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Your data matches 65 different statistics following compositions of up to 3 maps.
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Matching statistic: St000211
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(load all 7 compositions to match this statistic)
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000211: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00151: Permutations —to cycle type⟶ Set partitions
St000211: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => {{1}}
=> 0
[1,2] => [1,2] => {{1},{2}}
=> 0
[2,1] => [2,1] => {{1,2}}
=> 1
[1,2,3] => [1,2,3] => {{1},{2},{3}}
=> 0
[1,3,2] => [1,3,2] => {{1},{2,3}}
=> 1
[2,1,3] => [2,1,3] => {{1,2},{3}}
=> 1
[2,3,1] => [3,1,2] => {{1,2,3}}
=> 2
[3,1,2] => [3,2,1] => {{1,3},{2}}
=> 1
[3,2,1] => [2,3,1] => {{1,2,3}}
=> 2
[1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[1,2,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 1
[1,3,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 1
[1,3,4,2] => [1,4,2,3] => {{1},{2,3,4}}
=> 2
[1,4,2,3] => [1,4,3,2] => {{1},{2,4},{3}}
=> 1
[1,4,3,2] => [1,3,4,2] => {{1},{2,3,4}}
=> 2
[2,1,3,4] => [2,1,3,4] => {{1,2},{3},{4}}
=> 1
[2,1,4,3] => [2,1,4,3] => {{1,2},{3,4}}
=> 2
[2,3,1,4] => [3,1,2,4] => {{1,2,3},{4}}
=> 2
[2,3,4,1] => [4,1,2,3] => {{1,2,3,4}}
=> 3
[3,1,2,4] => [3,2,1,4] => {{1,3},{2},{4}}
=> 1
[3,2,1,4] => [2,3,1,4] => {{1,2,3},{4}}
=> 2
[3,2,4,1] => [2,4,1,3] => {{1,2,3,4}}
=> 3
[3,4,1,2] => [3,1,4,2] => {{1,2,3,4}}
=> 3
[4,2,3,1] => [2,3,4,1] => {{1,2,3,4}}
=> 3
[1,2,3,4,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 1
[1,2,4,5,3] => [1,2,5,3,4] => {{1},{2},{3,4,5}}
=> 2
[1,2,5,3,4] => [1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> 1
[1,2,5,4,3] => [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 2
[1,3,2,4,5] => [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 2
[1,3,4,2,5] => [1,4,2,3,5] => {{1},{2,3,4},{5}}
=> 2
[1,3,4,5,2] => [1,5,2,3,4] => {{1},{2,3,4,5}}
=> 3
[1,4,2,3,5] => [1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> 1
[1,4,3,2,5] => [1,3,4,2,5] => {{1},{2,3,4},{5}}
=> 2
[1,4,3,5,2] => [1,3,5,2,4] => {{1},{2,3,4,5}}
=> 3
[1,4,5,2,3] => [1,4,2,5,3] => {{1},{2,3,4,5}}
=> 3
[1,5,3,4,2] => [1,3,4,5,2] => {{1},{2,3,4,5}}
=> 3
[2,1,3,4,5] => [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> 1
[2,1,3,5,4] => [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 2
[2,1,4,3,5] => [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> 2
[2,1,4,5,3] => [2,1,5,3,4] => {{1,2},{3,4,5}}
=> 3
[2,1,5,4,3] => [2,1,4,5,3] => {{1,2},{3,4,5}}
=> 3
[2,3,1,4,5] => [3,1,2,4,5] => {{1,2,3},{4},{5}}
=> 2
[2,3,1,5,4] => [3,1,2,5,4] => {{1,2,3},{4,5}}
=> 3
[2,3,4,1,5] => [4,1,2,3,5] => {{1,2,3,4},{5}}
=> 3
[3,1,2,4,5] => [3,2,1,4,5] => {{1,3},{2},{4},{5}}
=> 1
[3,2,1,4,5] => [2,3,1,4,5] => {{1,2,3},{4},{5}}
=> 2
[3,2,1,5,4] => [2,3,1,5,4] => {{1,2,3},{4,5}}
=> 3
Description
The rank of the set partition.
