searching the database
Your data matches 37 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St001299
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
St001299: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 1
[1,0,1,0]
=> 2
[1,1,0,0]
=> 1
[1,0,1,0,1,0]
=> 6
[1,0,1,1,0,0]
=> 2
[1,1,0,0,1,0]
=> 2
[1,1,0,1,0,0]
=> 2
[1,1,1,0,0,0]
=> 1
[1,0,1,0,1,0,1,0]
=> 24
[1,0,1,0,1,1,0,0]
=> 6
[1,0,1,1,0,0,1,0]
=> 4
[1,0,1,1,0,1,0,0]
=> 6
[1,0,1,1,1,0,0,0]
=> 2
[1,1,0,0,1,0,1,0]
=> 6
[1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> 6
[1,1,0,1,0,1,0,0]
=> 6
[1,1,0,1,1,0,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> 2
[1,1,1,0,0,1,0,0]
=> 2
[1,1,1,0,1,0,0,0]
=> 2
[1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> 120
[1,0,1,0,1,0,1,1,0,0]
=> 24
[1,0,1,0,1,1,0,0,1,0]
=> 12
[1,0,1,0,1,1,0,1,0,0]
=> 24
[1,0,1,0,1,1,1,0,0,0]
=> 6
[1,0,1,1,0,0,1,0,1,0]
=> 12
[1,0,1,1,0,0,1,1,0,0]
=> 4
[1,0,1,1,0,1,0,0,1,0]
=> 24
[1,0,1,1,0,1,0,1,0,0]
=> 24
[1,0,1,1,0,1,1,0,0,0]
=> 6
[1,0,1,1,1,0,0,0,1,0]
=> 4
[1,0,1,1,1,0,0,1,0,0]
=> 4
[1,0,1,1,1,0,1,0,0,0]
=> 6
[1,0,1,1,1,1,0,0,0,0]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> 24
[1,1,0,0,1,0,1,1,0,0]
=> 6
[1,1,0,0,1,1,0,0,1,0]
=> 4
[1,1,0,0,1,1,0,1,0,0]
=> 6
[1,1,0,0,1,1,1,0,0,0]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> 24
[1,1,0,1,0,0,1,1,0,0]
=> 6
[1,1,0,1,0,1,0,0,1,0]
=> 24
[1,1,0,1,0,1,0,1,0,0]
=> 18
[1,1,0,1,0,1,1,0,0,0]
=> 6
[1,1,0,1,1,0,0,0,1,0]
=> 4
[1,1,0,1,1,0,0,1,0,0]
=> 6
[1,1,0,1,1,0,1,0,0,0]
=> 6
[1,1,0,1,1,1,0,0,0,0]
=> 2
Description
The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra.
Matching statistic: St000124
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000124: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000124: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [2,1] => 1
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [3,1,2] => 2
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 1
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => 2
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 2
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 2
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 6
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => 2
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => 6
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => 2
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,4] => 4
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => 2
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,1,2] => 2
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,5,1,2,4] => 6
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => 6
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => 6
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,5,1,2,3] => 6
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => 24
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,6,1,5] => 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [2,3,5,1,6,4] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,5,6,1,4] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,6,1,4,5] => 6
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [2,4,1,5,6,3] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [2,4,1,6,3,5] => 4
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [2,4,5,1,6,3] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [2,4,5,6,1,3] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,4,6,1,3,5] => 6
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [2,5,1,3,6,4] => 6
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [2,5,1,6,3,4] => 6
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,5,6,1,3,4] => 6
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,6,1,3,4,5] => 24
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,1,4,5,6,2] => 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,1,4,6,2,5] => 4
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [3,1,5,2,6,4] => 4
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,1,5,6,2,4] => 4
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,1,6,2,4,5] => 12
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [3,4,1,5,6,2] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [3,4,1,6,2,5] => 4
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [3,4,5,1,6,2] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,4,5,6,1,2] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [3,4,6,1,2,5] => 6
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [3,5,1,2,6,4] => 6
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [3,5,1,6,2,4] => 6
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [3,5,6,1,2,4] => 6
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [3,6,1,2,4,5] => 24
Description
The cardinality of the preimage of the Simion-Schmidt map.
The Simion-Schmidt bijection transforms a [3,1,2]-avoiding permutation into a [3,2,1]-avoiding permutation. More generally, it can be thought of as a map $S$ that turns any permutation into a [3,2,1]-avoiding permutation. This statistic is the size of $S^{-1}(\pi)$ for each permutation $\pi$.
The map $S$ can also be realized using the quotient of the $0$-Hecke Monoid of the symmetric group by the relation $\pi_i \pi_{i+1} \pi_i = \pi_{i+1} \pi_i$, sending each element of the fiber of the quotient to the unique [3,2,1]-avoiding element in that fiber.
Matching statistic: St001330
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 25% ●values known / values provided: 39%●distinct values known / distinct values provided: 25%
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 25% ●values known / values provided: 39%●distinct values known / distinct values provided: 25%
Values
[1,0]
=> [1,1,0,0]
=> [2] => ([],2)
=> 1
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> 2
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [3] => ([],3)
=> 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 6
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {4,6,6,6,6,6,24}
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {4,6,6,6,6,6,24}
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {4,6,6,6,6,6,24}
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {4,6,6,6,6,6,24}
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {4,6,6,6,6,6,24}
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {4,6,6,6,6,6,24}
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {4,6,6,6,6,6,24}
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => ([],5)
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,4] => ([(3,5),(4,5)],6)
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6] => ([],6)
=> 1
Description
The hat guessing number of a graph.
Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors.
Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
Matching statistic: St001624
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Values
[1,0]
=> ([],1)
=> ([(0,1)],2)
=> 1
[1,0,1,0]
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,0,0]
=> ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,0,1,1,0,0]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[1,1,0,0,1,0]
=> ([(0,1),(0,2)],3)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
[1,1,0,1,0,0]
=> ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,1,1,0,0,0]
=> ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,0,1,0,1,1,0,0]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2
[1,0,1,1,0,0,1,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2
[1,0,1,1,0,1,0,0]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2
[1,0,1,1,1,0,0,0]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> ? ∊ {4,6,6,6,6,6,24}
[1,1,0,0,1,0,1,0]
=> ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 2
[1,1,0,0,1,1,0,0]
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 2
[1,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(3,1)],4)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 2
[1,1,0,1,0,1,0,0]
=> ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? ∊ {4,6,6,6,6,6,24}
[1,1,0,1,1,0,0,0]
=> ([(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
=> ? ∊ {4,6,6,6,6,6,24}
[1,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3)],4)
=> ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
=> ? ∊ {4,6,6,6,6,6,24}
[1,1,1,0,0,1,0,0]
=> ([(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
=> ? ∊ {4,6,6,6,6,6,24}
[1,1,1,0,1,0,0,0]
=> ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? ∊ {4,6,6,6,6,6,24}
[1,1,1,1,0,0,0,0]
=> ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? ∊ {4,6,6,6,6,6,24}
[1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,0,1,1,1,0,0,0]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,2),(0,3),(0,4),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,1),(6,9),(7,9),(8,9),(9,5)],10)
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,1,0,0,1,0,1,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,1,0,1,0,1,0,0]
=> ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,1,0,1,1,0,0,0]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,7),(2,9),(3,7),(3,8),(4,6),(5,2),(5,3),(5,6),(6,8),(6,9),(7,10),(8,10),(9,10),(10,1)],11)
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(2,10),(3,6),(3,8),(4,6),(4,7),(5,2),(5,7),(5,8),(6,11),(7,9),(7,11),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,0,1,0,1,0,1,0]
=> ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,0,1,1,0,0,1,0]
=> ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,0,1,1,0,1,0,0]
=> ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(3,9),(4,7),(4,9),(5,7),(5,8),(7,10),(8,10),(9,10),(10,1),(10,2)],11)
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,1,0,0,1,1,0,0]
=> ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,1,0,1,0,1,0,0]
=> ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,1,0,1,1,0,0,0]
=> ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(3,10),(4,6),(4,10),(5,6),(5,7),(6,11),(7,11),(8,9),(10,2),(10,11),(11,1),(11,8)],12)
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,5),(1,8),(2,7),(2,9),(3,6),(3,9),(4,6),(4,7),(5,2),(5,3),(5,4),(6,10),(7,10),(9,1),(9,10),(10,8)],11)
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> ([(0,4),(0,5),(1,8),(2,7),(2,9),(3,7),(3,10),(4,6),(5,2),(5,3),(5,6),(6,9),(6,10),(7,11),(9,11),(10,1),(10,11),(11,8)],12)
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,8),(2,6),(2,7),(3,9),(3,11),(4,9),(4,10),(5,2),(5,10),(5,11),(6,13),(7,13),(9,12),(10,6),(10,12),(11,7),(11,12),(12,1),(12,13),(13,8)],14)
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,1,1,1,0,0,0,0]
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,7),(2,8),(2,9),(3,9),(3,11),(3,12),(4,8),(4,10),(4,12),(5,7),(5,10),(5,11),(7,13),(7,14),(8,13),(8,15),(9,14),(9,15),(10,13),(10,16),(11,14),(11,16),(12,15),(12,16),(13,17),(14,17),(15,17),(16,1),(16,17),(17,6)],18)
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,1,0,0,0,1,0,1,0]
=> ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,4),(0,5),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,9),(5,9),(6,10),(7,10),(8,10),(9,1),(9,2),(9,3)],11)
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,1,0,0,1,0,0,1,0]
=> ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,1,0,0,1,0,1,0,0]
=> ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,4),(0,5),(1,7),(1,9),(2,7),(2,8),(3,6),(4,10),(5,3),(5,10),(6,8),(6,9),(7,11),(8,11),(9,11),(10,1),(10,2),(10,6)],12)
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,1,0,0,1,1,0,0,0]
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,6),(1,8),(2,6),(2,7),(3,10),(3,11),(4,9),(4,11),(5,9),(5,10),(6,12),(7,12),(8,12),(9,13),(10,13),(11,1),(11,2),(11,13),(13,7),(13,8)],14)
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,1,0,1,0,0,0,1,0]
=> ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,5),(1,9),(1,10),(2,6),(2,8),(3,6),(3,7),(4,1),(4,7),(4,8),(5,2),(5,3),(5,4),(6,12),(7,9),(7,12),(8,10),(8,12),(9,11),(10,11),(12,11)],13)
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,1,0,1,0,0,1,0,0]
=> ([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,1),(0,2),(1,11),(2,4),(2,5),(2,11),(3,6),(3,7),(4,8),(4,10),(5,8),(5,9),(6,13),(7,13),(8,12),(9,6),(9,12),(10,7),(10,12),(11,3),(11,9),(11,10),(12,13)],14)
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,1,0,1,0,1,0,0,0]
=> ([(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,6),(1,7),(2,8),(2,10),(3,9),(3,11),(4,9),(4,12),(5,2),(5,11),(5,12),(6,14),(7,14),(8,13),(9,15),(10,6),(10,13),(11,8),(11,15),(12,1),(12,10),(12,15),(13,14),(15,7),(15,13)],16)
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,1,0,1,1,0,0,0,0]
=> ([(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,6),(1,7),(2,11),(2,12),(2,13),(3,9),(3,10),(3,13),(4,8),(4,10),(4,12),(5,8),(5,9),(5,11),(6,16),(7,16),(8,1),(8,17),(8,18),(9,14),(9,17),(10,15),(10,17),(11,14),(11,18),(12,15),(12,18),(13,14),(13,15),(14,19),(15,19),(17,6),(17,19),(18,7),(18,19),(19,16)],20)
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,1,1,0,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,1),(1,2),(1,3),(1,4),(1,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16)],17)
