- FindStat

Your data matches 125 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000882

Mapping

St000882: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => 1
[1,2] => 1
[2,1] => 1
[1,2,3] => 1
[1,3,2] => 1
[2,1,3] => 1
[2,3,1] => 1
[3,1,2] => 1
[3,2,1] => 2
[1,2,3,4] => 1
[1,2,4,3] => 1
[1,3,2,4] => 1
[1,3,4,2] => 1
[1,4,2,3] => 1
[1,4,3,2] => 2
[2,1,3,4] => 1
[2,1,4,3] => 1
[2,3,1,4] => 1
[2,3,4,1] => 1
[2,4,1,3] => 1
[2,4,3,1] => 2
[3,1,2,4] => 1
[3,1,4,2] => 1
[3,2,1,4] => 2
[3,2,4,1] => 2
[3,4,1,2] => 1
[3,4,2,1] => 3
[4,1,2,3] => 1
[4,1,3,2] => 2
[4,2,1,3] => 2
[4,2,3,1] => 3
[4,3,1,2] => 3
[4,3,2,1] => 8
[1,2,3,4,5] => 1
[1,2,3,5,4] => 1
[1,2,4,3,5] => 1
[1,2,4,5,3] => 1
[1,2,5,3,4] => 1
[1,2,5,4,3] => 2
[1,3,2,4,5] => 1
[1,3,2,5,4] => 1
[1,3,4,2,5] => 1
[1,3,4,5,2] => 1
[1,3,5,2,4] => 1
[1,3,5,4,2] => 2
[1,4,2,3,5] => 1
[1,4,2,5,3] => 1
[1,4,3,2,5] => 2
[1,4,3,5,2] => 2
[1,4,5,2,3] => 1

Description

The number of connected components of short braid edges in the graph of braid moves of a permutation. Given a permutation

$\pi$ , let

$\operatorname{Red}(\pi)$ denote the set of reduced words for

$\pi$ in terms of simple transpositions

$s_i = (i,i+1)$ . We now say that two reduced words are connected by a short braid move if they are obtained from each other by a modification of the form

$s_i s_j \leftrightarrow s_j s_i$ for

$|i-j| > 1$ as a consecutive subword of a reduced word. For example, the two reduced words

$s_1s_3s_2$ and

$s_3s_1s_2$ for

$(1243) = (12)(34)(23) = (34)(12)(23)$ share an edge because they are obtained from each other by interchanging

$s_1s_3 \leftrightarrow s_3s_1$ . This statistic counts the number connected components of such short braid moves among all reduced words.

Matching statistic: St001964
(load all 9 compositions to match this statistic)

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001964: Posets ⟶ ℤResult quality: 40% ●values known / values provided: 75%●distinct values known / distinct values provided: 40%

Values

[1] => [1,0]
 => [1] => ([],1)
 => 0 = 1 - 1
[1,2] => [1,0,1,0]
 => [1,2] => ([(0,1)],2)
 => 0 = 1 - 1
[2,1] => [1,1,0,0]
 => [2,1] => ([],2)
 => 0 = 1 - 1
[1,2,3] => [1,0,1,0,1,0]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => 0 = 1 - 1
[1,3,2] => [1,0,1,1,0,0]
 => [1,3,2] => ([(0,1),(0,2)],3)
 => 0 = 1 - 1
[2,1,3] => [1,1,0,0,1,0]
 => [2,1,3] => ([(0,2),(1,2)],3)
 => 0 = 1 - 1
[2,3,1] => [1,1,0,1,0,0]
 => [2,3,1] => ([(1,2)],3)
 => ? = 2 - 1
[3,1,2] => [1,1,1,0,0,0]
 => [3,2,1] => ([],3)
 => 0 = 1 - 1
[3,2,1] => [1,1,1,0,0,0]
 => [3,2,1] => ([],3)
 => 0 = 1 - 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => 1 = 2 - 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => 0 = 1 - 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => 1 = 2 - 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => 1 = 2 - 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => 1 = 2 - 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2 = 3 - 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => 0 = 1 - 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => ([(1,2),(2,3)],4)
 => 0 = 1 - 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [2,4,3,1] => ([(1,2),(1,3)],4)
 => ? ∊ {1,3,3,8} - 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [2,4,3,1] => ([(1,2),(1,3)],4)
 => ? ∊ {1,3,3,8} - 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => 1 = 2 - 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => ([(1,3),(2,3)],4)
 => 0 = 1 - 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => 1 = 2 - 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => ([(1,3),(2,3)],4)
 => 0 = 1 - 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [4,2,3,1] => ([(2,3)],4)
 => ? ∊ {1,3,3,8} - 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [4,2,3,1] => ([(2,3)],4)
 => ? ∊ {1,3,3,8} - 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => ([],4)
 => 0 = 1 - 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => ([],4)
 => 0 = 1 - 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => ([],4)
 => 0 = 1 - 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => ([],4)
 => 0 = 1 - 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => ([],4)
 => 0 = 1 - 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => ([],4)
 => 0 = 1 - 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
 => 1 = 2 - 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1 = 2 - 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => 1 = 2 - 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
 => 2 = 3 - 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
 => 2 = 3 - 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1 = 2 - 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 2 = 3 - 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 0 = 1 - 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => 0 = 1 - 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => 1 = 2 - 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => 1 = 2 - 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 1 = 2 - 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 1 = 2 - 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
 => 1 = 2 - 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
 => 1 = 2 - 1
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => 2 = 3 - 1
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => 2 = 3 - 1
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => 2 = 3 - 1
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => 2 = 3 - 1
[2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => ([(1,2),(1,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => ([(1,2),(1,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
 => [2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
 => [2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[2,5,3,1,4] => [1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
 => [3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[3,1,5,4,2] => [1,1,1,0,0,1,1,0,0,0]
 => [3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0]
 => [3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
 => [3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
 => [5,2,4,3,1] => ([(2,3),(2,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
 => [5,2,4,3,1] => ([(2,3),(2,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[3,5,2,1,4] => [1,1,1,0,1,1,0,0,0,0]
 => [5,2,4,3,1] => ([(2,3),(2,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
 => [5,2,4,3,1] => ([(2,3),(2,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
 => [5,2,4,3,1] => ([(2,3),(2,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
 => [5,2,4,3,1] => ([(2,3),(2,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
 => [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[4,1,3,5,2] => [1,1,1,1,0,0,0,1,0,0]
 => [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[4,2,1,5,3] => [1,1,1,1,0,0,0,1,0,0]
 => [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
 => [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
 => [5,3,4,2,1] => ([(3,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
 => [5,3,4,2,1] => ([(3,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
 => [5,3,4,2,1] => ([(3,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
 => [5,3,4,2,1] => ([(3,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
 => [5,3,4,2,1] => ([(3,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => [5,3,4,2,1] => ([(3,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1

Description

The interval resolution global dimension of a poset. This is the cardinality of the longest chain of right minimal approximations by interval modules of an indecomposable module over the incidence algebra.

