,15),(14,16),(15,19),(16,20),(17,13),(18,14),(19,8),(20,8)],21)
=> ? ∊ {4,6,8,8,10,12,12,14,16,16,18,20,24} + 1
[1,1,0,1,1,0,0,0]
=> 11011000 => 10001101 => ([(0,4),(0,5),(1,23),(1,24),(2,3),(2,16),(2,25),(3,8),(3,19),(4,1),(4,26),(4,27),(5,2),(5,26),(5,27),(7,11),(8,10),(9,7),(10,12),(11,13),(12,14),(13,15),(14,6),(15,6),(16,8),(17,11),(17,20),(18,20),(18,21),(19,10),(19,21),(20,13),(20,22),(21,12),(21,22),(22,14),(22,15),(23,7),(23,17),(24,17),(24,18),(25,18),(25,19),(26,9),(26,24),(26,25),(27,9),(27,16),(27,23)],28)
=> ? ∊ {4,6,8,8,10,12,12,14,16,16,18,20,24} + 1
[1,1,1,0,0,0,1,0]
=> 11100010 => 10110111 => ([(0,3),(0,4),(1,12),(2,1),(2,18),(2,22),(3,20),(3,21),(4,2),(4,20),(4,21),(6,8),(7,9),(8,10),(9,11),(10,5),(11,5),(12,7),(13,14),(13,15),(14,9),(14,16),(15,8),(15,16),(16,10),(16,11),(17,13),(17,23),(18,13),(18,19),(19,6),(19,15),(20,17),(20,18),(21,17),(21,22),(22,12),(22,19),(22,23),(23,6),(23,7),(23,14)],24)
=> ? ∊ {4,6,8,8,10,12,12,14,16,16,18,20,24} + 1
[1,1,1,0,0,1,0,0]
=> 11100100 => 10110001 => ([(0,4),(0,5),(1,23),(1,24),(2,3),(2,16),(2,25),(3,8),(3,19),(4,1),(4,26),(4,27),(5,2),(5,26),(5,27),(7,11),(8,10),(9,7),(10,12),(11,13),(12,14),(13,15),(14,6),(15,6),(16,8),(17,11),(17,20),(18,20),(18,21),(19,10),(19,21),(20,13),(20,22),(21,12),(21,22),(22,14),(22,15),(23,7),(23,17),(24,17),(24,18),(25,18),(25,19),(26,9),(26,24),(26,25),(27,9),(27,16),(27,23)],28)
=> ? ∊ {4,6,8,8,10,12,12,14,16,16,18,20,24} + 1
[1,1,1,0,1,0,0,0]
=> 11101000 => 10111101 => ([(0,4),(0,5),(1,3),(1,17),(1,18),(2,23),(2,24),(3,2),(3,21),(3,22),(4,19),(4,20),(5,1),(5,19),(5,20),(7,12),(8,11),(9,13),(10,14),(11,9),(12,10),(13,6),(14,6),(15,7),(15,8),(16,13),(16,14),(17,8),(17,21),(18,7),(18,22),(19,15),(19,17),(20,15),(20,18),(21,11),(21,23),(22,12),(22,24),(23,9),(23,16),(24,10),(24,16)],25)
=> ? ∊ {4,6,8,8,10,12,12,14,16,16,18,20,24} + 1
[1,1,1,1,0,0,0,0]
=> 11110000 => 10100101 => ([(0,2),(0,3),(1,17),(1,18),(2,22),(2,23),(3,1),(3,22),(3,23),(5,10),(6,9),(7,11),(8,12),(9,7),(10,8),(11,4),(12,4),(13,20),(13,21),(14,11),(14,12),(15,9),(15,20),(16,10),(16,21),(17,13),(17,15),(18,13),(18,16),(19,5),(19,6),(20,7),(20,14),(21,8),(21,14),(22,17),(22,19),(22,24),(23,18),(23,19),(23,24),(24,5),(24,6),(24,15),(24,16)],25)
=> ? ∊ {4,6,8,8,10,12,12,14,16,16,18,20,24} + 1
Description
The width of the poset.
This is the size of the poset's longest antichain, also called Dilworth number.
The following 53 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000909The number of maximal chains of maximal size in a poset. St001330The hat guessing number of a graph. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St000136The dinv of a parking function. St000194The number of primary dinversion pairs of a labelled dyck path corresponding to a parking function. St000217The number of occurrences of the pattern 312 in a permutation. St000219The number of occurrences of the pattern 231 in a permutation. St000226The convexity of a permutation. St000327The number of cover relations in a poset. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000425The number of occurrences of the pattern 132 or of the pattern 213 in a permutation. St000426The number of occurrences of the pattern 132 or of the pattern 312 in a permutation. St000427The number of occurrences of the pattern 123 or of the pattern 231 in a permutation. St000430The number of occurrences of the pattern 123 or of the pattern 312 in a permutation. St000434The number of occurrences of the pattern 213 or of the pattern 312 in a permutation. St000455The second largest eigenvalue of a graph if it is integral. St000555The number of occurrences of the pattern {{1,3},{2}} in a set partition. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000632The jump number of the poset. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001511The minimal number of transpositions needed to sort a permutation in either direction. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001645The pebbling number of a connected graph. St001684The reduced word complexity of a permutation. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001705The number of occurrences of the pattern 2413 in a permutation. St001766The number of cells which are not occupied by the same tile in all reduced pipe dreams corresponding to a permutation. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001964The interval resolution global dimension of a poset. St000075The orbit size of a standard tableau under promotion. St000255The number of reduced Kogan faces with the permutation as type. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000454The largest eigenvalue of a graph if it is integral. St000652The maximal difference between successive positions of a permutation. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001388The number of non-attacking neighbors of a permutation. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001768The number of reduced words of a signed permutation. St001958The degree of the polynomial interpolating the values of a permutation. St000748The major index of the permutation obtained by flattening the set partition. St001160The number of proper blocks (or intervals) of a permutations. St001375The pancake length of a permutation. St001565The number of arithmetic progressions of length 2 in a permutation. St000635The number of strictly order preserving maps of a poset into itself.
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