- FindStat

Your data matches 74 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001637
(load all 2 compositions to match this statistic)

Mapping

St001637: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],2)
 => 2
([(0,1)],2)
 => 1
([],3)
 => 0
([(1,2)],3)
 => 2
([(0,1),(0,2)],3)
 => 2
([(0,2),(2,1)],3)
 => 2
([(0,2),(1,2)],3)
 => 2
([],4)
 => 0
([(2,3)],4)
 => 0
([(1,2),(1,3)],4)
 => 1
([(0,1),(0,2),(0,3)],4)
 => 0
([(0,2),(0,3),(3,1)],4)
 => 2
([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
([(1,2),(2,3)],4)
 => 2
([(0,3),(3,1),(3,2)],4)
 => 3
([(1,3),(2,3)],4)
 => 1
([(0,3),(1,3),(3,2)],4)
 => 3
([(0,3),(1,3),(2,3)],4)
 => 0
([(0,3),(1,2)],4)
 => 2
([(0,3),(1,2),(1,3)],4)
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => 4
([(0,3),(2,1),(3,2)],4)
 => 3
([(0,3),(1,2),(2,3)],4)
 => 2
([],5)
 => 0
([(3,4)],5)
 => 0
([(2,3),(2,4)],5)
 => 0
([(1,2),(1,3),(1,4)],5)
 => 1
([(0,1),(0,2),(0,3),(0,4)],5)
 => 0
([(0,2),(0,3),(0,4),(4,1)],5)
 => 0
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 0
([(1,3),(1,4),(4,2)],5)
 => 1
([(0,3),(0,4),(4,1),(4,2)],5)
 => 1
([(1,2),(1,3),(2,4),(3,4)],5)
 => 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 3
([(0,3),(0,4),(3,2),(4,1)],5)
 => 2
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 4
([(2,3),(3,4)],5)
 => 0
([(1,4),(4,2),(4,3)],5)
 => 1
([(0,4),(4,1),(4,2),(4,3)],5)
 => 1
([(2,4),(3,4)],5)
 => 0
([(1,4),(2,4),(4,3)],5)
 => 1
([(0,4),(1,4),(4,2),(4,3)],5)
 => 4
([(1,4),(2,4),(3,4)],5)
 => 1
([(0,4),(1,4),(2,4),(4,3)],5)
 => 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => 0
([(0,4),(1,4),(2,3)],5)
 => 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => 2
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2

Description

The number of (upper) dissectors of a poset.

Matching statistic: St001615
(load all 2 compositions to match this statistic)

Mapping

Mp00282: Posets —Dedekind-MacNeille completion⟶ Lattices
St001615: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],2)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
([(0,1)],2)
 => ([(0,1)],2)
 => 1
([],3)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 0
([(1,2)],3)
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 2
([(0,1),(0,2)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2
([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 0
([(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
 => 0
([(1,2),(1,3)],4)
 => ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
 => 1
([(0,1),(0,2),(0,3)],4)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 0
([(0,2),(0,3),(3,1)],4)
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 2
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
([(1,2),(2,3)],4)
 => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
 => 2
([(0,3),(3,1),(3,2)],4)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 3
([(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => 1
([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 3
([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 0
([(0,3),(1,2)],4)
 => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
 => 2
([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 3
([(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)
 => 4
([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 3
([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 2
([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 0
([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
 => 0
([(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
 => 0
([(1,2),(1,3),(1,4)],5)
 => ([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
 => 1
([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 0
([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
 => 0
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 0
([(1,3),(1,4),(4,2)],5)
 => ([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
 => 1
([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
 => 1
([(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
 => 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 3
([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
 => 2
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => 4
([(2,3),(3,4)],5)
 => ([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
 => 0
([(1,4),(4,2),(4,3)],5)
 => ([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
 => 1
([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
 => 1
([(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => 0
([(1,4),(2,4),(4,3)],5)
 => ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
 => 1
([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => 4
([(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
 => 1
([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 0
([(0,4),(1,4),(2,3)],5)
 => ([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
 => 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => 2
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,2),(0,3),(0,4),(1,6),(2,7),(3,7),(4,5),(5,6),(7,1),(7,5)],8)
 => 2

Description

The number of join prime elements of a lattice. An element $x$ of a lattice $L$ is join-prime (or coprime) if $x \leq a \vee b$ implies $x \leq a$ or $x \leq b$ for every $a, b \in L$.

Matching statistic: St001636

Mapping

Mp00282: Posets —Dedekind-MacNeille completion⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St001636: Posets ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 100%

