Your data matches 6 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001958
(load all 19 compositions to match this statistic)
Mapping
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St001958: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => 0
[1,0,1,0]
 => [1,2] => 1
[1,1,0,0]
 => [2,1] => 1
[1,0,1,0,1,0]
 => [1,2,3] => 1
[1,0,1,1,0,0]
 => [1,3,2] => 2
[1,1,0,0,1,0]
 => [2,1,3] => 2
[1,1,0,1,0,0]
 => [2,3,1] => 2
[1,1,1,0,0,0]
 => [3,1,2] => 2
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 3
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 3
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 3
[1,0,1,1,1,0,0,0]
 => [1,4,2,3] => 3
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 3
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 3
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 3
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 3
[1,1,0,1,1,0,0,0]
 => [2,4,1,3] => 3
[1,1,1,0,0,0,1,0]
 => [3,1,2,4] => 3
[1,1,1,0,0,1,0,0]
 => [3,1,4,2] => 3
[1,1,1,0,1,0,0,0]
 => [3,4,1,2] => 3
[1,1,1,1,0,0,0,0]
 => [4,1,2,3] => 3
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => 4
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => 4
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => 3
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => 4
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => 4
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => 4
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => 4
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => 4
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => 4
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => 4
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => 4
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => 4
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => 4
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => 4
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => 3
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 4
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => 4
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => 4
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => 4
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => 4
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => 4
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => 4
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => 4
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => 4
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => 4
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => 4
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => 4
Description
The degree of the polynomial interpolating the values of a permutation. Given a permutation $\pi\in\mathfrak S_n$ there is a polynomial $p$ of minimal degree such that $p(n)=\pi(n)$ for $n\in\{1,\dots,n\}$. This statistic records the degree of $p$.
Matching statistic: St001879
(load all 8 compositions to match this statistic)
Mapping
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 71%
Values
[1,0]
 => [1,0]
 => [.,.]
 => ([],1)
 => ? = 0
[1,0,1,0]
 => [1,1,0,0]
 => [[.,.],.]
 => ([(0,1)],2)
 => ? = 1
[1,1,0,0]
 => [1,0,1,0]
 => [.,[.,.]]
 => ([(0,1)],2)
 => ? = 1
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ? = 1
[1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => 2
[1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => [[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => 2
[1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => [[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => 2
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => 2
[1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1
[1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ? = 3
[1,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 3
[1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => [.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3
[1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3
[1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => ? = 3
[1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 3
[1,1,0,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 3
[1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 3
[1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 3
[1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 3
[1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 3
[1,1,1,0,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 4
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 4
[1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 3
[1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 4
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 4
[1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 4
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 4
[1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 4
[1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 4
[1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [[[[.,.],.],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 4
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[.,[[.,.],[.,.]]],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 3
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [[[.,[.,.]],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 4
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [[.,[[[.,.],.],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [[.,[.,[[.,.],.]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [[[[.,.],[.,.]],.],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 4
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [[[.,[[.,.],.]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [[[[[.,.],.],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [[[[.,[.,.]],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [[[.,[.,[.,.]]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [[[.,.],[[.,.],.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 4
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [[[.,.],[.,[.,.]]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 4
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [[.,[[.,[.,.]],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [[.,[.,[.,[.,.]]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 3
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 4
[1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 4
[1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 4
[1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 4
[1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 4
[1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 4
[1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 4
[1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 4
[1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 4
[1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [[[.,.],[.,.]],[[.,.],.]]
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
 => ? = 5
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [[[.,.],[.,.]],[.,[.,.]]]