Matching statistic: St001514

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001514: Dyck paths ⟶ ℤResult quality: 40% ●values known / values provided: 66%●distinct values known / distinct values provided: 40%

Values

[1] => [1,0]
 => []
 => []
 => ? = 1
[1,2] => [1,0,1,0]
 => [1]
 => [1,0]
 => 1
[2,1] => [1,1,0,0]
 => []
 => []
 => ? = 1
[1,2,3] => [1,0,1,0,1,0]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2
[1,3,2] => [1,0,1,1,0,0]
 => [1,1]
 => [1,1,0,0]
 => 1
[2,1,3] => [1,1,0,0,1,0]
 => [2]
 => [1,0,1,0]
 => 1
[2,3,1] => [1,1,0,1,0,0]
 => [1]
 => [1,0]
 => 1
[3,1,2] => [1,1,1,0,0,0]
 => []
 => []
 => ? ∊ {1,1}
[3,2,1] => [1,1,1,0,0,0]
 => []
 => []
 => ? ∊ {1,1}
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 3
[1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 2
[1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
[1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
[1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [2,2]
 => [1,1,1,0,0,0]
 => 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 3
[2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2
[2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [1,1]
 => [1,1,0,0]
 => 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [1,1]
 => [1,1,0,0]
 => 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [3]
 => [1,0,1,0,1,0]
 => 2
[3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [2]
 => [1,0,1,0]
 => 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [3]
 => [1,0,1,0,1,0]
 => 2
[3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [2]
 => [1,0,1,0]
 => 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [1]
 => [1,0]
 => 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [1]
 => [1,0]
 => 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
 => []
 => []
 => ? ∊ {1,1,1,1,1,8}
[4,1,3,2] => [1,1,1,1,0,0,0,0]
 => []
 => []
 => ? ∊ {1,1,1,1,1,8}
[4,2,1,3] => [1,1,1,1,0,0,0,0]
 => []
 => []
 => ? ∊ {1,1,1,1,1,8}
[4,2,3,1] => [1,1,1,1,0,0,0,0]
 => []
 => []
 => ? ∊ {1,1,1,1,1,8}
[4,3,1,2] => [1,1,1,1,0,0,0,0]
 => []
 => []
 => ? ∊ {1,1,1,1,1,8}
[4,3,2,1] => [1,1,1,1,0,0,0,0]
 => []
 => []
 => ? ∊ {1,1,1,1,1,8}
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => 3
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => 3
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 3
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 3
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 2
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 2
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 2
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 2
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 2
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 2
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => 2
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => 1
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => 1
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => 2
[2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 3
[2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 2
[2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 2
[2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
[2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
[2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
[2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
[2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[2,5,3,1,4] => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}

Description

The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule.

Matching statistic: St001637

Mapping

Mp00065: Permutations —permutation poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St001637: Posets ⟶ ℤResult quality: 40% ●values known / values provided: 60%●distinct values known / distinct values provided: 40%

Values

[1] => ([],1)
 => ([],1)
 => ([],1)
 => ? = 1
[1,2] => ([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[2,1] => ([],2)
 => ([],1)
 => ([],1)
 => ? = 1
[1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2
[1,3,2] => ([(0,1),(0,2)],3)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1}
[2,1,3] => ([(0,2),(1,2)],3)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1}
[2,3,1] => ([(1,2)],3)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[3,1,2] => ([(1,2)],3)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[3,2,1] => ([],3)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1}
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 3
[1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,2,3,3,8}
[1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,2,3,3,8}
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,2,3,3,8}
[2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,2,3,3,8}
[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[2,3,4,1] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2
[2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2
[2,4,3,1] => ([(1,2),(1,3)],4)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,2,3,3,8}
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2
[3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,2,3,3,8}
[3,2,4,1] => ([(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,2,3,3,8}
[3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[3,4,2,1] => ([(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[4,1,2,3] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2
[4,1,3,2] => ([(1,2),(1,3)],4)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,2,3,3,8}
[4,2,1,3] => ([(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,2,3,3,8}
[4,2,3,1] => ([(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[4,3,1,2] => ([(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[4,3,2,1] => ([],4)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,2,3,3,8}
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2
[1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2
[1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2
[1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2
[1,5,2,4,3] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,5,3,2,4] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[1,5,4,2,3] => ([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2
[2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2
[2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2
[2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2
[2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 3
[2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 3
[2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2
[2,4,1,5,3] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 3
[2,4,3,1,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[2,4,3,5,1] => ([(1,2),(1,3),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 3
[2,4,5,3,1] => ([(1,3),(1,4),(4,2)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 3
[2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[2,5,3,1,4] => ([(0,4),(1,2),(1,3),(3,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[2,5,4,1,3] => ([(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2
[3,1,5,4,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[3,2,5,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[3,4,5,1,2] => ([(0,3),(1,4),(4,2)],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)
 => ? ∊ {1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[3,5,4,2,1] => ([(2,3),(2,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[4,1,3,2,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[4,1,3,5,2] => ([(0,4),(1,2),(1,3),(3,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[4,2,1,3,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[4,2,1,5,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[4,3,5,2,1] => ([(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}

Description

The number of (upper) dissectors of a poset.