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,1,1,0,0,0,1,0,0]
=> ([(1,2),(1,3),(1,4)],5)
=> ([(0,1),(0,2),(1,12),(2,3),(2,4),(2,5),(2,12),(3,8),(3,10),(3,11),(4,7),(4,9),(4,11),(5,6),(5,9),(5,10),(6,13),(6,14),(7,13),(7,15),(8,14),(8,15),(9,13),(9,16),(10,14),(10,16),(11,15),(11,16),(12,6),(12,7),(12,8),(13,17),(14,17),(15,17),(16,17)],18)
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,1,1,0,0,1,0,0,0]
=> ([(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(1,11),(1,13),(2,11),(2,12),(3,4),(3,5),(3,12),(3,13),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,15),(6,17),(7,15),(7,18),(8,16),(8,17),(9,16),(9,18),(10,15),(10,16),(11,14),(12,6),(12,7),(12,14),(13,8),(13,9),(13,14),(14,17),(14,18),(15,19),(16,19),(17,19),(18,19)],20)
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,1,1,0,1,0,0,0,0]
=> ([(3,4)],5)
=> ?
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,1,1,1,0,0,0,0,0]
=> ([],5)
=> ?
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
Description
The breadth of a lattice.
The '''breadth''' of a lattice is the least integer $b$ such that any join $x_1\vee x_2\vee\cdots\vee x_n$, with $n > b$, can be expressed as a join over a proper subset of $\{x_1,x_2,\ldots,x_n\}$.
Matching statistic: St000455
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
Mp00046: Ordered trees —to graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 25% ●values known / values provided: 28%●distinct values known / distinct values provided: 25%
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
Mp00046: Ordered trees —to graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 25% ●values known / values provided: 28%●distinct values known / distinct values provided: 25%
Values
[1,0]
=> [1,1,0,0]
=> [[[]]]
=> ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [[[],[]]]
=> ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [[[[]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 - 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [[[],[[]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {2,2,6} - 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [[[[]],[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {2,2,6} - 1
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {2,2,6} - 1
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [[[[[]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[[],[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [[[],[],[[]]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ? ∊ {2,4,6,6,6,6,6,24} - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [[[],[[]],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ? ∊ {2,4,6,6,6,6,6,24} - 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [[[],[[],[]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [[[],[[[]]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? ∊ {2,4,6,6,6,6,6,24} - 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[[[]],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ? ∊ {2,4,6,6,6,6,6,24} - 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[[[]],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[[[],[]],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[[[],[],[]]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ? ∊ {2,4,6,6,6,6,6,24} - 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [[[[],[[]]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[[[[]]],[]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? ∊ {2,4,6,6,6,6,6,24} - 1
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[[[[]],[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[[[[],[]]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? ∊ {2,4,6,6,6,6,6,24} - 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[[[[[]]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? ∊ {2,4,6,6,6,6,6,24} - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [[[],[],[],[],[]]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [[[],[],[],[[]]]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ? ∊ {2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120} - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [[[],[],[[]],[]]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ? ∊ {2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120} - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [[[],[],[[],[]]]]
=> ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ? ∊ {2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120} - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [[[],[],[[[]]]]]
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ? ∊ {2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120} - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [[[],[[]],[],[]]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ? ∊ {2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120} - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [[[],[[]],[[]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [[[],[[],[]],[]]]
=> ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ? ∊ {2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120} - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [[[],[[],[],[]]]]
=> ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ? ∊ {2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120} - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [[[],[[],[[]]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ? ∊ {2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120} - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [[[],[[[]]],[]]]
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ? ∊ {2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120} - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [[[],[[[]],[]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ? ∊ {2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120} - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [[[],[[[],[]]]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> ? ∊ {2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120} - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [[[],[[[[]]]]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> ? ∊ {2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120} - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [[[[]],[],[],[]]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ? ∊ {2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120} - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [[[[]],[],[[]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [[[[]],[[]],[]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [[[[]],[[],[]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ? ∊ {2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120} - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [[[[]],[[[]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ? ∊ {2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120} - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [[[[],[]],[],[]]]
=> ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ? ∊ {2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120} - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [[[[],[]],[[]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ? ∊ {2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120} - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [[[[],[],[]],[]]]
=> ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ? ∊ {2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120} - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [[[[],[],[],[]]]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ? ∊ {2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120} - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [[[[],[],[[]]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [[[[],[[]]],[]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ? ∊ {2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120} - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [[[[],[[]],[]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [[[[],[[],[]]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ? ∊ {2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120} - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [[[[],[[[]]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ? ∊ {2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120} - 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [[[[[]]],[],[]]]
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ? ∊ {2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120} - 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [[[[[]]],[[]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ? ∊ {2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120} - 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [[[[[]],[]],[]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ? ∊ {2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120} - 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [[[[[]],[],[]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> 1 = 2 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [[[[[]],[[]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [[[[[],[]]],[]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> ? ∊ {2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120} - 1
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [[[[[],[]],[]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ? ∊ {2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120} - 1
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [[[[[],[],[]]]]]
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ? ∊ {2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120} - 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [[[[[],[[]]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ? ∊ {2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120} - 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [[[[[[]]]],[]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> ? ∊ {2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120} - 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[[[[[]]],[]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ? ∊ {2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120} - 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [[[[[[]],[]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ? ∊ {2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120} - 1
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [[[[[[],[]]]]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> ? ∊ {2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120} - 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [[[[[[[]]]]]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ? ∊ {2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120} - 1
Description
The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
Matching statistic: St001232
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 25%
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 25%
Values
[1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {2,6}
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {2,6}
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? ∊ {2,2,2,4,6,6,6,6,6,24}
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {2,2,2,4,6,6,6,6,6,24}
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? ∊ {2,2,2,4,6,6,6,6,6,24}
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {2,2,2,4,6,6,6,6,6,24}
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? ∊ {2,2,2,4,6,6,6,6,6,24}
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {2,2,2,4,6,6,6,6,6,24}
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? ∊ {2,2,2,4,6,6,6,6,6,24}
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {2,2,2,4,6,6,6,6,6,24}
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? ∊ {2,2,2,4,6,6,6,6,6,24}
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {2,2,2,4,6,6,6,6,6,24}
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000260
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 25%
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 25%
Values
[1,0]
=> [1] => [1] => ([],1)
=> 0 = 1 - 1
[1,0,1,0]
=> [2,1] => [2] => ([],2)
=> ? ∊ {1,2} - 1
[1,1,0,0]
=> [1,2] => [2] => ([],2)
=> ? ∊ {1,2} - 1
[1,0,1,0,1,0]
=> [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[1,0,1,1,0,0]
=> [2,3,1] => [3] => ([],3)
=> ? ∊ {1,2,2,6} - 1
[1,1,0,0,1,0]
=> [3,1,2] => [3] => ([],3)
=> ? ∊ {1,2,2,6} - 1
[1,1,0,1,0,0]
=> [2,1,3] => [3] => ([],3)
=> ? ∊ {1,2,2,6} - 1
[1,1,1,0,0,0]
=> [1,2,3] => [3] => ([],3)
=> ? ∊ {1,2,2,6} - 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {1,2,2,2,4,6,6,6,6,6,24} - 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4] => ([],4)
=> ? ∊ {1,2,2,2,4,6,6,6,6,6,24} - 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,3] => ([(2,3)],4)