Values

([],2)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2 = 1 + 1
([],3)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
([(1,2)],3)
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 3 = 2 + 1
([(0,1),(0,2)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 1 = 0 + 1
([(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
 => ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
 => 1 = 0 + 1
([(1,2),(1,3)],4)
 => ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
 => ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
 => 2 = 1 + 1
([(0,1),(0,2),(0,3)],4)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
([(0,2),(0,3),(3,1)],4)
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 3 = 2 + 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
([(1,2),(2,3)],4)
 => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
 => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
 => 3 = 2 + 1
([(0,3),(3,1),(3,2)],4)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 4 = 3 + 1
([(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 4 = 3 + 1
([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
([(0,3),(1,2)],4)
 => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
 => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
 => 3 = 2 + 1
([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 4 = 3 + 1
([(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)
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => 5 = 4 + 1
([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 3 + 1
([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 3 = 2 + 1
([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 1 = 0 + 1
([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
 => ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4} + 1
([(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
 => ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
 => 1 = 0 + 1
([(1,2),(1,3),(1,4)],5)
 => ([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
 => ([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
 => 2 = 1 + 1
([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 1 = 0 + 1
([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
 => ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
 => 1 = 0 + 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
([(1,3),(1,4),(4,2)],5)
 => ([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
 => ([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4} + 1
([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
 => ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
 => 2 = 1 + 1
([(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
 => 3 = 2 + 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 4 = 3 + 1
([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
 => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
 => 3 = 2 + 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 4 = 3 + 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => 5 = 4 + 1
([(2,3),(3,4)],5)
 => ([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
 => ([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4} + 1
([(1,4),(4,2),(4,3)],5)
 => ([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
 => ([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4} + 1
([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
 => ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
 => 2 = 1 + 1
([(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => 1 = 0 + 1
([(1,4),(2,4),(4,3)],5)
 => ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
 => ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4} + 1
([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => 5 = 4 + 1
([(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
 => 2 = 1 + 1
([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => 2 = 1 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 1 = 0 + 1
([(0,4),(1,4),(2,3)],5)
 => ([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
 => ([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4} + 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => 3 = 2 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,2),(0,3),(0,4),(1,6),(2,7),(3,7),(4,5),(5,6),(7,1),(7,5)],8)
 => ([(0,2),(0,3),(0,4),(1,6),(2,7),(3,7),(4,5),(5,6),(7,1),(7,5)],8)
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4} + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,7),(4,7),(5,7),(7,1),(7,2)],8)
 => ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,7),(4,7),(5,7),(7,1),(7,2)],8)
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4} + 1
([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 5 = 4 + 1
([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
([(0,4),(1,4),(2,3),(2,4)],5)
 => ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,1),(4,6),(6,5)],7)
 => ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,1),(4,6),(6,5)],7)
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4} + 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
 => ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
 => 1 = 0 + 1
([(1,4),(2,3)],5)
 => ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1)],7)
 => ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1)],7)
 => 1 = 0 + 1
([(1,4),(2,3),(2,4)],5)
 => ([(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,5),(5,6)],7)
 => ([(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,5),(5,6)],7)
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4} + 1
([(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(4,3),(4,5),(5,6)],7)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(4,3),(4,5),(5,6)],7)
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4} + 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 4 = 3 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,1),(6,2)],8)
 => ([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,1),(6,2)],8)
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4} + 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,2),(7,1),(7,5)],8)
 => ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,2),(7,1),(7,5)],8)
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4} + 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => 5 = 4 + 1
([(0,4),(1,2),(1,4),(4,3)],5)
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,2),(4,6),(6,1)],7)
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,2),(4,6),(6,1)],7)
 => 4 = 3 + 1
([(0,4),(1,2),(1,3),(3,4)],5)
 => ([(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(5,6)],7)
 => ([(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(5,6)],7)
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4} + 1
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,6),(6,2),(6,3)],8)
 => ([(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,6),(6,2),(6,3)],8)
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4} + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,1),(6,2),(6,3)],8)
 => ([(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,1),(6,2),(6,3)],8)
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4} + 1
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4} + 1
([(0,3),(1,2),(1,4),(3,4)],5)
 => ([(0,3),(0,4),(1,6),(2,5),(3,2),(4,1),(4,5),(5,6)],7)
 => ([(0,3),(0,4),(1,6),(2,5),(3,2),(4,1),(4,5),(5,6)],7)
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4} + 1
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,4),(0,5),(1,7),(2,6),(3,6),(4,7),(5,1),(7,2),(7,3)],8)
 => ([(0,4),(0,5),(1,7),(2,6),(3,6),(4,7),(5,1),(7,2),(7,3)],8)
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4} + 1
([(1,4),(3,2),(4,3)],5)
 => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
 => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4} + 1
([(1,4),(2,3),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,5),(4,1),(5,6)],7)
 => ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,5),(4,1),(5,6)],7)
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4} + 1
([(0,3),(1,4),(4,2)],5)
 => ([(0,4),(0,5),(1,6),(2,6),(3,2),(4,3),(5,1)],7)
 => ([(0,4),(0,5),(1,6),(2,6),(3,2),(4,3),(5,1)],7)
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4} + 1
([(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,5),(4,3),(5,6)],7)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,5),(4,3),(5,6)],7)
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4} + 1

Description

The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset.

Matching statistic: St001632
(load all 2 compositions to match this statistic)

Mapping

Mp00198: Posets —incomparability graph⟶ Graphs
Mp00243: Graphs —weak duplicate order⟶ Posets
Mp00125: Posets —dual poset⟶ Posets
St001632: Posets ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 60%

Values

([],2)
 => ([(0,1)],2)
 => ([],2)
 => ([],2)
 => ? ∊ {1,2}
([(0,1)],2)
 => ([],2)
 => ([],1)
 => ([],1)
 => ? ∊ {1,2}
([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => ([],3)
 => ? ∊ {0,2,2}
([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([],2)
 => ([],2)
 => ? ∊ {0,2,2}
([(0,1),(0,2)],3)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2)],3)
 => 2
([(0,2),(2,1)],3)
 => ([],3)
 => ([],1)
 => ([],1)
 => ? ∊ {0,2,2}
([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2)],3)
 => 2
([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],4)
 => ([],4)
 => ? ∊ {0,0,1,1,3,3,3,3,4}
([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],3)
 => ([],3)
 => ? ∊ {0,0,1,1,3,3,3,3,4}
([(1,2),(1,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3)],4)
 => ? ∊ {0,0,1,1,3,3,3,3,4}
([(0,1),(0,2),(0,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3)],4)
 => 0
([(0,2),(0,3),(3,1)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2)],3)
 => 2
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2)],3)
 => 2
([(1,2),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([],2)
 => ([],2)
 => ? ∊ {0,0,1,1,3,3,3,3,4}
([(0,3),(3,1),(3,2)],4)
 => ([(2,3)],4)
 => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2)],3)
 => 2
([(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3)],4)
 => ? ∊ {0,0,1,1,3,3,3,3,4}
([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2)],3)
 => 2
([(0,3),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3)],4)
 => 0
([(0,3),(1,2)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([],2)
 => ([],2)
 => ? ∊ {0,0,1,1,3,3,3,3,4}
([(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => ? ∊ {0,0,1,1,3,3,3,3,4}
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2)],4)
 => ([],4)
 => ([],4)
 => ? ∊ {0,0,1,1,3,3,3,3,4}
([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,1,1,3,3,3,3,4}
([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2)],3)
 => 2
([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],5)
 => ([],5)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],4)
 => ([],4)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
([(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4)],5)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
([(1,2),(1,3),(1,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4)],5)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4)],5)
 => 0
([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3)],4)
 => 0
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(4,1),(4,2)],5)
 => 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3)],4)
 => 0
([(1,3),(1,4),(4,2)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3)],4)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(4,1),(4,2)],5)
 => 1
([(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3)],4)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2)],3)
 => 2
([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2)],3)
 => 2
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(3,2),(4,1)],5)
 => 2
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,4),(2,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4)],5)
 => 0
([(2,3),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],3)
 => ([],3)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
([(1,4),(4,2),(4,3)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3)],4)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3)],4)
 => 0
([(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ([(2,3),(2,4)],5)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
([(1,4),(2,4),(4,3)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3)],4)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(1,4),(2,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4)],5)
 => 0
([(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4)],5)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3)],4)
 => 0
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4)],5)
 => 0
([(0,4),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3)],4)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(3,4)],5)
 => ([(3,4)],5)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],5)
 => ([],5)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2)],3)
 => 2
([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(4,1),(4,2)],5)
 => 1
([(0,4),(1,4),(2,3),(2,4)],5)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3)],5)
 => ([(0,4),(1,2),(1,3)],5)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3)],4)
 => 0
([(1,4),(2,3)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([],3)
 => ([],3)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
([(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
([(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(3,2),(4,1)],5)
 => 2
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([],5)
 => ([],5)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => ([],4)
 => ([],4)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(1,4),(2,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4)],5)
 => 0
([(0,4),(1,2),(1,4),(4,3)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
([(0,4),(1,2),(1,3)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3)],4)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
([(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3)],5)
 => ([(0,4),(1,2),(1,3)],5)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2)],3)
 => 2
([(0,4),(1,2),(1,3),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4)],5)
 => 0
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2)],3)
 => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(4,1),(4,2)],5)
 => 1
([(0,3),(0,4),(1,2),(1,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(3,4)],5)
 => ([(3,4)],5)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],5)
 => ([],5)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(2,3),(2,4)],5)
 => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
([(0,3),(1,2),(1,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => ([],4)
 => ([],4)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
([(1,4),(3,2),(4,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([],2)
 => ([],2)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
([(0,3),(3,4),(4,1),(4,2)],5)
 => ([(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2)],3)
 => 2
([(1,4),(2,3),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ([(1,2),(1,3)],4)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2)],3)
 => 2
([(0,3),(1,4),(4,2)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],2)
 => ([],2)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2)],3)
 => 2
([(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2)],3)
 => 2
([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2)],3)
 => 2
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2)],3)
 => 2