Matching statistic: St001668

Mapping

Mp00065: Permutations —permutation poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St001668: Posets ⟶ ℤResult quality: 40% ●values known / values provided: 60%●distinct values known / distinct values provided: 40%

Values

[1] => ([],1)
 => ([],1)
 => ([],1)
 => ? = 1
[1,2] => ([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[2,1] => ([],2)
 => ([],1)
 => ([],1)
 => ? = 1
[1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2
[1,3,2] => ([(0,1),(0,2)],3)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1}
[2,1,3] => ([(0,2),(1,2)],3)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1}
[2,3,1] => ([(1,2)],3)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[3,1,2] => ([(1,2)],3)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[3,2,1] => ([],3)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1}
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 3
[1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,2,3,3,8}
[1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,2,3,3,8}
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,2,3,3,8}
[2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,2,3,3,8}
[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[2,3,4,1] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2
[2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2
[2,4,3,1] => ([(1,2),(1,3)],4)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,2,3,3,8}
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2
[3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,2,3,3,8}
[3,2,4,1] => ([(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,2,3,3,8}
[3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[3,4,2,1] => ([(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[4,1,2,3] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2
[4,1,3,2] => ([(1,2),(1,3)],4)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,2,3,3,8}
[4,2,1,3] => ([(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,2,3,3,8}
[4,2,3,1] => ([(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[4,3,1,2] => ([(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[4,3,2,1] => ([],4)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,2,3,3,8}
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2
[1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2
[1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2
[1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2
[1,5,2,4,3] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,5,3,2,4] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[1,5,4,2,3] => ([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2
[2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2
[2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2
[2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2
[2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 3
[2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 3
[2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2
[2,4,1,5,3] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 3
[2,4,3,1,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[2,4,3,5,1] => ([(1,2),(1,3),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 3
[2,4,5,3,1] => ([(1,3),(1,4),(4,2)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 3
[2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[2,5,3,1,4] => ([(0,4),(1,2),(1,3),(3,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[2,5,4,1,3] => ([(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2
[3,1,5,4,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[3,2,5,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[3,4,5,1,2] => ([(0,3),(1,4),(4,2)],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)
 => ? ∊ {1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[3,5,4,2,1] => ([(2,3),(2,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[4,1,3,2,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[4,1,3,5,2] => ([(0,4),(1,2),(1,3),(3,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[4,2,1,3,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[4,2,1,5,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[4,3,5,2,1] => ([(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}

Description

The number of points of the poset minus the width of the poset.

Matching statistic: St000456
(load all 18 compositions to match this statistic)

Mapping

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 40% ●values known / values provided: 59%●distinct values known / distinct values provided: 40%

Values

[1] => [1] => [1] => ([],1)
 => ? = 1
[1,2] => [1,2] => [2,1] => ([(0,1)],2)
 => 1
[2,1] => [1,2] => [2,1] => ([(0,1)],2)
 => 1
[1,2,3] => [1,2,3] => [2,3,1] => ([(0,2),(1,2)],3)
 => 1
[1,3,2] => [1,2,3] => [2,3,1] => ([(0,2),(1,2)],3)
 => 1
[2,1,3] => [1,2,3] => [2,3,1] => ([(0,2),(1,2)],3)
 => 1
[2,3,1] => [1,2,3] => [2,3,1] => ([(0,2),(1,2)],3)
 => 1
[3,1,2] => [1,3,2] => [2,1,3] => ([(1,2)],3)
 => ? ∊ {1,2}
[3,2,1] => [1,3,2] => [2,1,3] => ([(1,2)],3)
 => ? ∊ {1,2}
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[1,2,4,3] => [1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[1,3,2,4] => [1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[1,3,4,2] => [1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[1,4,2,3] => [1,2,4,3] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,2,2,3,3,3,8}
[1,4,3,2] => [1,2,4,3] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,2,2,3,3,3,8}
[2,1,3,4] => [1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[2,1,4,3] => [1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[2,3,1,4] => [1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[2,3,4,1] => [1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[2,4,1,3] => [1,2,4,3] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,2,2,3,3,3,8}
[2,4,3,1] => [1,2,4,3] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,2,2,3,3,3,8}
[3,1,2,4] => [1,3,2,4] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[3,1,4,2] => [1,3,4,2] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 1
[3,2,1,4] => [1,3,2,4] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[3,2,4,1] => [1,3,4,2] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 1
[3,4,1,2] => [1,3,2,4] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[3,4,2,1] => [1,3,2,4] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[4,1,2,3] => [1,4,3,2] => [2,1,3,4] => ([(2,3)],4)
 => ? ∊ {1,1,1,1,2,2,3,3,3,8}
[4,1,3,2] => [1,4,2,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ? ∊ {1,1,1,1,2,2,3,3,3,8}
[4,2,1,3] => [1,4,3,2] => [2,1,3,4] => ([(2,3)],4)
 => ? ∊ {1,1,1,1,2,2,3,3,3,8}
[4,2,3,1] => [1,4,2,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ? ∊ {1,1,1,1,2,2,3,3,3,8}
[4,3,1,2] => [1,4,2,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ? ∊ {1,1,1,1,2,2,3,3,3,8}
[4,3,2,1] => [1,4,2,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ? ∊ {1,1,1,1,2,2,3,3,3,8}
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[1,2,3,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[1,2,4,3,5] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[1,2,4,5,3] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[1,2,5,3,4] => [1,2,3,5,4] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,2,5,4,3] => [1,2,3,5,4] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,3,2,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[1,3,2,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[1,3,4,2,5] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[1,3,4,5,2] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[1,3,5,2,4] => [1,2,3,5,4] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,3,5,4,2] => [1,2,3,5,4] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,4,2,3,5] => [1,2,4,3,5] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,4,2,5,3] => [1,2,4,5,3] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1
[1,4,3,2,5] => [1,2,4,3,5] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,4,3,5,2] => [1,2,4,5,3] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1
[1,4,5,2,3] => [1,2,4,3,5] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,4,5,3,2] => [1,2,4,3,5] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,5,2,3,4] => [1,2,5,4,3] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,5,2,4,3] => [1,2,5,3,4] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,5,3,2,4] => [1,2,5,4,3] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,5,3,4,2] => [1,2,5,3,4] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,5,4,2,3] => [1,2,5,3,4] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,5,4,3,2] => [1,2,5,3,4] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[2,1,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[2,1,3,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[2,1,4,3,5] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[2,1,4,5,3] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[2,1,5,3,4] => [1,2,3,5,4] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[2,1,5,4,3] => [1,2,3,5,4] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[2,3,1,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[2,3,1,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[2,3,4,1,5] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[2,3,4,5,1] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[2,3,5,1,4] => [1,2,3,5,4] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[2,3,5,4,1] => [1,2,3,5,4] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[2,4,1,3,5] => [1,2,4,3,5] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[2,4,1,5,3] => [1,2,4,5,3] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1
[2,4,3,1,5] => [1,2,4,3,5] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[2,4,3,5,1] => [1,2,4,5,3] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1
[2,4,5,1,3] => [1,2,4,3,5] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[2,4,5,3,1] => [1,2,4,3,5] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[2,5,1,3,4] => [1,2,5,4,3] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[2,5,1,4,3] => [1,2,5,3,4] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[2,5,3,1,4] => [1,2,5,4,3] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[2,5,3,4,1] => [1,2,5,3,4] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[2,5,4,1,3] => [1,2,5,3,4] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[2,5,4,3,1] => [1,2,5,3,4] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[3,1,2,4,5] => [1,3,2,4,5] => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[3,1,2,5,4] => [1,3,2,4,5] => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[3,1,5,2,4] => [1,3,5,4,2] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[3,2,5,1,4] => [1,3,5,4,2] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[3,5,1,2,4] => [1,3,2,5,4] => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[3,5,1,4,2] => [1,3,2,5,4] => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[3,5,2,1,4] => [1,3,2,5,4] => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[3,5,2,4,1] => [1,3,2,5,4] => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[5,1,2,3,4] => [1,5,4,3,2] => [2,1,3,4,5] => ([(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[5,1,2,4,3] => [1,5,3,2,4] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[5,1,3,2,4] => [1,5,4,2,3] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[5,1,3,4,2] => [1,5,2,3,4] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[5,1,4,2,3] => [1,5,3,4,2] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[5,1,4,3,2] => [1,5,2,3,4] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[5,2,1,3,4] => [1,5,4,3,2] => [2,1,3,4,5] => ([(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[5,2,1,4,3] => [1,5,3,2,4] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[5,2,3,1,4] => [1,5,4,2,3] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[5,2,3,4,1] => [1,5,2,3,4] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[5,2,4,1,3] => [1,5,3,4,2] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}