=> ? ∊ {1,2,2,2,4,6,6,6,6,6,24} - 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4] => ([],4)
=> ? ∊ {1,2,2,2,4,6,6,6,6,6,24} - 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {1,2,2,2,4,6,6,6,6,6,24} - 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {1,2,2,2,4,6,6,6,6,6,24} - 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [4] => ([],4)
=> ? ∊ {1,2,2,2,4,6,6,6,6,6,24} - 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [4] => ([],4)
=> ? ∊ {1,2,2,2,4,6,6,6,6,6,24} - 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [4] => ([],4)
=> ? ∊ {1,2,2,2,4,6,6,6,6,6,24} - 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [4] => ([],4)
=> ? ∊ {1,2,2,2,4,6,6,6,6,6,24} - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4] => ([],4)
=> ? ∊ {1,2,2,2,4,6,6,6,6,6,24} - 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120} - 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120} - 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120} - 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {1,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120} - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5] => ([],5)
=> ? ∊ {1,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120} - 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120} - 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120} - 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,4] => ([(3,4)],5)
=> ? ∊ {1,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120} - 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,4] => ([(3,4)],5)
=> ? ∊ {1,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120} - 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [5] => ([],5)
=> ? ∊ {1,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120} - 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120} - 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120} - 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120} - 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120} - 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120} - 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120} - 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120} - 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {1,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120} - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [5] => ([],5)
=> ? ∊ {1,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120} - 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [1,4] => ([(3,4)],5)
=> ? ∊ {1,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120} - 1
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [5] => ([],5)
=> ? ∊ {1,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120} - 1
[1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,4] => ([(3,4)],5)
=> ? ∊ {1,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120} - 1
[1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => [1,4] => ([(3,4)],5)
=> ? ∊ {1,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120} - 1
[1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => [5] => ([],5)
=> ? ∊ {1,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120} - 1
[1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {1,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120} - 1
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {1,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120} - 1
[1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {1,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120} - 1
[1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => [5] => ([],5)
=> ? ∊ {1,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120} - 1
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [5] => ([],5)
=> ? ∊ {1,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120} - 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => [5] => ([],5)
=> ? ∊ {1,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120} - 1
[1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => [5] => ([],5)
=> ? ∊ {1,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120} - 1
[1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => [5] => ([],5)
=> ? ∊ {1,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120} - 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [5] => ([],5)
=> ? ∊ {1,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120} - 1
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
Matching statistic: St001621
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001621: Lattices ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 25%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001621: Lattices ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 25%
Values
[1,0]
=> [1] => ([],1)
=> ([(0,1)],2)
=> 1
[1,0,1,0]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,0,0]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1
[1,0,1,0,1,0]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,1,0,0]
=> [2,3,1] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,1,0,0,1,0]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,1,0,1,0,0]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[1,1,1,0,0,0]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? ∊ {2,2,2,4,6,6,6,6,6,24}
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? ∊ {2,2,2,4,6,6,6,6,6,24}
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? ∊ {2,2,2,4,6,6,6,6,6,24}
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => ([(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
=> ? ∊ {2,2,2,4,6,6,6,6,6,24}
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {2,2,2,4,6,6,6,6,6,24}
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? ∊ {2,2,2,4,6,6,6,6,6,24}
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? ∊ {2,2,2,4,6,6,6,6,6,24}
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => ([(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
=> ? ∊ {2,2,2,4,6,6,6,6,6,24}
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> ? ∊ {2,2,2,4,6,6,6,6,6,24}
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {2,2,2,4,6,6,6,6,6,24}
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => ([],5)
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => ([(3,4)],5)
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => ([(3,4)],5)