Description

The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset.

Matching statistic: St000443

Mapping

Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000443: Dyck paths ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 60%

Values

([],2)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
([(0,1)],2)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 3 = 2 + 1
([],3)
 => [2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => 3 = 2 + 1
([(1,2)],3)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => ? = 0 + 1
([(0,1),(0,2)],3)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
([(0,2),(2,1)],3)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 3 = 2 + 1
([(0,2),(1,2)],3)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
([],4)
 => [2,2,2,2,2,2,2,2]
 => [1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0,0]
 => ? ∊ {0,0,0,0,1,4} + 1
([(2,3)],4)
 => [6,6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => [1,1,1,0,1,0,1,0,1,1,0,0,0,0,1,0]
 => ? ∊ {0,0,0,0,1,4} + 1
([(1,2),(1,3)],4)
 => [6,2,2]
 => [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,1,0,0,0]
 => ? ∊ {0,0,0,0,1,4} + 1
([(0,1),(0,2),(0,3)],4)
 => [3,2,2,2]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => 4 = 3 + 1
([(0,2),(0,3),(3,1)],4)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,4} + 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 3 = 2 + 1
([(1,2),(2,3)],4)
 => [4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => 3 = 2 + 1
([(0,3),(3,1),(3,2)],4)
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 3 = 2 + 1
([(1,3),(2,3)],4)
 => [6,2,2]
 => [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,1,0,0,0]
 => ? ∊ {0,0,0,0,1,4} + 1
([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 3 = 2 + 1
([(0,3),(1,3),(2,3)],4)
 => [3,2,2,2]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => 4 = 3 + 1
([(0,3),(1,2)],4)
 => [3,3,3]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => 4 = 3 + 1
([(0,3),(1,2),(1,3)],4)
 => [5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => 3 = 2 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
([(0,3),(2,1),(3,2)],4)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => 4 = 3 + 1
([(0,3),(1,2),(2,3)],4)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,4} + 1
([],5)
 => [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => [1,1,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]
 => [1,0,1,1,1,1,1,1,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(3,4)],5)
 => [6,6,6,6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(2,3),(2,4)],5)
 => [6,6,2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,1,0,1,1,0,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(1,2),(1,3),(1,4)],5)
 => [6,2,2,2,2,2,2]
 => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [1,0,1,1,1,1,0,0,1,0,1,1,0,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,1),(0,2),(0,3),(0,4)],5)
 => [3,2,2,2,2,2,2,2]
 => [1,1,0,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,0,1,1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,2),(0,3),(0,4),(4,1)],5)
 => [7,6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [7,2,2]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,1,0,1,0,0,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => 3 = 2 + 1
([(1,3),(1,4),(4,2)],5)
 => [14]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,3),(0,4),(4,1),(4,2)],5)
 => [7,2,2]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,1,0,1,0,0,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(1,2),(1,3),(2,4),(3,4)],5)
 => [4,4,2,2]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 3 = 2 + 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 4 = 3 + 1
([(0,3),(0,4),(3,2),(4,1)],5)
 => [4,3,3]
 => [1,1,1,0,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 4 = 3 + 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
([(2,3),(3,4)],5)
 => [4,4,4,4]
 => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,1,0,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(1,4),(4,2),(4,3)],5)
 => [4,4,2,2]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 3 = 2 + 1
([(0,4),(4,1),(4,2),(4,3)],5)
 => [4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => 3 = 2 + 1
([(2,4),(3,4)],5)
 => [6,6,2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,1,0,1,1,0,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(1,4),(2,4),(4,3)],5)
 => [4,4,2,2]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 3 = 2 + 1
([(0,4),(1,4),(4,2),(4,3)],5)
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
([(1,4),(2,4),(3,4)],5)
 => [6,2,2,2,2,2,2]
 => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [1,0,1,1,1,1,0,0,1,0,1,1,0,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,4),(1,4),(2,4),(4,3)],5)
 => [4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => 3 = 2 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [3,2,2,2,2,2,2,2]
 => [1,1,0,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,0,1,1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,4),(1,4),(2,3)],5)
 => [6,3,3,3]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [8,3,2]
 => [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5,3,2,2]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 3 = 2 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,4),(1,4),(2,3),(4,2)],5)
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 4 = 3 + 1
([(0,4),(1,3),(2,3),(3,4)],5)
 => [7,2,2]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,1,0,1,0,0,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,4),(1,4),(2,3),(2,4)],5)
 => [6,5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,1,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [7,6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(1,4),(2,3)],5)
 => [6,6,6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(1,4),(2,3),(2,4)],5)
 => [10,6]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,1,1,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,4),(1,2),(1,4),(2,3)],5)
 => [8,3]
 => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
 => [1,1,1,1,0,1,1,0,1,0,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => [5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 4 = 3 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [6,2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,1,1,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => [7,2]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,1,0,1,0,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
([(0,4),(1,2),(1,4),(4,3)],5)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,4),(1,2),(1,3)],5)
 => [6,3,3,3]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,4),(1,2),(1,3),(1,4)],5)
 => [6,5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,1,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,2),(0,4),(3,1),(4,3)],5)
 => [5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 4 = 3 + 1
([(0,4),(1,2),(1,3),(3,4)],5)
 => [10,2]
 => [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [7,2,2]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,1,0,1,0,0,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,3),(0,4),(1,2),(1,4)],5)
 => [8,3,2]
 => [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [5,3,2,2]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 3 = 2 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [3,2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,3),(1,2),(1,4),(3,4)],5)
 => [8,3]
 => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
 => [1,1,1,1,0,1,1,0,1,0,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => [7,2]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,1,0,1,0,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(1,4),(3,2),(4,3)],5)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,3),(3,4),(4,1),(4,2)],5)
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 4 = 3 + 1
([(1,4),(2,3),(3,4)],5)
 => [14]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,4),(1,2),(2,4),(4,3)],5)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,3),(1,4),(4,2)],5)
 => [12]
 => [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,4),(3,2),(4,1),(4,3)],5)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,4),(1,2),(2,3),(2,4)],5)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,3),(1,2),(2,4),(3,4)],5)
 => [4,3,3]
 => [1,1,1,0,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
([(0,4),(1,2),(2,3),(3,4)],5)
 => [5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 4 = 3 + 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 4 = 3 + 1