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => ? = 5
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [.,[[[[.,.],.],.],[.,.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ? = 5
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [.,[.,[[[.,.],.],[.,.]]]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => ? = 5
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [[[[.,.],[.,.]],.],[.,.]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => ? = 5
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [.,[[.,[[.,.],.]],[.,.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ? = 5
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [[[[[.,.],.],.],.],[.,.]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ? = 5
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [.,[[.,[.,[.,.]]],[.,.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ? = 5
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,1,0,0,0]
 => [.,[.,[[[.,.],[.,.]],.]]]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => ? = 5
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [[[[.,.],.],[.,.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ? = 5
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [.,[[[.,[.,.]],.],[.,.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ? = 5
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [.,[.,[[.,[.,.]],[.,.]]]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => ? = 5
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [.,[.,[.,[[.,.],[.,.]]]]]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => ? = 5
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [.,[[[[[.,.],.],.],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => [.,[[[.,[[.,.],.]],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => [.,[[.,[[[.,.],.],.]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [.,[[.,[.,[[.,.],.]]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [.,[.,[[[[.,.],.],.],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [.,[.,[[.,[[.,.],.]],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [.,[.,[.,[[[.,.],.],.]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [.,[.,[.,[.,[[.,.],.]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0]
 => [[.,[[[[.,.],.],.],.]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[1,1,0,0,1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [[.,[[.,[[.,.],.]],.]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[1,1,0,0,1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => [[.,[.,[[[.,.],.],.]]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [[.,[.,[.,[[.,.],.]]]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [[[.,[[[.,.],.],.]],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[1,1,0,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,1,0,0]
 => [[[.,[.,[[.,.],.]]],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[1,1,0,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => [[[[.,[[.,.],.]],.],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [[[[[[.,.],.],.],.],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [[[[[.,[.,.]],.],.],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [[[[.,[.,[.,.]]],.],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [[[.,[[.,[.,.]],.]],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [[[.,[.,[.,[.,.]]]],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [[.,[[[.,[.,.]],.],.]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [[.,[[.,[.,[.,.]]],.]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
Description
The 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.
Matching statistic: St001232
(load all 3 compositions to match this statistic)
Mapping
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 86%
Values
[1,0]
 => [1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 0 + 1
[1,0,1,0]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => ? = 1 + 1
[1,1,0,0]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
[1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => ? = 1 + 1
[1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => ? = 2 + 1
[1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 3 = 2 + 1
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => ? = 2 + 1
[1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => ? = 1 + 1
[1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => ? = 3 + 1
[1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => ? = 3 + 1
[1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => ? = 3 + 1
[1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => ? = 3 + 1
[1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => ? = 3 + 1
[1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 4 = 3 + 1
[1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => ? = 3 + 1
[1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 4 = 3 + 1
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => ? = 3 + 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 4 = 3 + 1
[1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 4 = 3 + 1
[1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => ? = 3 + 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 4 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 4 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 3 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 4 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 4 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 4 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 4 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 4 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 4 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 4 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 4 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 4 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 4 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 4 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 3 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 5 = 4 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 4 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 5 = 4 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 4 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 4 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 5 = 4 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 4 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 5 = 4 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 4 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 4 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 5 = 4 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 4 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 3 + 1
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 5 = 4 + 1
[1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 5 = 4 + 1
[1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 5 = 4 + 1
[1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 5 = 4 + 1
[1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 4 + 1
[1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 5 = 4 + 1
[1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 5 = 4 + 1
[1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 4 + 1
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 5 = 4 + 1
[1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 5 = 4 + 1
[1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 5 = 4 + 1
[1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 4 + 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 5 = 4 + 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]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 5 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 5 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 5 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 5 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 5 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 5 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 5 + 1
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 5 + 1
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 5 + 1
[1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => 6 = 5 + 1
[1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => 6 = 5 + 1
[1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => 6 = 5 + 1
[1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => 6 = 5 + 1
[1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => 6 = 5 + 1
[1,1,0,1,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => 6 = 5 + 1
[1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => 6 = 5 + 1
[1,1,0,1,0,1,1,0,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => 6 = 5 + 1
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => 6 = 5 + 1
[1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => 6 = 5 + 1
[1,1,0,1,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => 6 = 5 + 1
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => 6 = 5 + 1
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => 6 = 5 + 1
[1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => 6 = 5 + 1
[1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => 6 = 5 + 1
[1,1,1,0,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => 6 = 5 + 1
[1,1,1,0,0,1,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => 6 = 5 + 1
[1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => 6 = 5 + 1
[1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => 6 = 5 + 1
[1,1,1,0,0,1,1,0,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => 6 = 5 + 1
[1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => 6 = 5 + 1
[1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => 6 = 5 + 1
[1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => 6 = 5 + 1
[1,1,1,0,1,0,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => 6 = 5 + 1
[1,1,1,0,1,0,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => 6 = 5 + 1
[1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => 6 = 5 + 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001880
(load all 8 compositions to match this statistic)
Mapping
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 71%
Values
[1,0]
 => [1,0]
 => [.,.]
 => ([],1)
 => ? = 0 + 1
[1,0,1,0]
 => [1,1,0,0]
 => [[.,.],.]