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => ([(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,6),(1,7),(2,11),(2,12),(2,13),(3,9),(3,10),(3,13),(4,8),(4,10),(4,12),(5,8),(5,9),(5,11),(6,16),(7,16),(8,1),(8,17),(8,18),(9,14),(9,17),(10,15),(10,17),(11,14),(11,18),(12,15),(12,18),(13,14),(13,15),(14,19),(15,19),(17,6),(17,19),(18,7),(18,19),(19,16)],20)
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => ([(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,6),(1,7),(2,1),(2,9),(2,10),(3,8),(3,12),(4,8),(4,11),(5,2),(5,11),(5,12),(6,14),(7,14),(8,13),(9,6),(9,15),(10,7),(10,15),(11,9),(11,13),(12,10),(12,13),(13,15),(15,14)],16)
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => ([(3,4)],5)
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => ([(1,4),(2,3)],5)
=> ([(0,3),(0,4),(0,5),(1,8),(1,10),(2,7),(2,9),(3,11),(3,12),(4,2),(4,11),(4,13),(5,1),(5,12),(5,13),(6,17),(7,15),(8,16),(9,6),(9,15),(10,6),(10,16),(11,7),(11,14),(12,8),(12,14),(13,9),(13,10),(13,14),(14,15),(14,16),(15,17),(16,17)],18)
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => ([(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,6),(1,7),(2,11),(2,12),(2,13),(3,9),(3,10),(3,13),(4,8),(4,10),(4,12),(5,8),(5,9),(5,11),(6,16),(7,16),(8,1),(8,17),(8,18),(9,14),(9,17),(10,15),(10,17),(11,14),(11,18),(12,15),(12,18),(13,14),(13,15),(14,19),(15,19),(17,6),(17,19),(18,7),(18,19),(19,16)],20)
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,7),(2,8),(2,9),(3,9),(3,11),(3,12),(4,8),(4,10),(4,12),(5,7),(5,10),(5,11),(7,13),(7,14),(8,13),(8,15),(9,14),(9,15),(10,13),(10,16),(11,14),(11,16),(12,15),(12,16),(13,17),(14,17),(15,17),(16,1),(16,17),(17,6)],18)
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,8),(2,6),(2,7),(3,9),(3,10),(4,9),(4,11),(5,2),(5,10),(5,11),(6,13),(7,1),(7,13),(9,12),(10,6),(10,12),(11,7),(11,12),(12,13),(13,8)],14)
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,6),(1,7),(2,1),(2,9),(2,10),(3,8),(3,12),(4,8),(4,11),(5,2),(5,11),(5,12),(6,14),(7,14),(8,13),(9,6),(9,15),(10,7),(10,15),(11,9),(11,13),(12,10),(12,13),(13,15),(15,14)],16)
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,8),(2,6),(2,7),(3,9),(3,10),(4,9),(4,11),(5,2),(5,10),(5,11),(6,13),(7,1),(7,13),(9,12),(10,6),(10,12),(11,7),(11,12),(12,13),(13,8)],14)
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
=> ([(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(3,7),(3,8),(4,6),(4,8),(5,1),(5,9),(6,11),(7,11),(8,5),(8,11),(9,10),(11,9)],12)
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => ([(3,4)],5)
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> ([(0,3),(0,4),(0,5),(1,8),(1,10),(2,7),(2,9),(3,11),(3,12),(4,2),(4,11),(4,13),(5,1),(5,12),(5,13),(6,17),(7,15),(8,16),(9,6),(9,15),(10,6),(10,16),(11,7),(11,14),(12,8),(12,14),(13,9),(13,10),(13,14),(14,15),(14,16),(15,17),(16,17)],18)
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => ([(1,4),(2,3)],5)
=> ([(0,3),(0,4),(0,5),(1,8),(1,10),(2,7),(2,9),(3,11),(3,12),(4,2),(4,11),(4,13),(5,1),(5,12),(5,13),(6,17),(7,15),(8,16),(9,6),(9,15),(10,6),(10,16),(11,7),(11,14),(12,8),(12,14),(13,9),(13,10),(13,14),(14,15),(14,16),(15,17),(16,17)],18)
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(0,5),(1,10),(2,7),(2,8),(3,9),(3,12),(4,9),(4,11),(5,2),(5,11),(5,12),(7,14),(8,14),(9,1),(9,13),(10,6),(11,7),(11,13),(12,8),(12,13),(13,10),(13,14),(14,6)],15)
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => ([(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,6),(1,7),(2,11),(2,12),(2,13),(3,9),(3,10),(3,13),(4,8),(4,10),(4,12),(5,8),(5,9),(5,11),(6,16),(7,16),(8,1),(8,17),(8,18),(9,14),(9,17),(10,15),(10,17),(11,14),(11,18),(12,15),(12,18),(13,14),(13,15),(14,19),(15,19),(17,6),(17,19),(18,7),(18,19),(19,16)],20)
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => ([(0,4),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(0,5),(1,10),(2,7),(2,8),(3,9),(3,12),(4,9),(4,11),(5,2),(5,11),(5,12),(7,14),(8,14),(9,1),(9,13),(10,6),(11,7),(11,13),(12,8),(12,13),(13,10),(13,14),(14,6)],15)
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,7),(2,8),(2,9),(3,9),(3,11),(3,12),(4,8),(4,10),(4,12),(5,7),(5,10),(5,11),(7,13),(7,14),(8,13),(8,15),(9,14),(9,15),(10,13),(10,16),(11,14),(11,16),(12,15),(12,16),(13,17),(14,17),(15,17),(16,1),(16,17),(17,6)],18)
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(2,10),(3,6),(3,8),(4,6),(4,7),(5,2),(5,7),(5,8),(6,11),(7,9),(7,11),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,8),(2,6),(2,7),(3,9),(3,10),(4,9),(4,11),(5,2),(5,10),(5,11),(6,13),(7,1),(7,13),(9,12),(10,6),(10,12),(11,7),(11,12),(12,13),(13,8)],14)
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(2,10),(3,6),(3,8),(4,6),(4,7),(5,2),(5,7),(5,8),(6,11),(7,9),(7,11),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => ([(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,6),(1,7),(2,1),(2,9),(2,10),(3,8),(3,12),(4,8),(4,11),(5,2),(5,11),(5,12),(6,14),(7,14),(8,13),(9,6),(9,15),(10,7),(10,15),(11,9),(11,13),(12,10),(12,13),(13,15),(15,14)],16)
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,8),(2,6),(2,7),(3,9),(3,10),(4,9),(4,11),(5,2),(5,10),(5,11),(6,13),(7,1),(7,13),(9,12),(10,6),(10,12),(11,7),(11,12),(12,13),(13,8)],14)
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(2,10),(3,6),(3,8),(4,6),(4,7),(5,2),(5,7),(5,8),(6,11),(7,9),(7,11),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5)
=> ([(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(3,7),(3,8),(4,6),(4,8),(5,1),(5,9),(6,11),(7,11),(8,5),(8,11),(9,10),(11,9)],12)
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,2),(0,3),(0,4),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,1),(6,9),(7,9),(8,9),(9,5)],10)
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
Description
The number of atoms of a lattice.
An element of a lattice is an '''atom''' if it covers the least element.
Matching statistic: St001605
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001605: Integer partitions ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 25%
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001605: Integer partitions ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 25%
Values
[1,0]
=> [1] => [1]
=> []
=> ? = 1
[1,0,1,0]
=> [2,1] => [2]
=> []
=> ? ∊ {1,2}
[1,1,0,0]
=> [1,2] => [1,1]
=> [1]
=> ? ∊ {1,2}
[1,0,1,0,1,0]
=> [3,2,1] => [3]
=> []
=> ? ∊ {1,2,2,2,6}
[1,0,1,1,0,0]
=> [2,3,1] => [2,1]
=> [1]
=> ? ∊ {1,2,2,2,6}
[1,1,0,0,1,0]
=> [3,1,2] => [2,1]
=> [1]
=> ? ∊ {1,2,2,2,6}
[1,1,0,1,0,0]
=> [2,1,3] => [2,1]
=> [1]