Description

The number of long tunnels of a Dyck path. A long tunnel of a Dyck path is a longest sequence of consecutive usual tunnels, i.e., a longest sequence of tunnels where the end point of one is the starting point of the next. See [1] for the definition of tunnels.

Matching statistic: St001187

Mapping

Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001187: Dyck paths ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 60%

Values

([],2)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
([(0,1)],2)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 3 = 2 + 1
([],3)
 => [2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => 3 = 2 + 1
([(1,2)],3)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => ? = 0 + 1
([(0,1),(0,2)],3)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
([(0,2),(2,1)],3)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 3 = 2 + 1
([(0,2),(1,2)],3)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
([],4)
 => [2,2,2,2,2,2,2,2]
 => [1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0,0]
 => ? ∊ {0,0,0,0,1,4} + 1
([(2,3)],4)
 => [6,6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => [1,1,1,0,1,0,1,0,1,1,0,0,0,0,1,0]
 => ? ∊ {0,0,0,0,1,4} + 1
([(1,2),(1,3)],4)
 => [6,2,2]
 => [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,1,0,0,0]
 => ? ∊ {0,0,0,0,1,4} + 1
([(0,1),(0,2),(0,3)],4)
 => [3,2,2,2]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => 4 = 3 + 1
([(0,2),(0,3),(3,1)],4)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,4} + 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 3 = 2 + 1
([(1,2),(2,3)],4)
 => [4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => 3 = 2 + 1
([(0,3),(3,1),(3,2)],4)
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 3 = 2 + 1
([(1,3),(2,3)],4)
 => [6,2,2]
 => [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,1,0,0,0]
 => ? ∊ {0,0,0,0,1,4} + 1
([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 3 = 2 + 1
([(0,3),(1,3),(2,3)],4)
 => [3,2,2,2]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => 4 = 3 + 1
([(0,3),(1,2)],4)
 => [3,3,3]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => 4 = 3 + 1
([(0,3),(1,2),(1,3)],4)
 => [5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => 3 = 2 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
([(0,3),(2,1),(3,2)],4)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => 4 = 3 + 1
([(0,3),(1,2),(2,3)],4)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,4} + 1
([],5)
 => [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => [1,1,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]
 => [1,0,1,1,1,1,1,1,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(3,4)],5)
 => [6,6,6,6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(2,3),(2,4)],5)
 => [6,6,2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,1,0,1,1,0,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(1,2),(1,3),(1,4)],5)
 => [6,2,2,2,2,2,2]
 => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [1,0,1,1,1,1,0,0,1,0,1,1,0,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,1),(0,2),(0,3),(0,4)],5)
 => [3,2,2,2,2,2,2,2]
 => [1,1,0,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,0,1,1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,2),(0,3),(0,4),(4,1)],5)
 => [7,6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [7,2,2]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,1,0,1,0,0,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => 3 = 2 + 1
([(1,3),(1,4),(4,2)],5)
 => [14]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,3),(0,4),(4,1),(4,2)],5)
 => [7,2,2]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,1,0,1,0,0,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(1,2),(1,3),(2,4),(3,4)],5)
 => [4,4,2,2]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 3 = 2 + 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 4 = 3 + 1
([(0,3),(0,4),(3,2),(4,1)],5)
 => [4,3,3]
 => [1,1,1,0,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 4 = 3 + 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
([(2,3),(3,4)],5)
 => [4,4,4,4]
 => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,1,0,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(1,4),(4,2),(4,3)],5)
 => [4,4,2,2]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 3 = 2 + 1
([(0,4),(4,1),(4,2),(4,3)],5)
 => [4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => 3 = 2 + 1
([(2,4),(3,4)],5)
 => [6,6,2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,1,0,1,1,0,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(1,4),(2,4),(4,3)],5)
 => [4,4,2,2]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 3 = 2 + 1
([(0,4),(1,4),(4,2),(4,3)],5)
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
([(1,4),(2,4),(3,4)],5)
 => [6,2,2,2,2,2,2]
 => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [1,0,1,1,1,1,0,0,1,0,1,1,0,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,4),(1,4),(2,4),(4,3)],5)
 => [4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => 3 = 2 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [3,2,2,2,2,2,2,2]
 => [1,1,0,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,0,1,1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,4),(1,4),(2,3)],5)
 => [6,3,3,3]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [8,3,2]
 => [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5,3,2,2]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 3 = 2 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,4),(1,4),(2,3),(4,2)],5)
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 4 = 3 + 1
([(0,4),(1,3),(2,3),(3,4)],5)
 => [7,2,2]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,1,0,1,0,0,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,4),(1,4),(2,3),(2,4)],5)
 => [6,5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,1,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [7,6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(1,4),(2,3)],5)
 => [6,6,6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(1,4),(2,3),(2,4)],5)
 => [10,6]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,1,1,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,4),(1,2),(1,4),(2,3)],5)
 => [8,3]
 => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
 => [1,1,1,1,0,1,1,0,1,0,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => [5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 4 = 3 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [6,2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,1,1,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => [7,2]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,1,0,1,0,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
([(0,4),(1,2),(1,4),(4,3)],5)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,4),(1,2),(1,3)],5)
 => [6,3,3,3]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,4),(1,2),(1,3),(1,4)],5)
 => [6,5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,1,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,2),(0,4),(3,1),(4,3)],5)
 => [5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 4 = 3 + 1
([(0,4),(1,2),(1,3),(3,4)],5)
 => [10,2]
 => [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [7,2,2]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,1,0,1,0,0,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,3),(0,4),(1,2),(1,4)],5)
 => [8,3,2]
 => [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [5,3,2,2]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 3 = 2 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [3,2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,3),(1,2),(1,4),(3,4)],5)
 => [8,3]
 => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
 => [1,1,1,1,0,1,1,0,1,0,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => [7,2]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,1,0,1,0,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(1,4),(3,2),(4,3)],5)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,3),(3,4),(4,1),(4,2)],5)
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 4 = 3 + 1
([(1,4),(2,3),(3,4)],5)
 => [14]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,4),(1,2),(2,4),(4,3)],5)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,3),(1,4),(4,2)],5)
 => [12]
 => [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,4),(3,2),(4,1),(4,3)],5)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,4),(1,2),(2,3),(2,4)],5)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,3),(1,2),(2,4),(3,4)],5)
 => [4,3,3]
 => [1,1,1,0,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
([(0,4),(1,2),(2,3),(3,4)],5)
 => [5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 4 = 3 + 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 4 = 3 + 1