Description

The monochromatic index of a connected graph. This is the maximal number of colours such that there is a colouring of the edges where any two vertices can be joined by a monochromatic path. For example, a circle graph other than the triangle can be coloured with at most two colours: one edge blue, all the others red.

Matching statistic: St000771
(load all 36 compositions to match this statistic)

Mapping

Mp00160: Permutations —graph of inversions⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 40% ●values known / values provided: 58%●distinct values known / distinct values provided: 40%

Values

[1] => ([],1)
 => 1
[1,2] => ([],2)
 => ? = 1
[2,1] => ([(0,1)],2)
 => 1
[1,2,3] => ([],3)
 => ? ∊ {1,1,1}
[1,3,2] => ([(1,2)],3)
 => ? ∊ {1,1,1}
[2,1,3] => ([(1,2)],3)
 => ? ∊ {1,1,1}
[2,3,1] => ([(0,2),(1,2)],3)
 => 1
[3,1,2] => ([(0,2),(1,2)],3)
 => 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,2,3,4] => ([],4)
 => ? ∊ {1,1,1,1,1,1,1,1,3,3,8}
[1,2,4,3] => ([(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,3,3,8}
[1,3,2,4] => ([(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,3,3,8}
[1,3,4,2] => ([(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,3,3,8}
[1,4,2,3] => ([(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,3,3,8}
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,3,3,8}
[2,1,3,4] => ([(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,3,3,8}
[2,1,4,3] => ([(0,3),(1,2)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,3,3,8}
[2,3,1,4] => ([(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,3,3,8}
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[3,1,2,4] => ([(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,3,3,8}
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,3,3,8}
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,2,3,4,5] => ([],5)
 => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,2,3,5,4] => ([(3,4)],5)
 => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,2,4,3,5] => ([(3,4)],5)
 => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,3,2,4,5] => ([(3,4)],5)
 => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[2,1,3,4,5] => ([(3,4)],5)
 => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[2,1,3,5,4] => ([(1,4),(2,3)],5)
 => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[2,1,4,3,5] => ([(1,4),(2,3)],5)
 => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[2,3,1,4,5] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1
[2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1
[2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 1
[2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1
[2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 1
[2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 1
[2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => 1
[2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1
[3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1
[3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 1
[3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 1
[3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 2
[3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 1
[3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 1
[3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 1
[3,5,2,1,4] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => 1
[3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 1
[3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1
[4,1,3,5,2] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 1
[4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 1

Description

The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. The distance Laplacian of a graph is the (symmetric) matrix with row and column sums

$0$ , which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue. For example, the cycle on four vertices has distance Laplacian

$\left(\begin{array}{rrrr} 4 & -1 & -2 & -1 \\ -1 & 4 & -1 & -2 \\ -2 & -1 & 4 & -1 \\ -1 & -2 & -1 & 4 \end{array}\right).$ Its eigenvalues are

$0,4,4,6$ , so the statistic is

$2$ . The path on four vertices has eigenvalues

$0, 4.7\dots, 6, 9.2\dots$ and therefore statistic

$1$ .

Matching statistic: St000772
(load all 45 compositions to match this statistic)

Mapping

Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00264: Graphs —delete endpoints⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 40% ●values known / values provided: 58%●distinct values known / distinct values provided: 40%