 => ([(0,1)],2)
 => ? = 1 + 1
[1,1,0,0]
 => [1,0,1,0]
 => [.,[.,.]]
 => ([(0,1)],2)
 => ? = 1 + 1
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ? = 1 + 1
[1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => [[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => [[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 1
[1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ? = 3 + 1
[1,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 3 + 1
[1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => [.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 3 + 1
[1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 3 + 1
[1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => ? = 3 + 1
[1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 3 + 1
[1,1,0,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 3 + 1
[1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 3 + 1
[1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 3 + 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 3 + 1
[1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 3 + 1
[1,1,1,0,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 3 + 1
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 4 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 4 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 3 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 4 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 4 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 4 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 4 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 4 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 4 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [[[[.,.],.],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 4 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[.,[[.,.],[.,.]]],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 3 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [[[.,[.,.]],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 4 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [[.,[[[.,.],.],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [[.,[.,[[.,.],.]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [[[[.,.],[.,.]],.],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 4 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [[[.,[[.,.],.]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [[[[[.,.],.],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [[[[.,[.,.]],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [[[.,[.,[.,.]]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [[[.,.],[[.,.],.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 4 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [[[.,.],[.,[.,.]]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 4 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [[.,[[.,[.,.]],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [[.,[.,[.,[.,.]]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 3 + 1
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 4 + 1
[1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 4 + 1
[1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 4 + 1
[1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 4 + 1
[1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 4 + 1
[1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 4 + 1
[1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 4 + 1
[1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 4 + 1
[1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 4 + 1
[1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [[[.,.],[.,.]],[[.,.],.]]
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
 => ? = 5 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [[[.,.],[.,.]],[.,[.,.]]]
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => ? = 5 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [.,[[[[.,.],.],.],[.,.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ? = 5 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [.,[.,[[[.,.],.],[.,.]]]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => ? = 5 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [[[[.,.],[.,.]],.],[.,.]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => ? = 5 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [.,[[.,[[.,.],.]],[.,.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ? = 5 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [[[[[.,.],.],.],.],[.,.]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ? = 5 + 1
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [.,[[.,[.,[.,.]]],[.,.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ? = 5 + 1
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,1,0,0,0]
 => [.,[.,[[[.,.],[.,.]],.]]]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => ? = 5 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [[[[.,.],.],[.,.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ? = 5 + 1
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [.,[[[.,[.,.]],.],[.,.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ? = 5 + 1
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [.,[.,[[.,[.,.]],[.,.]]]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => ? = 5 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [.,[.,[.,[[.,.],[.,.]]]]]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => ? = 5 + 1
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [.,[[[[[.,.],.],.],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => [.,[[[.,[[.,.],.]],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => [.,[[.,[[[.,.],.],.]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [.,[[.,[.,[[.,.],.]]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [.,[.,[[[[.,.],.],.],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [.,[.,[[.,[[.,.],.]],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [.,[.,[.,[[[.,.],.],.]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [.,[.,[.,[.,[[.,.],.]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0]