Description

The number of simple modules with grade at least one in the corresponding Nakayama algebra.

Matching statistic: St001224

Mapping

Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001224: Dyck paths ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 60%

Values

([],2)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
([(0,1)],2)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 3 = 2 + 1
([],3)
 => [2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => 3 = 2 + 1
([(1,2)],3)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => ? = 0 + 1
([(0,1),(0,2)],3)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
([(0,2),(2,1)],3)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 3 = 2 + 1
([(0,2),(1,2)],3)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
([],4)
 => [2,2,2,2,2,2,2,2]
 => [1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0,0]
 => ? ∊ {0,0,0,0,1,4} + 1
([(2,3)],4)
 => [6,6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => [1,1,1,0,1,0,1,0,1,1,0,0,0,0,1,0]
 => ? ∊ {0,0,0,0,1,4} + 1
([(1,2),(1,3)],4)
 => [6,2,2]
 => [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,1,0,0,0]
 => ? ∊ {0,0,0,0,1,4} + 1
([(0,1),(0,2),(0,3)],4)
 => [3,2,2,2]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => 4 = 3 + 1
([(0,2),(0,3),(3,1)],4)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,4} + 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 3 = 2 + 1
([(1,2),(2,3)],4)
 => [4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => 3 = 2 + 1
([(0,3),(3,1),(3,2)],4)
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 3 = 2 + 1
([(1,3),(2,3)],4)
 => [6,2,2]
 => [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,1,0,0,0]
 => ? ∊ {0,0,0,0,1,4} + 1
([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 3 = 2 + 1
([(0,3),(1,3),(2,3)],4)
 => [3,2,2,2]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => 4 = 3 + 1
([(0,3),(1,2)],4)
 => [3,3,3]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => 4 = 3 + 1
([(0,3),(1,2),(1,3)],4)
 => [5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => 3 = 2 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
([(0,3),(2,1),(3,2)],4)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => 4 = 3 + 1
([(0,3),(1,2),(2,3)],4)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,4} + 1
([],5)
 => [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => [1,1,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]
 => [1,0,1,1,1,1,1,1,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(3,4)],5)
 => [6,6,6,6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(2,3),(2,4)],5)
 => [6,6,2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,1,0,1,1,0,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(1,2),(1,3),(1,4)],5)
 => [6,2,2,2,2,2,2]
 => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [1,0,1,1,1,1,0,0,1,0,1,1,0,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,1),(0,2),(0,3),(0,4)],5)
 => [3,2,2,2,2,2,2,2]
 => [1,1,0,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,0,1,1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,2),(0,3),(0,4),(4,1)],5)
 => [7,6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [7,2,2]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,1,0,1,0,0,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => 3 = 2 + 1
([(1,3),(1,4),(4,2)],5)
 => [14]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,3),(0,4),(4,1),(4,2)],5)
 => [7,2,2]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,1,0,1,0,0,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(1,2),(1,3),(2,4),(3,4)],5)
 => [4,4,2,2]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 3 = 2 + 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 4 = 3 + 1
([(0,3),(0,4),(3,2),(4,1)],5)
 => [4,3,3]
 => [1,1,1,0,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 4 = 3 + 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
([(2,3),(3,4)],5)
 => [4,4,4,4]
 => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,1,0,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(1,4),(4,2),(4,3)],5)
 => [4,4,2,2]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 3 = 2 + 1
([(0,4),(4,1),(4,2),(4,3)],5)
 => [4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => 3 = 2 + 1
([(2,4),(3,4)],5)
 => [6,6,2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,1,0,1,1,0,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(1,4),(2,4),(4,3)],5)
 => [4,4,2,2]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 3 = 2 + 1
([(0,4),(1,4),(4,2),(4,3)],5)
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
([(1,4),(2,4),(3,4)],5)
 => [6,2,2,2,2,2,2]
 => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [1,0,1,1,1,1,0,0,1,0,1,1,0,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,4),(1,4),(2,4),(4,3)],5)
 => [4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => 3 = 2 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [3,2,2,2,2,2,2,2]
 => [1,1,0,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,0,1,1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,4),(1,4),(2,3)],5)
 => [6,3,3,3]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [8,3,2]
 => [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5,3,2,2]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 3 = 2 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,4),(1,4),(2,3),(4,2)],5)
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 4 = 3 + 1
([(0,4),(1,3),(2,3),(3,4)],5)
 => [7,2,2]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,1,0,1,0,0,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,4),(1,4),(2,3),(2,4)],5)
 => [6,5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,1,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [7,6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(1,4),(2,3)],5)
 => [6,6,6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(1,4),(2,3),(2,4)],5)
 => [10,6]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,1,1,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,4),(1,2),(1,4),(2,3)],5)
 => [8,3]
 => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
 => [1,1,1,1,0,1,1,0,1,0,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => [5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 4 = 3 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [6,2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,1,1,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => [7,2]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,1,0,1,0,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
([(0,4),(1,2),(1,4),(4,3)],5)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,4),(1,2),(1,3)],5)
 => [6,3,3,3]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,4),(1,2),(1,3),(1,4)],5)
 => [6,5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,1,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,2),(0,4),(3,1),(4,3)],5)
 => [5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 4 = 3 + 1
([(0,4),(1,2),(1,3),(3,4)],5)
 => [10,2]
 => [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [7,2,2]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,1,0,1,0,0,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,3),(0,4),(1,2),(1,4)],5)
 => [8,3,2]
 => [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [5,3,2,2]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 3 = 2 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [3,2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,3),(1,2),(1,4),(3,4)],5)
 => [8,3]
 => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
 => [1,1,1,1,0,1,1,0,1,0,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => [7,2]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,1,0,1,0,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(1,4),(3,2),(4,3)],5)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,3),(3,4),(4,1),(4,2)],5)
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 4 = 3 + 1
([(1,4),(2,3),(3,4)],5)
 => [14]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,4),(1,2),(2,4),(4,3)],5)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,3),(1,4),(4,2)],5)
 => [12]
 => [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,4),(3,2),(4,1),(4,3)],5)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,4),(1,2),(2,3),(2,4)],5)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4} + 1
([(0,3),(1,2),(2,4),(3,4)],5)
 => [4,3,3]
 => [1,1,1,0,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
([(0,4),(1,2),(2,3),(3,4)],5)
 => [5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 4 = 3 + 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 4 = 3 + 1