Values

[1] => ([],1)
 => ([],1)
 => 1
[1,2] => ([],2)
 => ([],2)
 => ? = 1
[2,1] => ([(0,1)],2)
 => ([],1)
 => 1
[1,2,3] => ([],3)
 => ([],3)
 => ? ∊ {1,1,1}
[1,3,2] => ([(1,2)],3)
 => ([],2)
 => ? ∊ {1,1,1}
[2,1,3] => ([(1,2)],3)
 => ([],2)
 => ? ∊ {1,1,1}
[2,3,1] => ([(0,2),(1,2)],3)
 => ([],1)
 => 1
[3,1,2] => ([(0,2),(1,2)],3)
 => ([],1)
 => 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,2,3,4] => ([],4)
 => ([],4)
 => ? ∊ {1,1,1,1,1,1,1,2,3,3,8}
[1,2,4,3] => ([(2,3)],4)
 => ([],3)
 => ? ∊ {1,1,1,1,1,1,1,2,3,3,8}
[1,3,2,4] => ([(2,3)],4)
 => ([],3)
 => ? ∊ {1,1,1,1,1,1,1,2,3,3,8}
[1,3,4,2] => ([(1,3),(2,3)],4)
 => ([],2)
 => ? ∊ {1,1,1,1,1,1,1,2,3,3,8}
[1,4,2,3] => ([(1,3),(2,3)],4)
 => ([],2)
 => ? ∊ {1,1,1,1,1,1,1,2,3,3,8}
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,2,3,3,8}
[2,1,3,4] => ([(2,3)],4)
 => ([],3)
 => ? ∊ {1,1,1,1,1,1,1,2,3,3,8}
[2,1,4,3] => ([(0,3),(1,2)],4)
 => ([],2)
 => ? ∊ {1,1,1,1,1,1,1,2,3,3,8}
[2,3,1,4] => ([(1,3),(2,3)],4)
 => ([],2)
 => ? ∊ {1,1,1,1,1,1,1,2,3,3,8}
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ([],1)
 => 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => ([],1)
 => 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[3,1,2,4] => ([(1,3),(2,3)],4)
 => ([],2)
 => ? ∊ {1,1,1,1,1,1,1,2,3,3,8}
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => ([],1)
 => 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,2,3,3,8}
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([],1)
 => 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,2,3,4,5] => ([],5)
 => ([],5)
 => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,2,3,5,4] => ([(3,4)],5)
 => ([],4)
 => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,2,4,3,5] => ([(3,4)],5)
 => ([],4)
 => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([],3)
 => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([],3)
 => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,3,2,4,5] => ([(3,4)],5)
 => ([],4)
 => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([],3)
 => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ([],3)
 => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ([],2)
 => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ([],2)
 => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => ([],3)
 => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => ([],2)
 => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => ([],2)
 => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[2,1,3,4,5] => ([(3,4)],5)
 => ([],4)
 => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[2,1,3,5,4] => ([(1,4),(2,3)],5)
 => ([],3)
 => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[2,1,4,3,5] => ([(1,4),(2,3)],5)
 => ([],3)
 => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => ([],2)
 => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => ([],2)
 => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[2,3,1,4,5] => ([(2,4),(3,4)],5)
 => ([],3)
 => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ([],2)
 => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ([],2)
 => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => 1
[2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([],1)
 => 1
[2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => ([],2)
 => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([],1)
 => 1
[2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62}
[2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([],1)
 => 1
[2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([],1)
 => 1
[3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([],1)
 => 1
[3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 1
[3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[3,4,5,1,2] => ([(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)
 => 2
[3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 1
[3,5,2,1,4] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 1
[3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([],1)
 => 1
[4,1,3,5,2] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2

Description

The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. The distance Laplacian of a graph is the (symmetric) matrix with row and column sums

$\left(\begin{array}{rrrr} 4 & -1 & -2 & -1 \\ -1 & 4 & -1 & -2 \\ -2 & -1 & 4 & -1 \\ -1 & -2 & -1 & 4 \end{array}\right).$ Its eigenvalues are

$0,4,4,6$ , so the statistic is

$1$ . The path on four vertices has eigenvalues

$0, 4.7\dots, 6, 9.2\dots$ and therefore also statistic

$1$ . The graphs with statistic

$n-1$ ,

$n-2$ and

$n-3$ have been characterised, see [1].

Matching statistic: St000319

Mapping

Mp00065: Permutations —permutation poset⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 20% ●values known / values provided: 58%●distinct values known / distinct values provided: 20%

