Processing math: 66%

Your data matches 11 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000741
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000741: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => [1] => ([],1)
 => 0
{{1,2}}
 => [2,1] => [1,2] => ([],2)
 => 1
{{1},{2}}
 => [1,2] => [1,2] => ([],2)
 => 1
{{1,2,3}}
 => [2,3,1] => [1,2,3] => ([],3)
 => 1
{{1,2},{3}}
 => [2,1,3] => [1,2,3] => ([],3)
 => 1
{{1},{2,3}}
 => [1,3,2] => [1,2,3] => ([],3)
 => 1
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => ([],3)
 => 1
{{1,2,3,4}}
 => [2,3,4,1] => [1,2,3,4] => ([],4)
 => 1
{{1,2,3},{4}}
 => [2,3,1,4] => [1,2,3,4] => ([],4)
 => 1
{{1,2},{3,4}}
 => [2,1,4,3] => [1,2,3,4] => ([],4)
 => 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [1,2,3,4] => ([],4)
 => 1
{{1,3,4},{2}}
 => [3,2,4,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => 1
{{1,4},{2,3}}
 => [4,3,2,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => 1
{{1},{2,3,4}}
 => [1,3,4,2] => [1,2,3,4] => ([],4)
 => 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,2,3,4] => ([],4)
 => 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => 1
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,3,4] => ([],4)
 => 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => ([],4)
 => 1
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [1,2,3,4,5] => ([],5)
 => 1
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [1,2,3,4,5] => ([],5)
 => 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [1,2,3,4,5] => ([],5)
 => 1
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [1,2,3,4,5] => ([],5)
 => 1
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => 1
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => 1
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [1,2,3,4,5] => ([],5)
 => 1
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [1,2,3,4,5] => ([],5)
 => 1
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => 1
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [1,2,3,4,5] => ([],5)
 => 1
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [1,2,3,4,5] => ([],5)
 => 1
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => 1
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => 1
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => 1
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => 1
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
 => 1
{{1,4},{2,3},{5}}
 => [4,3,2,1,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
 => 1
{{1},{2,3,4,5}}
 => [1,3,4,5,2] => [1,2,3,4,5] => ([],5)
 => 1
{{1},{2,3,4},{5}}
 => [1,3,4,2,5] => [1,2,3,4,5] => ([],5)
 => 1
{{1},{2,3},{4,5}}
 => [1,3,2,5,4] => [1,2,3,4,5] => ([],5)
 => 1
{{1},{2,3},{4},{5}}
 => [1,3,2,4,5] => [1,2,3,4,5] => ([],5)
 => 1
{{1,4},{2,5},{3}}
 => [4,5,3,1,2] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => 1
{{1,4},{2},{3,5}}
 => [4,2,5,1,3] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
 => 1
{{1,4},{2},{3},{5}}
 => [4,2,3,1,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
 => 1
{{1,5},{2,4},{3}}
 => [5,4,3,2,1] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 2
{{1},{2,4,5},{3}}
 => [1,4,3,5,2] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => 1
{{1},{2,5},{3,4}}
 => [1,5,4,3,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => 1
{{1},{2},{3,4,5}}
 => [1,2,4,5,3] => [1,2,3,4,5] => ([],5)
 => 1
{{1},{2},{3,4},{5}}
 => [1,2,4,3,5] => [1,2,3,4,5] => ([],5)
 => 1
{{1},{2,5},{3},{4}}
 => [1,5,3,4,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => 1
{{1},{2},{3},{4,5}}
 => [1,2,3,5,4] => [1,2,3,4,5] => ([],5)
 => 1
{{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => 1
Description
The Colin de Verdière graph invariant.
Matching statistic: St000779
(load all 3 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000779: Permutations ⟶ ℤResult quality: 67% ●values known / values provided: 99%●distinct values known / distinct values provided: 67%
Values
{{1}}
 => [1] => [1] => [1] => ? = 0 - 1
{{1,2}}
 => [2,1] => [1,2] => [1,2] => 0 = 1 - 1
{{1},{2}}
 => [1,2] => [1,2] => [1,2] => 0 = 1 - 1
{{1,2,3}}
 => [2,3,1] => [1,2,3] => [1,2,3] => 0 = 1 - 1
{{1,2},{3}}
 => [2,1,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
{{1},{2,3}}
 => [1,3,2] => [1,2,3] => [1,2,3] => 0 = 1 - 1
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
{{1,2,3,4}}
 => [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
{{1,2,3},{4}}
 => [2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
{{1,2},{3,4}}
 => [2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
{{1,3,4},{2}}
 => [3,2,4,1] => [1,3,4,2] => [1,4,2,3] => 0 = 1 - 1
{{1,4},{2,3}}
 => [4,3,2,1] => [1,4,2,3] => [1,4,3,2] => 0 = 1 - 1
{{1},{2,3,4}}
 => [1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [1,4,2,3] => [1,4,3,2] => 0 = 1 - 1
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [1,2,4,5,3] => [1,2,5,3,4] => 0 = 1 - 1
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [1,2,5,3,4] => [1,2,5,4,3] => 0 = 1 - 1
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [1,2,5,3,4] => [1,2,5,4,3] => 0 = 1 - 1
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [1,3,4,2,5] => [1,4,2,3,5] => 0 = 1 - 1
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [1,3,4,2,5] => [1,4,2,3,5] => 0 = 1 - 1
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [1,3,5,2,4] => [1,5,4,2,3] => 0 = 1 - 1
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [1,3,5,2,4] => [1,5,4,2,3] => 0 = 1 - 1
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [1,4,2,3,5] => [1,4,3,2,5] => 0 = 1 - 1
{{1,4},{2,3},{5}}
 => [4,3,2,1,5] => [1,4,2,3,5] => [1,4,3,2,5] => 0 = 1 - 1
{{1},{2,3,4,5}}
 => [1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
{{1},{2,3,4},{5}}
 => [1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
{{1},{2,3},{4,5}}
 => [1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
{{1},{2,3},{4},{5}}
 => [1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
{{1,4},{2,5},{3}}
 => [4,5,3,1,2] => [1,4,2,5,3] => [1,5,3,2,4] => 0 = 1 - 1
{{1,4},{2},{3,5}}
 => [4,2,5,1,3] => [1,4,2,3,5] => [1,4,3,2,5] => 0 = 1 - 1
{{1,4},{2},{3},{5}}
 => [4,2,3,1,5] => [1,4,2,3,5] => [1,4,3,2,5] => 0 = 1 - 1
{{1,5},{2,4},{3}}
 => [5,4,3,2,1] => [1,5,2,4,3] => [1,4,5,3,2] => 1 = 2 - 1
{{1},{2,4,5},{3}}
 => [1,4,3,5,2] => [1,2,4,5,3] => [1,2,5,3,4] => 0 = 1 - 1
{{1},{2,5},{3,4}}
 => [1,5,4,3,2] => [1,2,5,3,4] => [1,2,5,4,3] => 0 = 1 - 1
{{1},{2},{3,4,5}}
 => [1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
{{1},{2},{3,4},{5}}
 => [1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
{{1},{2,5},{3},{4}}
 => [1,5,3,4,2] => [1,2,5,3,4] => [1,2,5,4,3] => 0 = 1 - 1
{{1},{2},{3},{4,5}}
 => [1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
{{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
{{1,2,3,4,5,6}}
 => [2,3,4,5,6,1] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0 = 1 - 1
Description
The tier of a permutation. This is the number of elements i such that [i+1,k,i] is an occurrence of the pattern [2,3,1]. For example, [3,5,6,1,2,4] has tier 2, with witnesses [3,5,2] (or [3,6,2]) and [5,6,4]. According to [1], this is the number of passes minus one needed to sort the permutation using a single stack. The generating function for this statistic appears as [[OEIS:A122890]] and [[OEIS:A158830]] in the form of triangles read by rows, see [sec. 4, 1].
Matching statistic: St000872
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000872: Permutations ⟶ ℤResult quality: 67% ●values known / values provided: 99%●distinct values known / distinct values provided: 67%
Values
{{1}}
 => [1] => [1] => [1] => ? = 0 - 1
{{1,2}}
 => [2,1] => [1,2] => [1,2] => 0 = 1 - 1
{{1},{2}}
 => [1,2] => [1,2] => [1,2] => 0 = 1 - 1
{{1,2,3}}
 => [2,3,1] => [1,2,3] => [1,2,3] => 0 = 1 - 1
{{1,2},{3}}
 => [2,1,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
{{1},{2,3}}
 => [1,3,2] => [1,2,3] => [1,2,3] => 0 = 1 - 1
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
{{1,2,3,4}}
 => [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
{{1,2,3},{4}}
 => [2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
{{1,2},{3,4}}
 => [2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
{{1,3,4},{2}}
 => [3,2,4,1] => [1,3,4,2] => [1,4,3,2] => 0 = 1 - 1
{{1,4},{2,3}}
 => [4,3,2,1] => [1,4,2,3] => [1,4,2,3] => 0 = 1 - 1
{{1},{2,3,4}}
 => [1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [1,4,2,3] => [1,4,2,3] => 0 = 1 - 1
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [1,2,4,5,3] => [1,2,5,4,3] => 0 = 1 - 1
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [1,2,5,3,4] => [1,2,5,3,4] => 0 = 1 - 1
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [1,2,5,3,4] => [1,2,5,3,4] => 0 = 1 - 1
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [1,3,4,2,5] => [1,4,3,2,5] => 0 = 1 - 1
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [1,3,4,2,5] => [1,4,3,2,5] => 0 = 1 - 1
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [1,3,5,2,4] => [1,5,3,2,4] => 0 = 1 - 1
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [1,3,5,2,4] => [1,5,3,2,4] => 0 = 1 - 1
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [1,4,2,3,5] => [1,4,2,3,5] => 0 = 1 - 1
{{1,4},{2,3},{5}}
 => [4,3,2,1,5] => [1,4,2,3,5] => [1,4,2,3,5] => 0 = 1 - 1
{{1},{2,3,4,5}}
 => [1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
{{1},{2,3,4},{5}}
 => [1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
{{1},{2,3},{4,5}}
 => [1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
{{1},{2,3},{4},{5}}
 => [1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
{{1,4},{2,5},{3}}
 => [4,5,3,1,2] => [1,4,2,5,3] => [1,5,4,2,3] => 0 = 1 - 1
{{1,4},{2},{3,5}}
 => [4,2,5,1,3] => [1,4,2,3,5] => [1,4,2,3,5] => 0 = 1 - 1
{{1,4},{2},{3},{5}}
 => [4,2,3,1,5] => [1,4,2,3,5] => [1,4,2,3,5] => 0 = 1 - 1
{{1,5},{2,4},{3}}
 => [5,4,3,2,1] => [1,5,2,4,3] => [1,4,5,2,3] => 1 = 2 - 1
{{1},{2,4,5},{3}}
 => [1,4,3,5,2] => [1,2,4,5,3] => [1,2,5,4,3] => 0 = 1 - 1
{{1},{2,5},{3,4}}
 => [1,5,4,3,2] => [1,2,5,3,4] => [1,2,5,3,4] => 0 = 1 - 1
{{1},{2},{3,4,5}}
 => [1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
{{1},{2},{3,4},{5}}
 => [1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
{{1},{2,5},{3},{4}}
 => [1,5,3,4,2] => [1,2,5,3,4] => [1,2,5,3,4] => 0 = 1 - 1
{{1},{2},{3},{4,5}}
 => [1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
{{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
{{1,2,3,4,5,6}}
 => [2,3,4,5,6,1] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0 = 1 - 1
Description
The number of very big descents of a permutation. A very big descent of a permutation π is an index i such that πi−πi+1>2. For the number of descents, see [[St000021]] and for the number of big descents, see [[St000647]]. General r-descents were for example be studied in [1, Section 2].
Matching statistic: St001890
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00310: Permutations —toric promotion⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001890: Posets ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 67%
Values
{{1}}
 => [1] => [1] => ([],1)
 => ? = 0
{{1,2}}
 => [2,1] => [2,1] => ([],2)
 => 1
{{1},{2}}
 => [1,2] => [1,2] => ([(0,1)],2)
 => 1
{{1,2,3}}
 => [2,3,1] => [1,3,2] => ([(0,1),(0,2)],3)
 => 1
{{1,2},{3}}
 => [2,1,3] => [3,1,2] => ([(1,2)],3)
 => 1
{{1},{2,3}}
 => [1,3,2] => [2,3,1] => ([(1,2)],3)
 => 1
{{1},{2},{3}}
 => [1,2,3] => [3,2,1] => ([],3)
 => 1
{{1,2,3,4}}
 => [2,3,4,1] => [4,2,1,3] => ([(1,3),(2,3)],4)
 => 1
{{1,2,3},{4}}
 => [2,3,1,4] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => 1
{{1,2},{3,4}}
 => [2,1,4,3] => [4,1,3,2] => ([(1,2),(1,3)],4)
 => 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [4,1,2,3] => ([(1,2),(2,3)],4)
 => 1
{{1,3,4},{2}}
 => [3,2,4,1] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 1
{{1,4},{2,3}}
 => [4,3,2,1] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
{{1},{2,3,4}}
 => [1,3,4,2] => [2,3,4,1] => ([(1,2),(2,3)],4)
 => 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => 1
{{1},{2},{3,4}}
 => [1,2,4,3] => [4,3,1,2] => ([(2,3)],4)
 => 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [4,2,3,1] => ([(2,3)],4)
 => 1
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5)
 => 1
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5)
 => 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [1,5,2,4,3] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => 1
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => 1
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5)
 => 1
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => 1
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [5,1,3,4,2] => ([(1,3),(1,4),(4,2)],5)
 => 1
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [5,1,3,2,4] => ([(1,2),(1,3),(2,4),(3,4)],5)
 => 1
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [5,4,2,1,3] => ([(2,4),(3,4)],5)
 => 1
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [5,1,2,4,3] => ([(1,4),(4,2),(4,3)],5)
 => 1
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
 => 1
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [2,4,3,1,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 1
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => 1
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 1
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => 1
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
 => 1
{{1,4},{2,3},{5}}
 => [4,3,2,1,5] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 1
{{1},{2,3,4,5}}
 => [1,3,4,5,2] => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
 => 1
{{1},{2,3,4},{5}}
 => [1,3,4,2,5] => [2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5)
 => 1
{{1},{2,3},{4,5}}
 => [1,3,2,5,4] => [2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => 1
{{1},{2,3},{4},{5}}
 => [1,3,2,4,5] => [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
 => 1
{{1,4},{2,5},{3}}
 => [4,5,3,1,2] => [3,4,2,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1
{{1,4},{2},{3,5}}
 => [4,2,5,1,3] => [3,5,4,2,1] => ([(2,3),(2,4)],5)
 => 1
{{1,4},{2},{3},{5}}
 => [4,2,3,1,5] => [3,1,5,2,4] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => 1
{{1,5},{2,4},{3}}
 => [5,4,3,2,1] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 2
{{1},{2,4,5},{3}}
 => [1,4,3,5,2] => [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 1
{{1},{2,5},{3,4}}
 => [1,5,4,3,2] => [4,1,3,2,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 1
{{1},{2},{3,4,5}}
 => [1,2,4,5,3] => [5,3,4,1,2] => ([(1,4),(2,3)],5)
 => 1
{{1},{2},{3,4},{5}}
 => [1,2,4,3,5] => [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
 => 1
{{1},{2,5},{3},{4}}
 => [1,5,3,4,2] => [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 1
{{1},{2},{3},{4,5}}
 => [1,2,3,5,4] => [5,2,4,1,3] => ([(1,4),(2,3),(2,4)],5)
 => 1
{{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
 => 1
{{1,2,3,4,5,6}}
 => [2,3,4,5,6,1] => [6,2,3,4,1,5] => ([(1,5),(2,3),(3,4),(4,5)],6)
 => ? = 1
{{1,2,3,4,5},{6}}
 => [2,3,4,5,1,6] => [6,2,3,1,4,5] => ([(1,5),(2,3),(3,5),(5,4)],6)
 => ? = 1
{{1,2,3,4},{5,6}}
 => [2,3,4,1,6,5] => [6,2,1,3,5,4] => ([(1,5),(2,5),(5,3),(5,4)],6)
 => ? = 1
{{1,2,3,4},{5},{6}}
 => [2,3,4,1,5,6] => [6,2,1,3,4,5] => ([(1,5),(2,5),(3,4),(5,3)],6)
 => ? = 1
{{1,2,3,5,6},{4}}
 => [2,3,5,4,6,1] => [6,2,4,3,1,5] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ? = 1
{{1,2,3,6},{4,5}}
 => [2,3,6,5,4,1] => [6,2,1,5,4,3] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ? = 1
{{1,2,3},{4,5,6}}
 => [2,3,1,5,6,4] => [1,6,2,4,5,3] => ([(0,3),(0,5),(4,2),(5,1),(5,4)],6)
 => ? = 1
{{1,2,3},{4,5},{6}}
 => [2,3,1,5,4,6] => [1,6,2,4,3,5] => ([(0,3),(0,4),(1,5),(2,5),(4,1),(4,2)],6)
 => ? = 1
{{1,2,3,6},{4},{5}}
 => [2,3,6,4,5,1] => [6,2,5,3,1,4] => ([(1,5),(2,3),(2,4),(4,5)],6)
 => ? = 1
{{1,2,3},{4},{5,6}}
 => [2,3,1,4,6,5] => [1,6,2,3,5,4] => ([(0,3),(0,4),(4,5),(5,1),(5,2)],6)
 => ? = 1
{{1,2,3},{4},{5},{6}}
 => [2,3,1,4,5,6] => [1,6,2,3,4,5] => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
 => ? = 1
{{1,2,4,5},{3,6}}
 => [2,4,6,5,1,3] => [6,3,5,4,2,1] => ([(3,4),(3,5)],6)
 => ? = 1
{{1,2,4,5},{3},{6}}
 => [2,4,3,5,1,6] => [6,3,2,1,4,5] => ([(1,5),(2,5),(3,5),(5,4)],6)
 => ? = 1
{{1,2,4,6},{3,5}}
 => [2,4,5,6,3,1] => [6,3,4,1,5,2] => ([(1,4),(2,3),(2,5),(4,5)],6)
 => ? = 1
{{1,2,4,6},{3},{5}}
 => [2,4,3,6,5,1] => [6,3,2,1,5,4] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 1
{{1,2,5},{3,4,6}}
 => [2,5,4,6,1,3] => [6,4,3,5,2,1] => ([(3,5),(4,5)],6)
 => ? = 1
{{1,2,5},{3,4},{6}}
 => [2,5,4,3,1,6] => [1,6,4,3,2,5] => ([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
 => ? = 1
{{1,2},{3,4,5,6}}
 => [2,1,4,5,6,3] => [6,1,3,4,5,2] => ([(1,3),(1,5),(4,2),(5,4)],6)
 => ? = 1
{{1,2},{3,4,5},{6}}
 => [2,1,4,5,3,6] => [6,1,3,4,2,5] => ([(1,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 1
{{1,2},{3,4},{5,6}}
 => [2,1,4,3,6,5] => [6,1,3,2,5,4] => ([(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 1
{{1,2},{3,4},{5},{6}}
 => [2,1,4,3,5,6] => [6,1,3,2,4,5] => ([(1,2),(1,3),(2,5),(3,5),(5,4)],6)
 => ? = 1
{{1,2,5},{3,6},{4}}
 => [2,5,6,4,1,3] => [6,4,5,3,2,1] => ([(4,5)],6)
 => ? = 1
{{1,2,5},{3},{4,6}}
 => [2,5,3,6,1,4] => [6,4,2,5,3,1] => ([(2,5),(3,4),(3,5)],6)
 => ? = 1
{{1,2,5},{3},{4},{6}}
 => [2,5,3,4,1,6] => [6,4,2,1,3,5] => ([(1,5),(2,4),(3,4),(4,5)],6)
 => ? = 1
{{1,2,6},{3,5},{4}}
 => [2,6,5,4,3,1] => [1,6,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
 => ? = 2
{{1,2},{3,5,6},{4}}
 => [2,1,5,4,6,3] => [6,1,4,3,5,2] => ([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
 => ? = 1
{{1,2},{3,6},{4,5}}
 => [2,1,6,5,4,3] => [6,1,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5)],6)
 => ? = 1
{{1,2},{3},{4,5,6}}
 => [2,1,3,5,6,4] => [6,1,2,4,5,3] => ([(1,5),(4,3),(5,2),(5,4)],6)
 => ? = 1
{{1,2},{3},{4,5},{6}}
 => [2,1,3,5,4,6] => [6,1,2,4,3,5] => ([(1,4),(2,5),(3,5),(4,2),(4,3)],6)
 => ? = 1
{{1,2},{3,6},{4},{5}}
 => [2,1,6,4,5,3] => [6,1,5,3,4,2] => ([(1,3),(1,4),(1,5),(5,2)],6)
 => ? = 1
{{1,2},{3},{4},{5,6}}
 => [2,1,3,4,6,5] => [6,1,2,3,5,4] => ([(1,4),(4,5),(5,2),(5,3)],6)
 => ? = 1
{{1,2},{3},{4},{5},{6}}
 => [2,1,3,4,5,6] => [6,1,2,3,4,5] => ([(1,5),(3,4),(4,2),(5,3)],6)
 => ? = 1
{{1,3,4},{2,5,6}}
 => [3,5,4,1,6,2] => [2,1,4,3,5,6] => ([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
 => ? = 1
{{1,3,4},{2,5},{6}}
 => [3,5,4,1,2,6] => [2,4,3,6,5,1] => ([(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 1
{{1,3,4},{2,6},{5}}
 => [3,6,4,1,5,2] => [2,1,5,3,4,6] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(4,2),(5,3)],6)
 => ? = 1
{{1,3,4},{2},{5,6}}
 => [3,2,4,1,6,5] => [2,1,6,3,5,4] => ([(0,4),(0,5),(1,4),(1,5),(5,2),(5,3)],6)
 => ? = 1
{{1,3,4},{2},{5},{6}}
 => [3,2,4,1,5,6] => [2,1,6,3,4,5] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6)
 => ? = 1
{{1,3,5},{2,4,6}}
 => [3,4,5,6,1,2] => [2,3,4,5,1,6] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ? = 1
{{1,3,5},{2,4},{6}}
 => [3,4,5,2,1,6] => [2,3,1,4,6,5] => ([(0,5),(1,2),(2,5),(5,3),(5,4)],6)
 => ? = 1
{{1,3,5},{2,6},{4}}
 => [3,6,5,4,1,2] => [2,5,4,3,1,6] => ([(0,5),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
 => ? = 1
{{1,3,5},{2},{4,6}}
 => [3,2,5,6,1,4] => [2,6,4,5,3,1] => ([(1,3),(1,4),(1,5),(5,2)],6)
 => ? = 1
{{1,3,5},{2},{4},{6}}
 => [3,2,5,4,1,6] => [2,1,6,4,3,5] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(4,2),(5,2)],6)
 => ? = 1
{{1,3,6},{2,5},{4}}
 => [3,5,6,4,2,1] => [2,4,1,5,3,6] => ([(0,2),(0,5),(1,4),(1,5),(2,4),(4,3),(5,3)],6)
 => ? = 2
{{1,3},{2,5,6},{4}}
 => [3,5,1,4,6,2] => [2,4,3,5,6,1] => ([(1,2),(1,3),(2,5),(3,5),(5,4)],6)
 => ? = 1
{{1,3},{2,6},{4,5}}
 => [3,6,1,5,4,2] => [2,5,4,1,3,6] => ([(0,4),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
 => ? = 1
{{1,3},{2,6},{4},{5}}
 => [3,6,1,4,5,2] => [2,5,3,4,6,1] => ([(1,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 1
{{1,4},{2,3,5,6}}
 => [4,3,5,1,6,2] => [3,2,1,4,5,6] => ([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
 => ? = 1
{{1,4},{2,3,5},{6}}
 => [4,3,5,1,2,6] => [3,2,4,6,5,1] => ([(1,5),(2,5),(5,3),(5,4)],6)
 => ? = 1
{{1,4},{2,3,6},{5}}
 => [4,3,6,1,5,2] => [3,2,5,4,6,1] => ([(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ? = 1
Description
The maximum magnitude of the Möbius function of a poset. The '''Möbius function''' of a poset is the multiplicative inverse of the zeta function in the incidence algebra. The Möbius value μ(x,y) is equal to the signed sum of chains from x to y, where odd-length chains are counted with a minus sign, so this statistic is bounded above by the total number of chains in the poset.
Matching statistic: St001200
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001200: Dyck paths ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 33%
Values
{{1}}
 => [1] => [1] => [1,0]
 => ? = 0 + 1
{{1,2}}
 => [2,1] => [2,1] => [1,1,0,0]
 => ? = 1 + 1
{{1},{2}}
 => [1,2] => [1,2] => [1,0,1,0]
 => 2 = 1 + 1
{{1,2,3}}
 => [2,3,1] => [2,3,1] => [1,1,0,1,0,0]
 => 2 = 1 + 1
{{1,2},{3}}
 => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => 2 = 1 + 1
{{1},{2,3}}
 => [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
 => 2 = 1 + 1
{{1},{2},{3}}
 => [1,2,3] => [1,3,2] => [1,0,1,1,0,0]
 => 2 = 1 + 1
{{1,2,3,4}}
 => [2,3,4,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
{{1,2,3},{4}}
 => [2,3,1,4] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
{{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
{{1,3,4},{2}}
 => [3,2,4,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
{{1,4},{2,3}}
 => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? = 1 + 1
{{1},{2,3,4}}
 => [1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => ? = 1 + 1
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => 2 = 1 + 1
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
 => 2 = 1 + 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
 => 2 = 1 + 1
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
 => 2 = 1 + 1
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => 2 = 1 + 1
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => 2 = 1 + 1
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2 = 1 + 1
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2 = 1 + 1
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => 2 = 1 + 1
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2 = 1 + 1
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2 = 1 + 1
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
 => 2 = 1 + 1
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0]
 => 2 = 1 + 1
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
 => 2 = 1 + 1
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
 => 2 = 1 + 1
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
 => 2 = 1 + 1
{{1,4},{2,3},{5}}
 => [4,3,2,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
{{1},{2,3,4,5}}
 => [1,3,4,5,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
{{1},{2,3,4},{5}}
 => [1,3,4,2,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
{{1},{2,3},{4,5}}
 => [1,3,2,5,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
{{1},{2,3},{4},{5}}
 => [1,3,2,4,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
{{1,4},{2,5},{3}}
 => [4,5,3,1,2] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
 => 2 = 1 + 1
{{1,4},{2},{3,5}}
 => [4,2,5,1,3] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
 => 2 = 1 + 1
{{1,4},{2},{3},{5}}
 => [4,2,3,1,5] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
 => 2 = 1 + 1
{{1,5},{2,4},{3}}
 => [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
{{1},{2,4,5},{3}}
 => [1,4,3,5,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
{{1},{2,5},{3,4}}
 => [1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
{{1},{2},{3,4,5}}
 => [1,2,4,5,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
{{1},{2},{3,4},{5}}
 => [1,2,4,3,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
{{1},{2,5},{3},{4}}
 => [1,5,3,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
{{1},{2},{3},{4,5}}
 => [1,2,3,5,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
{{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
{{1,2,3,4,5,6}}
 => [2,3,4,5,6,1] => [2,6,5,4,3,1] => [1,1,0,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
{{1,2,3,4,5},{6}}
 => [2,3,4,5,1,6] => [2,6,5,4,1,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
{{1,2,3,4},{5,6}}
 => [2,3,4,1,6,5] => [2,6,5,1,4,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
{{1,2,3,4},{5},{6}}
 => [2,3,4,1,5,6] => [2,6,5,1,4,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
{{1,2,3,5,6},{4}}
 => [2,3,5,4,6,1] => [2,6,5,4,3,1] => [1,1,0,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
{{1,2,3,6},{4,5}}
 => [2,3,6,5,4,1] => [2,6,5,4,3,1] => [1,1,0,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
{{1,2,3},{4,5,6}}
 => [2,3,1,5,6,4] => [2,6,1,5,4,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
{{1,2,3},{4,5},{6}}
 => [2,3,1,5,4,6] => [2,6,1,5,4,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
{{1,2,3,6},{4},{5}}
 => [2,3,6,4,5,1] => [2,6,5,4,3,1] => [1,1,0,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
{{1,2,3},{4},{5,6}}
 => [2,3,1,4,6,5] => [2,6,1,5,4,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
{{1,2,3},{4},{5},{6}}
 => [2,3,1,4,5,6] => [2,6,1,5,4,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
{{1,2,4,5},{3,6}}
 => [2,4,6,5,1,3] => [2,6,5,4,1,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
{{1,2,4,5},{3},{6}}
 => [2,4,3,5,1,6] => [2,6,5,4,1,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
{{1,2,4,6},{3,5}}
 => [2,4,5,6,3,1] => [2,6,5,4,3,1] => [1,1,0,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
{{1,2,4,6},{3},{5}}
 => [2,4,3,6,5,1] => [2,6,5,4,3,1] => [1,1,0,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
{{1,2,5},{3,4,6}}
 => [2,5,4,6,1,3] => [2,6,5,4,1,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
{{1,2,5},{3,4},{6}}
 => [2,5,4,3,1,6] => [2,6,5,4,1,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
{{1,2},{3,4,5,6}}
 => [2,1,4,5,6,3] => [2,1,6,5,4,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 1 + 1
{{1,2},{3,4,5},{6}}
 => [2,1,4,5,3,6] => [2,1,6,5,4,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 1 + 1
{{1,2},{3,4},{5,6}}
 => [2,1,4,3,6,5] => [2,1,6,5,4,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 1 + 1
{{1,2},{3,4},{5},{6}}
 => [2,1,4,3,5,6] => [2,1,6,5,4,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 1 + 1
{{1,2,5},{3,6},{4}}
 => [2,5,6,4,1,3] => [2,6,5,4,1,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
{{1,2,5},{3},{4,6}}
 => [2,5,3,6,1,4] => [2,6,5,4,1,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
{{1,2,5},{3},{4},{6}}
 => [2,5,3,4,1,6] => [2,6,5,4,1,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
{{1,2,6},{3,5},{4}}
 => [2,6,5,4,3,1] => [2,6,5,4,3,1] => [1,1,0,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
{{1,2},{3,5,6},{4}}
 => [2,1,5,4,6,3] => [2,1,6,5,4,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 1 + 1
{{1,2},{3,6},{4,5}}
 => [2,1,6,5,4,3] => [2,1,6,5,4,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 1 + 1
{{1,2},{3},{4,5,6}}
 => [2,1,3,5,6,4] => [2,1,6,5,4,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 1 + 1
{{1,2},{3},{4,5},{6}}
 => [2,1,3,5,4,6] => [2,1,6,5,4,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 1 + 1
{{1,2},{3,6},{4},{5}}
 => [2,1,6,4,5,3] => [2,1,6,5,4,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 1 + 1
{{1,2},{3},{4},{5,6}}
 => [2,1,3,4,6,5] => [2,1,6,5,4,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 1 + 1
{{1,2},{3},{4},{5},{6}}
 => [2,1,3,4,5,6] => [2,1,6,5,4,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 1 + 1
{{1,3,4},{2,5,6}}
 => [3,5,4,1,6,2] => [3,6,5,1,4,2] => [1,1,1,0,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
{{1,3,4},{2,5},{6}}
 => [3,5,4,1,2,6] => [3,6,5,1,4,2] => [1,1,1,0,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
{{1,3,4},{2,6},{5}}
 => [3,6,4,1,5,2] => [3,6,5,1,4,2] => [1,1,1,0,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
{{1,3,4},{2},{5,6}}
 => [3,2,4,1,6,5] => [3,2,6,1,5,4] => [1,1,1,0,0,1,1,1,0,0,0,0]
 => ? = 1 + 1
{{1,3,4},{2},{5},{6}}
 => [3,2,4,1,5,6] => [3,2,6,1,5,4] => [1,1,1,0,0,1,1,1,0,0,0,0]
 => ? = 1 + 1
{{1,3,5},{2,4,6}}
 => [3,4,5,6,1,2] => [3,6,5,4,1,2] => [1,1,1,0,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
{{1,3,5},{2,4},{6}}
 => [3,4,5,2,1,6] => [3,6,5,2,1,4] => [1,1,1,0,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
{{1,3,5},{2,6},{4}}
 => [3,6,5,4,1,2] => [3,6,5,4,1,2] => [1,1,1,0,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
{{1,3,5},{2},{4,6}}
 => [3,2,5,6,1,4] => [3,2,6,5,1,4] => [1,1,1,0,0,1,1,1,0,0,0,0]
 => ? = 1 + 1
{{1,3,5},{2},{4},{6}}
 => [3,2,5,4,1,6] => [3,2,6,5,1,4] => [1,1,1,0,0,1,1,1,0,0,0,0]
 => ? = 1 + 1
{{1,3,6},{2,5},{4}}
 => [3,5,6,4,2,1] => [3,6,5,4,2,1] => [1,1,1,0,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
{{1,3},{2,5,6},{4}}
 => [3,5,1,4,6,2] => [3,6,1,5,4,2] => [1,1,1,0,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
{{1,3},{2,6},{4,5}}
 => [3,6,1,5,4,2] => [3,6,1,5,4,2] => [1,1,1,0,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
Description
The number of simple modules in eAe with projective dimension at most 2 in the corresponding Nakayama algebra A with minimal faithful projective-injective module eA.
Matching statistic: St001870
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00167: Signed permutations —inverse Kreweras complement⟶ Signed permutations
St001870: Signed permutations ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 67%
Values
{{1}}
 => [1] => [1] => [-1] => 1 = 0 + 1
{{1,2}}
 => [2,1] => [2,1] => [1,-2] => 2 = 1 + 1
{{1},{2}}
 => [1,2] => [1,2] => [2,-1] => 2 = 1 + 1
{{1,2,3}}
 => [2,3,1] => [2,3,1] => [1,2,-3] => 2 = 1 + 1
{{1,2},{3}}
 => [2,1,3] => [2,1,3] => [1,3,-2] => 2 = 1 + 1
{{1},{2,3}}
 => [1,3,2] => [1,3,2] => [3,2,-1] => 2 = 1 + 1
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => [2,3,-1] => 2 = 1 + 1
{{1,2,3,4}}
 => [2,3,4,1] => [2,3,4,1] => [1,2,3,-4] => 2 = 1 + 1
{{1,2,3},{4}}
 => [2,3,1,4] => [2,3,1,4] => [1,2,4,-3] => 2 = 1 + 1
{{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => [1,4,3,-2] => 2 = 1 + 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,3,4] => [1,3,4,-2] => 2 = 1 + 1
{{1,3,4},{2}}
 => [3,2,4,1] => [3,2,4,1] => [2,1,3,-4] => 2 = 1 + 1
{{1,4},{2,3}}
 => [4,3,2,1] => [4,3,2,1] => [3,2,1,-4] => 2 = 1 + 1
{{1},{2,3,4}}
 => [1,3,4,2] => [1,3,4,2] => [4,2,3,-1] => 2 = 1 + 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,3,2,4] => [3,2,4,-1] => 2 = 1 + 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [4,2,3,1] => [2,3,1,-4] => 2 = 1 + 1
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,4,3] => [2,4,3,-1] => 2 = 1 + 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => [2,3,4,-1] => 2 = 1 + 1
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [2,3,4,5,1] => [1,2,3,4,-5] => 2 = 1 + 1
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [2,3,4,1,5] => [1,2,3,5,-4] => 2 = 1 + 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [2,3,1,5,4] => [1,2,5,4,-3] => 2 = 1 + 1
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [2,3,1,4,5] => [1,2,4,5,-3] => 2 = 1 + 1
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [2,4,3,5,1] => [1,3,2,4,-5] => 2 = 1 + 1
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [2,5,4,3,1] => [1,4,3,2,-5] => 2 = 1 + 1
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [2,1,4,5,3] => [1,5,3,4,-2] => 2 = 1 + 1
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [2,1,4,3,5] => [1,4,3,5,-2] => 2 = 1 + 1
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [2,5,3,4,1] => [1,3,4,2,-5] => 2 = 1 + 1
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [2,1,3,5,4] => [1,3,5,4,-2] => 2 = 1 + 1
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [2,1,3,4,5] => [1,3,4,5,-2] => 2 = 1 + 1
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [3,5,4,1,2] => [5,1,3,2,-4] => ? = 1 + 1
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [3,2,4,1,5] => [2,1,3,5,-4] => ? = 1 + 1
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [3,4,5,2,1] => [4,1,2,3,-5] => ? = 1 + 1
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [3,2,5,4,1] => [2,1,4,3,-5] => ? = 1 + 1
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [4,3,5,1,2] => [5,2,1,3,-4] => ? = 1 + 1
{{1,4},{2,3},{5}}
 => [4,3,2,1,5] => [4,3,2,1,5] => [3,2,1,5,-4] => ? = 1 + 1
{{1},{2,3,4,5}}
 => [1,3,4,5,2] => [1,3,4,5,2] => [5,2,3,4,-1] => ? = 1 + 1
{{1},{2,3,4},{5}}
 => [1,3,4,2,5] => [1,3,4,2,5] => [4,2,3,5,-1] => ? = 1 + 1
{{1},{2,3},{4,5}}
 => [1,3,2,5,4] => [1,3,2,5,4] => [3,2,5,4,-1] => ? = 1 + 1
{{1},{2,3},{4},{5}}
 => [1,3,2,4,5] => [1,3,2,4,5] => [3,2,4,5,-1] => ? = 1 + 1
{{1,4},{2,5},{3}}
 => [4,5,3,1,2] => [4,5,3,1,2] => [5,3,1,2,-4] => ? = 1 + 1
{{1,4},{2},{3,5}}
 => [4,2,5,1,3] => [4,2,5,1,3] => [2,5,1,3,-4] => ? = 1 + 1
{{1,4},{2},{3},{5}}
 => [4,2,3,1,5] => [4,2,3,1,5] => [2,3,1,5,-4] => ? = 1 + 1
{{1,5},{2,4},{3}}
 => [5,4,3,2,1] => [5,4,3,2,1] => [4,3,2,1,-5] => ? = 2 + 1
{{1},{2,4,5},{3}}
 => [1,4,3,5,2] => [1,4,3,5,2] => [5,3,2,4,-1] => ? = 1 + 1
{{1},{2,5},{3,4}}
 => [1,5,4,3,2] => [1,5,4,3,2] => [5,4,3,2,-1] => ? = 1 + 1
{{1},{2},{3,4,5}}
 => [1,2,4,5,3] => [1,2,4,5,3] => [2,5,3,4,-1] => ? = 1 + 1
{{1},{2},{3,4},{5}}
 => [1,2,4,3,5] => [1,2,4,3,5] => [2,4,3,5,-1] => ? = 1 + 1
{{1},{2,5},{3},{4}}
 => [1,5,3,4,2] => [1,5,3,4,2] => [5,3,4,2,-1] => ? = 1 + 1
{{1},{2},{3},{4,5}}
 => [1,2,3,5,4] => [1,2,3,5,4] => [2,3,5,4,-1] => ? = 1 + 1
{{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,-1] => ? = 1 + 1
{{1,2,3,4,5,6}}
 => [2,3,4,5,6,1] => [2,3,4,5,6,1] => [1,2,3,4,5,-6] => ? = 1 + 1
{{1,2,3,4,5},{6}}
 => [2,3,4,5,1,6] => [2,3,4,5,1,6] => [1,2,3,4,6,-5] => ? = 1 + 1
{{1,2,3,4},{5,6}}
 => [2,3,4,1,6,5] => [2,3,4,1,6,5] => [1,2,3,6,5,-4] => ? = 1 + 1
{{1,2,3,4},{5},{6}}
 => [2,3,4,1,5,6] => [2,3,4,1,5,6] => [1,2,3,5,6,-4] => ? = 1 + 1
{{1,2,3,5,6},{4}}
 => [2,3,5,4,6,1] => [2,3,5,4,6,1] => [1,2,4,3,5,-6] => ? = 1 + 1
{{1,2,3,6},{4,5}}
 => [2,3,6,5,4,1] => [2,3,6,5,4,1] => [1,2,5,4,3,-6] => ? = 1 + 1
{{1,2,3},{4,5,6}}
 => [2,3,1,5,6,4] => [2,3,1,5,6,4] => [1,2,6,4,5,-3] => ? = 1 + 1
{{1,2,3},{4,5},{6}}
 => [2,3,1,5,4,6] => [2,3,1,5,4,6] => [1,2,5,4,6,-3] => ? = 1 + 1
{{1,2,3,6},{4},{5}}
 => [2,3,6,4,5,1] => [2,3,6,4,5,1] => [1,2,4,5,3,-6] => ? = 1 + 1
{{1,2,3},{4},{5,6}}
 => [2,3,1,4,6,5] => [2,3,1,4,6,5] => [1,2,4,6,5,-3] => ? = 1 + 1
{{1,2,3},{4},{5},{6}}
 => [2,3,1,4,5,6] => [2,3,1,4,5,6] => [1,2,4,5,6,-3] => ? = 1 + 1
{{1,2,4,5},{3,6}}
 => [2,4,6,5,1,3] => [2,4,6,5,1,3] => [1,6,2,4,3,-5] => ? = 1 + 1
{{1,2,4,5},{3},{6}}
 => [2,4,3,5,1,6] => [2,4,3,5,1,6] => [1,3,2,4,6,-5] => ? = 1 + 1
{{1,2,4,6},{3,5}}
 => [2,4,5,6,3,1] => [2,4,5,6,3,1] => [1,5,2,3,4,-6] => ? = 1 + 1
{{1,2,4,6},{3},{5}}
 => [2,4,3,6,5,1] => [2,4,3,6,5,1] => [1,3,2,5,4,-6] => ? = 1 + 1
{{1,2,5},{3,4,6}}
 => [2,5,4,6,1,3] => [2,5,4,6,1,3] => [1,6,3,2,4,-5] => ? = 1 + 1
{{1,2,5},{3,4},{6}}
 => [2,5,4,3,1,6] => [2,5,4,3,1,6] => [1,4,3,2,6,-5] => ? = 1 + 1
{{1,2},{3,4,5,6}}
 => [2,1,4,5,6,3] => [2,1,4,5,6,3] => [1,6,3,4,5,-2] => ? = 1 + 1
{{1,2},{3,4,5},{6}}
 => [2,1,4,5,3,6] => [2,1,4,5,3,6] => [1,5,3,4,6,-2] => ? = 1 + 1
{{1,2},{3,4},{5,6}}
 => [2,1,4,3,6,5] => [2,1,4,3,6,5] => [1,4,3,6,5,-2] => ? = 1 + 1
{{1,2},{3,4},{5},{6}}
 => [2,1,4,3,5,6] => [2,1,4,3,5,6] => [1,4,3,5,6,-2] => ? = 1 + 1
{{1,2,5},{3,6},{4}}
 => [2,5,6,4,1,3] => [2,5,6,4,1,3] => [1,6,4,2,3,-5] => ? = 1 + 1
{{1,2,5},{3},{4,6}}
 => [2,5,3,6,1,4] => [2,5,3,6,1,4] => [1,3,6,2,4,-5] => ? = 1 + 1
{{1,2,5},{3},{4},{6}}
 => [2,5,3,4,1,6] => [2,5,3,4,1,6] => [1,3,4,2,6,-5] => ? = 1 + 1
{{1,2,6},{3,5},{4}}
 => [2,6,5,4,3,1] => [2,6,5,4,3,1] => [1,5,4,3,2,-6] => ? = 2 + 1
{{1,2},{3,5,6},{4}}
 => [2,1,5,4,6,3] => [2,1,5,4,6,3] => [1,6,4,3,5,-2] => ? = 1 + 1
{{1,2},{3,6},{4,5}}
 => [2,1,6,5,4,3] => [2,1,6,5,4,3] => [1,6,5,4,3,-2] => ? = 1 + 1
{{1,2},{3},{4,5,6}}
 => [2,1,3,5,6,4] => [2,1,3,5,6,4] => [1,3,6,4,5,-2] => ? = 1 + 1
{{1,2},{3},{4,5},{6}}
 => [2,1,3,5,4,6] => [2,1,3,5,4,6] => [1,3,5,4,6,-2] => ? = 1 + 1
Description
The number of positive entries followed by a negative entry in a signed permutation. For a signed permutation π∈Hn, this is the number of positive entries followed by a negative entry in π(−n),…,π(−1),π(1),…,π(n).
Matching statistic: St001491
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 33%
Values
{{1}}
 => [1] => [1] => => ? = 0
{{1,2}}
 => [2,1] => [2,1] => 0 => ? = 1
{{1},{2}}
 => [1,2] => [1,2] => 1 => 1
{{1,2,3}}
 => [2,3,1] => [2,3,1] => 00 => ? = 1
{{1,2},{3}}
 => [2,1,3] => [2,1,3] => 01 => 1
{{1},{2,3}}
 => [1,3,2] => [1,3,2] => 10 => 1
{{1},{2},{3}}
 => [1,2,3] => [1,3,2] => 10 => 1
{{1,2,3,4}}
 => [2,3,4,1] => [2,4,3,1] => 000 => ? = 1
{{1,2,3},{4}}
 => [2,3,1,4] => [2,4,1,3] => 000 => ? = 1
{{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => 010 => 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,4,3] => 010 => 1
{{1,3,4},{2}}
 => [3,2,4,1] => [3,2,4,1] => 000 => ? = 1
{{1,4},{2,3}}
 => [4,3,2,1] => [4,3,2,1] => 000 => ? = 1
{{1},{2,3,4}}
 => [1,3,4,2] => [1,4,3,2] => 100 => 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,4,3,2] => 100 => 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [4,2,3,1] => 000 => ? = 1
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,4,3,2] => 100 => 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,4,3,2] => 100 => 1
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [2,5,4,3,1] => 0000 => ? = 1
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [2,5,4,1,3] => 0000 => ? = 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [2,5,1,4,3] => 0000 => ? = 1
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [2,5,1,4,3] => 0000 => ? = 1
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [2,5,4,3,1] => 0000 => ? = 1
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [2,5,4,3,1] => 0000 => ? = 1
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [2,1,5,4,3] => 0100 => 1
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [2,1,5,4,3] => 0100 => 1
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [2,5,4,3,1] => 0000 => ? = 1
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [2,1,5,4,3] => 0100 => 1
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [2,1,5,4,3] => 0100 => 1
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [3,5,4,1,2] => 0000 => ? = 1
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [3,2,5,1,4] => 0000 => ? = 1
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [3,5,4,2,1] => 0000 => ? = 1
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [3,2,5,4,1] => 0000 => ? = 1
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [4,3,5,1,2] => 0000 => ? = 1
{{1,4},{2,3},{5}}
 => [4,3,2,1,5] => [4,3,2,1,5] => 0001 => 1
{{1},{2,3,4,5}}
 => [1,3,4,5,2] => [1,5,4,3,2] => 1000 => 1
{{1},{2,3,4},{5}}
 => [1,3,4,2,5] => [1,5,4,3,2] => 1000 => 1
{{1},{2,3},{4,5}}
 => [1,3,2,5,4] => [1,5,4,3,2] => 1000 => 1
{{1},{2,3},{4},{5}}
 => [1,3,2,4,5] => [1,5,4,3,2] => 1000 => 1
{{1,4},{2,5},{3}}
 => [4,5,3,1,2] => [4,5,3,1,2] => 0000 => ? = 1
{{1,4},{2},{3,5}}
 => [4,2,5,1,3] => [4,2,5,1,3] => 0000 => ? = 1
{{1,4},{2},{3},{5}}
 => [4,2,3,1,5] => [4,2,5,1,3] => 0000 => ? = 1
{{1,5},{2,4},{3}}
 => [5,4,3,2,1] => [5,4,3,2,1] => 0000 => ? = 2
{{1},{2,4,5},{3}}
 => [1,4,3,5,2] => [1,5,4,3,2] => 1000 => 1
{{1},{2,5},{3,4}}
 => [1,5,4,3,2] => [1,5,4,3,2] => 1000 => 1
{{1},{2},{3,4,5}}
 => [1,2,4,5,3] => [1,5,4,3,2] => 1000 => 1
{{1},{2},{3,4},{5}}
 => [1,2,4,3,5] => [1,5,4,3,2] => 1000 => 1
{{1},{2,5},{3},{4}}
 => [1,5,3,4,2] => [1,5,4,3,2] => 1000 => 1
{{1},{2},{3},{4,5}}
 => [1,2,3,5,4] => [1,5,4,3,2] => 1000 => 1
{{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => [1,5,4,3,2] => 1000 => 1
{{1,2,3,4,5,6}}
 => [2,3,4,5,6,1] => [2,6,5,4,3,1] => 00000 => ? = 1
{{1,2,3,4,5},{6}}
 => [2,3,4,5,1,6] => [2,6,5,4,1,3] => 00000 => ? = 1
{{1,2,3,4},{5,6}}
 => [2,3,4,1,6,5] => [2,6,5,1,4,3] => 00000 => ? = 1
{{1,2,3,4},{5},{6}}
 => [2,3,4,1,5,6] => [2,6,5,1,4,3] => 00000 => ? = 1
{{1,2,3,5,6},{4}}
 => [2,3,5,4,6,1] => [2,6,5,4,3,1] => 00000 => ? = 1
{{1,2,3,6},{4,5}}
 => [2,3,6,5,4,1] => [2,6,5,4,3,1] => 00000 => ? = 1
{{1,2,3},{4,5,6}}
 => [2,3,1,5,6,4] => [2,6,1,5,4,3] => 00000 => ? = 1
{{1,2,3},{4,5},{6}}
 => [2,3,1,5,4,6] => [2,6,1,5,4,3] => 00000 => ? = 1
{{1,2,3,6},{4},{5}}
 => [2,3,6,4,5,1] => [2,6,5,4,3,1] => 00000 => ? = 1
{{1,2,3},{4},{5,6}}
 => [2,3,1,4,6,5] => [2,6,1,5,4,3] => 00000 => ? = 1
{{1,2,3},{4},{5},{6}}
 => [2,3,1,4,5,6] => [2,6,1,5,4,3] => 00000 => ? = 1
{{1,2,4,5},{3,6}}
 => [2,4,6,5,1,3] => [2,6,5,4,1,3] => 00000 => ? = 1
{{1,2,4,5},{3},{6}}
 => [2,4,3,5,1,6] => [2,6,5,4,1,3] => 00000 => ? = 1
{{1,2,4,6},{3,5}}
 => [2,4,5,6,3,1] => [2,6,5,4,3,1] => 00000 => ? = 1
{{1,2,4,6},{3},{5}}
 => [2,4,3,6,5,1] => [2,6,5,4,3,1] => 00000 => ? = 1
{{1,2,5},{3,4,6}}
 => [2,5,4,6,1,3] => [2,6,5,4,1,3] => 00000 => ? = 1
{{1,2,5},{3,4},{6}}
 => [2,5,4,3,1,6] => [2,6,5,4,1,3] => 00000 => ? = 1
{{1,2},{3,4,5,6}}
 => [2,1,4,5,6,3] => [2,1,6,5,4,3] => 01000 => ? = 1
{{1,2},{3,4,5},{6}}
 => [2,1,4,5,3,6] => [2,1,6,5,4,3] => 01000 => ? = 1
{{1,2},{3,4},{5,6}}
 => [2,1,4,3,6,5] => [2,1,6,5,4,3] => 01000 => ? = 1
{{1,2},{3,4},{5},{6}}
 => [2,1,4,3,5,6] => [2,1,6,5,4,3] => 01000 => ? = 1
{{1,2,5},{3,6},{4}}
 => [2,5,6,4,1,3] => [2,6,5,4,1,3] => 00000 => ? = 1
{{1,2,5},{3},{4,6}}
 => [2,5,3,6,1,4] => [2,6,5,4,1,3] => 00000 => ? = 1
{{1,2,5},{3},{4},{6}}
 => [2,5,3,4,1,6] => [2,6,5,4,1,3] => 00000 => ? = 1
{{1,2,6},{3,5},{4}}
 => [2,6,5,4,3,1] => [2,6,5,4,3,1] => 00000 => ? = 2
{{1,2},{3,5,6},{4}}
 => [2,1,5,4,6,3] => [2,1,6,5,4,3] => 01000 => ? = 1
Description
The number of indecomposable projective-injective modules in the algebra corresponding to a subset. Let An=K[x]/(xn). We associate to a nonempty subset S of an (n-1)-set the module MS, which is the direct sum of An-modules with indecomposable non-projective direct summands of dimension i when i is in S (note that such modules have vector space dimension at most n-1). Then the corresponding algebra associated to S is the stable endomorphism ring of MS. We decode the subset as a binary word so that for example the subset S={1,3} of {1,2,3} is decoded as 101.
Matching statistic: St000914
(load all 9 compositions to match this statistic)
Mapping
Mp00218: Set partitions —inverse Wachs-White-rho⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000914: Posets ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 33%
Values
{{1}}
 => {{1}}
 => [1] => ([],1)
 => ? = 0
{{1,2}}
 => {{1,2}}
 => [2,1] => ([(0,1)],2)
 => 1
{{1},{2}}
 => {{1},{2}}
 => [1,2] => ([(0,1)],2)
 => 1
{{1,2,3}}
 => {{1,2,3}}
 => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
{{1,2},{3}}
 => {{1,2},{3}}
 => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
{{1},{2,3}}
 => {{1},{2,3}}
 => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
{{1},{2},{3}}
 => {{1},{2},{3}}
 => [1,2,3] => ([(0,2),(2,1)],3)
 => 1
{{1,2,3,4}}
 => {{1,2,3,4}}
 => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1
{{1,2,3},{4}}
 => {{1,2,3},{4}}
 => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 1
{{1,2},{3,4}}
 => {{1,2},{3,4}}
 => [2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => 1
{{1,2},{3},{4}}
 => {{1,2},{3},{4}}
 => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1
{{1,3,4},{2}}
 => {{1,3,4},{2}}
 => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 1
{{1,4},{2,3}}
 => {{1,3},{2,4}}
 => [3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => 1
{{1},{2,3,4}}
 => {{1},{2,3,4}}
 => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 1
{{1},{2,3},{4}}
 => {{1},{2,3},{4}}
 => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 1
{{1,4},{2},{3}}
 => {{1,4},{2},{3}}
 => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 1
{{1},{2},{3,4}}
 => {{1},{2},{3,4}}
 => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1
{{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1
{{1,2,3,4,5}}
 => {{1,2,3,4,5}}
 => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
{{1,2,3,4},{5}}
 => {{1,2,3,4},{5}}
 => [2,3,4,1,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 1
{{1,2,3},{4,5}}
 => {{1,2,3},{4,5}}
 => [2,3,1,5,4] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ? = 1
{{1,2,3},{4},{5}}
 => {{1,2,3},{4},{5}}
 => [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 1
{{1,2,4,5},{3}}
 => {{1,2,4,5},{3}}
 => [2,4,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
 => ? = 1
{{1,2,5},{3,4}}
 => {{1,2,4},{3,5}}
 => [2,4,5,1,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
 => ? = 1
{{1,2},{3,4,5}}
 => {{1,2},{3,4,5}}
 => [2,1,4,5,3] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ? = 1
{{1,2},{3,4},{5}}
 => {{1,2},{3,4},{5}}
 => [2,1,4,3,5] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
 => ? = 1
{{1,2,5},{3},{4}}
 => {{1,2,5},{3},{4}}
 => [2,5,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
 => ? = 1
{{1,2},{3},{4,5}}
 => {{1,2},{3},{4,5}}
 => [2,1,3,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
 => ? = 1
{{1,2},{3},{4},{5}}
 => {{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
{{1,3,4},{2,5}}
 => {{1,5},{2,3,4}}
 => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 1
{{1,3,4},{2},{5}}
 => {{1,3,4},{2},{5}}
 => [3,2,4,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
 => ? = 1
{{1,3,5},{2,4}}
 => {{1,4},{2,3,5}}
 => [4,3,5,1,2] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ? = 1
{{1,3,5},{2},{4}}
 => {{1,3,5},{2},{4}}
 => [3,2,5,4,1] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
 => ? = 1
{{1,4},{2,3,5}}
 => {{1,3,4},{2,5}}
 => [3,5,4,1,2] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ? = 1
{{1,4},{2,3},{5}}
 => {{1,3},{2,4},{5}}
 => [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
 => ? = 1
{{1},{2,3,4,5}}
 => {{1},{2,3,4,5}}
 => [1,3,4,5,2] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 1
{{1},{2,3,4},{5}}
 => {{1},{2,3,4},{5}}
 => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
 => ? = 1
{{1},{2,3},{4,5}}
 => {{1},{2,3},{4,5}}
 => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
 => ? = 1
{{1},{2,3},{4},{5}}
 => {{1},{2,3},{4},{5}}
 => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 1
{{1,4},{2,5},{3}}
 => {{1,5},{2,4},{3}}
 => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
{{1,4},{2},{3,5}}
 => {{1,5},{2},{3,4}}
 => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
 => ? = 1
{{1,4},{2},{3},{5}}
 => {{1,4},{2},{3},{5}}
 => [4,2,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
 => ? = 1
{{1,5},{2,4},{3}}
 => {{1,4},{2,5},{3}}
 => [4,5,3,1,2] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
 => ? = 2
{{1},{2,4,5},{3}}
 => {{1},{2,4,5},{3}}
 => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
 => ? = 1
{{1},{2,5},{3,4}}
 => {{1},{2,4},{3,5}}
 => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
 => ? = 1
{{1},{2},{3,4,5}}
 => {{1},{2},{3,4,5}}
 => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 1
{{1},{2},{3,4},{5}}
 => {{1},{2},{3,4},{5}}
 => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 1
{{1},{2,5},{3},{4}}
 => {{1},{2,5},{3},{4}}
 => [1,5,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
 => ? = 1
{{1},{2},{3},{4,5}}
 => {{1},{2},{3},{4,5}}
 => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
{{1},{2},{3},{4},{5}}
 => {{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
{{1,2,3,4,5,6}}
 => {{1,2,3,4,5,6}}
 => [2,3,4,5,6,1] => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? = 1
{{1,2,3,4,5},{6}}
 => {{1,2,3,4,5},{6}}
 => [2,3,4,5,1,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
 => ? = 1
{{1,2,3,4},{5,6}}
 => {{1,2,3,4},{5,6}}
 => [2,3,4,1,6,5] => ([(0,3),(0,4),(0,5),(1,8),(1,14),(2,6),(2,7),(3,10),(3,11),(4,2),(4,11),(4,12),(5,1),(5,10),(5,12),(6,13),(6,15),(7,13),(7,15),(8,13),(8,15),(10,14),(11,7),(11,14),(12,6),(12,8),(12,14),(13,9),(14,15),(15,9)],16)
 => ? = 1
{{1,2,3,4},{5},{6}}
 => {{1,2,3,4},{5},{6}}
 => [2,3,4,1,5,6] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
 => ? = 1
{{1,2,3,5,6},{4}}
 => {{1,2,3,5,6},{4}}
 => [2,3,5,4,6,1] => ([(0,1),(0,3),(0,4),(0,5),(1,6),(1,15),(2,7),(2,8),(2,13),(3,10),(3,12),(3,15),(4,2),(4,11),(4,12),(4,15),(5,6),(5,10),(5,11),(6,16),(7,17),(8,17),(8,18),(10,14),(10,16),(11,8),(11,14),(11,16),(12,7),(12,13),(12,14),(13,17),(13,18),(14,17),(14,18),(15,13),(15,16),(16,18),(17,9),(18,9)],19)
 => ? = 1
{{1,2,3,6},{4,5}}
 => {{1,2,3,5},{4,6}}
 => [2,3,5,6,1,4] => ([(0,3),(0,4),(0,5),(0,6),(1,11),(1,18),(2,12),(2,17),(2,18),(3,7),(3,14),(4,1),(4,10),(4,13),(4,14),(5,2),(5,9),(5,13),(5,14),(6,7),(6,9),(6,10),(7,17),(9,15),(9,17),(10,15),(10,17),(10,18),(11,16),(11,19),(12,16),(12,19),(13,11),(13,12),(13,15),(13,18),(14,17),(14,18),(15,16),(15,19),(16,8),(17,19),(18,16),(18,19),(19,8)],20)
 => ? = 1
{{1,2,3},{4,5,6}}
 => {{1,2,3},{4,5,6}}
 => [2,3,1,5,6,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(1,10),(1,12),(2,8),(2,10),(2,11),(3,7),(3,8),(3,12),(4,7),(4,9),(4,11),(5,15),(5,16),(7,17),(8,13),(8,17),(9,14),(9,17),(10,13),(10,14),(11,5),(11,14),(11,17),(12,5),(12,13),(12,17),(13,15),(13,16),(14,15),(14,16),(15,6),(16,6),(17,16)],18)
 => ? = 1
{{1,2,3},{4,5},{6}}
 => {{1,2,3},{4,5},{6}}
 => [2,3,1,5,4,6] => ([(0,2),(0,3),(0,4),(0,5),(1,14),(1,16),(2,7),(2,8),(2,9),(3,7),(3,10),(3,12),(4,8),(4,11),(4,12),(5,9),(5,10),(5,11),(7,17),(8,15),(8,17),(9,1),(9,15),(9,17),(10,13),(10,17),(11,13),(11,15),(12,13),(12,15),(12,17),(13,14),(13,16),(14,6),(15,14),(15,16),(16,6),(17,16)],18)
 => ? = 1
{{1,2,3,6},{4},{5}}
 => {{1,2,3,6},{4},{5}}
 => [2,3,6,4,5,1] => ([(0,1),(0,3),(0,4),(0,5),(1,6),(1,15),(2,7),(2,8),(2,13),(3,10),(3,12),(3,15),(4,2),(4,11),(4,12),(4,15),(5,6),(5,10),(5,11),(6,16),(7,17),(8,17),(8,18),(10,14),(10,16),(11,8),(11,14),(11,16),(12,7),(12,13),(12,14),(13,17),(13,18),(14,17),(14,18),(15,13),(15,16),(16,18),(17,9),(18,9)],19)
 => ? = 1
{{1,2,3},{4},{5,6}}
 => {{1,2,3},{4},{5,6}}
 => [2,3,1,4,6,5] => ([(0,2),(0,3),(0,4),(0,5),(1,13),(1,16),(2,6),(2,14),(3,10),(3,11),(3,14),(4,9),(4,11),(4,14),(5,6),(5,9),(5,10),(6,15),(7,13),(7,16),(9,12),(9,15),(10,1),(10,12),(10,15),(11,7),(11,12),(12,13),(12,16),(13,8),(14,7),(14,15),(15,16),(16,8)],17)
 => ? = 1
{{1,2,3},{4},{5},{6}}
 => {{1,2,3},{4},{5},{6}}
 => [2,3,1,4,5,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
 => ? = 1
{{1,2,4,5},{3,6}}
 => {{1,2,6},{3,4,5}}
 => [2,6,4,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,15),(2,10),(2,12),(2,16),(3,11),(3,13),(3,15),(3,16),(4,8),(4,10),(4,11),(4,15),(5,8),(5,9),(5,13),(5,16),(6,18),(6,19),(8,14),(8,17),(8,20),(9,17),(9,21),(10,20),(10,21),(11,14),(11,20),(12,21),(13,6),(13,14),(13,17),(14,19),(15,17),(15,20),(15,21),(16,6),(16,20),(16,21),(17,18),(17,19),(18,7),(19,7),(20,18),(20,19),(21,18)],22)
 => ? = 1
{{1,2,4,5},{3},{6}}
 => {{1,2,4,5},{3},{6}}
 => [2,4,3,5,1,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,15),(2,10),(2,12),(2,16),(3,11),(3,13),(3,15),(3,16),(4,8),(4,10),(4,11),(4,15),(5,8),(5,9),(5,13),(5,16),(6,18),(6,19),(8,14),(8,17),(8,20),(9,17),(9,21),(10,20),(10,21),(11,14),(11,20),(12,21),(13,6),(13,14),(13,17),(14,19),(15,17),(15,20),(15,21),(16,6),(16,20),(16,21),(17,18),(17,19),(18,7),(19,7),(20,18),(20,19),(21,18)],22)
 => ? = 1
{{1,2,4,6},{3,5}}
 => {{1,2,5},{3,4,6}}
 => [2,5,4,6,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(1,8),(1,18),(2,11),(2,12),(2,14),(2,18),(3,10),(3,13),(3,14),(3,18),(4,7),(4,9),(4,12),(4,13),(5,8),(5,9),(5,10),(5,11),(7,16),(7,23),(8,16),(8,19),(9,16),(9,17),(9,20),(9,23),(10,19),(10,20),(11,19),(11,20),(11,23),(12,15),(12,23),(13,15),(13,20),(13,23),(14,15),(14,17),(14,20),(15,22),(16,21),(16,22),(17,21),(17,22),(18,17),(18,19),(18,23),(19,21),(20,21),(20,22),(21,6),(22,6),(23,21),(23,22)],24)
 => ? = 1
{{1,2,4,6},{3},{5}}
 => {{1,2,4,6},{3},{5}}
 => [2,4,3,6,5,1] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(1,10),(1,15),(2,8),(2,11),(2,15),(3,7),(3,8),(3,9),(4,7),(4,10),(4,11),(4,15),(5,17),(7,12),(7,13),(7,16),(8,13),(8,16),(9,12),(9,16),(10,12),(10,14),(11,5),(11,13),(11,14),(12,18),(13,17),(13,18),(14,17),(14,18),(15,5),(15,14),(15,16),(16,17),(16,18),(17,6),(18,6)],19)
 => ? = 1
{{1,2,5},{3,4,6}}
 => {{1,2,4,5},{3,6}}
 => [2,4,6,5,1,3] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,21),(1,22),(2,11),(2,13),(2,18),(3,12),(3,14),(3,18),(4,8),(4,9),(4,13),(4,18),(5,8),(5,10),(5,14),(5,18),(6,9),(6,10),(6,11),(6,12),(8,17),(8,19),(9,15),(9,19),(9,20),(10,1),(10,16),(10,19),(10,20),(11,15),(11,20),(12,16),(12,20),(13,15),(13,19),(14,16),(14,17),(15,21),(16,21),(16,22),(17,22),(18,17),(18,19),(18,20),(19,21),(19,22),(20,21),(20,22),(21,7),(22,7)],23)
 => ? = 1
{{1,2,5},{3,4},{6}}
 => {{1,2,4},{3,5},{6}}
 => [2,4,5,1,3,6] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,20),(1,22),(2,13),(2,14),(2,17),(3,11),(3,12),(3,17),(4,8),(4,9),(4,11),(4,17),(5,1),(5,8),(5,10),(5,14),(5,17),(6,9),(6,10),(6,12),(6,13),(8,15),(8,20),(8,22),(9,15),(9,19),(9,22),(10,15),(10,16),(10,19),(10,20),(11,22),(12,19),(12,22),(13,16),(13,19),(14,16),(14,20),(14,22),(15,18),(15,21),(16,18),(16,21),(17,19),(17,20),(17,22),(18,7),(19,18),(19,21),(20,18),(20,21),(21,7),(22,21)],23)
 => ? = 1
{{1,2},{3,4,5,6}}
 => {{1,2},{3,4,5,6}}
 => [2,1,4,5,6,3] => ([(0,3),(0,4),(0,5),(1,8),(1,14),(2,6),(2,7),(3,10),(3,11),(4,2),(4,11),(4,12),(5,1),(5,10),(5,12),(6,13),(6,15),(7,13),(7,15),(8,13),(8,15),(10,14),(11,7),(11,14),(12,6),(12,8),(12,14),(13,9),(14,15),(15,9)],16)
 => ? = 1
{{1,2},{3,4,5},{6}}
 => {{1,2},{3,4,5},{6}}
 => [2,1,4,5,3,6] => ([(0,2),(0,3),(0,4),(0,5),(1,10),(1,15),(1,17),(2,7),(2,13),(3,8),(3,9),(3,13),(4,8),(4,11),(4,13),(5,1),(5,7),(5,9),(5,11),(7,17),(8,12),(8,15),(9,12),(9,15),(9,17),(10,14),(10,16),(11,10),(11,12),(11,17),(12,14),(12,16),(13,15),(13,17),(14,6),(15,14),(15,16),(16,6),(17,16)],18)
 => ? = 1
{{1,4},{2,5},{3,6}}
 => {{1,6},{2,5},{3,4}}
 => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
{{1},{2},{3},{4},{5},{6}}
 => {{1},{2},{3},{4},{5},{6}}
 => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
Description
The sum of the values of the Möbius function of a poset. The Möbius function μ of a finite poset is defined as μ(x,y)={1if x=y−∑z:x≤z<yμ(x,z)for x<y0otherwise. Since μ(x,y)=0 whenever x≰, this statistic is \sum_{x\leq y} \mu(x,y). If the poset has a minimal or a maximal element, then the definition implies immediately that the statistic equals 1. Moreover, the statistic equals the sum of the statistics of the connected components. This statistic is also called the magnitude of a poset.
Matching statistic: St001195
(load all 2 compositions to match this statistic)
Mapping
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001195: Dyck paths ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 33%
Values
{{1}}
 => [1] => [1,0]
 => [1,1,0,0]
 => ? = 0
{{1,2}}
 => [2] => [1,1,0,0]
 => [1,1,1,0,0,0]
 => 1
{{1},{2}}
 => [1,1] => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 1
{{1,2,3}}
 => [3] => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 1
{{1,2},{3}}
 => [2,1] => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1
{{1},{2,3}}
 => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 1
{{1},{2},{3}}
 => [1,1,1] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 1
{{1,2,3,4}}
 => [4] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1
{{1,2,3},{4}}
 => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1
{{1,2},{3,4}}
 => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 1
{{1,2},{3},{4}}
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1
{{1,3,4},{2}}
 => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1
{{1,4},{2,3}}
 => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 1
{{1},{2,3,4}}
 => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1
{{1},{2,3},{4}}
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 1
{{1,4},{2},{3}}
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1
{{1},{2},{3,4}}
 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 1
{{1},{2},{3},{4}}
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
{{1,2,3,4,5}}
 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
{{1,2,3,4},{5}}
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => ? = 1
{{1,2,3},{4,5}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => ? = 1
{{1,2,3},{4},{5}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => ? = 1
{{1,2,4,5},{3}}
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => ? = 1
{{1,2,5},{3,4}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => ? = 1
{{1,2},{3,4,5}}
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => ? = 1
{{1,2},{3,4},{5}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => ? = 1
{{1,2,5},{3},{4}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => ? = 1
{{1,2},{3},{4,5}}
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => ? = 1
{{1,2},{3},{4},{5}}
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => ? = 1
{{1,3,4},{2,5}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => ? = 1
{{1,3,4},{2},{5}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => ? = 1
{{1,3,5},{2,4}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => ? = 1
{{1,3,5},{2},{4}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => ? = 1
{{1,4},{2,3,5}}
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => ? = 1
{{1,4},{2,3},{5}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => ? = 1
{{1},{2,3,4,5}}
 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => ? = 1
{{1},{2,3,4},{5}}
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => ? = 1
{{1},{2,3},{4,5}}
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => ? = 1
{{1},{2,3},{4},{5}}
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => ? = 1
{{1,4},{2,5},{3}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => ? = 1
{{1,4},{2},{3,5}}
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => ? = 1
{{1,4},{2},{3},{5}}
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => ? = 1
{{1,5},{2,4},{3}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => ? = 2
{{1},{2,4,5},{3}}
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => ? = 1
{{1},{2,5},{3,4}}
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => ? = 1
{{1},{2},{3,4,5}}
 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 1
{{1},{2},{3,4},{5}}
 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => ? = 1
{{1},{2,5},{3},{4}}
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => ? = 1
{{1},{2},{3},{4,5}}
 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 1
{{1},{2},{3},{4},{5}}
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1
{{1,2,3,4,5,6}}
 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
{{1,2,3,4,5},{6}}
 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => ? = 1
{{1,2,3,4},{5,6}}
 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
 => ? = 1
{{1,2,3,4},{5},{6}}
 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
 => ? = 1
{{1,2,3,5,6},{4}}
 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => ? = 1
{{1,2,3,6},{4,5}}
 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
 => ? = 1
{{1,2,3},{4,5,6}}
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
 => ? = 1
{{1,2,3},{4,5},{6}}
 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
 => ? = 1
{{1,2,3,6},{4},{5}}
 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
 => ? = 1
{{1,2,3},{4},{5,6}}
 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,0,1,1,0,0,0]
 => ? = 1
{{1,2,3},{4},{5},{6}}
 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => ? = 1
{{1,2,4,5},{3,6}}
 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
 => ? = 1
{{1,2,4,5},{3},{6}}
 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
 => ? = 1
{{1,2,4,6},{3,5}}
 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
 => ? = 1
{{1,2,4,6},{3},{5}}
 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
 => ? = 1
{{1,2,5},{3,4,6}}
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
 => ? = 1
{{1,2,5},{3,4},{6}}
 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
 => ? = 1
Description
The global dimension of the algebra A/AfA of the corresponding Nakayama algebra A with minimal left faithful projective-injective module Af.
Matching statistic: St001927
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00244: Signed permutations —bar⟶ Signed permutations
St001927: Signed permutations ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 33%
Values
{{1}}
 => [1] => [1] => [-1] => ? = 0 - 1
{{1,2}}
 => [2,1] => [2,1] => [-2,-1] => 0 = 1 - 1
{{1},{2}}
 => [1,2] => [1,2] => [-1,-2] => 0 = 1 - 1
{{1,2,3}}
 => [2,3,1] => [2,3,1] => [-2,-3,-1] => 0 = 1 - 1
{{1,2},{3}}
 => [2,1,3] => [2,1,3] => [-2,-1,-3] => 0 = 1 - 1
{{1},{2,3}}
 => [1,3,2] => [1,3,2] => [-1,-3,-2] => 0 = 1 - 1
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => [-1,-2,-3] => 0 = 1 - 1
{{1,2,3,4}}
 => [2,3,4,1] => [2,3,4,1] => [-2,-3,-4,-1] => 0 = 1 - 1
{{1,2,3},{4}}
 => [2,3,1,4] => [2,3,1,4] => [-2,-3,-1,-4] => 0 = 1 - 1
{{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => [-2,-1,-4,-3] => 0 = 1 - 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,3,4] => [-2,-1,-3,-4] => 0 = 1 - 1
{{1,3,4},{2}}
 => [3,2,4,1] => [3,2,4,1] => [-3,-2,-4,-1] => 0 = 1 - 1
{{1,4},{2,3}}
 => [4,3,2,1] => [4,3,2,1] => [-4,-3,-2,-1] => 0 = 1 - 1
{{1},{2,3,4}}
 => [1,3,4,2] => [1,3,4,2] => [-1,-3,-4,-2] => 0 = 1 - 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,3,2,4] => [-1,-3,-2,-4] => 0 = 1 - 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [4,2,3,1] => [-4,-2,-3,-1] => 0 = 1 - 1
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,4,3] => [-1,-2,-4,-3] => 0 = 1 - 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => 0 = 1 - 1
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [2,3,4,5,1] => [-2,-3,-4,-5,-1] => ? = 1 - 1
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [2,3,4,1,5] => [-2,-3,-4,-1,-5] => ? = 1 - 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [2,3,1,5,4] => [-2,-3,-1,-5,-4] => ? = 1 - 1
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [2,3,1,4,5] => [-2,-3,-1,-4,-5] => ? = 1 - 1
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [2,4,3,5,1] => [-2,-4,-3,-5,-1] => ? = 1 - 1
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [2,5,4,3,1] => [-2,-5,-4,-3,-1] => ? = 1 - 1
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [2,1,4,5,3] => [-2,-1,-4,-5,-3] => ? = 1 - 1
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [2,1,4,3,5] => [-2,-1,-4,-3,-5] => ? = 1 - 1
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [2,5,3,4,1] => [-2,-5,-3,-4,-1] => ? = 1 - 1
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => ? = 1 - 1
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => ? = 1 - 1
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [3,5,4,1,2] => [-3,-5,-4,-1,-2] => ? = 1 - 1
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [3,2,4,1,5] => [-3,-2,-4,-1,-5] => ? = 1 - 1
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [3,4,5,2,1] => [-3,-4,-5,-2,-1] => ? = 1 - 1
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [3,2,5,4,1] => [-3,-2,-5,-4,-1] => ? = 1 - 1
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [4,3,5,1,2] => [-4,-3,-5,-1,-2] => ? = 1 - 1
{{1,4},{2,3},{5}}
 => [4,3,2,1,5] => [4,3,2,1,5] => [-4,-3,-2,-1,-5] => ? = 1 - 1
{{1},{2,3,4,5}}
 => [1,3,4,5,2] => [1,3,4,5,2] => [-1,-3,-4,-5,-2] => ? = 1 - 1
{{1},{2,3,4},{5}}
 => [1,3,4,2,5] => [1,3,4,2,5] => [-1,-3,-4,-2,-5] => ? = 1 - 1
{{1},{2,3},{4,5}}
 => [1,3,2,5,4] => [1,3,2,5,4] => [-1,-3,-2,-5,-4] => ? = 1 - 1
{{1},{2,3},{4},{5}}
 => [1,3,2,4,5] => [1,3,2,4,5] => [-1,-3,-2,-4,-5] => ? = 1 - 1
{{1,4},{2,5},{3}}
 => [4,5,3,1,2] => [4,5,3,1,2] => [-4,-5,-3,-1,-2] => ? = 1 - 1
{{1,4},{2},{3,5}}
 => [4,2,5,1,3] => [4,2,5,1,3] => [-4,-2,-5,-1,-3] => ? = 1 - 1
{{1,4},{2},{3},{5}}
 => [4,2,3,1,5] => [4,2,3,1,5] => [-4,-2,-3,-1,-5] => ? = 1 - 1
{{1,5},{2,4},{3}}
 => [5,4,3,2,1] => [5,4,3,2,1] => [-5,-4,-3,-2,-1] => ? = 2 - 1
{{1},{2,4,5},{3}}
 => [1,4,3,5,2] => [1,4,3,5,2] => [-1,-4,-3,-5,-2] => ? = 1 - 1
{{1},{2,5},{3,4}}
 => [1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => ? = 1 - 1
{{1},{2},{3,4,5}}
 => [1,2,4,5,3] => [1,2,4,5,3] => [-1,-2,-4,-5,-3] => ? = 1 - 1
{{1},{2},{3,4},{5}}
 => [1,2,4,3,5] => [1,2,4,3,5] => [-1,-2,-4,-3,-5] => ? = 1 - 1
{{1},{2,5},{3},{4}}
 => [1,5,3,4,2] => [1,5,3,4,2] => [-1,-5,-3,-4,-2] => ? = 1 - 1
{{1},{2},{3},{4,5}}
 => [1,2,3,5,4] => [1,2,3,5,4] => [-1,-2,-3,-5,-4] => ? = 1 - 1
{{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => [1,2,3,4,5] => [-1,-2,-3,-4,-5] => ? = 1 - 1
{{1,2,3,4,5,6}}
 => [2,3,4,5,6,1] => [2,3,4,5,6,1] => [-2,-3,-4,-5,-6,-1] => ? = 1 - 1
{{1,2,3,4,5},{6}}
 => [2,3,4,5,1,6] => [2,3,4,5,1,6] => [-2,-3,-4,-5,-1,-6] => ? = 1 - 1
{{1,2,3,4},{5,6}}
 => [2,3,4,1,6,5] => [2,3,4,1,6,5] => [-2,-3,-4,-1,-6,-5] => ? = 1 - 1
{{1,2,3,4},{5},{6}}
 => [2,3,4,1,5,6] => [2,3,4,1,5,6] => [-2,-3,-4,-1,-5,-6] => ? = 1 - 1
{{1,2,3,5,6},{4}}
 => [2,3,5,4,6,1] => [2,3,5,4,6,1] => [-2,-3,-5,-4,-6,-1] => ? = 1 - 1
{{1,2,3,6},{4,5}}
 => [2,3,6,5,4,1] => [2,3,6,5,4,1] => [-2,-3,-6,-5,-4,-1] => ? = 1 - 1
{{1,2,3},{4,5,6}}
 => [2,3,1,5,6,4] => [2,3,1,5,6,4] => [-2,-3,-1,-5,-6,-4] => ? = 1 - 1
{{1,2,3},{4,5},{6}}
 => [2,3,1,5,4,6] => [2,3,1,5,4,6] => [-2,-3,-1,-5,-4,-6] => ? = 1 - 1
{{1,2,3,6},{4},{5}}
 => [2,3,6,4,5,1] => [2,3,6,4,5,1] => [-2,-3,-6,-4,-5,-1] => ? = 1 - 1
{{1,2,3},{4},{5,6}}
 => [2,3,1,4,6,5] => [2,3,1,4,6,5] => [-2,-3,-1,-4,-6,-5] => ? = 1 - 1
{{1,2,3},{4},{5},{6}}
 => [2,3,1,4,5,6] => [2,3,1,4,5,6] => [-2,-3,-1,-4,-5,-6] => ? = 1 - 1
{{1,2,4,5},{3,6}}
 => [2,4,6,5,1,3] => [2,4,6,5,1,3] => ? => ? = 1 - 1
{{1,2,4,5},{3},{6}}
 => [2,4,3,5,1,6] => [2,4,3,5,1,6] => [-2,-4,-3,-5,-1,-6] => ? = 1 - 1
{{1,2,4,6},{3,5}}
 => [2,4,5,6,3,1] => [2,4,5,6,3,1] => [-2,-4,-5,-6,-3,-1] => ? = 1 - 1
{{1,2,4,6},{3},{5}}
 => [2,4,3,6,5,1] => [2,4,3,6,5,1] => [-2,-4,-3,-6,-5,-1] => ? = 1 - 1
{{1,2,5},{3,4,6}}
 => [2,5,4,6,1,3] => [2,5,4,6,1,3] => ? => ? = 1 - 1
{{1,2,5},{3,4},{6}}
 => [2,5,4,3,1,6] => [2,5,4,3,1,6] => [-2,-5,-4,-3,-1,-6] => ? = 1 - 1
Description
Sparre Andersen's number of positives of a signed permutation. For \pi a signed permutation of length n, first create the tuple x = (x_1, \dots, x_n), where x_i = c_{|\pi_1|} \operatorname{sgn}(\pi_{|\pi_1|}) + \cdots + c_{|\pi_i|} \operatorname{sgn}(\pi_{|\pi_i|}) and (c_1, \dots ,c_n) = (1, 2, \dots, 2^{n-1}). The actual value of the c-tuple for Andersen's statistic does not matter so long as no sums or differences of any subset of the c_i's is zero. The choice of powers of 2 is just a convenient choice. This returns the number of strictly positive values in the x-tuple. This is related to the ''discrete arcsin distribution''. The number of signed permutations with value equal to k is given by \binom{2k}{k} \binom{2n-2k}{n-k} \frac{n!}{2^n}. This statistic is equidistributed with Sparre Andersen's `Position of Maximum' statistic.
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