=> ? ∊ {1,2,2,2,6}
[1,1,1,0,0,0]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> ? ∊ {1,2,2,2,6}
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4]
=> []
=> ? ∊ {1,2,2,2,2,2,4,6,6,6,6,6,24}
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [3,1]
=> [1]
=> ? ∊ {1,2,2,2,2,2,4,6,6,6,6,6,24}
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [3,1]
=> [1]
=> ? ∊ {1,2,2,2,2,2,4,6,6,6,6,6,24}
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [3,1]
=> [1]
=> ? ∊ {1,2,2,2,2,2,4,6,6,6,6,6,24}
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,1,1]
=> [1,1]
=> ? ∊ {1,2,2,2,2,2,4,6,6,6,6,6,24}
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [3,1]
=> [1]
=> ? ∊ {1,2,2,2,2,2,4,6,6,6,6,6,24}
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,1]
=> [1,1]
=> ? ∊ {1,2,2,2,2,2,4,6,6,6,6,6,24}
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [3,1]
=> [1]
=> ? ∊ {1,2,2,2,2,2,4,6,6,6,6,6,24}
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,1]
=> [1]
=> ? ∊ {1,2,2,2,2,2,4,6,6,6,6,6,24}
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [2,1,1]
=> [1,1]
=> ? ∊ {1,2,2,2,2,2,4,6,6,6,6,6,24}
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,1,1]
=> [1,1]
=> ? ∊ {1,2,2,2,2,2,4,6,6,6,6,6,24}
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [2,1,1]
=> [1,1]
=> ? ∊ {1,2,2,2,2,2,4,6,6,6,6,6,24}
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,1]
=> [1,1]
=> ? ∊ {1,2,2,2,2,2,4,6,6,6,6,6,24}
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [5]
=> []
=> ? ∊ {1,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [4,1]
=> [1]
=> ? ∊ {1,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [4,1]
=> [1]
=> ? ∊ {1,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [4,1]
=> [1]
=> ? ∊ {1,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [3,1,1]
=> [1,1]
=> ? ∊ {1,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [4,1]
=> [1]
=> ? ∊ {1,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [3,1,1]
=> [1,1]
=> ? ∊ {1,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [4,1]
=> [1]
=> ? ∊ {1,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [4,1]
=> [1]
=> ? ∊ {1,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,1,1]
=> [1,1]
=> ? ∊ {1,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [3,1,1]
=> [1,1]
=> ? ∊ {1,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [3,1,1]
=> [1,1]
=> ? ∊ {1,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [3,1,1]
=> [1,1]
=> ? ∊ {1,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [4,1]
=> [1]
=> ? ∊ {1,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [3,1,1]
=> [1,1]
=> ? ∊ {1,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [3,1,1]
=> [1,1]
=> ? ∊ {1,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [3,1,1]
=> [1,1]
=> ? ∊ {1,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [4,1]
=> [1]
=> ? ∊ {1,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [3,1,1]
=> [1,1]
=> ? ∊ {1,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [4,1]
=> [1]
=> ? ∊ {1,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [4,1]
=> [1]
=> ? ∊ {1,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [3,1,1]
=> [1,1]
=> ? ∊ {1,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [3,1,1]
=> [1,1]
=> ? ∊ {1,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [3,1,1]
=> [1,1]
=> ? ∊ {1,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [3,1,1]
=> [1,1]
=> ? ∊ {1,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [3,1,1]
=> [1,1]
=> ? ∊ {1,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [3,1,1]
=> [1,1]
=> ? ∊ {1,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => [3,1,1]
=> [1,1]
=> ? ∊ {1,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [3,1,1]
=> [1,1]
=> ? ∊ {1,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 6
Description
The number of colourings of a cycle such that the multiplicities of colours are given by a partition.
Two colourings are considered equal, if they are obtained by an action of the cyclic group.
This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Matching statistic: St001878
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 25%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 25%
Values
[1,0]
=> [1] => ([],1)
=> ([(0,1)],2)
=> ? = 1
[1,0,1,0]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,0,0]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1
[1,0,1,0,1,0]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,1,0,0]
=> [2,3,1] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,1,0,0,1,0]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,1,0,1,0,0]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[1,1,1,0,0,0]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? ∊ {2,2,2,4,6,6,6,6,6,24}
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? ∊ {2,2,2,4,6,6,6,6,6,24}
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? ∊ {2,2,2,4,6,6,6,6,6,24}
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => ([(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
=> ? ∊ {2,2,2,4,6,6,6,6,6,24}
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {2,2,2,4,6,6,6,6,6,24}
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? ∊ {2,2,2,4,6,6,6,6,6,24}
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? ∊ {2,2,2,4,6,6,6,6,6,24}
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => ([(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
=> ? ∊ {2,2,2,4,6,6,6,6,6,24}
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> ? ∊ {2,2,2,4,6,6,6,6,6,24}
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {2,2,2,4,6,6,6,6,6,24}
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => ([],5)
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => ([(3,4)],5)
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => ([(3,4)],5)
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => ([(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,6),(1,7),(2,11),(2,12),(2,13),(3,9),(3,10),(3,13),(4,8),(4,10),(4,12),(5,8),(5,9),(5,11),(6,16),(7,16),(8,1),(8,17),(8,18),(9,14),(9,17),(10,15),(10,17),(11,14),(11,18),(12,15),(12,18),(13,14),(13,15),(14,19),(15,19),(17,6),(17,19),(18,7),(18,19),(19,16)],20)
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => ([(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,6),(1,7),(2,1),(2,9),(2,10),(3,8),(3,12),(4,8),(4,11),(5,2),(5,11),(5,12),(6,14),(7,14),(8,13),(9,6),(9,15),(10,7),(10,15),(11,9),(11,13),(12,10),(12,13),(13,15),(15,14)],16)