 => [[.,[[[[.,.],.],.],.]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[1,1,0,0,1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [[.,[[.,[[.,.],.]],.]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[1,1,0,0,1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => [[.,[.,[[[.,.],.],.]]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [[.,[.,[.,[[.,.],.]]]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [[[.,[[[.,.],.],.]],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[1,1,0,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,1,0,0]
 => [[[.,[.,[[.,.],.]]],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[1,1,0,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => [[[[.,[[.,.],.]],.],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [[[[[[.,.],.],.],.],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [[[[[.,[.,.]],.],.],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [[[[.,[.,[.,.]]],.],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [[[.,[[.,[.,.]],.]],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [[[.,[.,[.,[.,.]]]],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [[.,[[[.,[.,.]],.],.]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [[.,[[.,[.,[.,.]]],.]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Matching statistic: St001645
(load all 4 compositions to match this statistic)
Mapping
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 86%
Values
[1,0]
 => [2,1] => [2,1] => ([(0,1)],2)
 => 2 = 0 + 2
[1,0,1,0]
 => [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 1 + 2
[1,1,0,0]
 => [2,3,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 1 + 2
[1,0,1,0,1,0]
 => [4,1,2,3] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 + 2
[1,0,1,1,0,0]
 => [3,1,4,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2 + 2
[1,1,0,0,1,0]
 => [2,4,1,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 4 = 2 + 2
[1,1,0,1,0,0]
 => [4,3,1,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 2 + 2
[1,1,1,0,0,0]
 => [2,3,4,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2 + 2
[1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 2
[1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 2
[1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 2
[1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 2
[1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 2
[1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 2
[1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 2
[1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 3 + 2
[1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 3 + 2
[1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 2
[1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 2
[1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 5 = 3 + 2
[1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 3 + 2
[1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 2
[1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => [6,2,3,4,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 2
[1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => [6,2,3,4,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 + 2
[1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => [5,2,3,6,1,4] => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 + 2
[1,0,1,0,1,1,0,1,0,0]
 => [6,1,2,5,3,4] => [6,2,3,5,4,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 + 2
[1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => [6,2,3,4,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 + 2
[1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => [4,2,6,1,5,3] => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
 => ? = 4 + 2
[1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => [4,2,6,1,5,3] => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
 => ? = 4 + 2
[1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => [6,2,5,4,3,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 + 2
[1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => [6,2,5,4,3,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 + 2
[1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => [6,2,4,3,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 + 2
[1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => [5,2,3,6,1,4] => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 + 2
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => [5,2,6,4,1,3] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 4 + 2
[1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => [6,2,5,4,3,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 + 2
[1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => [6,2,3,4,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 + 2
[1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => [3,6,1,4,5,2] => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 + 2
[1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => [3,6,1,4,5,2] => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 + 2
[1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => [3,5,1,6,2,4] => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => ? = 4 + 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => [3,6,1,5,4,2] => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 + 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => [3,6,1,4,5,2] => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 + 2
[1,1,0,1,0,0,1,0,1,0]
 => [6,3,1,2,4,5] => [6,4,3,2,5,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 + 2
[1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => [6,4,3,2,5,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 + 2
[1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [6,5,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 4 + 2
[1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => [6,5,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 4 + 2
[1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => [6,4,3,2,5,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 + 2
[1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => [5,3,2,6,1,4] => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 + 2
[1,1,0,1,1,0,0,1,0,0]
 => [6,3,1,5,2,4] => [6,5,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 4 + 2
[1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => [6,5,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 4 + 2
[1,1,0,1,1,1,0,0,0,0]
 => [4,3,1,5,6,2] => [6,3,2,4,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 + 2