Values

[1] => ([],1)
 => [1]
 => [1]
 => 0 = 1 - 1
[1,2] => ([(0,1)],2)
 => [1]
 => [1]
 => 0 = 1 - 1
[2,1] => ([],2)
 => [2]
 => [1,1]
 => 0 = 1 - 1
[1,2,3] => ([(0,2),(2,1)],3)
 => [1]
 => [1]
 => 0 = 1 - 1
[1,3,2] => ([(0,1),(0,2)],3)
 => [2]
 => [1,1]
 => 0 = 1 - 1
[2,1,3] => ([(0,2),(1,2)],3)
 => [2]
 => [1,1]
 => 0 = 1 - 1
[2,3,1] => ([(1,2)],3)
 => [3]
 => [1,1,1]
 => 0 = 1 - 1
[3,1,2] => ([(1,2)],3)
 => [3]
 => [1,1,1]
 => 0 = 1 - 1
[3,2,1] => ([],3)
 => [3,3]
 => [2,2,2]
 => 1 = 2 - 1
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => [1]
 => [1]
 => 0 = 1 - 1
[1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => [2]
 => [1,1]
 => 0 = 1 - 1
[1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => [2]
 => [1,1]
 => 0 = 1 - 1
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => [3]
 => [1,1,1]
 => 0 = 1 - 1
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => [3]
 => [1,1,1]
 => 0 = 1 - 1
[1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => [3,3]
 => [2,2,2]
 => 1 = 2 - 1
[2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => [2]
 => [1,1]
 => 0 = 1 - 1
[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [2,2]
 => 1 = 2 - 1
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => [3]
 => [1,1,1]
 => 0 = 1 - 1
[2,3,4,1] => ([(1,2),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => 0 = 1 - 1
[2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => [3,2]
 => [2,2,1]
 => 1 = 2 - 1
[2,4,3,1] => ([(1,2),(1,3)],4)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => 0 = 1 - 1
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => [3]
 => [1,1,1]
 => 0 = 1 - 1
[3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => [3,2]
 => [2,2,1]
 => 1 = 2 - 1
[3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => [3,3]
 => [2,2,2]
 => 1 = 2 - 1
[3,2,4,1] => ([(1,3),(2,3)],4)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => 0 = 1 - 1
[3,4,1,2] => ([(0,3),(1,2)],4)
 => [4,2]
 => [2,2,1,1]
 => 1 = 2 - 1
[3,4,2,1] => ([(2,3)],4)
 => [4,4,4]
 => [3,3,3,3]
 => ? ∊ {3,3,3,8} - 1
[4,1,2,3] => ([(1,2),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => 0 = 1 - 1
[4,1,3,2] => ([(1,2),(1,3)],4)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => 0 = 1 - 1
[4,2,1,3] => ([(1,3),(2,3)],4)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => 0 = 1 - 1
[4,2,3,1] => ([(2,3)],4)
 => [4,4,4]
 => [3,3,3,3]
 => ? ∊ {3,3,3,8} - 1
[4,3,1,2] => ([(2,3)],4)
 => [4,4,4]
 => [3,3,3,3]
 => ? ∊ {3,3,3,8} - 1
[4,3,2,1] => ([],4)
 => [4,4,4,4,4,4]
 => [6,6,6,6]
 => ? ∊ {3,3,3,8} - 1
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [1]
 => [1]
 => 0 = 1 - 1
[1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
 => [2]
 => [1,1]
 => 0 = 1 - 1
[1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [2]
 => [1,1]
 => 0 = 1 - 1
[1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => [3]
 => [1,1,1]
 => 0 = 1 - 1
[1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => [3]
 => [1,1,1]
 => 0 = 1 - 1
[1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
 => [3,3]
 => [2,2,2]
 => 1 = 2 - 1
[1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [2]
 => [1,1]
 => 0 = 1 - 1
[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2]
 => [2,2]
 => 1 = 2 - 1
[1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [3]
 => [1,1,1]
 => 0 = 1 - 1
[1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => [4]
 => [1,1,1,1]
 => 0 = 1 - 1
[1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [3,2]
 => [2,2,1]
 => 1 = 2 - 1
[1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => 0 = 1 - 1
[1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [3]
 => [1,1,1]
 => 0 = 1 - 1
[1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [3,2]
 => [2,2,1]
 => 1 = 2 - 1
[1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [2,2,2]
 => 1 = 2 - 1
[1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => 0 = 1 - 1
[1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
 => [4,2]
 => [2,2,1,1]
 => 1 = 2 - 1
[1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
 => [4,4,4]
 => [3,3,3,3]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => [4]
 => [1,1,1,1]
 => 0 = 1 - 1
[1,5,2,4,3] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => 0 = 1 - 1
[1,5,3,2,4] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => 0 = 1 - 1
[1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
 => [4,4,4]
 => [3,3,3,3]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[1,5,4,2,3] => ([(0,2),(0,3),(0,4),(4,1)],5)
 => [4,4,4]
 => [3,3,3,3]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => [4,4,4,4,4,4]
 => [6,6,6,6]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
 => [2]
 => [1,1]
 => 0 = 1 - 1
[2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [6,6]
 => [2,2,2,2,2,2]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[2,4,5,3,1] => ([(1,3),(1,4),(4,2)],5)
 => [15]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [14]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[2,5,3,1,4] => ([(0,4),(1,2),(1,3),(3,4)],5)
 => [4,4,3]
 => [3,3,3,2]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
 => [15]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[2,5,4,1,3] => ([(0,4),(1,2),(1,3),(1,4)],5)
 => [10,4,4]
 => [3,3,3,3,1,1,1,1,1,1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
 => [15,15]
 => [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[3,1,5,4,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [14]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6,6]
 => [2,2,2,2,2,2]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[3,2,5,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [14]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [5,5,5,5]
 => [4,4,4,4,4]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[3,4,2,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [4,4,4]
 => [3,3,3,3]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
 => [15]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[3,4,5,2,1] => ([(2,3),(3,4)],5)
 => [5,5,5,5]
 => [4,4,4,4,4]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
 => [12,4]
 => [2,2,2,2,1,1,1,1,1,1,1,1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[3,5,2,1,4] => ([(0,4),(1,4),(2,3),(2,4)],5)
 => [10,4,4]
 => [3,3,3,3,1,1,1,1,1,1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[3,5,2,4,1] => ([(1,4),(2,3),(2,4)],5)
 => [15,5,5]
 => [3,3,3,3,3,1,1,1,1,1,1,1,1,1,1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[3,5,4,1,2] => ([(0,4),(1,2),(1,3)],5)
 => [10,10]
 => [2,2,2,2,2,2,2,2,2,2]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[3,5,4,2,1] => ([(2,3),(2,4)],5)
 => [10,10,10,10]
 => [4,4,4,4,4,4,4,4,4,4]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[4,1,3,5,2] => ([(0,4),(1,2),(1,3),(3,4)],5)
 => [4,4,3]
 => [3,3,3,2]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[4,1,5,3,2] => ([(0,4),(1,2),(1,3),(1,4)],5)
 => [10,4,4]
 => [3,3,3,3,1,1,1,1,1,1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[4,2,1,5,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [14]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [4,4,4]
 => [3,3,3,3]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
 => [15]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [12,4]
 => [2,2,2,2,1,1,1,1,1,1,1,1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
 => [15,5,5]
 => [3,3,3,3,3,1,1,1,1,1,1,1,1,1,1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[4,3,1,2,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [4,4,4]
 => [3,3,3,3]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[4,3,1,5,2] => ([(0,4),(1,4),(2,3),(2,4)],5)
 => [10,4,4]
 => [3,3,3,3,1,1,1,1,1,1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [4,4,4,4,4,4]
 => [6,6,6,6]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
 => [15,15]
 => [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
 => [10,10]
 => [2,2,2,2,2,2,2,2,2,2]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[4,3,5,2,1] => ([(2,4),(3,4)],5)
 => [10,10,10,10]
 => [4,4,4,4,4,4,4,4,4,4]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[4,5,1,3,2] => ([(0,4),(1,2),(1,3)],5)
 => [10,10]
 => [2,2,2,2,2,2,2,2,2,2]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[4,5,2,1,3] => ([(0,4),(1,4),(2,3)],5)
 => [10,10]
 => [2,2,2,2,2,2,2,2,2,2]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[4,5,2,3,1] => ([(1,4),(2,3)],5)
 => [5,5,5,5,5,5]
 => [6,6,6,6,6]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[4,5,3,1,2] => ([(1,4),(2,3)],5)
 => [5,5,5,5,5,5]
 => [6,6,6,6,6]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[4,5,3,2,1] => ([(3,4)],5)
 => [5,5,5,5,5,5,5,5,5,5,5,5]
 => [12,12,12,12,12]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[5,1,3,4,2] => ([(1,3),(1,4),(4,2)],5)
 => [15]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[5,1,4,2,3] => ([(1,3),(1,4),(4,2)],5)
 => [15]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[5,1,4,3,2] => ([(1,2),(1,3),(1,4)],5)
 => [15,15]
 => [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[5,2,1,4,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [5,5,5,5]
 => [4,4,4,4,4]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5)
 => [15]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1

Description

The spin of an integer partition. The Ferrers shape of an integer partition

$\lambda$ can be decomposed into border strips. The spin is then defined to be the total number of crossings of border strips of

$\lambda$ with the vertical lines in the Ferrers shape. The following example is taken from Appendix B in [1]: Let

$\lambda = (5,5,4,4,2,1)$ . Removing the border strips successively yields the sequence of partitions

$(5,5,4,4,2,1), (4,3,3,1), (2,2), (1), ().$ The first strip

$(5,5,4,4,2,1) \setminus (4,3,3,1)$ crosses

$4$ times, the second strip

$(4,3,3,1) \setminus (2,2)$ crosses

$3$ times, the strip

$(2,2) \setminus (1)$ crosses

$1$ time, and the remaining strip

$(1) \setminus ()$ does not cross. This yields the spin of

$(5,5,4,4,2,1)$ to be

$4+3+1 = 8$ .