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => ([(3,4)],5)
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => ([(1,4),(2,3)],5)
=> ([(0,3),(0,4),(0,5),(1,8),(1,10),(2,7),(2,9),(3,11),(3,12),(4,2),(4,11),(4,13),(5,1),(5,12),(5,13),(6,17),(7,15),(8,16),(9,6),(9,15),(10,6),(10,16),(11,7),(11,14),(12,8),(12,14),(13,9),(13,10),(13,14),(14,15),(14,16),(15,17),(16,17)],18)
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => ([(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,6),(1,7),(2,11),(2,12),(2,13),(3,9),(3,10),(3,13),(4,8),(4,10),(4,12),(5,8),(5,9),(5,11),(6,16),(7,16),(8,1),(8,17),(8,18),(9,14),(9,17),(10,15),(10,17),(11,14),(11,18),(12,15),(12,18),(13,14),(13,15),(14,19),(15,19),(17,6),(17,19),(18,7),(18,19),(19,16)],20)
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,7),(2,8),(2,9),(3,9),(3,11),(3,12),(4,8),(4,10),(4,12),(5,7),(5,10),(5,11),(7,13),(7,14),(8,13),(8,15),(9,14),(9,15),(10,13),(10,16),(11,14),(11,16),(12,15),(12,16),(13,17),(14,17),(15,17),(16,1),(16,17),(17,6)],18)
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,8),(2,6),(2,7),(3,9),(3,10),(4,9),(4,11),(5,2),(5,10),(5,11),(6,13),(7,1),(7,13),(9,12),(10,6),(10,12),(11,7),(11,12),(12,13),(13,8)],14)
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,6),(1,7),(2,1),(2,9),(2,10),(3,8),(3,12),(4,8),(4,11),(5,2),(5,11),(5,12),(6,14),(7,14),(8,13),(9,6),(9,15),(10,7),(10,15),(11,9),(11,13),(12,10),(12,13),(13,15),(15,14)],16)
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,8),(2,6),(2,7),(3,9),(3,10),(4,9),(4,11),(5,2),(5,10),(5,11),(6,13),(7,1),(7,13),(9,12),(10,6),(10,12),(11,7),(11,12),(12,13),(13,8)],14)
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
=> ([(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(3,7),(3,8),(4,6),(4,8),(5,1),(5,9),(6,11),(7,11),(8,5),(8,11),(9,10),(11,9)],12)
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => ([(3,4)],5)
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> ([(0,3),(0,4),(0,5),(1,8),(1,10),(2,7),(2,9),(3,11),(3,12),(4,2),(4,11),(4,13),(5,1),(5,12),(5,13),(6,17),(7,15),(8,16),(9,6),(9,15),(10,6),(10,16),(11,7),(11,14),(12,8),(12,14),(13,9),(13,10),(13,14),(14,15),(14,16),(15,17),(16,17)],18)
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => ([(1,4),(2,3)],5)
=> ([(0,3),(0,4),(0,5),(1,8),(1,10),(2,7),(2,9),(3,11),(3,12),(4,2),(4,11),(4,13),(5,1),(5,12),(5,13),(6,17),(7,15),(8,16),(9,6),(9,15),(10,6),(10,16),(11,7),(11,14),(12,8),(12,14),(13,9),(13,10),(13,14),(14,15),(14,16),(15,17),(16,17)],18)
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(0,5),(1,10),(2,7),(2,8),(3,9),(3,12),(4,9),(4,11),(5,2),(5,11),(5,12),(7,14),(8,14),(9,1),(9,13),(10,6),(11,7),(11,13),(12,8),(12,13),(13,10),(13,14),(14,6)],15)
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => ([(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,6),(1,7),(2,11),(2,12),(2,13),(3,9),(3,10),(3,13),(4,8),(4,10),(4,12),(5,8),(5,9),(5,11),(6,16),(7,16),(8,1),(8,17),(8,18),(9,14),(9,17),(10,15),(10,17),(11,14),(11,18),(12,15),(12,18),(13,14),(13,15),(14,19),(15,19),(17,6),(17,19),(18,7),(18,19),(19,16)],20)
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => ([(0,4),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(0,5),(1,10),(2,7),(2,8),(3,9),(3,12),(4,9),(4,11),(5,2),(5,11),(5,12),(7,14),(8,14),(9,1),(9,13),(10,6),(11,7),(11,13),(12,8),(12,13),(13,10),(13,14),(14,6)],15)
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,7),(2,8),(2,9),(3,9),(3,11),(3,12),(4,8),(4,10),(4,12),(5,7),(5,10),(5,11),(7,13),(7,14),(8,13),(8,15),(9,14),(9,15),(10,13),(10,16),(11,14),(11,16),(12,15),(12,16),(13,17),(14,17),(15,17),(16,1),(16,17),(17,6)],18)
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(2,10),(3,6),(3,8),(4,6),(4,7),(5,2),(5,7),(5,8),(6,11),(7,9),(7,11),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,8),(2,6),(2,7),(3,9),(3,10),(4,9),(4,11),(5,2),(5,10),(5,11),(6,13),(7,1),(7,13),(9,12),(10,6),(10,12),(11,7),(11,12),(12,13),(13,8)],14)
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(2,10),(3,6),(3,8),(4,6),(4,7),(5,2),(5,7),(5,8),(6,11),(7,9),(7,11),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => ([(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,6),(1,7),(2,1),(2,9),(2,10),(3,8),(3,12),(4,8),(4,11),(5,2),(5,11),(5,12),(6,14),(7,14),(8,13),(9,6),(9,15),(10,7),(10,15),(11,9),(11,13),(12,10),(12,13),(13,15),(15,14)],16)
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,8),(2,6),(2,7),(3,9),(3,10),(4,9),(4,11),(5,2),(5,10),(5,11),(6,13),(7,1),(7,13),(9,12),(10,6),(10,12),(11,7),(11,12),(12,13),(13,8)],14)
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(2,10),(3,6),(3,8),(4,6),(4,7),(5,2),(5,7),(5,8),(6,11),(7,9),(7,11),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5)
=> ([(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(3,7),(3,8),(4,6),(4,8),(5,1),(5,9),(6,11),(7,11),(8,5),(8,11),(9,10),(11,9)],12)
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,2),(0,3),(0,4),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,1),(6,9),(7,9),(8,9),(9,5)],10)
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? ∊ {2,2,2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,12,12,18,24,24,24,24,24,24,24,120}
[1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
Description
The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
The following 27 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001545The second Elser number of a connected graph. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000706The product of the factorials of the multiplicities of an integer partition. St000707The product of the factorials of the parts. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001378The product of the cohook lengths of the integer partition. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St000454The largest eigenvalue of a graph if it is integral. St000307The number of rowmotion orbits of a poset. St000558The number of occurrences of the pattern {{1,2}} in a set partition. St000886The number of permutations with the same antidiagonal sums. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001735The number of permutations with the same set of runs. St001537The number of cyclic crossings of a permutation. St001399The distinguishing number of a poset. St001644The dimension of a graph. St001638The book thickness of a graph. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!