Description

Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. Then the statistic gives the vector space dimension of the first Ext-group between X and the regular module.

Matching statistic: St001233

Mapping

Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001233: Dyck paths ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 60%

Values

([],2)
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 0 = 1 - 1
([(0,1)],2)
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1 = 2 - 1
([],3)
 => [2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => 1 = 2 - 1
([(1,2)],3)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => ? = 0 - 1
([(0,1),(0,2)],3)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
([(0,2),(2,1)],3)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 1 = 2 - 1
([(0,2),(1,2)],3)
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
([],4)
 => [2,2,2,2,2,2,2,2]
 => [1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0,0]
 => ? ∊ {0,0,0,0,1,4} - 1
([(2,3)],4)
 => [6,6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => [1,1,1,0,1,0,1,0,1,1,0,0,0,0,1,0]
 => ? ∊ {0,0,0,0,1,4} - 1
([(1,2),(1,3)],4)
 => [6,2,2]
 => [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,1,0,0,0]
 => ? ∊ {0,0,0,0,1,4} - 1
([(0,1),(0,2),(0,3)],4)
 => [3,2,2,2]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => 2 = 3 - 1
([(0,2),(0,3),(3,1)],4)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,4} - 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 1 = 2 - 1
([(1,2),(2,3)],4)
 => [4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => 1 = 2 - 1
([(0,3),(3,1),(3,2)],4)
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 1 = 2 - 1
([(1,3),(2,3)],4)
 => [6,2,2]
 => [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,1,0,0,0]
 => ? ∊ {0,0,0,0,1,4} - 1
([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
 => [3,2,2,2]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => 2 = 3 - 1
([(0,3),(1,2)],4)
 => [3,3,3]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => 2 = 3 - 1
([(0,3),(1,2),(1,3)],4)
 => [5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 0 = 1 - 1
([(0,3),(2,1),(3,2)],4)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => 2 = 3 - 1
([(0,3),(1,2),(2,3)],4)
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,4} - 1
([],5)
 => [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => [1,1,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]
 => [1,0,1,1,1,1,1,1,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4} - 1
([(3,4)],5)
 => [6,6,6,6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4} - 1
([(2,3),(2,4)],5)
 => [6,6,2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,1,0,1,1,0,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4} - 1
([(1,2),(1,3),(1,4)],5)
 => [6,2,2,2,2,2,2]
 => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [1,0,1,1,1,1,0,0,1,0,1,1,0,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4} - 1
([(0,1),(0,2),(0,3),(0,4)],5)
 => [3,2,2,2,2,2,2,2]
 => [1,1,0,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,0,1,1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4} - 1
([(0,2),(0,3),(0,4),(4,1)],5)
 => [7,6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4} - 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [7,2,2]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,1,0,1,0,0,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4} - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => 1 = 2 - 1
([(1,3),(1,4),(4,2)],5)
 => [14]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4} - 1
([(0,3),(0,4),(4,1),(4,2)],5)
 => [7,2,2]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,1,0,1,0,0,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4} - 1
([(1,2),(1,3),(2,4),(3,4)],5)
 => [4,4,2,2]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 0 = 1 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 1 = 2 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
 => [4,3,3]
 => [1,1,1,0,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => 1 = 2 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 2 = 3 - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 0 = 1 - 1
([(2,3),(3,4)],5)
 => [4,4,4,4]
 => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,1,0,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4} - 1
([(1,4),(4,2),(4,3)],5)
 => [4,4,2,2]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 0 = 1 - 1
([(0,4),(4,1),(4,2),(4,3)],5)
 => [4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => 1 = 2 - 1
([(2,4),(3,4)],5)
 => [6,6,2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,1,0,1,1,0,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4} - 1
([(1,4),(2,4),(4,3)],5)
 => [4,4,2,2]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 0 = 1 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 0 = 1 - 1
([(1,4),(2,4),(3,4)],5)
 => [6,2,2,2,2,2,2]
 => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [1,0,1,1,1,1,0,0,1,0,1,1,0,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4} - 1
([(0,4),(1,4),(2,4),(4,3)],5)
 => [4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => 1 = 2 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [3,2,2,2,2,2,2,2]
 => [1,1,0,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,0,1,1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4} - 1
([(0,4),(1,4),(2,3)],5)
 => [6,3,3,3]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4} - 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [8,3,2]
 => [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4} - 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5,3,2,2]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 0 = 1 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4} - 1
([(0,4),(1,4),(2,3),(4,2)],5)
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 1 = 2 - 1
([(0,4),(1,3),(2,3),(3,4)],5)
 => [7,2,2]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,1,0,1,0,0,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4} - 1
([(0,4),(1,4),(2,3),(2,4)],5)
 => [6,5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,1,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4} - 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [7,6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4} - 1
([(1,4),(2,3)],5)
 => [6,6,6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4} - 1
([(1,4),(2,3),(2,4)],5)
 => [10,6]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,1,1,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4} - 1
([(0,4),(1,2),(1,4),(2,3)],5)
 => [8,3]
 => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
 => [1,1,1,1,0,1,1,0,1,0,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4} - 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => [5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 2 = 3 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [6,2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,1,1,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4} - 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => [7,2]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,1,0,1,0,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4} - 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 0 = 1 - 1
([(0,4),(1,2),(1,4),(4,3)],5)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4} - 1
([(0,4),(1,2),(1,3)],5)
 => [6,3,3,3]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4} - 1
([(0,4),(1,2),(1,3),(1,4)],5)
 => [6,5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,1,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4} - 1
([(0,2),(0,4),(3,1),(4,3)],5)
 => [5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 2 = 3 - 1
([(0,4),(1,2),(1,3),(3,4)],5)
 => [10,2]
 => [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4} - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4} - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [7,2,2]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,1,0,1,0,0,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4} - 1
([(0,3),(0,4),(1,2),(1,4)],5)
 => [8,3,2]
 => [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4} - 1
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [5,3,2,2]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 0 = 1 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [3,2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4} - 1
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4} - 1
([(0,3),(1,2),(1,4),(3,4)],5)
 => [8,3]
 => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
 => [1,1,1,1,0,1,1,0,1,0,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4} - 1
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => [7,2]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,1,0,1,0,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4} - 1
([(1,4),(3,2),(4,3)],5)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4} - 1
([(0,3),(3,4),(4,1),(4,2)],5)
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 1 = 2 - 1
([(1,4),(2,3),(3,4)],5)
 => [14]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4} - 1
([(0,4),(1,2),(2,4),(4,3)],5)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4} - 1
([(0,3),(1,4),(4,2)],5)
 => [12]
 => [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4} - 1
([(0,4),(3,2),(4,1),(4,3)],5)
 => [8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4} - 1
([(0,4),(1,2),(2,3),(2,4)],5)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4} - 1
([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4} - 1
([(0,3),(1,2),(2,4),(3,4)],5)
 => [4,3,3]
 => [1,1,1,0,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => 1 = 2 - 1
([(0,4),(1,2),(2,3),(3,4)],5)
 => [5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 2 = 3 - 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 1 = 2 - 1