[1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => [4,2,6,1,5,3] => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
 => ? = 3 + 2
[1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => [4,2,6,1,5,3] => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
 => ? = 4 + 2
[1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => [4,6,5,1,3,2] => ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 4 + 2
[1,1,1,0,0,1,0,1,0,0]
 => [2,6,5,1,3,4] => [4,6,5,1,3,2] => ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 4 + 2
[1,1,1,0,0,1,1,0,0,0]
 => [2,5,4,1,6,3] => [4,6,3,1,5,2] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 4 + 2
[1,1,1,0,1,0,0,0,1,0]
 => [6,3,4,1,2,5] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 4 + 2
[1,1,1,0,1,0,0,1,0,0]
 => [6,3,5,1,2,4] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 4 + 2
[1,1,1,0,1,0,1,0,0,0]
 => [6,5,4,1,2,3] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 4 + 2
[1,1,1,0,1,1,0,0,0,0]
 => [5,3,4,1,6,2] => [6,4,3,2,5,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 + 2
[1,1,1,1,0,0,0,0,1,0]
 => [2,3,4,6,1,5] => [5,2,3,6,1,4] => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 + 2
[1,1,1,1,0,0,0,1,0,0]
 => [2,3,6,5,1,4] => [5,2,6,4,1,3] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 4 + 2
[1,1,1,1,0,0,1,0,0,0]
 => [2,6,4,5,1,3] => [5,6,4,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 4 + 2
[1,1,1,1,0,1,0,0,0,0]
 => [6,3,4,5,1,2] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 4 + 2
[1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => [6,2,3,4,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 + 2
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => [7,2,3,4,5,6,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 2
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [6,1,2,3,4,7,5] => [7,2,3,4,5,6,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 5 + 2
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [5,1,2,3,7,4,6] => [6,2,3,4,7,1,5] => ([(0,1),(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 5 + 2
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [7,1,2,3,6,4,5] => [7,2,3,4,6,5,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 5 + 2
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [5,1,2,3,6,7,4] => [7,2,3,4,5,6,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 5 + 2
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [4,1,2,7,3,5,6] => [5,2,3,7,1,6,4] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(3,4),(3,6),(4,5),(5,6)],7)
 => ? = 5 + 2
[1,1,0,1,0,1,0,1,0,0,1,0]
 => [5,7,1,2,3,4,6] => [6,7,3,4,5,1,2] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => 7 = 5 + 2
[1,1,0,1,0,1,0,1,0,1,0,0]
 => [7,6,1,2,3,4,5] => [7,6,3,4,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 5 + 2
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [7,4,1,2,6,3,5] => [7,6,3,4,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 5 + 2
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [5,7,1,2,6,3,4] => [7,6,3,4,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 5 + 2
[1,1,0,1,1,0,0,1,0,0,1,0]
 => [7,3,1,5,2,4,6] => [7,5,3,6,2,4,1] => ([(0,1),(0,4),(0,5),(0,6),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 5 + 2
[1,1,0,1,1,0,0,1,0,1,0,0]
 => [7,3,1,6,2,4,5] => [7,5,3,6,2,4,1] => ([(0,1),(0,4),(0,5),(0,6),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 5 + 2
[1,1,0,1,1,0,1,0,0,0,1,0]
 => [7,4,1,5,2,3,6] => [7,6,3,5,4,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 5 + 2
[1,1,0,1,1,0,1,0,0,1,0,0]
 => [7,4,1,6,2,3,5] => [7,6,3,5,4,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 5 + 2
[1,1,0,1,1,0,1,0,1,0,0,0]
 => [6,7,1,5,2,3,4] => [7,6,3,5,4,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 5 + 2
[1,1,0,1,1,1,0,0,1,0,0,0]
 => [7,3,1,5,6,2,4] => [7,6,3,5,4,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 5 + 2
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [7,4,1,5,6,2,3] => [7,6,3,5,4,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 5 + 2
[1,1,1,0,0,1,0,1,0,0,1,0]
 => [2,7,5,1,3,4,6] => [4,7,6,1,5,3,2] => ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 5 + 2
[1,1,1,0,0,1,0,1,0,1,0,0]
 => [2,6,7,1,3,4,5] => [4,7,6,1,5,3,2] => ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 5 + 2
[1,1,1,0,0,1,1,0,0,1,0,0]
 => [2,7,4,1,6,3,5] => [4,7,6,1,5,3,2] => ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 5 + 2
[1,1,1,0,0,1,1,0,1,0,0,0]
 => [2,7,5,1,6,3,4] => [4,7,6,1,5,3,2] => ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 5 + 2
[1,1,1,0,1,0,0,1,0,0,1,0]
 => [7,3,5,1,2,4,6] => [7,5,6,4,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 5 + 2
[1,1,1,0,1,0,0,1,0,1,0,0]
 => [6,3,7,1,2,4,5] => [7,5,6,4,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 5 + 2
[1,1,1,0,1,0,1,0,0,0,1,0]
 => [7,5,4,1,2,3,6] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 5 + 2
[1,1,1,0,1,0,1,0,0,1,0,0]
 => [6,7,4,1,2,3,5] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 5 + 2
[1,1,1,0,1,0,1,0,1,0,0,0]
 => [6,7,5,1,2,3,4] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 5 + 2
[1,1,1,0,1,1,0,0,0,1,0,0]
 => [7,3,4,1,6,2,5] => [7,6,4,3,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 5 + 2
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [7,3,5,1,6,2,4] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 5 + 2
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [7,5,4,1,6,2,3] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 5 + 2
[1,1,1,1,0,0,1,0,0,0,1,0]
 => [2,7,4,5,1,3,6] => [5,7,6,4,1,3,2] => ([(0,1),(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 5 + 2
[1,1,1,1,0,0,1,0,0,1,0,0]
 => [2,7,4,6,1,3,5] => [5,7,6,4,1,3,2] => ([(0,1),(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 5 + 2
[1,1,1,1,0,0,1,0,1,0,0,0]
 => [2,7,6,5,1,3,4] => [5,7,6,4,1,3,2] => ([(0,1),(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 5 + 2
[1,1,1,1,0,1,0,0,0,0,1,0]
 => [7,3,4,5,1,2,6] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 5 + 2
[1,1,1,1,0,1,0,0,0,1,0,0]
 => [7,3,4,6,1,2,5] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 5 + 2
[1,1,1,1,0,1,0,0,1,0,0,0]
 => [7,3,6,5,1,2,4] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 5 + 2
[1,1,1,1,0,1,0,1,0,0,0,0]
 => [7,6,4,5,1,2,3] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 5 + 2
Description
The pebbling number of a connected graph.