Matching statistic: St000320

Mapping

Mp00065: Permutations —permutation poset⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 20% ●values known / values provided: 58%●distinct values known / distinct values provided: 20%

Values

[1] => ([],1)
 => [1]
 => [1]
 => 0 = 1 - 1
[1,2] => ([(0,1)],2)
 => [1]
 => [1]
 => 0 = 1 - 1
[2,1] => ([],2)
 => [2]
 => [1,1]
 => 0 = 1 - 1
[1,2,3] => ([(0,2),(2,1)],3)
 => [1]
 => [1]
 => 0 = 1 - 1
[1,3,2] => ([(0,1),(0,2)],3)
 => [2]
 => [1,1]
 => 0 = 1 - 1
[2,1,3] => ([(0,2),(1,2)],3)
 => [2]
 => [1,1]
 => 0 = 1 - 1
[2,3,1] => ([(1,2)],3)
 => [3]
 => [1,1,1]
 => 0 = 1 - 1
[3,1,2] => ([(1,2)],3)
 => [3]
 => [1,1,1]
 => 0 = 1 - 1
[3,2,1] => ([],3)
 => [3,3]
 => [2,2,2]
 => 1 = 2 - 1
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => [1]
 => [1]
 => 0 = 1 - 1
[1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => [2]
 => [1,1]
 => 0 = 1 - 1
[1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => [2]
 => [1,1]
 => 0 = 1 - 1
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => [3]
 => [1,1,1]
 => 0 = 1 - 1
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => [3]
 => [1,1,1]
 => 0 = 1 - 1
[1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => [3,3]
 => [2,2,2]
 => 1 = 2 - 1
[2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => [2]
 => [1,1]
 => 0 = 1 - 1
[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [2,2]
 => 1 = 2 - 1
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => [3]
 => [1,1,1]
 => 0 = 1 - 1
[2,3,4,1] => ([(1,2),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => 0 = 1 - 1
[2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => [3,2]
 => [2,2,1]
 => 1 = 2 - 1
[2,4,3,1] => ([(1,2),(1,3)],4)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => 0 = 1 - 1
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => [3]
 => [1,1,1]
 => 0 = 1 - 1
[3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => [3,2]
 => [2,2,1]
 => 1 = 2 - 1
[3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => [3,3]
 => [2,2,2]
 => 1 = 2 - 1
[3,2,4,1] => ([(1,3),(2,3)],4)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => 0 = 1 - 1
[3,4,1,2] => ([(0,3),(1,2)],4)
 => [4,2]
 => [2,2,1,1]
 => 1 = 2 - 1
[3,4,2,1] => ([(2,3)],4)
 => [4,4,4]
 => [3,3,3,3]
 => ? ∊ {3,3,3,8} - 1
[4,1,2,3] => ([(1,2),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => 0 = 1 - 1
[4,1,3,2] => ([(1,2),(1,3)],4)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => 0 = 1 - 1
[4,2,1,3] => ([(1,3),(2,3)],4)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => 0 = 1 - 1
[4,2,3,1] => ([(2,3)],4)
 => [4,4,4]
 => [3,3,3,3]
 => ? ∊ {3,3,3,8} - 1
[4,3,1,2] => ([(2,3)],4)
 => [4,4,4]
 => [3,3,3,3]
 => ? ∊ {3,3,3,8} - 1
[4,3,2,1] => ([],4)
 => [4,4,4,4,4,4]
 => [6,6,6,6]
 => ? ∊ {3,3,3,8} - 1
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [1]
 => [1]
 => 0 = 1 - 1
[1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
 => [2]
 => [1,1]
 => 0 = 1 - 1
[1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [2]
 => [1,1]
 => 0 = 1 - 1
[1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => [3]
 => [1,1,1]
 => 0 = 1 - 1
[1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => [3]
 => [1,1,1]
 => 0 = 1 - 1
[1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
 => [3,3]
 => [2,2,2]
 => 1 = 2 - 1
[1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [2]
 => [1,1]
 => 0 = 1 - 1
[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2]
 => [2,2]
 => 1 = 2 - 1
[1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [3]
 => [1,1,1]
 => 0 = 1 - 1
[1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => [4]
 => [1,1,1,1]
 => 0 = 1 - 1
[1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [3,2]
 => [2,2,1]
 => 1 = 2 - 1
[1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => 0 = 1 - 1
[1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [3]
 => [1,1,1]
 => 0 = 1 - 1
[1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [3,2]
 => [2,2,1]
 => 1 = 2 - 1
[1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [2,2,2]
 => 1 = 2 - 1
[1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => 0 = 1 - 1
[1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
 => [4,2]
 => [2,2,1,1]
 => 1 = 2 - 1
[1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
 => [4,4,4]
 => [3,3,3,3]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => [4]
 => [1,1,1,1]
 => 0 = 1 - 1
[1,5,2,4,3] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => 0 = 1 - 1
[1,5,3,2,4] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => 0 = 1 - 1
[1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
 => [4,4,4]
 => [3,3,3,3]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[1,5,4,2,3] => ([(0,2),(0,3),(0,4),(4,1)],5)
 => [4,4,4]
 => [3,3,3,3]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => [4,4,4,4,4,4]
 => [6,6,6,6]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
 => [2]
 => [1,1]
 => 0 = 1 - 1
[2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [6,6]
 => [2,2,2,2,2,2]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[2,4,5,3,1] => ([(1,3),(1,4),(4,2)],5)
 => [15]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [14]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[2,5,3,1,4] => ([(0,4),(1,2),(1,3),(3,4)],5)
 => [4,4,3]
 => [3,3,3,2]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
 => [15]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[2,5,4,1,3] => ([(0,4),(1,2),(1,3),(1,4)],5)
 => [10,4,4]
 => [3,3,3,3,1,1,1,1,1,1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
 => [15,15]
 => [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[3,1,5,4,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [14]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6,6]
 => [2,2,2,2,2,2]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[3,2,5,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [14]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [5,5,5,5]
 => [4,4,4,4,4]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[3,4,2,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [4,4,4]
 => [3,3,3,3]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
 => [15]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[3,4,5,2,1] => ([(2,3),(3,4)],5)
 => [5,5,5,5]
 => [4,4,4,4,4]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
 => [12,4]
 => [2,2,2,2,1,1,1,1,1,1,1,1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[3,5,2,1,4] => ([(0,4),(1,4),(2,3),(2,4)],5)
 => [10,4,4]
 => [3,3,3,3,1,1,1,1,1,1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[3,5,2,4,1] => ([(1,4),(2,3),(2,4)],5)
 => [15,5,5]
 => [3,3,3,3,3,1,1,1,1,1,1,1,1,1,1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[3,5,4,1,2] => ([(0,4),(1,2),(1,3)],5)
 => [10,10]
 => [2,2,2,2,2,2,2,2,2,2]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[3,5,4,2,1] => ([(2,3),(2,4)],5)
 => [10,10,10,10]
 => [4,4,4,4,4,4,4,4,4,4]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[4,1,3,5,2] => ([(0,4),(1,2),(1,3),(3,4)],5)
 => [4,4,3]
 => [3,3,3,2]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[4,1,5,3,2] => ([(0,4),(1,2),(1,3),(1,4)],5)
 => [10,4,4]
 => [3,3,3,3,1,1,1,1,1,1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[4,2,1,5,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [14]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [4,4,4]
 => [3,3,3,3]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
 => [15]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [12,4]
 => [2,2,2,2,1,1,1,1,1,1,1,1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
 => [15,5,5]
 => [3,3,3,3,3,1,1,1,1,1,1,1,1,1,1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[4,3,1,2,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [4,4,4]
 => [3,3,3,3]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[4,3,1,5,2] => ([(0,4),(1,4),(2,3),(2,4)],5)
 => [10,4,4]
 => [3,3,3,3,1,1,1,1,1,1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [4,4,4,4,4,4]
 => [6,6,6,6]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
 => [15,15]
 => [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
 => [10,10]
 => [2,2,2,2,2,2,2,2,2,2]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[4,3,5,2,1] => ([(2,4),(3,4)],5)
 => [10,10,10,10]
 => [4,4,4,4,4,4,4,4,4,4]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[4,5,1,3,2] => ([(0,4),(1,2),(1,3)],5)
 => [10,10]
 => [2,2,2,2,2,2,2,2,2,2]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[4,5,2,1,3] => ([(0,4),(1,4),(2,3)],5)
 => [10,10]
 => [2,2,2,2,2,2,2,2,2,2]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[4,5,2,3,1] => ([(1,4),(2,3)],5)
 => [5,5,5,5,5,5]
 => [6,6,6,6,6]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[4,5,3,1,2] => ([(1,4),(2,3)],5)
 => [5,5,5,5,5,5]
 => [6,6,6,6,6]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[4,5,3,2,1] => ([(3,4)],5)
 => [5,5,5,5,5,5,5,5,5,5,5,5]
 => [12,12,12,12,12]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[5,1,3,4,2] => ([(1,3),(1,4),(4,2)],5)
 => [15]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[5,1,4,2,3] => ([(1,3),(1,4),(4,2)],5)
 => [15]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[5,1,4,3,2] => ([(1,2),(1,3),(1,4)],5)
 => [15,15]
 => [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[5,2,1,4,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [5,5,5,5]
 => [4,4,4,4,4]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1
[5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5)
 => [15]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,8,8,8,8,8,8,11,11,11,11,11,11,20,20,20,20,62} - 1