Description

The number of indecomposable 2-dimensional modules with projective dimension one.

Matching statistic: St001570

Mapping

Mp00074: Posets —to graph⟶ Graphs
Mp00264: Graphs —delete endpoints⟶ Graphs
Mp00111: Graphs —complement⟶ Graphs
St001570: Graphs ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 60%

Values

([],2)
 => ([],2)
 => ([],2)
 => ([(0,1)],2)
 => ? ∊ {1,2}
([(0,1)],2)
 => ([(0,1)],2)
 => ([],1)
 => ([],1)
 => ? ∊ {1,2}
([],3)
 => ([],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(1,2)],3)
 => ([(1,2)],3)
 => ([],2)
 => ([(0,1)],2)
 => ? ∊ {2,2,2,2}
([(0,1),(0,2)],3)
 => ([(0,2),(1,2)],3)
 => ([],1)
 => ([],1)
 => ? ∊ {2,2,2,2}
([(0,2),(2,1)],3)
 => ([(0,2),(1,2)],3)
 => ([],1)
 => ([],1)
 => ? ∊ {2,2,2,2}
([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([],1)
 => ([],1)
 => ? ∊ {2,2,2,2}
([],4)
 => ([],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
([(2,3)],4)
 => ([(2,3)],4)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(1,2),(1,3)],4)
 => ([(1,3),(2,3)],4)
 => ([],2)
 => ([(0,1)],2)
 => ? ∊ {0,0,1,1,2,2,2,3,3,3,3,4}
([(0,1),(0,2),(0,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,1,1,2,2,2,3,3,3,3,4}
([(0,2),(0,3),(3,1)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,1,1,2,2,2,3,3,3,3,4}
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2)],4)
 => 2
([(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([],2)
 => ([(0,1)],2)
 => ? ∊ {0,0,1,1,2,2,2,3,3,3,3,4}
([(0,3),(3,1),(3,2)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,1,1,2,2,2,3,3,3,3,4}
([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([],2)
 => ([(0,1)],2)
 => ? ∊ {0,0,1,1,2,2,2,3,3,3,3,4}
([(0,3),(1,3),(3,2)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,1,1,2,2,2,3,3,3,3,4}
([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,1,1,2,2,2,3,3,3,3,4}
([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => ([],2)
 => ([(0,1)],2)
 => ? ∊ {0,0,1,1,2,2,2,3,3,3,3,4}
([(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,1,1,2,2,2,3,3,3,3,4}
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2)],4)
 => 2
([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,1,1,2,2,2,3,3,3,3,4}
([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,1,1,2,2,2,3,3,3,3,4}
([],5)
 => ([],5)
 => ([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
([(3,4)],5)
 => ([(3,4)],5)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
([(2,3),(2,4)],5)
 => ([(2,4),(3,4)],5)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(1,2),(1,3),(1,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([],2)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2)],4)
 => 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => 2
([(1,3),(1,4),(4,2)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ([],2)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
([(1,2),(1,3),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2)],4)
 => 2
([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2)],4)
 => 2
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => 2
([(2,3),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(1,4),(4,2),(4,3)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([],2)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(1,4),(2,4),(4,3)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([],2)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([],2)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
([(0,4),(1,4),(2,3)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => ([],2)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2)],4)
 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => 2
([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
([(0,4),(1,4),(2,3),(2,4)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(1,4),(2,3),(2,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ([],2)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
([(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2)],4)
 => 2
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2)],4)
 => 2
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => 2
([(0,4),(1,2),(1,4),(4,3)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
([(0,4),(1,2),(1,3)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => ([],2)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
([(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
([(0,4),(1,2),(1,3),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2)],4)
 => 2
([(0,3),(0,4),(1,2),(1,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2)],4)
 => 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => 2
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 0
([(0,3),(1,2),(1,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2)],4)
 => 2
([(1,4),(3,2),(4,3)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ([],2)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
([(0,3),(3,4),(4,1),(4,2)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ([],2)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4}
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2)],4)
 => 2

Description

The minimal number of edges to add to make a graph Hamiltonian. A graph is Hamiltonian if it contains a cycle as a subgraph, which contains all vertices.

