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Your data matches 23 different statistics following compositions of up to 3 maps.
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Matching statistic: St000977
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
St000977: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00143: Dyck paths —inverse promotion⟶ 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]
=> [1,0,1,0,1,0]
=> 0
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> 4
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 8
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 4
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> 6
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 12
[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]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 12
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 8
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 10
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 20
[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]
=> 4
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 16
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 6
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 8
[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]
=> 18
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 12
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 14
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 16
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 24
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 10
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
[1,1,0,1,0,1,0,1,0,0]
=> 1101010100 => [2,1,1,1,1,1,1,2] => [1,8,1] => 10
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: 96% ●values known / values provided: 96%●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: 96% ●values known / values provided: 96%●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
[1,1,0,1,0,1,0,1,0,0]
=> 1101010100 => 0100000001 => 1000000010 => ? = 10
Description
The sum of the positions of the ones in a binary word.
Matching statistic: St001232
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 18% ●values known / values provided: 22%●distinct values known / distinct values provided: 18%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 18% ●values known / values provided: 22%●distinct values known / distinct values provided: 18%
Values
[1,0]
=> [1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,0,0]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 4 + 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 8 + 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? = 4 + 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 6 + 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 12 + 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 12 + 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 8 + 1
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 10 + 1
[1,0,1,1,1,0,0,0]
=> [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]
=> ? = 20 + 1
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 5 = 4 + 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 16 + 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 6 + 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 8 + 1
[1,1,0,1,1,0,0,0]
=> [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]
=> ? = 18 + 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? = 12 + 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 14 + 1
[1,1,1,0,1,0,0,0]
=> [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]
=> ? = 16 + 1
[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]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 24 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> ? = 10 + 1
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: 17%●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: 17%●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)
=> ? = 8 + 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 + 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)
=> ? = 6 + 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)
=> ? = 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)
=> ? = 12 + 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)
=> ? = 8 + 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)
=> ? = 10 + 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)
=> ? = 20 + 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 + 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)
=> ? = 16 + 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)
=> ? = 6 + 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)
=> ? = 8 + 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)
=> ? = 18 + 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)
=> ? = 12 + 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)
=> ? = 14 + 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)
=> ? = 16 + 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)
=> ? = 24 + 1
[1,1,0,1,0,1,0,1,0,0]
=> 1101010100 => 1000000001 => ([(0,8),(0,9),(1,7),(1,11),(1,12),(2,25),(2,26),(3,4),(3,15),(3,16),(4,6),(4,17),(4,18),(5,3),(5,19),(5,20),(6,2),(6,21),(6,22),(7,5),(7,23),(7,24),(8,13),(8,14),(9,1),(9,13),(9,14),(11,24),(12,23),(13,12),(14,11),(15,17),(16,18),(17,21),(18,22),(19,15),(20,16),(21,25),(22,26),(23,19),(24,20),(25,10),(26,10)],27)
=> ? = 10 + 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: 17%●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: 17%●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)
=> ? = 8 + 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 + 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)
=> ? = 6 + 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)
=> ? = 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)
=> ? = 12 + 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)
=> ? = 8 + 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)
=> ? = 10 + 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)
=> ? = 20 + 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 + 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)
=> ? = 16 + 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)
=> ? = 6 + 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)
=> ? = 8 + 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)
=> ? = 18 + 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)
=> ? = 12 + 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)
=> ? = 14 + 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)
=> ? = 16 + 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)
=> ? = 24 + 1
[1,1,0,1,0,1,0,1,0,0]
=> 1101010100 => 1000000001 => ([(0,8),(0,9),(1,7),(1,11),(1,12),(2,25),(2,26),(3,4),(3,15),(3,16),(4,6),(4,17),(4,18),(5,3),(5,19),(5,20),(6,2),(6,21),(6,22),(7,5),(7,23),(7,24),(8,13),(8,14),(9,1),(9,13),(9,14),(11,24),(12,23),(13,12),(14,11),(15,17),(16,18),(17,21),(18,22),(19,15),(20,16),(21,25),(22,26),(23,19),(24,20),(25,10),(26,10)],27)
=> ? = 10 + 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: 17%●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: 17%●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)
=> ? = 8 + 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 + 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)
=> ? = 6 + 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)
=> ? = 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)
=> ? = 12 + 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)
=> ? = 8 + 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)
=> ? = 10 + 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)
=> ? = 20 + 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 + 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)
=> ? = 16 + 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)
=> ? = 6 + 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)
=> ? = 8 + 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)
=> ? = 18 + 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)
=> ? = 12 + 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)
=> ? = 14 + 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)
=> ? = 16 + 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)
=> ? = 24 + 1
[1,1,0,1,0,1,0,1,0,0]
=> 1101010100 => 1000000001 => ([(0,8),(0,9),(1,7),(1,11),(1,12),(2,25),(2,26),(3,4),(3,15),(3,16),(4,6),(4,17),(4,18),(5,3),(5,19),(5,20),(6,2),(6,21),(6,22),(7,5),(7,23),(7,24),(8,13),(8,14),(9,1),(9,13),(9,14),(11,24),(12,23),(13,12),(14,11),(15,17),(16,18),(17,21),(18,22),(19,15),(20,16),(21,25),(22,26),(23,19),(24,20),(25,10),(26,10)],27)
=> ? = 10 + 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: 17%●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: 17%●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)
=> ? = 8 + 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 + 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)
=> ? = 6 + 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)
=> ? = 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)
=> ? = 12 + 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)
=> ? = 8 + 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)
=> ? = 10 + 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)
=> ? = 20 + 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 + 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)
=> ? = 16 + 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)
=> ? = 6 + 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)
=> ? = 8 + 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)
=> ? = 18 + 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)
=> ? = 12 + 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)
=> ? = 14 + 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)
=> ? = 16 + 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)
=> ? = 24 + 1
[1,1,0,1,0,1,0,1,0,0]
=> 1101010100 => 1000000001 => ([(0,8),(0,9),(1,7),(1,11),(1,12),(2,25),(2,26),(3,4),(3,15),(3,16),(4,6),(4,17),(4,18),(5,3),(5,19),(5,20),(6,2),(6,21),(6,22),(7,5),(7,23),(7,24),(8,13),(8,14),(9,1),(9,13),(9,14),(11,24),(12,23),(13,12),(14,11),(15,17),(16,18),(17,21),(18,22),(19,15),(20,16),(21,25),(22,26),(23,19),(24,20),(25,10),(26,10)],27)
=> ? = 10 + 1
Description
The width of the poset.
This is the size of the poset's longest antichain, also called Dilworth number.
Matching statistic: St000909
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00158: Binary words —alternating inverse⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St000909: Posets ⟶ ℤResult quality: 9% ●values known / values provided: 17%●distinct values known / distinct values provided: 9%
Mp00158: Binary words —alternating inverse⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St000909: Posets ⟶ ℤResult quality: 9% ●values known / values provided: 17%●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)
=> ? = 8 + 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 + 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)
=> ? = 6 + 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)
=> ? = 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)
=> ? = 12 + 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)
=> ? = 8 + 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)
=> ? = 10 + 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)
=> ? = 20 + 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 + 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)
=> ? = 16 + 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)
=> ? = 6 + 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)
=> ? = 8 + 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)
=> ? = 18 + 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)
=> ? = 12 + 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)
=> ? = 14 + 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)
=> ? = 16 + 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)
=> ? = 24 + 1
[1,1,0,1,0,1,0,1,0,0]
=> 1101010100 => 1000000001 => ([(0,8),(0,9),(1,7),(1,11),(1,12),(2,25),(2,26),(3,4),(3,15),(3,16),(4,6),(4,17),(4,18),(5,3),(5,19),(5,20),(6,2),(6,21),(6,22),(7,5),(7,23),(7,24),(8,13),(8,14),(9,1),(9,13),(9,14),(11,24),(12,23),(13,12),(14,11),(15,17),(16,18),(17,21),(18,22),(19,15),(20,16),(21,25),(22,26),(23,19),(24,20),(25,10),(26,10)],27)
=> ? = 10 + 1
Description
The number of maximal chains of maximal size in a poset.
Matching statistic: St000632
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00158: Binary words —alternating inverse⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St000632: Posets ⟶ ℤResult quality: 9% ●values known / values provided: 13%●distinct values known / distinct values provided: 9%
Mp00158: Binary words —alternating inverse⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St000632: Posets ⟶ ℤResult quality: 9% ●values known / values provided: 13%●distinct values known / distinct values provided: 9%
Values
[1,0]
=> 10 => 11 => ([(0,2),(2,1)],3)
=> 0
[1,0,1,0]
=> 1010 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[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
[1,0,1,0,1,0]
=> 101010 => 111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 0
[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)
=> ? = 8
[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
[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)
=> ? = 6
[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)
=> ? = 12
[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)
=> ? = 0
[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)
=> ? = 12
[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)
=> ? = 8
[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)
=> ? = 10
[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)
=> ? = 20
[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
[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)
=> ? = 16
[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)
=> ? = 6
[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)
=> ? = 8
[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)
=> ? = 18
[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)
=> ? = 12
[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)
=> ? = 14
[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)
=> ? = 16
[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)
=> ? = 24
[1,1,0,1,0,1,0,1,0,0]
=> 1101010100 => 1000000001 => ([(0,8),(0,9),(1,7),(1,11),(1,12),(2,25),(2,26),(3,4),(3,15),(3,16),(4,6),(4,17),(4,18),(5,3),(5,19),(5,20),(6,2),(6,21),(6,22),(7,5),(7,23),(7,24),(8,13),(8,14),(9,1),(9,13),(9,14),(11,24),(12,23),(13,12),(14,11),(15,17),(16,18),(17,21),(18,22),(19,15),(20,16),(21,25),(22,26),(23,19),(24,20),(25,10),(26,10)],27)
=> ? = 10
Description
The jump number of the poset.
A jump in a linear extension $e_1, \dots, e_n$ of a poset $P$ is a pair $(e_i, e_{i+1})$ so that $e_{i+1}$ does not cover $e_i$ in $P$. The jump number of a poset is the minimal number of jumps in linear extensions of a poset.
The following 13 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. 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. St000456The monochromatic index of a connected graph. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000748The major index of the permutation obtained by flattening the set partition. St001160The number of proper blocks (or intervals) of a permutations. St001330The hat guessing number of a graph. St001565The number of arithmetic progressions of length 2 in a permutation.
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