Description

The dinv adjustment of an integer partition. The Ferrers shape of an integer partition

$\lambda = (\lambda_1,\ldots,\lambda_k)$ can be decomposed into border strips. For

$0 \leq j < \lambda_1$ let

$n_j$ be the length of the border strip starting at

$(\lambda_1-j,0)$ . The dinv adjustment is then defined by

$\sum_{j:n_j > 0}(\lambda_1-1-j).$ The following example is taken from Appendix B in [2]: Let

$\lambda=(5,5,4,4,2,1)$ . Removing the border strips successively yields the sequence of partitions

$(5,5,4,4,2,1),(4,3,3,1),(2,2),(1),(),$ and we obtain

$(n_0,\ldots,n_4) = (10,7,0,3,1)$ . The dinv adjustment is thus

$4+3+1+0 = 8$ .

The following 115 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001389The number of partitions of the same length below the given integer partition. St001933The largest multiplicity of a part in an integer partition. St000815The number of semistandard Young tableaux of partition weight of given shape. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000668The least common multiple of the parts of the partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. 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. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001432The order dimension of the partition. St000454The largest eigenvalue of a graph if it is integral. St001118The acyclic chromatic index of a graph. St001568The smallest positive integer that does not appear twice in the partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000460The hook length of the last cell along the main diagonal of an integer partition. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001360The number of covering relations in Young's lattice below a partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001498The normalised height of a Nakayama algebra with magnitude 1. 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. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001438The number of missing boxes of a skew partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000707The product of the factorials of the parts. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. St001378The product of the cohook lengths of the integer partition. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001845The number of join irreducibles minus the rank of a lattice. St000909The number of maximal chains of maximal size in a poset. St001435The number of missing boxes in the first row. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001846The number of elements which do not have a complement in the lattice. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St000527The width of the poset. St001060The distinguishing index of a graph. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000181The number of connected components of the Hasse diagram for the poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000908The length of the shortest maximal antichain in a poset. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001510The number of self-evacuating linear extensions of a finite poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001645The pebbling number of a connected graph. St001779The order of promotion on the set of linear extensions of a poset. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000632The jump number of the poset. St001301The first Betti number of the order complex associated with the poset. St001330The hat guessing number of a graph. St001396Number of triples of incomparable elements in a finite poset. St001397Number of pairs of incomparable elements in a finite poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001902The number of potential covers of a poset. St001472The permanent of the Coxeter matrix of the poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001877Number of indecomposable injective modules with projective dimension 2. St000284The Plancherel distribution on integer partitions. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001128The exponens consonantiae of a partition. St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. 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. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000706The product of the factorials of the multiplicities of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000929The constant term of the character polynomial of an integer partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. 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. St001487The number of inner corners of a skew partition. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St000914The sum of the values of the Möbius function of a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001890The maximum magnitude of the Möbius function of a poset. St001875The number of simple modules with projective dimension at most 1. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001095The number of non-isomorphic posets with precisely one further covering relation. St000264The girth of a graph, which is not a tree.