Matching statistic: St000717
Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000717: Posets ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 71%
Values
[1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => ([(0,1)],2)
 => 2 = 0 + 2
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => ([(0,2),(2,1)],3)
 => 3 = 1 + 2
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => 3 = 1 + 2
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 3 = 1 + 2
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 2 + 2
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 2 + 2
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 2 + 2
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 2 + 2
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1 + 2
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => ? = 3 + 2
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 5 = 3 + 2
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => ? = 3 + 2
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ? = 3 + 2
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ? = 3 + 2
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 3 + 2
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ? = 1 + 2
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
 => ? = 4 + 2
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
 => ? = 4 + 2
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
 => ? = 3 + 2
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 4 + 2
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ? = 4 + 2
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => ? = 4 + 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
 => ? = 4 + 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
 => ? = 4 + 2
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 4 + 2
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 4 + 2
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => ? = 4 + 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 4 + 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 4 + 2
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => ? = 4 + 2
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0,1,0]
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ? = 3 + 2
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => 6 = 4 + 2
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ? = 4 + 2
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 4 + 2
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => ? = 4 + 2
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0,1,0]
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ? = 4 + 2
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
 => ? = 4 + 2
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? = 4 + 2
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
 => ? = 4 + 2
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => 6 = 4 + 2
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ? = 4 + 2
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
 => ? = 4 + 2
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 4 + 2
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ? = 3 + 2
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 4 + 2
[1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ? = 4 + 2
[1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => ? = 4 + 2
[1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 4 + 2
[1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ? = 4 + 2
[1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => ? = 4 + 2
[1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? = 4 + 2
[1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 4 + 2
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 4 + 2
[1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 4 + 2
[1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 4 + 2
[1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 4 + 2
[1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 4 + 2
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
 => ? = 1 + 2
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
 => ? = 5 + 2
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
 => ([(0,4),(0,6),(2,8),(3,9),(4,10),(5,3),(5,7),(6,5),(6,10),(7,8),(7,9),(8,11),(9,11),(10,2),(10,7),(11,1)],12)
 => ? = 5 + 2
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
 => ? = 5 + 2
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
 => ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
 => ? = 5 + 2
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,1,1,0,0,0,1,0,0,0]
 => ([(0,4),(0,6),(1,12),(2,8),(3,10),(4,11),(5,1),(5,7),(6,5),(6,11),(7,10),(7,12),(9,8),(10,9),(11,3),(11,7),(12,2),(12,9)],13)
 => ? = 5 + 2
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0,1,0]
 => ([(0,4),(0,6),(1,8),(3,7),(4,9),(5,2),(6,3),(6,9),(7,8),(8,5),(9,1),(9,7)],10)
 => ? = 5 + 2
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
 => ([(0,4),(0,6),(1,12),(2,8),(3,10),(4,11),(5,3),(5,7),(6,5),(6,11),(7,10),(7,12),(9,8),(10,9),(11,1),(11,7),(12,2),(12,9)],13)
 => ? = 5 + 2
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => ([(0,5),(0,6),(1,11),(2,4),(2,13),(3,7),(4,10),(5,1),(5,12),(6,2),(6,12),(8,9),(9,7),(10,3),(10,9),(11,8),(12,11),(12,13),(13,8),(13,10)],14)
 => ? = 5 + 2
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
 => ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
 => ? = 5 + 2
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,1,0,1,0,0]
 => ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
 => ? = 5 + 2
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
 => ([(0,4),(0,6),(1,8),(3,7),(4,9),(5,2),(6,3),(6,9),(7,8),(8,5),(9,1),(9,7)],10)
 => ? = 5 + 2
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
 => ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
 => ? = 5 + 2
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
 => ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
 => ? = 5 + 2
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
 => ([(0,6),(1,11),(2,8),(3,9),(4,3),(4,7),(5,1),(5,7),(6,4),(6,5),(7,9),(7,11),(9,10),(10,8),(11,2),(11,10)],12)
 => ? = 5 + 2
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0,1,0]
 => ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
 => ? = 5 + 2
Description
The number of ordinal summands of a poset. The ordinal sum of two posets $P$ and $Q$ is the poset having elements $(p,0)$ and $(q,1)$ for $p\in P$ and $q\in Q$, and relations $(a,0) < (b,0)$ if $a < b$ in $P$, $(a,1) < (b,1)$ if $a < b$ in $Q$, and $(a,0) < (b,1)$. This statistic is the length of the longest ordinal decomposition of a poset.