Matching statistic: St001199
(load all 2 compositions to match this statistic)

Mapping

Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 60%

Values

([],2)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => ? ∊ {1,2}
([(0,1)],2)
 => [1]
 => [1,0]
 => [1,0]
 => ? ∊ {1,2}
([],3)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 2
([(1,2)],3)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => ? ∊ {0,2,2,2}
([(0,1),(0,2)],3)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => ? ∊ {0,2,2,2}
([(0,2),(2,1)],3)
 => [1]
 => [1,0]
 => [1,0]
 => ? ∊ {0,2,2,2}
([(0,2),(1,2)],3)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => ? ∊ {0,2,2,2}
([],4)
 => [4,4,4,4,4,4]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {0,0,0,0,2,2,3,3,3,3}
([(2,3)],4)
 => [4,4,4]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 4
([(1,2),(1,3)],4)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,2,2,3,3,3,3}
([(0,1),(0,2),(0,3)],4)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 2
([(0,2),(0,3),(3,1)],4)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => ? ∊ {0,0,0,0,2,2,3,3,3,3}
([(0,1),(0,2),(1,3),(2,3)],4)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => ? ∊ {0,0,0,0,2,2,3,3,3,3}
([(1,2),(2,3)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,2,2,3,3,3,3}
([(0,3),(3,1),(3,2)],4)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => ? ∊ {0,0,0,0,2,2,3,3,3,3}
([(1,3),(2,3)],4)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,2,2,3,3,3,3}
([(0,3),(1,3),(3,2)],4)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => ? ∊ {0,0,0,0,2,2,3,3,3,3}
([(0,3),(1,3),(2,3)],4)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 2
([(0,3),(1,2)],4)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1
([(0,3),(1,2),(1,3)],4)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 2
([(0,3),(2,1),(3,2)],4)
 => [1]
 => [1,0]
 => [1,0]
 => ? ∊ {0,0,0,0,2,2,3,3,3,3}
([(0,3),(1,2),(2,3)],4)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => ? ∊ {0,0,0,0,2,2,3,3,3,3}
([],5)
 => [5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5]
 => [1,1,1,1,1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
([(3,4)],5)
 => [5,5,5,5,5,5,5,5,5,5,5,5]
 => [1,1,1,1,1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
([(2,3),(2,4)],5)
 => [10,10,10,10]
 => [1,1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
([(1,2),(1,3),(1,4)],5)
 => [15,15]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
([(0,1),(0,2),(0,3),(0,4)],5)
 => [4,4,4,4,4,4]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
([(0,2),(0,3),(0,4),(4,1)],5)
 => [4,4,4]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 4
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 2
([(1,3),(1,4),(4,2)],5)
 => [15]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
([(0,3),(0,4),(4,1),(4,2)],5)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
([(1,2),(1,3),(2,4),(3,4)],5)
 => [5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
([(0,3),(0,4),(3,2),(4,1)],5)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 2
([(2,3),(3,4)],5)
 => [5,5,5,5]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
([(1,4),(4,2),(4,3)],5)
 => [5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 2
([(0,4),(4,1),(4,2),(4,3)],5)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 2
([(2,4),(3,4)],5)
 => [10,10,10,10]
 => [1,1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
([(1,4),(2,4),(4,3)],5)
 => [5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 2
([(0,4),(1,4),(4,2),(4,3)],5)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 2
([(1,4),(2,4),(3,4)],5)
 => [15,15]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
([(0,4),(1,4),(2,4),(4,3)],5)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4,4,4,4,4,4]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
([(0,4),(1,4),(2,3)],5)
 => [10,10]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
([(0,4),(1,3),(2,3),(2,4)],5)
 => [12,4]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [14]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6,6]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => 2
([(0,4),(1,4),(2,3),(4,2)],5)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
([(0,4),(1,3),(2,3),(3,4)],5)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
([(0,4),(1,4),(2,3),(2,4)],5)
 => [10,4,4]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
([(0,4),(1,4),(2,3),(3,4)],5)
 => [4,4,4]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 4
([(1,4),(2,3)],5)
 => [5,5,5,5,5,5]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
([(1,4),(2,3),(2,4)],5)
 => [15,5,5]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,1,1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
([(0,4),(1,2),(1,4),(2,3)],5)
 => [5,4]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [5,5,5,5]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 2
([(0,4),(1,2),(1,4),(4,3)],5)
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
([(0,4),(1,2),(1,3)],5)
 => [10,10]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
([(0,4),(1,2),(1,3),(1,4)],5)
 => [10,4,4]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
([(0,2),(0,4),(3,1),(4,3)],5)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
([(0,4),(1,2),(1,3),(3,4)],5)
 => [4,4,3]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
([(0,3),(0,4),(1,2),(1,4)],5)
 => [12,4]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [14]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [6,6]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => 2
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => [5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => 1
([(0,3),(1,2),(1,4),(3,4)],5)
 => [5,4]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 1
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
([(1,4),(3,2),(4,3)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
([(0,3),(3,4),(4,1),(4,2)],5)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
([(0,3),(1,4),(4,2)],5)
 => [5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 2
([(0,3),(1,2),(2,4),(3,4)],5)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1

Description

The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.

The following 64 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. 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 $eA$. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000680The Grundy value for Hackendot on posets. St000717The number of ordinal summands of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000906The length of the shortest maximal chain in a poset. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000941The number of characters of the symmetric group whose value on the partition is even. St000993The multiplicity of the largest part of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000456The monochromatic index of a connected graph. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001623The number of doubly irreducible elements of a lattice. St001119The length of a shortest maximal path in a graph. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000478Another weight of a partition according to Alladi. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001645The pebbling number of a connected graph. St001118The acyclic chromatic index of a graph. St000455The second largest eigenvalue of a graph if it is integral. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000264The girth of a graph, which is not a tree. St000284The Plancherel distribution on integer partitions. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition.