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Your data matches 79 different statistics following compositions of up to 3 maps.
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Matching statistic: St001418
Mp00108: Permutations ācycle typeā¶ Integer partitions
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
St001418: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
St001418: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1,2,3] => [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,2,4,3] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,3,2,4] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,1,4,3] => [2,2]
=> [2]
=> [1,0,1,0]
=> 0
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[3,4,1,2] => [2,2]
=> [2]
=> [1,0,1,0]
=> 0
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[4,3,2,1] => [2,2]
=> [2]
=> [1,0,1,0]
=> 0
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,2,4,5,3] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,2,5,3,4] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,3,4,2,5] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,4,2,3,5] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,4,5,2,3] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,5,4,3,2] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,1,4,3,5] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,1,4,5,3] => [3,2]
=> [2]
=> [1,0,1,0]
=> 0
[2,1,5,3,4] => [3,2]
=> [2]
=> [1,0,1,0]
=> 0
[2,1,5,4,3] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,3,1,4,5] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,3,1,5,4] => [3,2]
=> [2]
=> [1,0,1,0]
=> 0
[2,4,3,1,5] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,4,5,1,3] => [3,2]
=> [2]
=> [1,0,1,0]
=> 0
[2,5,3,4,1] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,5,4,3,1] => [3,2]
=> [2]
=> [1,0,1,0]
=> 0
[3,1,2,4,5] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[3,1,2,5,4] => [3,2]
=> [2]
=> [1,0,1,0]
=> 0
[3,2,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[3,2,1,5,4] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[3,2,4,1,5] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[3,2,5,4,1] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[3,4,1,2,5] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[3,4,1,5,2] => [3,2]
=> [2]
=> [1,0,1,0]
=> 0
[3,4,5,2,1] => [3,2]
=> [2]
=> [1,0,1,0]
=> 0
Description
Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path.
The stable Auslander algebra is by definition the stable endomorphism ring of the direct sum of all indecomposable modules.
Matching statistic: St000098
Mp00151: Permutations āto cycle typeā¶ Set partitions
Mp00128: Set partitions āto compositionā¶ Integer compositions
Mp00184: Integer compositions āto threshold graphā¶ Graphs
St000098: Graphs ā¶ ā¤Result quality: 37% āvalues known / values provided: 37%ādistinct values known / distinct values provided: 100%
Mp00128: Set partitions āto compositionā¶ Integer compositions
Mp00184: Integer compositions āto threshold graphā¶ Graphs
St000098: Graphs ā¶ ā¤Result quality: 37% āvalues known / values provided: 37%ādistinct values known / distinct values provided: 100%
Values
[1,2,3] => {{1},{2},{3}}
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[1,2,3,4] => {{1},{2},{3},{4}}
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 2 + 2
[1,2,4,3] => {{1},{2},{3,4}}
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[1,3,2,4] => {{1},{2,3},{4}}
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[1,4,3,2] => {{1},{2,4},{3}}
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[2,1,3,4] => {{1,2},{3},{4}}
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[2,1,4,3] => {{1,2},{3,4}}
=> [2,2] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[3,2,1,4] => {{1,3},{2},{4}}
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[3,4,1,2] => {{1,3},{2,4}}
=> [2,2] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[4,2,3,1] => {{1,4},{2},{3}}
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[4,3,2,1] => {{1,4},{2,3}}
=> [2,2] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 3 + 2
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[1,2,4,5,3] => {{1},{2},{3,4,5}}
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[1,2,5,3,4] => {{1},{2},{3,4,5}}
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[1,3,2,5,4] => {{1},{2,3},{4,5}}
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[1,3,4,2,5] => {{1},{2,3,4},{5}}
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[1,3,5,4,2] => {{1},{2,3,5},{4}}
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[1,4,2,3,5] => {{1},{2,3,4},{5}}
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[1,4,3,5,2] => {{1},{2,4,5},{3}}
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[1,4,5,2,3] => {{1},{2,4},{3,5}}
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[1,5,2,4,3] => {{1},{2,3,5},{4}}
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[1,5,3,2,4] => {{1},{2,4,5},{3}}
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[1,5,3,4,2] => {{1},{2,5},{3},{4}}
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[1,5,4,3,2] => {{1},{2,5},{3,4}}
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[2,1,3,5,4] => {{1,2},{3},{4,5}}
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[2,1,4,3,5] => {{1,2},{3,4},{5}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[2,1,4,5,3] => {{1,2},{3,4,5}}
=> [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,1,5,3,4] => {{1,2},{3,4,5}}
=> [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,1,5,4,3] => {{1,2},{3,5},{4}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[2,3,1,4,5] => {{1,2,3},{4},{5}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[2,3,1,5,4] => {{1,2,3},{4,5}}
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,4,3,1,5] => {{1,2,4},{3},{5}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[2,4,5,1,3] => {{1,2,4},{3,5}}
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,5,3,4,1] => {{1,2,5},{3},{4}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[2,5,4,3,1] => {{1,2,5},{3,4}}
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[3,1,2,4,5] => {{1,2,3},{4},{5}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[3,1,2,5,4] => {{1,2,3},{4,5}}
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[3,2,1,4,5] => {{1,3},{2},{4},{5}}
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[3,2,1,5,4] => {{1,3},{2},{4,5}}
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[3,2,4,1,5] => {{1,3,4},{2},{5}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[3,2,5,4,1] => {{1,3,5},{2},{4}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[3,4,1,2,5] => {{1,3},{2,4},{5}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[3,4,1,5,2] => {{1,3},{2,4,5}}
=> [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[3,4,5,2,1] => {{1,3,5},{2,4}}
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[8,7,6,5,4,3,2,1] => {{1,8},{2,7},{3,6},{4,5}}
=> [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1 + 2
[7,8,6,5,4,3,2,1] => {{1,2,7,8},{3,6},{4,5}}
=> ? => ?
=> ? = 1 + 2
[8,6,7,5,4,3,2,1] => {{1,8},{2,3,6,7},{4,5}}
=> [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1 + 2
[7,6,8,5,4,3,2,1] => {{1,2,3,6,7,8},{4,5}}
=> [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 2
[6,7,8,5,4,3,2,1] => {{1,3,6,8},{2,7},{4,5}}
=> [4,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1 + 2
[8,7,5,6,4,3,2,1] => {{1,8},{2,7},{3,4,5,6}}
=> ? => ?
=> ? = 1 + 2
[7,8,5,6,4,3,2,1] => {{1,2,7,8},{3,4,5,6}}
=> [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 2
[8,6,5,7,4,3,2,1] => {{1,8},{2,3,4,5,6,7}}
=> [2,6] => ([(5,7),(6,7)],8)
=> ? = 0 + 2
[8,5,6,7,4,3,2,1] => {{1,8},{2,4,5,7},{3,6}}
=> ? => ?
=> ? = 1 + 2
[6,7,5,8,4,3,2,1] => {{1,3,4,5,6,8},{2,7}}
=> [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 2
[7,5,6,8,4,3,2,1] => {{1,2,4,5,7,8},{3,6}}
=> [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 2
[5,6,7,8,4,3,2,1] => {{1,4,5,8},{2,3,6,7}}
=> [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 2
[8,7,6,4,5,3,2,1] => {{1,8},{2,7},{3,6},{4},{5}}
=> ? => ?
=> ? = 3 + 2
[8,7,4,5,6,3,2,1] => {{1,8},{2,7},{3,4,5,6}}
=> ? => ?
=> ? = 1 + 2
[8,6,4,5,7,3,2,1] => {{1,8},{2,3,4,5,6,7}}
=> [2,6] => ([(5,7),(6,7)],8)
=> ? = 0 + 2
[8,4,5,6,7,3,2,1] => {{1,8},{2,3,4,5,6,7}}
=> [2,6] => ([(5,7),(6,7)],8)
=> ? = 0 + 2
[5,6,7,4,8,3,2,1] => {{1,5,8},{2,3,6,7},{4}}
=> ? => ?
=> ? = 1 + 2
[6,7,4,5,8,3,2,1] => {{1,3,4,5,6,8},{2,7}}
=> [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 2
[7,5,4,6,8,3,2,1] => {{1,2,5,7,8},{3,4,6}}
=> [5,3] => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 2
[5,6,4,7,8,3,2,1] => {{1,5,8},{2,3,4,6,7}}
=> [3,5] => ([(4,7),(5,7),(6,7)],8)
=> ? = 0 + 2
[6,4,5,7,8,3,2,1] => {{1,3,5,6,8},{2,4,7}}
=> [5,3] => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 2
[5,4,6,7,8,3,2,1] => {{1,5,8},{2,4,7},{3,6}}
=> ? => ?
=> ? = 1 + 2
[4,5,6,7,8,3,2,1] => {{1,2,4,5,7,8},{3,6}}
=> [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 2
[8,7,6,5,3,4,2,1] => {{1,8},{2,7},{3,4,5,6}}
=> ? => ?
=> ? = 1 + 2
[7,8,6,5,3,4,2,1] => {{1,2,7,8},{3,4,5,6}}
=> [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 2
[8,6,7,5,3,4,2,1] => {{1,8},{2,3,4,5,6,7}}
=> [2,6] => ([(5,7),(6,7)],8)
=> ? = 0 + 2
[8,7,5,6,3,4,2,1] => {{1,8},{2,7},{3,5},{4,6}}
=> [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1 + 2
[7,8,5,6,3,4,2,1] => {{1,2,7,8},{3,5},{4,6}}
=> [4,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1 + 2
[8,5,6,7,3,4,2,1] => {{1,8},{2,3,4,5,6,7}}
=> [2,6] => ([(5,7),(6,7)],8)
=> ? = 0 + 2
[7,6,5,8,3,4,2,1] => {{1,2,4,6,7,8},{3,5}}
=> [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 2
[6,5,7,8,3,4,2,1] => {{1,4,6,8},{2,3,5,7}}
=> [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 2
[7,8,6,4,3,5,2,1] => {{1,2,7,8},{3,5,6},{4}}
=> ? => ?
=> ? = 1 + 2
[8,7,6,3,4,5,2,1] => {{1,8},{2,7},{3,4,5,6}}
=> ? => ?
=> ? = 1 + 2
[7,8,6,3,4,5,2,1] => {{1,2,7,8},{3,4,5,6}}
=> [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 2
[8,6,7,3,4,5,2,1] => {{1,8},{2,3,4,5,6,7}}
=> [2,6] => ([(5,7),(6,7)],8)
=> ? = 0 + 2
[6,7,8,3,4,5,2,1] => {{1,3,4,5,6,8},{2,7}}
=> [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 2
[8,7,5,4,3,6,2,1] => {{1,8},{2,7},{3,5},{4},{6}}
=> ? => ?
=> ? = 3 + 2
[7,8,5,3,4,6,2,1] => {{1,2,7,8},{3,4,5},{6}}
=> ? => ?
=> ? = 1 + 2
[8,5,4,3,6,7,2,1] => {{1,8},{2,5,6,7},{3,4}}
=> ? => ?
=> ? = 1 + 2
[8,3,4,5,6,7,2,1] => {{1,8},{2,3,4,5,6,7}}
=> [2,6] => ([(5,7),(6,7)],8)
=> ? = 0 + 2
[6,5,7,4,3,8,2,1] => {{1,6,8},{2,3,5,7},{4}}
=> ? => ?
=> ? = 1 + 2
[6,7,4,5,3,8,2,1] => {{1,6,8},{2,7},{3,4,5}}
=> ? => ?
=> ? = 1 + 2
[7,4,5,6,3,8,2,1] => {{1,2,4,6,7,8},{3,5}}
=> [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 2
[6,5,4,7,3,8,2,1] => {{1,6,8},{2,3,4,5,7}}
=> [3,5] => ([(4,7),(5,7),(6,7)],8)
=> ? = 0 + 2
[6,4,5,7,3,8,2,1] => {{1,6,8},{2,4,7},{3,5}}
=> [3,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1 + 2
[5,4,6,7,3,8,2,1] => {{1,3,5,6,8},{2,4,7}}
=> [5,3] => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 2
[7,6,5,3,4,8,2,1] => {{1,2,6,7,8},{3,4,5}}
=> [5,3] => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 2
[7,6,4,3,5,8,2,1] => {{1,2,6,7,8},{3,4},{5}}
=> ? => ?
=> ? = 1 + 2
[7,5,4,3,6,8,2,1] => {{1,2,5,6,7,8},{3,4}}
=> [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 0 + 2
[6,5,4,3,7,8,2,1] => {{1,6,8},{2,5,7},{3,4}}
=> [3,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1 + 2
Description
The chromatic number of a graph.
The minimal number of colors needed to color the vertices of the graph such that no two vertices which share an edge have the same color.
Matching statistic: St001227
Mp00108: Permutations ācycle typeā¶ Integer partitions
Mp00043: Integer partitions āto Dyck pathā¶ Dyck paths
Mp00199: Dyck paths āprime Dyck pathā¶ Dyck paths
St001227: Dyck paths ā¶ ā¤Result quality: 35% āvalues known / values provided: 35%ādistinct values known / distinct values provided: 43%
Mp00043: Integer partitions āto Dyck pathā¶ Dyck paths
Mp00199: Dyck paths āprime Dyck pathā¶ Dyck paths
St001227: Dyck paths ā¶ ā¤Result quality: 35% āvalues known / values provided: 35%ādistinct values known / distinct values provided: 43%
Values
[1,2,3] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3 = 1 + 2
[1,2,3,4] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 4 = 2 + 2
[1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3 = 1 + 2
[1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3 = 1 + 2
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3 = 1 + 2
[2,1,3,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3 = 1 + 2
[2,1,4,3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2 = 0 + 2
[3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3 = 1 + 2
[3,4,1,2] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2 = 0 + 2
[4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3 = 1 + 2
[4,3,2,1] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2 = 0 + 2
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 + 2
[1,2,3,5,4] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 4 = 2 + 2
[1,2,4,3,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 4 = 2 + 2
[1,2,4,5,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 1 + 2
[1,2,5,3,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 1 + 2
[1,2,5,4,3] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 4 = 2 + 2
[1,3,2,4,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 4 = 2 + 2
[1,3,2,5,4] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3 = 1 + 2
[1,3,4,2,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 1 + 2
[1,3,5,4,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 1 + 2
[1,4,2,3,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 1 + 2
[1,4,3,2,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 4 = 2 + 2
[1,4,3,5,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 1 + 2
[1,4,5,2,3] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3 = 1 + 2
[1,5,2,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 1 + 2
[1,5,3,2,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 1 + 2
[1,5,3,4,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 4 = 2 + 2
[1,5,4,3,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3 = 1 + 2
[2,1,3,4,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 4 = 2 + 2
[2,1,3,5,4] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3 = 1 + 2
[2,1,4,3,5] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3 = 1 + 2
[2,1,4,5,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2 = 0 + 2
[2,1,5,3,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2 = 0 + 2
[2,1,5,4,3] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3 = 1 + 2
[2,3,1,4,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 1 + 2
[2,3,1,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2 = 0 + 2
[2,4,3,1,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 1 + 2
[2,4,5,1,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2 = 0 + 2
[2,5,3,4,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 1 + 2
[2,5,4,3,1] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2 = 0 + 2
[3,1,2,4,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 1 + 2
[3,1,2,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2 = 0 + 2
[3,2,1,4,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 4 = 2 + 2
[3,2,1,5,4] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3 = 1 + 2
[3,2,4,1,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 1 + 2
[3,2,5,4,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 1 + 2
[3,4,1,2,5] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3 = 1 + 2
[3,4,1,5,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2 = 0 + 2
[3,4,5,2,1] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2 = 0 + 2
[3,5,1,2,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2 = 0 + 2
[1,2,3,4,5,6] => [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 4 + 2
[1,2,3,4,6,5] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 3 + 2
[1,2,3,5,4,6] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 3 + 2
[1,2,3,6,5,4] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 3 + 2
[1,2,4,3,5,6] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 3 + 2
[1,2,5,4,3,6] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 3 + 2
[1,2,6,4,5,3] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 3 + 2
[1,3,2,4,5,6] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 3 + 2
[1,4,3,2,5,6] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 3 + 2
[1,5,3,4,2,6] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 3 + 2
[1,6,3,4,5,2] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 3 + 2
[2,1,3,4,5,6] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 3 + 2
[3,2,1,4,5,6] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 3 + 2
[4,2,3,1,5,6] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 3 + 2
[5,2,3,4,1,6] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 3 + 2
[6,2,3,4,5,1] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 3 + 2
[1,2,3,4,5,6,7] => [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 5 + 2
[1,2,3,4,5,7,6] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 4 + 2
[1,2,3,4,6,5,7] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 4 + 2
[1,2,3,4,6,7,5] => [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 3 + 2
[1,2,3,4,7,5,6] => [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 3 + 2
[1,2,3,4,7,6,5] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 4 + 2
[1,2,3,5,4,6,7] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 4 + 2
[1,2,3,5,4,7,6] => [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 3 + 2
[1,2,3,5,6,4,7] => [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 3 + 2
[1,2,3,5,7,6,4] => [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 3 + 2
[1,2,3,6,4,5,7] => [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 3 + 2
[1,2,3,6,5,4,7] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 4 + 2
[1,2,3,6,5,7,4] => [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 3 + 2
[1,2,3,6,7,4,5] => [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 3 + 2
[1,2,3,7,4,6,5] => [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 3 + 2
[1,2,3,7,5,4,6] => [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 3 + 2
[1,2,3,7,5,6,4] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 4 + 2
[1,2,3,7,6,5,4] => [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 3 + 2
[1,2,4,3,5,6,7] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 4 + 2
[1,2,4,3,5,7,6] => [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 3 + 2
[1,2,4,3,6,5,7] => [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 3 + 2
[1,2,4,3,7,6,5] => [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 3 + 2
[1,2,4,5,3,6,7] => [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 3 + 2
[1,2,4,5,6,7,3] => [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> ? = 1 + 2
[1,2,4,5,7,3,6] => [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> ? = 1 + 2
[1,2,4,6,3,7,5] => [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> ? = 1 + 2
[1,2,4,6,5,3,7] => [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 3 + 2
[1,2,4,6,7,5,3] => [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> ? = 1 + 2
[1,2,4,7,3,5,6] => [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> ? = 1 + 2
[1,2,4,7,5,6,3] => [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 3 + 2
[1,2,4,7,6,3,5] => [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> ? = 1 + 2
[1,2,5,3,4,6,7] => [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 3 + 2
[1,2,5,3,6,7,4] => [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> ? = 1 + 2
Description
The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra.
Matching statistic: St000925
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00108: Permutations ācycle typeā¶ Integer partitions
Mp00042: Integer partitions āinitial tableauā¶ Standard tableaux
Mp00284: Standard tableaux ārowsā¶ Set partitions
St000925: Set partitions ā¶ ā¤Result quality: 23% āvalues known / values provided: 23%ādistinct values known / distinct values provided: 86%
Mp00042: Integer partitions āinitial tableauā¶ Standard tableaux
Mp00284: Standard tableaux ārowsā¶ Set partitions
St000925: Set partitions ā¶ ā¤Result quality: 23% āvalues known / values provided: 23%ādistinct values known / distinct values provided: 86%
Values
[1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 3 = 1 + 2
[1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 4 = 2 + 2
[1,2,4,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 3 = 1 + 2
[1,3,2,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 3 = 1 + 2
[1,4,3,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 3 = 1 + 2
[2,1,3,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 3 = 1 + 2
[2,1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 2 = 0 + 2
[3,2,1,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 3 = 1 + 2
[3,4,1,2] => [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 2 = 0 + 2
[4,2,3,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 3 = 1 + 2
[4,3,2,1] => [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 2 = 0 + 2
[1,2,3,4,5] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> 5 = 3 + 2
[1,2,3,5,4] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 4 = 2 + 2
[1,2,4,3,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 4 = 2 + 2
[1,2,4,5,3] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 3 = 1 + 2
[1,2,5,3,4] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 3 = 1 + 2
[1,2,5,4,3] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 4 = 2 + 2
[1,3,2,4,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 4 = 2 + 2
[1,3,2,5,4] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> 3 = 1 + 2
[1,3,4,2,5] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 3 = 1 + 2
[1,3,5,4,2] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 3 = 1 + 2
[1,4,2,3,5] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 3 = 1 + 2
[1,4,3,2,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 4 = 2 + 2
[1,4,3,5,2] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 3 = 1 + 2
[1,4,5,2,3] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> 3 = 1 + 2
[1,5,2,4,3] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 3 = 1 + 2
[1,5,3,2,4] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 3 = 1 + 2
[1,5,3,4,2] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 4 = 2 + 2
[1,5,4,3,2] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> 3 = 1 + 2
[2,1,3,4,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 4 = 2 + 2
[2,1,3,5,4] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> 3 = 1 + 2
[2,1,4,3,5] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> 3 = 1 + 2
[2,1,4,5,3] => [3,2]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 2 = 0 + 2
[2,1,5,3,4] => [3,2]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 2 = 0 + 2
[2,1,5,4,3] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> 3 = 1 + 2
[2,3,1,4,5] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 3 = 1 + 2
[2,3,1,5,4] => [3,2]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 2 = 0 + 2
[2,4,3,1,5] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 3 = 1 + 2
[2,4,5,1,3] => [3,2]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 2 = 0 + 2
[2,5,3,4,1] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 3 = 1 + 2
[2,5,4,3,1] => [3,2]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 2 = 0 + 2
[3,1,2,4,5] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 3 = 1 + 2
[3,1,2,5,4] => [3,2]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 2 = 0 + 2
[3,2,1,4,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 4 = 2 + 2
[3,2,1,5,4] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> 3 = 1 + 2
[3,2,4,1,5] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 3 = 1 + 2
[3,2,5,4,1] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 3 = 1 + 2
[3,4,1,2,5] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> 3 = 1 + 2
[3,4,1,5,2] => [3,2]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 2 = 0 + 2
[3,4,5,2,1] => [3,2]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 2 = 0 + 2
[8,7,6,5,4,3,2,1] => [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> {{1,2},{3,4},{5,6},{7,8}}
=> ? = 1 + 2
[7,8,6,5,4,3,2,1] => [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> {{1,2,3,4},{5,6},{7,8}}
=> ? = 1 + 2
[8,6,7,5,4,3,2,1] => [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> {{1,2,3,4},{5,6},{7,8}}
=> ? = 1 + 2
[6,7,8,5,4,3,2,1] => [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> {{1,2,3,4},{5,6},{7,8}}
=> ? = 1 + 2
[8,7,5,6,4,3,2,1] => [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> {{1,2,3,4},{5,6},{7,8}}
=> ? = 1 + 2
[7,8,5,6,4,3,2,1] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 0 + 2
[8,5,6,7,4,3,2,1] => [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> {{1,2,3,4},{5,6},{7,8}}
=> ? = 1 + 2
[5,6,7,8,4,3,2,1] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 0 + 2
[8,7,6,4,5,3,2,1] => [2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> {{1,2},{3,4},{5,6},{7},{8}}
=> ? = 3 + 2
[7,8,6,4,5,3,2,1] => [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> {{1,2,3,4},{5,6},{7},{8}}
=> ? = 2 + 2
[8,6,7,4,5,3,2,1] => [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> {{1,2,3,4},{5,6},{7},{8}}
=> ? = 2 + 2
[7,6,8,4,5,3,2,1] => [6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 1 + 2
[8,7,5,4,6,3,2,1] => [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> {{1,2,3},{4,5},{6,7},{8}}
=> ? = 2 + 2
[8,7,4,5,6,3,2,1] => [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> {{1,2,3,4},{5,6},{7,8}}
=> ? = 1 + 2
[8,6,5,4,7,3,2,1] => [5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> {{1,2,3,4,5},{6,7},{8}}
=> ? = 1 + 2
[8,5,6,4,7,3,2,1] => [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> {{1,2,3},{4,5},{6,7},{8}}
=> ? = 2 + 2
[6,7,5,4,8,3,2,1] => [5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> {{1,2,3,4,5},{6,7},{8}}
=> ? = 1 + 2
[7,5,6,4,8,3,2,1] => [5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> {{1,2,3,4,5},{6,7},{8}}
=> ? = 1 + 2
[5,6,7,4,8,3,2,1] => [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> {{1,2,3,4},{5,6,7},{8}}
=> ? = 1 + 2
[7,5,4,6,8,3,2,1] => [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 0 + 2
[5,6,4,7,8,3,2,1] => [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 0 + 2
[6,4,5,7,8,3,2,1] => [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 0 + 2
[5,4,6,7,8,3,2,1] => [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> {{1,2,3},{4,5,6},{7,8}}
=> ? = 1 + 2
[8,7,6,5,3,4,2,1] => [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> {{1,2,3,4},{5,6},{7,8}}
=> ? = 1 + 2
[7,8,6,5,3,4,2,1] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 0 + 2
[8,7,5,6,3,4,2,1] => [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> {{1,2},{3,4},{5,6},{7,8}}
=> ? = 1 + 2
[7,8,5,6,3,4,2,1] => [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> {{1,2,3,4},{5,6},{7,8}}
=> ? = 1 + 2
[6,5,7,8,3,4,2,1] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 0 + 2
[8,7,6,4,3,5,2,1] => [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> {{1,2,3},{4,5},{6,7},{8}}
=> ? = 2 + 2
[7,8,6,4,3,5,2,1] => [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> {{1,2,3,4},{5,6,7},{8}}
=> ? = 1 + 2
[8,6,7,4,3,5,2,1] => [5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> {{1,2,3,4,5},{6,7},{8}}
=> ? = 1 + 2
[6,7,8,4,3,5,2,1] => [5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> {{1,2,3,4,5},{6,7},{8}}
=> ? = 1 + 2
[8,7,6,3,4,5,2,1] => [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> {{1,2,3,4},{5,6},{7,8}}
=> ? = 1 + 2
[7,8,6,3,4,5,2,1] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 0 + 2
[8,7,5,4,3,6,2,1] => [2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> {{1,2},{3,4},{5,6},{7},{8}}
=> ? = 3 + 2
[7,8,5,4,3,6,2,1] => [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> {{1,2,3,4},{5,6},{7},{8}}
=> ? = 2 + 2
[8,7,4,5,3,6,2,1] => [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> {{1,2,3},{4,5},{6,7},{8}}
=> ? = 2 + 2
[7,8,5,3,4,6,2,1] => [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> {{1,2,3,4},{5,6,7},{8}}
=> ? = 1 + 2
[8,7,4,3,5,6,2,1] => [2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> {{1,2},{3,4},{5,6},{7},{8}}
=> ? = 3 + 2
[8,7,3,4,5,6,2,1] => [2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> {{1,2},{3,4},{5},{6},{7},{8}}
=> ? = 4 + 2
[7,8,3,4,5,6,2,1] => [4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> {{1,2,3,4},{5},{6},{7},{8}}
=> ? = 3 + 2
[8,6,5,4,3,7,2,1] => [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> {{1,2,3},{4,5},{6,7},{8}}
=> ? = 2 + 2
[8,6,4,3,5,7,2,1] => [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> {{1,2,3},{4,5},{6,7},{8}}
=> ? = 2 + 2
[8,5,4,3,6,7,2,1] => [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> {{1,2,3,4},{5,6},{7,8}}
=> ? = 1 + 2
[8,5,3,4,6,7,2,1] => [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> {{1,2,3,4},{5,6},{7},{8}}
=> ? = 2 + 2
[8,4,3,5,6,7,2,1] => [5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> {{1,2,3,4,5},{6,7},{8}}
=> ? = 1 + 2
[7,6,5,4,3,8,2,1] => [5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> {{1,2,3,4,5},{6,7},{8}}
=> ? = 1 + 2
[6,7,5,4,3,8,2,1] => [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> {{1,2,3},{4,5},{6,7},{8}}
=> ? = 2 + 2
[6,5,7,4,3,8,2,1] => [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> {{1,2,3,4},{5,6,7},{8}}
=> ? = 1 + 2
[6,7,4,5,3,8,2,1] => [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> {{1,2,3},{4,5,6},{7,8}}
=> ? = 1 + 2
Description
The number of topologically connected components of a set partition.
For example, the set partition $\{\{1,5\},\{2,3\},\{4,6\}\}$ has the two connected components $\{1,4,5,6\}$ and $\{2,3\}$.
The number of set partitions with only one block is [[oeis:A099947]].
Matching statistic: St000105
(load all 172 compositions to match this statistic)
(load all 172 compositions to match this statistic)
Mp00151: Permutations āto cycle typeā¶ Set partitions
St000105: Set partitions ā¶ ā¤Result quality: 15% āvalues known / values provided: 15%ādistinct values known / distinct values provided: 86%
St000105: Set partitions ā¶ ā¤Result quality: 15% āvalues known / values provided: 15%ādistinct values known / distinct values provided: 86%
Values
[1,2,3] => {{1},{2},{3}}
=> 3 = 1 + 2
[1,2,3,4] => {{1},{2},{3},{4}}
=> 4 = 2 + 2
[1,2,4,3] => {{1},{2},{3,4}}
=> 3 = 1 + 2
[1,3,2,4] => {{1},{2,3},{4}}
=> 3 = 1 + 2
[1,4,3,2] => {{1},{2,4},{3}}
=> 3 = 1 + 2
[2,1,3,4] => {{1,2},{3},{4}}
=> 3 = 1 + 2
[2,1,4,3] => {{1,2},{3,4}}
=> 2 = 0 + 2
[3,2,1,4] => {{1,3},{2},{4}}
=> 3 = 1 + 2
[3,4,1,2] => {{1,3},{2,4}}
=> 2 = 0 + 2
[4,2,3,1] => {{1,4},{2},{3}}
=> 3 = 1 + 2
[4,3,2,1] => {{1,4},{2,3}}
=> 2 = 0 + 2
[1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 5 = 3 + 2
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 4 = 2 + 2
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 4 = 2 + 2
[1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 3 = 1 + 2
[1,2,5,3,4] => {{1},{2},{3,4,5}}
=> 3 = 1 + 2
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> 4 = 2 + 2
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 4 = 2 + 2
[1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 3 = 1 + 2
[1,3,4,2,5] => {{1},{2,3,4},{5}}
=> 3 = 1 + 2
[1,3,5,4,2] => {{1},{2,3,5},{4}}
=> 3 = 1 + 2
[1,4,2,3,5] => {{1},{2,3,4},{5}}
=> 3 = 1 + 2
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> 4 = 2 + 2
[1,4,3,5,2] => {{1},{2,4,5},{3}}
=> 3 = 1 + 2
[1,4,5,2,3] => {{1},{2,4},{3,5}}
=> 3 = 1 + 2
[1,5,2,4,3] => {{1},{2,3,5},{4}}
=> 3 = 1 + 2
[1,5,3,2,4] => {{1},{2,4,5},{3}}
=> 3 = 1 + 2
[1,5,3,4,2] => {{1},{2,5},{3},{4}}
=> 4 = 2 + 2
[1,5,4,3,2] => {{1},{2,5},{3,4}}
=> 3 = 1 + 2
[2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> 4 = 2 + 2
[2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 3 = 1 + 2
[2,1,4,3,5] => {{1,2},{3,4},{5}}
=> 3 = 1 + 2
[2,1,4,5,3] => {{1,2},{3,4,5}}
=> 2 = 0 + 2
[2,1,5,3,4] => {{1,2},{3,4,5}}
=> 2 = 0 + 2
[2,1,5,4,3] => {{1,2},{3,5},{4}}
=> 3 = 1 + 2
[2,3,1,4,5] => {{1,2,3},{4},{5}}
=> 3 = 1 + 2
[2,3,1,5,4] => {{1,2,3},{4,5}}
=> 2 = 0 + 2
[2,4,3,1,5] => {{1,2,4},{3},{5}}
=> 3 = 1 + 2
[2,4,5,1,3] => {{1,2,4},{3,5}}
=> 2 = 0 + 2
[2,5,3,4,1] => {{1,2,5},{3},{4}}
=> 3 = 1 + 2
[2,5,4,3,1] => {{1,2,5},{3,4}}
=> 2 = 0 + 2
[3,1,2,4,5] => {{1,2,3},{4},{5}}
=> 3 = 1 + 2
[3,1,2,5,4] => {{1,2,3},{4,5}}
=> 2 = 0 + 2
[3,2,1,4,5] => {{1,3},{2},{4},{5}}
=> 4 = 2 + 2
[3,2,1,5,4] => {{1,3},{2},{4,5}}
=> 3 = 1 + 2
[3,2,4,1,5] => {{1,3,4},{2},{5}}
=> 3 = 1 + 2
[3,2,5,4,1] => {{1,3,5},{2},{4}}
=> 3 = 1 + 2
[3,4,1,2,5] => {{1,3},{2,4},{5}}
=> 3 = 1 + 2
[3,4,1,5,2] => {{1,3},{2,4,5}}
=> 2 = 0 + 2
[3,4,5,2,1] => {{1,3,5},{2,4}}
=> 2 = 0 + 2
[8,7,6,5,4,3,2,1] => {{1,8},{2,7},{3,6},{4,5}}
=> ? = 1 + 2
[7,8,6,5,4,3,2,1] => {{1,2,7,8},{3,6},{4,5}}
=> ? = 1 + 2
[8,6,7,5,4,3,2,1] => {{1,8},{2,3,6,7},{4,5}}
=> ? = 1 + 2
[7,6,8,5,4,3,2,1] => {{1,2,3,6,7,8},{4,5}}
=> ? = 0 + 2
[6,7,8,5,4,3,2,1] => {{1,3,6,8},{2,7},{4,5}}
=> ? = 1 + 2
[8,7,5,6,4,3,2,1] => {{1,8},{2,7},{3,4,5,6}}
=> ? = 1 + 2
[7,8,5,6,4,3,2,1] => {{1,2,7,8},{3,4,5,6}}
=> ? = 0 + 2
[8,6,5,7,4,3,2,1] => {{1,8},{2,3,4,5,6,7}}
=> ? = 0 + 2
[8,5,6,7,4,3,2,1] => {{1,8},{2,4,5,7},{3,6}}
=> ? = 1 + 2
[6,7,5,8,4,3,2,1] => {{1,3,4,5,6,8},{2,7}}
=> ? = 0 + 2
[7,5,6,8,4,3,2,1] => {{1,2,4,5,7,8},{3,6}}
=> ? = 0 + 2
[5,6,7,8,4,3,2,1] => {{1,4,5,8},{2,3,6,7}}
=> ? = 0 + 2
[8,7,6,4,5,3,2,1] => {{1,8},{2,7},{3,6},{4},{5}}
=> ? = 3 + 2
[7,8,6,4,5,3,2,1] => {{1,2,7,8},{3,6},{4},{5}}
=> ? = 2 + 2
[8,6,7,4,5,3,2,1] => {{1,8},{2,3,6,7},{4},{5}}
=> ? = 2 + 2
[7,6,8,4,5,3,2,1] => {{1,2,3,6,7,8},{4},{5}}
=> ? = 1 + 2
[8,7,5,4,6,3,2,1] => {{1,8},{2,7},{3,5,6},{4}}
=> ? = 2 + 2
[8,7,4,5,6,3,2,1] => {{1,8},{2,7},{3,4,5,6}}
=> ? = 1 + 2
[8,6,5,4,7,3,2,1] => {{1,8},{2,3,5,6,7},{4}}
=> ? = 1 + 2
[8,5,6,4,7,3,2,1] => {{1,8},{2,5,7},{3,6},{4}}
=> ? = 2 + 2
[8,6,4,5,7,3,2,1] => {{1,8},{2,3,4,5,6,7}}
=> ? = 0 + 2
[8,4,5,6,7,3,2,1] => {{1,8},{2,3,4,5,6,7}}
=> ? = 0 + 2
[6,7,5,4,8,3,2,1] => {{1,3,5,6,8},{2,7},{4}}
=> ? = 1 + 2
[7,5,6,4,8,3,2,1] => {{1,2,5,7,8},{3,6},{4}}
=> ? = 1 + 2
[5,6,7,4,8,3,2,1] => {{1,5,8},{2,3,6,7},{4}}
=> ? = 1 + 2
[6,7,4,5,8,3,2,1] => {{1,3,4,5,6,8},{2,7}}
=> ? = 0 + 2
[7,5,4,6,8,3,2,1] => {{1,2,5,7,8},{3,4,6}}
=> ? = 0 + 2
[5,6,4,7,8,3,2,1] => {{1,5,8},{2,3,4,6,7}}
=> ? = 0 + 2
[6,4,5,7,8,3,2,1] => {{1,3,5,6,8},{2,4,7}}
=> ? = 0 + 2
[5,4,6,7,8,3,2,1] => {{1,5,8},{2,4,7},{3,6}}
=> ? = 1 + 2
[4,5,6,7,8,3,2,1] => {{1,2,4,5,7,8},{3,6}}
=> ? = 0 + 2
[8,7,6,5,3,4,2,1] => {{1,8},{2,7},{3,4,5,6}}
=> ? = 1 + 2
[7,8,6,5,3,4,2,1] => {{1,2,7,8},{3,4,5,6}}
=> ? = 0 + 2
[8,6,7,5,3,4,2,1] => {{1,8},{2,3,4,5,6,7}}
=> ? = 0 + 2
[8,7,5,6,3,4,2,1] => {{1,8},{2,7},{3,5},{4,6}}
=> ? = 1 + 2
[7,8,5,6,3,4,2,1] => {{1,2,7,8},{3,5},{4,6}}
=> ? = 1 + 2
[8,5,6,7,3,4,2,1] => {{1,8},{2,3,4,5,6,7}}
=> ? = 0 + 2
[7,6,5,8,3,4,2,1] => {{1,2,4,6,7,8},{3,5}}
=> ? = 0 + 2
[6,5,7,8,3,4,2,1] => {{1,4,6,8},{2,3,5,7}}
=> ? = 0 + 2
[8,7,6,4,3,5,2,1] => {{1,8},{2,7},{3,5,6},{4}}
=> ? = 2 + 2
[7,8,6,4,3,5,2,1] => {{1,2,7,8},{3,5,6},{4}}
=> ? = 1 + 2
[8,6,7,4,3,5,2,1] => {{1,8},{2,3,5,6,7},{4}}
=> ? = 1 + 2
[6,7,8,4,3,5,2,1] => {{1,3,5,6,8},{2,7},{4}}
=> ? = 1 + 2
[8,7,6,3,4,5,2,1] => {{1,8},{2,7},{3,4,5,6}}
=> ? = 1 + 2
[7,8,6,3,4,5,2,1] => {{1,2,7,8},{3,4,5,6}}
=> ? = 0 + 2
[8,6,7,3,4,5,2,1] => {{1,8},{2,3,4,5,6,7}}
=> ? = 0 + 2
[6,7,8,3,4,5,2,1] => {{1,3,4,5,6,8},{2,7}}
=> ? = 0 + 2
[8,7,5,4,3,6,2,1] => {{1,8},{2,7},{3,5},{4},{6}}
=> ? = 3 + 2
[7,8,5,4,3,6,2,1] => {{1,2,7,8},{3,5},{4},{6}}
=> ? = 2 + 2
[8,7,4,5,3,6,2,1] => {{1,8},{2,7},{3,4,5},{6}}
=> ? = 2 + 2
Description
The number of blocks in the set partition.
The generating function of this statistic yields the famous [[wiki:Stirling numbers of the second kind|Stirling numbers of the second kind]] $S_2(n,k)$ given by the number of [[SetPartitions|set partitions]] of $\{ 1,\ldots,n\}$ into $k$ blocks, see [1].
Matching statistic: St000053
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00087: Permutations āinverse first fundamental transformationā¶ Permutations
Mp00127: Permutations āleft-to-right-maxima to Dyck pathā¶ Dyck paths
St000053: Dyck paths ā¶ ā¤Result quality: 15% āvalues known / values provided: 15%ādistinct values known / distinct values provided: 86%
Mp00127: Permutations āleft-to-right-maxima to Dyck pathā¶ Dyck paths
St000053: Dyck paths ā¶ ā¤Result quality: 15% āvalues known / values provided: 15%ādistinct values known / distinct values provided: 86%
Values
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 2 = 1 + 1
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,4,3,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[3,2,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[3,4,1,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[4,2,3,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[4,3,2,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,2,4,5,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,2,5,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,2,5,4,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,3,4,2,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,3,5,4,2] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,4,2,3,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,4,3,2,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[1,4,3,5,2] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,4,5,2,3] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,5,2,4,3] => [1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,5,3,2,4] => [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,5,3,4,2] => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[1,5,4,3,2] => [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[2,1,3,5,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[2,1,4,3,5] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[2,1,4,5,3] => [2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[2,1,5,3,4] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[2,1,5,4,3] => [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[2,3,1,4,5] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 1 + 1
[2,3,1,5,4] => [3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[2,4,3,1,5] => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> 2 = 1 + 1
[2,4,5,1,3] => [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> 1 = 0 + 1
[2,5,3,4,1] => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> 2 = 1 + 1
[2,5,4,3,1] => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[3,1,2,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 1 + 1
[3,1,2,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[3,2,1,4,5] => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[3,2,1,5,4] => [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[3,2,4,1,5] => [2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[3,2,5,4,1] => [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[3,4,1,2,5] => [3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0]
=> 2 = 1 + 1
[3,4,1,5,2] => [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
=> 1 = 0 + 1
[3,4,5,2,1] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[8,7,6,5,4,3,2,1] => [5,4,6,3,7,2,8,1] => [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 1 + 1
[7,8,6,5,4,3,2,1] => [5,4,6,3,8,1,7,2] => [1,1,1,1,1,0,0,1,0,0,1,1,0,0,0,0]
=> ? = 1 + 1
[8,6,7,5,4,3,2,1] => [5,4,7,2,6,3,8,1] => [1,1,1,1,1,0,0,1,1,0,0,0,0,1,0,0]
=> ? = 1 + 1
[7,6,8,5,4,3,2,1] => [5,4,8,1,7,2,6,3] => [1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[6,7,8,5,4,3,2,1] => [5,4,7,2,8,1,6,3] => [1,1,1,1,1,0,0,1,1,0,0,1,0,0,0,0]
=> ? = 1 + 1
[8,7,5,6,4,3,2,1] => [6,3,5,4,7,2,8,1] => [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> ? = 1 + 1
[7,8,5,6,4,3,2,1] => [6,3,5,4,8,1,7,2] => [1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0]
=> ? = 0 + 1
[8,6,5,7,4,3,2,1] => [7,2,6,3,5,4,8,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 0 + 1
[8,5,6,7,4,3,2,1] => [6,3,7,2,5,4,8,1] => [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> ? = 1 + 1
[6,7,5,8,4,3,2,1] => [7,2,8,1,6,3,5,4] => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 0 + 1
[7,5,6,8,4,3,2,1] => [6,3,8,1,7,2,5,4] => [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[5,6,7,8,4,3,2,1] => [7,2,6,3,8,1,5,4] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> ? = 0 + 1
[8,7,6,4,5,3,2,1] => [4,5,6,3,7,2,8,1] => [1,1,1,1,0,1,0,1,0,0,1,0,0,1,0,0]
=> ? = 3 + 1
[7,8,6,4,5,3,2,1] => [4,5,6,3,8,1,7,2] => [1,1,1,1,0,1,0,1,0,0,1,1,0,0,0,0]
=> ? = 2 + 1
[8,6,7,4,5,3,2,1] => [4,5,7,2,6,3,8,1] => [1,1,1,1,0,1,0,1,1,0,0,0,0,1,0,0]
=> ? = 2 + 1
[7,6,8,4,5,3,2,1] => [4,5,8,1,7,2,6,3] => [1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[8,7,5,4,6,3,2,1] => [4,6,3,5,7,2,8,1] => [1,1,1,1,0,1,1,0,0,0,1,0,0,1,0,0]
=> ? = 2 + 1
[8,7,4,5,6,3,2,1] => [6,3,4,5,7,2,8,1] => [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> ? = 1 + 1
[8,6,5,4,7,3,2,1] => [4,7,2,6,3,5,8,1] => ?
=> ? = 1 + 1
[8,5,6,4,7,3,2,1] => [4,6,3,7,2,5,8,1] => [1,1,1,1,0,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 2 + 1
[8,6,4,5,7,3,2,1] => [7,2,6,3,4,5,8,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 0 + 1
[8,4,5,6,7,3,2,1] => [7,2,4,6,3,5,8,1] => ?
=> ? = 0 + 1
[6,7,5,4,8,3,2,1] => [4,7,2,8,1,6,3,5] => [1,1,1,1,0,1,1,1,0,0,1,0,0,0,0,0]
=> ? = 1 + 1
[7,5,6,4,8,3,2,1] => [4,6,3,8,1,7,2,5] => [1,1,1,1,0,1,1,0,0,1,1,0,0,0,0,0]
=> ? = 1 + 1
[5,6,7,4,8,3,2,1] => [4,7,2,6,3,8,1,5] => [1,1,1,1,0,1,1,1,0,0,0,0,1,0,0,0]
=> ? = 1 + 1
[6,7,4,5,8,3,2,1] => [7,2,8,1,6,3,4,5] => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 0 + 1
[7,5,4,6,8,3,2,1] => [6,3,4,8,1,7,2,5] => [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
=> ? = 0 + 1
[5,6,4,7,8,3,2,1] => [7,2,6,3,4,8,1,5] => ?
=> ? = 0 + 1
[6,4,5,7,8,3,2,1] => [7,2,4,8,1,6,3,5] => ?
=> ? = 0 + 1
[5,4,6,7,8,3,2,1] => [6,3,7,2,4,8,1,5] => [1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0]
=> ? = 1 + 1
[4,5,6,7,8,3,2,1] => [6,3,8,1,4,7,2,5] => [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[8,7,6,5,3,4,2,1] => [6,4,5,3,7,2,8,1] => [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> ? = 1 + 1
[7,8,6,5,3,4,2,1] => [6,4,5,3,8,1,7,2] => [1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0]
=> ? = 0 + 1
[8,6,7,5,3,4,2,1] => [7,2,6,4,5,3,8,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 0 + 1
[8,7,5,6,3,4,2,1] => [5,3,6,4,7,2,8,1] => [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 1 + 1
[7,8,5,6,3,4,2,1] => [5,3,6,4,8,1,7,2] => ?
=> ? = 1 + 1
[8,5,6,7,3,4,2,1] => [7,2,5,3,6,4,8,1] => ?
=> ? = 0 + 1
[7,6,5,8,3,4,2,1] => [5,3,8,1,7,2,6,4] => [1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[6,5,7,8,3,4,2,1] => [7,2,5,3,8,1,6,4] => ?
=> ? = 0 + 1
[8,7,6,4,3,5,2,1] => [4,6,5,3,7,2,8,1] => [1,1,1,1,0,1,1,0,0,0,1,0,0,1,0,0]
=> ? = 2 + 1
[7,8,6,4,3,5,2,1] => [4,6,5,3,8,1,7,2] => [1,1,1,1,0,1,1,0,0,0,1,1,0,0,0,0]
=> ? = 1 + 1
[8,6,7,4,3,5,2,1] => [4,7,2,6,5,3,8,1] => ?
=> ? = 1 + 1
[6,7,8,4,3,5,2,1] => [4,7,2,8,1,6,5,3] => [1,1,1,1,0,1,1,1,0,0,1,0,0,0,0,0]
=> ? = 1 + 1
[8,7,6,3,4,5,2,1] => [6,5,4,3,7,2,8,1] => [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> ? = 1 + 1
[7,8,6,3,4,5,2,1] => [6,5,4,3,8,1,7,2] => [1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0]
=> ? = 0 + 1
[8,6,7,3,4,5,2,1] => [7,2,6,5,4,3,8,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 0 + 1
[6,7,8,3,4,5,2,1] => [7,2,8,1,6,5,4,3] => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 0 + 1
[8,7,5,4,3,6,2,1] => [4,5,3,6,7,2,8,1] => [1,1,1,1,0,1,0,0,1,0,1,0,0,1,0,0]
=> ? = 3 + 1
[7,8,5,4,3,6,2,1] => [4,5,3,6,8,1,7,2] => [1,1,1,1,0,1,0,0,1,0,1,1,0,0,0,0]
=> ? = 2 + 1
[8,7,4,5,3,6,2,1] => [5,3,4,6,7,2,8,1] => ?
=> ? = 2 + 1
Description
The number of valleys of the Dyck path.
Matching statistic: St001068
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00087: Permutations āinverse first fundamental transformationā¶ Permutations
Mp00127: Permutations āleft-to-right-maxima to Dyck pathā¶ Dyck paths
St001068: Dyck paths ā¶ ā¤Result quality: 15% āvalues known / values provided: 15%ādistinct values known / distinct values provided: 86%
Mp00127: Permutations āleft-to-right-maxima to Dyck pathā¶ Dyck paths
St001068: Dyck paths ā¶ ā¤Result quality: 15% āvalues known / values provided: 15%ādistinct values known / distinct values provided: 86%
Values
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 3 = 1 + 2
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4 = 2 + 2
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 3 = 1 + 2
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 3 = 1 + 2
[1,4,3,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 3 = 1 + 2
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 3 = 1 + 2
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2 = 0 + 2
[3,2,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 3 = 1 + 2
[3,4,1,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 2 = 0 + 2
[4,2,3,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[4,3,2,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 2 = 0 + 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 5 = 3 + 2
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 4 = 2 + 2
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 4 = 2 + 2
[1,2,4,5,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 3 = 1 + 2
[1,2,5,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3 = 1 + 2
[1,2,5,4,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> 4 = 2 + 2
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 4 = 2 + 2
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 3 = 1 + 2
[1,3,4,2,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> 3 = 1 + 2
[1,3,5,4,2] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
[1,4,2,3,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> 3 = 1 + 2
[1,4,3,2,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> 4 = 2 + 2
[1,4,3,5,2] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> 3 = 1 + 2
[1,4,5,2,3] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> 3 = 1 + 2
[1,5,2,4,3] => [1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
[1,5,3,2,4] => [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> 3 = 1 + 2
[1,5,3,4,2] => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> 4 = 2 + 2
[1,5,4,3,2] => [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> 3 = 1 + 2
[2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 4 = 2 + 2
[2,1,3,5,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> 3 = 1 + 2
[2,1,4,3,5] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 1 + 2
[2,1,4,5,3] => [2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> 2 = 0 + 2
[2,1,5,3,4] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2 = 0 + 2
[2,1,5,4,3] => [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> 3 = 1 + 2
[2,3,1,4,5] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 3 = 1 + 2
[2,3,1,5,4] => [3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[2,4,3,1,5] => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> 3 = 1 + 2
[2,4,5,1,3] => [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> 2 = 0 + 2
[2,5,3,4,1] => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> 3 = 1 + 2
[2,5,4,3,1] => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> 2 = 0 + 2
[3,1,2,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 3 = 1 + 2
[3,1,2,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[3,2,1,4,5] => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> 4 = 2 + 2
[3,2,1,5,4] => [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> 3 = 1 + 2
[3,2,4,1,5] => [2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
=> 3 = 1 + 2
[3,2,5,4,1] => [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> 3 = 1 + 2
[3,4,1,2,5] => [3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0]
=> 3 = 1 + 2
[3,4,1,5,2] => [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
=> 2 = 0 + 2
[3,4,5,2,1] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 2 = 0 + 2
[8,7,6,5,4,3,2,1] => [5,4,6,3,7,2,8,1] => [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 1 + 2
[7,8,6,5,4,3,2,1] => [5,4,6,3,8,1,7,2] => [1,1,1,1,1,0,0,1,0,0,1,1,0,0,0,0]
=> ? = 1 + 2
[8,6,7,5,4,3,2,1] => [5,4,7,2,6,3,8,1] => [1,1,1,1,1,0,0,1,1,0,0,0,0,1,0,0]
=> ? = 1 + 2
[7,6,8,5,4,3,2,1] => [5,4,8,1,7,2,6,3] => [1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[6,7,8,5,4,3,2,1] => [5,4,7,2,8,1,6,3] => [1,1,1,1,1,0,0,1,1,0,0,1,0,0,0,0]
=> ? = 1 + 2
[8,7,5,6,4,3,2,1] => [6,3,5,4,7,2,8,1] => [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> ? = 1 + 2
[7,8,5,6,4,3,2,1] => [6,3,5,4,8,1,7,2] => [1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0]
=> ? = 0 + 2
[8,6,5,7,4,3,2,1] => [7,2,6,3,5,4,8,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 0 + 2
[8,5,6,7,4,3,2,1] => [6,3,7,2,5,4,8,1] => [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> ? = 1 + 2
[6,7,5,8,4,3,2,1] => [7,2,8,1,6,3,5,4] => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 0 + 2
[7,5,6,8,4,3,2,1] => [6,3,8,1,7,2,5,4] => [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[5,6,7,8,4,3,2,1] => [7,2,6,3,8,1,5,4] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> ? = 0 + 2
[8,7,6,4,5,3,2,1] => [4,5,6,3,7,2,8,1] => [1,1,1,1,0,1,0,1,0,0,1,0,0,1,0,0]
=> ? = 3 + 2
[7,8,6,4,5,3,2,1] => [4,5,6,3,8,1,7,2] => [1,1,1,1,0,1,0,1,0,0,1,1,0,0,0,0]
=> ? = 2 + 2
[8,6,7,4,5,3,2,1] => [4,5,7,2,6,3,8,1] => [1,1,1,1,0,1,0,1,1,0,0,0,0,1,0,0]
=> ? = 2 + 2
[7,6,8,4,5,3,2,1] => [4,5,8,1,7,2,6,3] => [1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 2
[8,7,5,4,6,3,2,1] => [4,6,3,5,7,2,8,1] => [1,1,1,1,0,1,1,0,0,0,1,0,0,1,0,0]
=> ? = 2 + 2
[8,7,4,5,6,3,2,1] => [6,3,4,5,7,2,8,1] => [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> ? = 1 + 2
[8,6,5,4,7,3,2,1] => [4,7,2,6,3,5,8,1] => ?
=> ? = 1 + 2
[8,5,6,4,7,3,2,1] => [4,6,3,7,2,5,8,1] => [1,1,1,1,0,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 2 + 2
[8,6,4,5,7,3,2,1] => [7,2,6,3,4,5,8,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 0 + 2
[8,4,5,6,7,3,2,1] => [7,2,4,6,3,5,8,1] => ?
=> ? = 0 + 2
[6,7,5,4,8,3,2,1] => [4,7,2,8,1,6,3,5] => [1,1,1,1,0,1,1,1,0,0,1,0,0,0,0,0]
=> ? = 1 + 2
[7,5,6,4,8,3,2,1] => [4,6,3,8,1,7,2,5] => [1,1,1,1,0,1,1,0,0,1,1,0,0,0,0,0]
=> ? = 1 + 2
[5,6,7,4,8,3,2,1] => [4,7,2,6,3,8,1,5] => [1,1,1,1,0,1,1,1,0,0,0,0,1,0,0,0]
=> ? = 1 + 2
[6,7,4,5,8,3,2,1] => [7,2,8,1,6,3,4,5] => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 0 + 2
[7,5,4,6,8,3,2,1] => [6,3,4,8,1,7,2,5] => [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
=> ? = 0 + 2
[5,6,4,7,8,3,2,1] => [7,2,6,3,4,8,1,5] => ?
=> ? = 0 + 2
[6,4,5,7,8,3,2,1] => [7,2,4,8,1,6,3,5] => ?
=> ? = 0 + 2
[5,4,6,7,8,3,2,1] => [6,3,7,2,4,8,1,5] => [1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0]
=> ? = 1 + 2
[4,5,6,7,8,3,2,1] => [6,3,8,1,4,7,2,5] => [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[8,7,6,5,3,4,2,1] => [6,4,5,3,7,2,8,1] => [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> ? = 1 + 2
[7,8,6,5,3,4,2,1] => [6,4,5,3,8,1,7,2] => [1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0]
=> ? = 0 + 2
[8,6,7,5,3,4,2,1] => [7,2,6,4,5,3,8,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 0 + 2
[8,7,5,6,3,4,2,1] => [5,3,6,4,7,2,8,1] => [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 1 + 2
[7,8,5,6,3,4,2,1] => [5,3,6,4,8,1,7,2] => ?
=> ? = 1 + 2
[8,5,6,7,3,4,2,1] => [7,2,5,3,6,4,8,1] => ?
=> ? = 0 + 2
[7,6,5,8,3,4,2,1] => [5,3,8,1,7,2,6,4] => [1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[6,5,7,8,3,4,2,1] => [7,2,5,3,8,1,6,4] => ?
=> ? = 0 + 2
[8,7,6,4,3,5,2,1] => [4,6,5,3,7,2,8,1] => [1,1,1,1,0,1,1,0,0,0,1,0,0,1,0,0]
=> ? = 2 + 2
[7,8,6,4,3,5,2,1] => [4,6,5,3,8,1,7,2] => [1,1,1,1,0,1,1,0,0,0,1,1,0,0,0,0]
=> ? = 1 + 2
[8,6,7,4,3,5,2,1] => [4,7,2,6,5,3,8,1] => ?
=> ? = 1 + 2
[6,7,8,4,3,5,2,1] => [4,7,2,8,1,6,5,3] => [1,1,1,1,0,1,1,1,0,0,1,0,0,0,0,0]
=> ? = 1 + 2
[8,7,6,3,4,5,2,1] => [6,5,4,3,7,2,8,1] => [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> ? = 1 + 2
[7,8,6,3,4,5,2,1] => [6,5,4,3,8,1,7,2] => [1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0]
=> ? = 0 + 2
[8,6,7,3,4,5,2,1] => [7,2,6,5,4,3,8,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 0 + 2
[6,7,8,3,4,5,2,1] => [7,2,8,1,6,5,4,3] => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 0 + 2
[8,7,5,4,3,6,2,1] => [4,5,3,6,7,2,8,1] => [1,1,1,1,0,1,0,0,1,0,1,0,0,1,0,0]
=> ? = 3 + 2
[7,8,5,4,3,6,2,1] => [4,5,3,6,8,1,7,2] => [1,1,1,1,0,1,0,0,1,0,1,1,0,0,0,0]
=> ? = 2 + 2
[8,7,4,5,3,6,2,1] => [5,3,4,6,7,2,8,1] => ?
=> ? = 2 + 2
Description
Number of torsionless simple modules in the corresponding Nakayama algebra.
Matching statistic: St000024
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00087: Permutations āinverse first fundamental transformationā¶ Permutations
Mp00127: Permutations āleft-to-right-maxima to Dyck pathā¶ Dyck paths
Mp00327: Dyck paths āinverse Kreweras complementā¶ Dyck paths
St000024: Dyck paths ā¶ ā¤Result quality: 15% āvalues known / values provided: 15%ādistinct values known / distinct values provided: 86%
Mp00127: Permutations āleft-to-right-maxima to Dyck pathā¶ Dyck paths
Mp00327: Dyck paths āinverse Kreweras complementā¶ Dyck paths
St000024: Dyck paths ā¶ ā¤Result quality: 15% āvalues known / values provided: 15%ādistinct values known / distinct values provided: 86%
Values
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 2 = 1 + 1
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 3 = 2 + 1
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,4,3,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[3,2,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[3,4,1,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1 = 0 + 1
[4,2,3,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[4,3,2,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4 = 3 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 3 = 2 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 3 = 2 + 1
[1,2,4,5,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,2,5,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,2,5,4,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 2 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3 = 2 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,3,4,2,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,3,5,4,2] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,4,2,3,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,4,3,2,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,4,3,5,2] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,4,5,2,3] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,5,2,4,3] => [1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,5,3,2,4] => [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,5,3,4,2] => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 3 = 2 + 1
[1,5,4,3,2] => [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2 = 1 + 1
[2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 3 = 2 + 1
[2,1,3,5,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[2,1,4,3,5] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[2,1,4,5,3] => [2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[2,1,5,3,4] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[2,1,5,4,3] => [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[2,3,1,4,5] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[2,3,1,5,4] => [3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[2,4,3,1,5] => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[2,4,5,1,3] => [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1 = 0 + 1
[2,5,3,4,1] => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[2,5,4,3,1] => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1 = 0 + 1
[3,1,2,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[3,1,2,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[3,2,1,4,5] => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[3,2,1,5,4] => [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[3,2,4,1,5] => [2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[3,2,5,4,1] => [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[3,4,1,2,5] => [3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[3,4,1,5,2] => [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[3,4,5,2,1] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1 = 0 + 1
[8,7,6,5,4,3,2,1] => [5,4,6,3,7,2,8,1] => [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 1 + 1
[7,8,6,5,4,3,2,1] => [5,4,6,3,8,1,7,2] => [1,1,1,1,1,0,0,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> ? = 1 + 1
[8,6,7,5,4,3,2,1] => [5,4,7,2,6,3,8,1] => [1,1,1,1,1,0,0,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> ? = 1 + 1
[7,6,8,5,4,3,2,1] => [5,4,8,1,7,2,6,3] => [1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 0 + 1
[6,7,8,5,4,3,2,1] => [5,4,7,2,8,1,6,3] => [1,1,1,1,1,0,0,1,1,0,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> ? = 1 + 1
[8,7,5,6,4,3,2,1] => [6,3,5,4,7,2,8,1] => [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 1 + 1
[7,8,5,6,4,3,2,1] => [6,3,5,4,8,1,7,2] => [1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 0 + 1
[8,6,5,7,4,3,2,1] => [7,2,6,3,5,4,8,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 0 + 1
[8,5,6,7,4,3,2,1] => [6,3,7,2,5,4,8,1] => [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> ? = 1 + 1
[6,7,5,8,4,3,2,1] => [7,2,8,1,6,3,5,4] => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> ? = 0 + 1
[7,5,6,8,4,3,2,1] => [6,3,8,1,7,2,5,4] => [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> ? = 0 + 1
[5,6,7,8,4,3,2,1] => [7,2,6,3,8,1,5,4] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 0 + 1
[8,7,6,4,5,3,2,1] => [4,5,6,3,7,2,8,1] => [1,1,1,1,0,1,0,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> ? = 3 + 1
[7,8,6,4,5,3,2,1] => [4,5,6,3,8,1,7,2] => [1,1,1,1,0,1,0,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> ? = 2 + 1
[8,6,7,4,5,3,2,1] => [4,5,7,2,6,3,8,1] => [1,1,1,1,0,1,0,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> ? = 2 + 1
[7,6,8,4,5,3,2,1] => [4,5,8,1,7,2,6,3] => [1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 1
[8,7,5,4,6,3,2,1] => [4,6,3,5,7,2,8,1] => [1,1,1,1,0,1,1,0,0,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 2 + 1
[8,7,4,5,6,3,2,1] => [6,3,4,5,7,2,8,1] => [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 1 + 1
[8,6,5,4,7,3,2,1] => [4,7,2,6,3,5,8,1] => ?
=> ?
=> ? = 1 + 1
[8,5,6,4,7,3,2,1] => [4,6,3,7,2,5,8,1] => [1,1,1,1,0,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> ? = 2 + 1
[8,6,4,5,7,3,2,1] => [7,2,6,3,4,5,8,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 0 + 1
[8,4,5,6,7,3,2,1] => [7,2,4,6,3,5,8,1] => ?
=> ?
=> ? = 0 + 1
[6,7,5,4,8,3,2,1] => [4,7,2,8,1,6,3,5] => [1,1,1,1,0,1,1,1,0,0,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 1 + 1
[7,5,6,4,8,3,2,1] => [4,6,3,8,1,7,2,5] => [1,1,1,1,0,1,1,0,0,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> ? = 1 + 1
[5,6,7,4,8,3,2,1] => [4,7,2,6,3,8,1,5] => [1,1,1,1,0,1,1,1,0,0,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> ? = 1 + 1
[6,7,4,5,8,3,2,1] => [7,2,8,1,6,3,4,5] => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> ? = 0 + 1
[7,5,4,6,8,3,2,1] => [6,3,4,8,1,7,2,5] => [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 0 + 1
[5,6,4,7,8,3,2,1] => [7,2,6,3,4,8,1,5] => ?
=> ?
=> ? = 0 + 1
[6,4,5,7,8,3,2,1] => [7,2,4,8,1,6,3,5] => ?
=> ?
=> ? = 0 + 1
[5,4,6,7,8,3,2,1] => [6,3,7,2,4,8,1,5] => [1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> ? = 1 + 1
[4,5,6,7,8,3,2,1] => [6,3,8,1,4,7,2,5] => [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> ? = 0 + 1
[8,7,6,5,3,4,2,1] => [6,4,5,3,7,2,8,1] => [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 1 + 1
[7,8,6,5,3,4,2,1] => [6,4,5,3,8,1,7,2] => [1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 0 + 1
[8,6,7,5,3,4,2,1] => [7,2,6,4,5,3,8,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 0 + 1
[8,7,5,6,3,4,2,1] => [5,3,6,4,7,2,8,1] => [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 1 + 1
[7,8,5,6,3,4,2,1] => [5,3,6,4,8,1,7,2] => ?
=> ?
=> ? = 1 + 1
[8,5,6,7,3,4,2,1] => [7,2,5,3,6,4,8,1] => ?
=> ?
=> ? = 0 + 1
[7,6,5,8,3,4,2,1] => [5,3,8,1,7,2,6,4] => [1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 0 + 1
[6,5,7,8,3,4,2,1] => [7,2,5,3,8,1,6,4] => ?
=> ?
=> ? = 0 + 1
[8,7,6,4,3,5,2,1] => [4,6,5,3,7,2,8,1] => [1,1,1,1,0,1,1,0,0,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 2 + 1
[7,8,6,4,3,5,2,1] => [4,6,5,3,8,1,7,2] => [1,1,1,1,0,1,1,0,0,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> ? = 1 + 1
[8,6,7,4,3,5,2,1] => [4,7,2,6,5,3,8,1] => ?
=> ?
=> ? = 1 + 1
[6,7,8,4,3,5,2,1] => [4,7,2,8,1,6,5,3] => [1,1,1,1,0,1,1,1,0,0,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 1 + 1
[8,7,6,3,4,5,2,1] => [6,5,4,3,7,2,8,1] => [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 1 + 1
[7,8,6,3,4,5,2,1] => [6,5,4,3,8,1,7,2] => [1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 0 + 1
[8,6,7,3,4,5,2,1] => [7,2,6,5,4,3,8,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 0 + 1
[6,7,8,3,4,5,2,1] => [7,2,8,1,6,5,4,3] => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> ? = 0 + 1
[8,7,5,4,3,6,2,1] => [4,5,3,6,7,2,8,1] => [1,1,1,1,0,1,0,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,1,0,0,1,0]
=> ? = 3 + 1
[7,8,5,4,3,6,2,1] => [4,5,3,6,8,1,7,2] => [1,1,1,1,0,1,0,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0,1,0,1,0]
=> ? = 2 + 1
[8,7,4,5,3,6,2,1] => [5,3,4,6,7,2,8,1] => ?
=> ?
=> ? = 2 + 1
Description
The number of double up and double down steps of a Dyck path.
In other words, this is the number of double rises (and, equivalently, the number of double falls) of a Dyck path.
Matching statistic: St001197
Mp00151: Permutations āto cycle typeā¶ Set partitions
Mp00128: Set partitions āto compositionā¶ Integer compositions
Mp00231: Integer compositions ābounce pathā¶ Dyck paths
St001197: Dyck paths ā¶ ā¤Result quality: 15% āvalues known / values provided: 15%ādistinct values known / distinct values provided: 86%
Mp00128: Set partitions āto compositionā¶ Integer compositions
Mp00231: Integer compositions ābounce pathā¶ Dyck paths
St001197: Dyck paths ā¶ ā¤Result quality: 15% āvalues known / values provided: 15%ādistinct values known / distinct values provided: 86%
Values
[1,2,3] => {{1},{2},{3}}
=> [1,1,1] => [1,0,1,0,1,0]
=> 2 = 1 + 1
[1,2,3,4] => {{1},{2},{3},{4}}
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[1,2,4,3] => {{1},{2},{3,4}}
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,3,2,4] => {{1},{2,3},{4}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,4,3,2] => {{1},{2,4},{3}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[2,1,3,4] => {{1,2},{3},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[2,1,4,3] => {{1,2},{3,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[3,2,1,4] => {{1,3},{2},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[3,4,1,2] => {{1,3},{2,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[4,2,3,1] => {{1,4},{2},{3}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[4,3,2,1] => {{1,4},{2,3}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,2,4,5,3] => {{1},{2},{3,4,5}}
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,2,5,3,4] => {{1},{2},{3,4,5}}
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,3,2,5,4] => {{1},{2,3},{4,5}}
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,3,4,2,5] => {{1},{2,3,4},{5}}
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,3,5,4,2] => {{1},{2,3,5},{4}}
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,4,2,3,5] => {{1},{2,3,4},{5}}
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,4,3,5,2] => {{1},{2,4,5},{3}}
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,4,5,2,3] => {{1},{2,4},{3,5}}
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,5,2,4,3] => {{1},{2,3,5},{4}}
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,5,3,2,4] => {{1},{2,4,5},{3}}
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,5,3,4,2] => {{1},{2,5},{3},{4}}
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,5,4,3,2] => {{1},{2,5},{3,4}}
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[2,1,3,5,4] => {{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[2,1,4,3,5] => {{1,2},{3,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[2,1,4,5,3] => {{1,2},{3,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[2,1,5,3,4] => {{1,2},{3,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[2,1,5,4,3] => {{1,2},{3,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[2,3,1,4,5] => {{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 1 + 1
[2,3,1,5,4] => {{1,2,3},{4,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[2,4,3,1,5] => {{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 1 + 1
[2,4,5,1,3] => {{1,2,4},{3,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[2,5,3,4,1] => {{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 1 + 1
[2,5,4,3,1] => {{1,2,5},{3,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[3,1,2,4,5] => {{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 1 + 1
[3,1,2,5,4] => {{1,2,3},{4,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[3,2,1,4,5] => {{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[3,2,1,5,4] => {{1,3},{2},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[3,2,4,1,5] => {{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 1 + 1
[3,2,5,4,1] => {{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 1 + 1
[3,4,1,2,5] => {{1,3},{2,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[3,4,1,5,2] => {{1,3},{2,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[3,4,5,2,1] => {{1,3,5},{2,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[8,7,6,5,4,3,2,1] => {{1,8},{2,7},{3,6},{4,5}}
=> [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 1 + 1
[7,8,6,5,4,3,2,1] => {{1,2,7,8},{3,6},{4,5}}
=> ? => ?
=> ? = 1 + 1
[8,6,7,5,4,3,2,1] => {{1,8},{2,3,6,7},{4,5}}
=> [2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 1 + 1
[7,6,8,5,4,3,2,1] => {{1,2,3,6,7,8},{4,5}}
=> [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 0 + 1
[6,7,8,5,4,3,2,1] => {{1,3,6,8},{2,7},{4,5}}
=> [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 1 + 1
[8,7,5,6,4,3,2,1] => {{1,8},{2,7},{3,4,5,6}}
=> ? => ?
=> ? = 1 + 1
[7,8,5,6,4,3,2,1] => {{1,2,7,8},{3,4,5,6}}
=> [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0 + 1
[8,6,5,7,4,3,2,1] => {{1,8},{2,3,4,5,6,7}}
=> [2,6] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[8,5,6,7,4,3,2,1] => {{1,8},{2,4,5,7},{3,6}}
=> ? => ?
=> ? = 1 + 1
[6,7,5,8,4,3,2,1] => {{1,3,4,5,6,8},{2,7}}
=> [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 0 + 1
[7,5,6,8,4,3,2,1] => {{1,2,4,5,7,8},{3,6}}
=> [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 0 + 1
[5,6,7,8,4,3,2,1] => {{1,4,5,8},{2,3,6,7}}
=> [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0 + 1
[8,7,6,4,5,3,2,1] => {{1,8},{2,7},{3,6},{4},{5}}
=> ? => ?
=> ? = 3 + 1
[7,8,6,4,5,3,2,1] => {{1,2,7,8},{3,6},{4},{5}}
=> [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 2 + 1
[8,6,7,4,5,3,2,1] => {{1,8},{2,3,6,7},{4},{5}}
=> [2,4,1,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 2 + 1
[7,6,8,4,5,3,2,1] => {{1,2,3,6,7,8},{4},{5}}
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1 + 1
[8,7,5,4,6,3,2,1] => {{1,8},{2,7},{3,5,6},{4}}
=> [2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 2 + 1
[8,7,4,5,6,3,2,1] => {{1,8},{2,7},{3,4,5,6}}
=> ? => ?
=> ? = 1 + 1
[8,6,5,4,7,3,2,1] => {{1,8},{2,3,5,6,7},{4}}
=> [2,5,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1 + 1
[8,5,6,4,7,3,2,1] => {{1,8},{2,5,7},{3,6},{4}}
=> [2,3,2,1] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 2 + 1
[8,6,4,5,7,3,2,1] => {{1,8},{2,3,4,5,6,7}}
=> [2,6] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[8,4,5,6,7,3,2,1] => {{1,8},{2,3,4,5,6,7}}
=> [2,6] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[6,7,5,4,8,3,2,1] => {{1,3,5,6,8},{2,7},{4}}
=> [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 1 + 1
[7,5,6,4,8,3,2,1] => {{1,2,5,7,8},{3,6},{4}}
=> [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 1 + 1
[5,6,7,4,8,3,2,1] => {{1,5,8},{2,3,6,7},{4}}
=> ? => ?
=> ? = 1 + 1
[6,7,4,5,8,3,2,1] => {{1,3,4,5,6,8},{2,7}}
=> [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 0 + 1
[7,5,4,6,8,3,2,1] => {{1,2,5,7,8},{3,4,6}}
=> [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 0 + 1
[5,6,4,7,8,3,2,1] => {{1,5,8},{2,3,4,6,7}}
=> [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 1
[6,4,5,7,8,3,2,1] => {{1,3,5,6,8},{2,4,7}}
=> [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 0 + 1
[5,4,6,7,8,3,2,1] => {{1,5,8},{2,4,7},{3,6}}
=> ? => ?
=> ? = 1 + 1
[4,5,6,7,8,3,2,1] => {{1,2,4,5,7,8},{3,6}}
=> [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 0 + 1
[8,7,6,5,3,4,2,1] => {{1,8},{2,7},{3,4,5,6}}
=> ? => ?
=> ? = 1 + 1
[7,8,6,5,3,4,2,1] => {{1,2,7,8},{3,4,5,6}}
=> [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0 + 1
[8,6,7,5,3,4,2,1] => {{1,8},{2,3,4,5,6,7}}
=> [2,6] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[8,7,5,6,3,4,2,1] => {{1,8},{2,7},{3,5},{4,6}}
=> [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 1 + 1
[7,8,5,6,3,4,2,1] => {{1,2,7,8},{3,5},{4,6}}
=> [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 1 + 1
[8,5,6,7,3,4,2,1] => {{1,8},{2,3,4,5,6,7}}
=> [2,6] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[7,6,5,8,3,4,2,1] => {{1,2,4,6,7,8},{3,5}}
=> [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 0 + 1
[6,5,7,8,3,4,2,1] => {{1,4,6,8},{2,3,5,7}}
=> [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0 + 1
[8,7,6,4,3,5,2,1] => {{1,8},{2,7},{3,5,6},{4}}
=> [2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 2 + 1
[7,8,6,4,3,5,2,1] => {{1,2,7,8},{3,5,6},{4}}
=> ? => ?
=> ? = 1 + 1
[8,6,7,4,3,5,2,1] => {{1,8},{2,3,5,6,7},{4}}
=> [2,5,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1 + 1
[6,7,8,4,3,5,2,1] => {{1,3,5,6,8},{2,7},{4}}
=> [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 1 + 1
[8,7,6,3,4,5,2,1] => {{1,8},{2,7},{3,4,5,6}}
=> ? => ?
=> ? = 1 + 1
[7,8,6,3,4,5,2,1] => {{1,2,7,8},{3,4,5,6}}
=> [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0 + 1
[8,6,7,3,4,5,2,1] => {{1,8},{2,3,4,5,6,7}}
=> [2,6] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[6,7,8,3,4,5,2,1] => {{1,3,4,5,6,8},{2,7}}
=> [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 0 + 1
[8,7,5,4,3,6,2,1] => {{1,8},{2,7},{3,5},{4},{6}}
=> ? => ?
=> ? = 3 + 1
[7,8,5,4,3,6,2,1] => {{1,2,7,8},{3,5},{4},{6}}
=> [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 2 + 1
[8,7,4,5,3,6,2,1] => {{1,8},{2,7},{3,4,5},{6}}
=> [2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 2 + 1
Description
The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St001199
Mp00151: Permutations āto cycle typeā¶ Set partitions
Mp00128: Set partitions āto compositionā¶ Integer compositions
Mp00231: Integer compositions ābounce pathā¶ Dyck paths
St001199: Dyck paths ā¶ ā¤Result quality: 15% āvalues known / values provided: 15%ādistinct values known / distinct values provided: 86%
Mp00128: Set partitions āto compositionā¶ Integer compositions
Mp00231: Integer compositions ābounce pathā¶ Dyck paths
St001199: Dyck paths ā¶ ā¤Result quality: 15% āvalues known / values provided: 15%ādistinct values known / distinct values provided: 86%
Values
[1,2,3] => {{1},{2},{3}}
=> [1,1,1] => [1,0,1,0,1,0]
=> 2 = 1 + 1
[1,2,3,4] => {{1},{2},{3},{4}}
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[1,2,4,3] => {{1},{2},{3,4}}
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,3,2,4] => {{1},{2,3},{4}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,4,3,2] => {{1},{2,4},{3}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[2,1,3,4] => {{1,2},{3},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[2,1,4,3] => {{1,2},{3,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[3,2,1,4] => {{1,3},{2},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[3,4,1,2] => {{1,3},{2,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[4,2,3,1] => {{1,4},{2},{3}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[4,3,2,1] => {{1,4},{2,3}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,2,4,5,3] => {{1},{2},{3,4,5}}
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,2,5,3,4] => {{1},{2},{3,4,5}}
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,3,2,5,4] => {{1},{2,3},{4,5}}
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,3,4,2,5] => {{1},{2,3,4},{5}}
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,3,5,4,2] => {{1},{2,3,5},{4}}
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,4,2,3,5] => {{1},{2,3,4},{5}}
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,4,3,5,2] => {{1},{2,4,5},{3}}
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,4,5,2,3] => {{1},{2,4},{3,5}}
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,5,2,4,3] => {{1},{2,3,5},{4}}
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,5,3,2,4] => {{1},{2,4,5},{3}}
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,5,3,4,2] => {{1},{2,5},{3},{4}}
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,5,4,3,2] => {{1},{2,5},{3,4}}
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[2,1,3,5,4] => {{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[2,1,4,3,5] => {{1,2},{3,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[2,1,4,5,3] => {{1,2},{3,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[2,1,5,3,4] => {{1,2},{3,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[2,1,5,4,3] => {{1,2},{3,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[2,3,1,4,5] => {{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 1 + 1
[2,3,1,5,4] => {{1,2,3},{4,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[2,4,3,1,5] => {{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 1 + 1
[2,4,5,1,3] => {{1,2,4},{3,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[2,5,3,4,1] => {{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 1 + 1
[2,5,4,3,1] => {{1,2,5},{3,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[3,1,2,4,5] => {{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 1 + 1
[3,1,2,5,4] => {{1,2,3},{4,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[3,2,1,4,5] => {{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[3,2,1,5,4] => {{1,3},{2},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[3,2,4,1,5] => {{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 1 + 1
[3,2,5,4,1] => {{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 1 + 1
[3,4,1,2,5] => {{1,3},{2,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[3,4,1,5,2] => {{1,3},{2,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[3,4,5,2,1] => {{1,3,5},{2,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[8,7,6,5,4,3,2,1] => {{1,8},{2,7},{3,6},{4,5}}
=> [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 1 + 1
[7,8,6,5,4,3,2,1] => {{1,2,7,8},{3,6},{4,5}}
=> ? => ?
=> ? = 1 + 1
[8,6,7,5,4,3,2,1] => {{1,8},{2,3,6,7},{4,5}}
=> [2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 1 + 1
[7,6,8,5,4,3,2,1] => {{1,2,3,6,7,8},{4,5}}
=> [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 0 + 1
[6,7,8,5,4,3,2,1] => {{1,3,6,8},{2,7},{4,5}}
=> [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 1 + 1
[8,7,5,6,4,3,2,1] => {{1,8},{2,7},{3,4,5,6}}
=> ? => ?
=> ? = 1 + 1
[7,8,5,6,4,3,2,1] => {{1,2,7,8},{3,4,5,6}}
=> [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0 + 1
[8,6,5,7,4,3,2,1] => {{1,8},{2,3,4,5,6,7}}
=> [2,6] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[8,5,6,7,4,3,2,1] => {{1,8},{2,4,5,7},{3,6}}
=> ? => ?
=> ? = 1 + 1
[6,7,5,8,4,3,2,1] => {{1,3,4,5,6,8},{2,7}}
=> [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 0 + 1
[7,5,6,8,4,3,2,1] => {{1,2,4,5,7,8},{3,6}}
=> [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 0 + 1
[5,6,7,8,4,3,2,1] => {{1,4,5,8},{2,3,6,7}}
=> [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0 + 1
[8,7,6,4,5,3,2,1] => {{1,8},{2,7},{3,6},{4},{5}}
=> ? => ?
=> ? = 3 + 1
[7,8,6,4,5,3,2,1] => {{1,2,7,8},{3,6},{4},{5}}
=> [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 2 + 1
[8,6,7,4,5,3,2,1] => {{1,8},{2,3,6,7},{4},{5}}
=> [2,4,1,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 2 + 1
[7,6,8,4,5,3,2,1] => {{1,2,3,6,7,8},{4},{5}}
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1 + 1
[8,7,5,4,6,3,2,1] => {{1,8},{2,7},{3,5,6},{4}}
=> [2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 2 + 1
[8,7,4,5,6,3,2,1] => {{1,8},{2,7},{3,4,5,6}}
=> ? => ?
=> ? = 1 + 1
[8,6,5,4,7,3,2,1] => {{1,8},{2,3,5,6,7},{4}}
=> [2,5,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1 + 1
[8,5,6,4,7,3,2,1] => {{1,8},{2,5,7},{3,6},{4}}
=> [2,3,2,1] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 2 + 1
[8,6,4,5,7,3,2,1] => {{1,8},{2,3,4,5,6,7}}
=> [2,6] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[8,4,5,6,7,3,2,1] => {{1,8},{2,3,4,5,6,7}}
=> [2,6] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[6,7,5,4,8,3,2,1] => {{1,3,5,6,8},{2,7},{4}}
=> [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 1 + 1
[7,5,6,4,8,3,2,1] => {{1,2,5,7,8},{3,6},{4}}
=> [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 1 + 1
[5,6,7,4,8,3,2,1] => {{1,5,8},{2,3,6,7},{4}}
=> ? => ?
=> ? = 1 + 1
[6,7,4,5,8,3,2,1] => {{1,3,4,5,6,8},{2,7}}
=> [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 0 + 1
[7,5,4,6,8,3,2,1] => {{1,2,5,7,8},{3,4,6}}
=> [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 0 + 1
[5,6,4,7,8,3,2,1] => {{1,5,8},{2,3,4,6,7}}
=> [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 1
[6,4,5,7,8,3,2,1] => {{1,3,5,6,8},{2,4,7}}
=> [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 0 + 1
[5,4,6,7,8,3,2,1] => {{1,5,8},{2,4,7},{3,6}}
=> ? => ?
=> ? = 1 + 1
[4,5,6,7,8,3,2,1] => {{1,2,4,5,7,8},{3,6}}
=> [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 0 + 1
[8,7,6,5,3,4,2,1] => {{1,8},{2,7},{3,4,5,6}}
=> ? => ?
=> ? = 1 + 1
[7,8,6,5,3,4,2,1] => {{1,2,7,8},{3,4,5,6}}
=> [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0 + 1
[8,6,7,5,3,4,2,1] => {{1,8},{2,3,4,5,6,7}}
=> [2,6] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[8,7,5,6,3,4,2,1] => {{1,8},{2,7},{3,5},{4,6}}
=> [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 1 + 1
[7,8,5,6,3,4,2,1] => {{1,2,7,8},{3,5},{4,6}}
=> [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 1 + 1
[8,5,6,7,3,4,2,1] => {{1,8},{2,3,4,5,6,7}}
=> [2,6] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[7,6,5,8,3,4,2,1] => {{1,2,4,6,7,8},{3,5}}
=> [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 0 + 1
[6,5,7,8,3,4,2,1] => {{1,4,6,8},{2,3,5,7}}
=> [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0 + 1
[8,7,6,4,3,5,2,1] => {{1,8},{2,7},{3,5,6},{4}}
=> [2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 2 + 1
[7,8,6,4,3,5,2,1] => {{1,2,7,8},{3,5,6},{4}}
=> ? => ?
=> ? = 1 + 1
[8,6,7,4,3,5,2,1] => {{1,8},{2,3,5,6,7},{4}}
=> [2,5,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1 + 1
[6,7,8,4,3,5,2,1] => {{1,3,5,6,8},{2,7},{4}}
=> [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 1 + 1
[8,7,6,3,4,5,2,1] => {{1,8},{2,7},{3,4,5,6}}
=> ? => ?
=> ? = 1 + 1
[7,8,6,3,4,5,2,1] => {{1,2,7,8},{3,4,5,6}}
=> [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0 + 1
[8,6,7,3,4,5,2,1] => {{1,8},{2,3,4,5,6,7}}
=> [2,6] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[6,7,8,3,4,5,2,1] => {{1,3,4,5,6,8},{2,7}}
=> [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 0 + 1
[8,7,5,4,3,6,2,1] => {{1,8},{2,7},{3,5},{4},{6}}
=> ? => ?
=> ? = 3 + 1
[7,8,5,4,3,6,2,1] => {{1,2,7,8},{3,5},{4},{6}}
=> [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 2 + 1
[8,7,4,5,3,6,2,1] => {{1,8},{2,7},{3,4,5},{6}}
=> [2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 2 + 1
Description
The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
The following 69 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St001494The Alon-Tarsi number of a graph. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000536The pathwidth of a graph. St000172The Grundy number of a graph. St001029The size of the core of a graph. St001580The acyclic chromatic number of a graph. St001670The connected partition number of a graph. St001331The size of the minimal feedback vertex set. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001963The tree-depth of a graph. St000052The number of valleys of a Dyck path not on the x-axis. St000292The number of ascents of a binary word. St000157The number of descents of a standard tableau. St000164The number of short pairs. St000291The number of descents of a binary word. St000390The number of runs of ones in a binary word. St000702The number of weak deficiencies of a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000007The number of saliances of the permutation. St000314The number of left-to-right-maxima of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St000015The number of peaks of a Dyck path. St000542The number of left-to-right-minima of a permutation. St000991The number of right-to-left minima of a permutation. St000021The number of descents of a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St000155The number of exceedances (also excedences) of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000354The number of recoils of a permutation. St000619The number of cyclic descents of a permutation. St000831The number of indices that are either descents or recoils. St001061The number of indices that are both descents and recoils of a permutation. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001489The maximum of the number of descents and the number of inverse descents. St000061The number of nodes on the left branch of a binary tree. St000062The length of the longest increasing subsequence of the permutation. St000084The number of subtrees. St000213The number of weak exceedances (also weak excedences) of a permutation. St000239The number of small weak excedances. St000325The width of the tree associated to a permutation. St000443The number of long tunnels of a Dyck path. St000470The number of runs in a permutation. St000822The Hadwiger number of the graph. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{nā1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001180Number of indecomposable injective modules with projective dimension at most 1. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001812The biclique partition number of a graph. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St001330The hat guessing number of a graph. St001960The number of descents of a permutation minus one if its first entry is not one. St001769The reflection length of a signed permutation.
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