This is defined as the number of elements in the set partition minus the number of blocks, or, equivalently, the number of arcs in the one-line diagram associated to the set partition.
Matching statistic: St000010
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
=> []
=> 0
[1,2] => [1,2] => ([],2)
=> []
=> 0
[2,1] => [2,1] => ([(0,1)],2)
=> [1]
=> 1
[1,2,3] => [1,2,3] => ([],3)
=> []
=> 0
[1,3,2] => [1,3,2] => ([(1,2)],3)
=> [1]
=> 1
[2,1,3] => [2,1,3] => ([(1,2)],3)
=> [1]
=> 1
[2,3,1] => [3,1,2] => ([(0,2),(1,2)],3)
=> [1,1]
=> 2
[3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> [1,1]
=> 2
[1,2,3,4] => [1,2,3,4] => ([],4)
=> []
=> 0
[1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> [1]
=> 1
[1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> [1]
=> 1
[1,3,4,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> [1,1]
=> 2
[1,4,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 1
[1,4,3,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> [1,1]
=> 2
[2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> [1]
=> 1
[2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> [1,1]
=> 2
[2,3,1,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> [1,1]
=> 2
[2,3,4,1] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> 3
[3,1,2,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 1
[3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> [1,1]
=> 2
[3,2,4,1] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> 3
[3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> 3
[4,2,3,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> 3
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> []
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
=> [1]
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> [1]
=> 1
[1,2,4,5,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[1,2,5,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> [3]
=> 1
[1,2,5,4,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> [1]
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> [1,1]
=> 2
[1,3,4,2,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[1,3,4,5,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[1,4,2,3,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> [3]
=> 1
[1,4,3,2,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[1,4,3,5,2] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> 3
[1,4,5,2,3] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> 3
[1,5,3,4,2] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
=> [1]
=> 1
[2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> [1,1]
=> 2
[2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> [1,1]
=> 2
[2,1,4,5,3] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[2,1,5,4,3] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[2,3,1,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[2,3,1,5,4] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[2,3,4,1,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[3,1,2,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> [3]
=> 1
[3,2,1,4,5] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[3,2,1,5,4] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
Description
The length of the partition.
Matching statistic: St000024
Mp00239: Permutations —Corteel⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1,0]
=> 0
[1,2] => [1,2] => [1,2] => [1,0,1,0]
=> 0
[2,1] => [2,1] => [2,1] => [1,1,0,0]
=> 1
[1,2,3] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0
[1,3,2] => [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 1
[2,1,3] => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 1
[2,3,1] => [3,2,1] => [3,1,2] => [1,1,1,0,0,0]
=> 2
[3,1,2] => [3,1,2] => [2,3,1] => [1,1,0,1,0,0]
=> 1
[3,2,1] => [2,3,1] => [3,2,1] => [1,1,1,0,0,0]
=> 2
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 1
[1,3,4,2] => [1,4,3,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2
[1,4,2,3] => [1,4,2,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 1
[1,4,3,2] => [1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2
[2,3,1,4] => [3,2,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 2
[2,3,4,1] => [4,2,3,1] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 3
[3,1,2,4] => [3,1,2,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 1
[3,2,1,4] => [2,3,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2
[3,2,4,1] => [2,4,3,1] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 3
[3,4,1,2] => [4,3,2,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 3
[4,2,3,1] => [2,3,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 3
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,2,5,3,4] => [1,2,5,3,4] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,2,5,4,3] => [1,2,4,5,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,3,4,2,5] => [1,4,3,2,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,3,4,5,2] => [1,5,3,4,2] => [1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,4,2,3,5] => [1,4,2,3,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,4,3,2,5] => [1,3,4,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,4,3,5,2] => [1,3,5,4,2] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,4,5,2,3] => [1,5,4,3,2] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,5,3,4,2] => [1,3,4,5,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> 3
[2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 1
[2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[2,1,4,5,3] => [2,1,5,4,3] => [2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> 3
[2,1,5,4,3] => [2,1,4,5,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
[2,3,1,4,5] => [3,2,1,4,5] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 2
[2,3,1,5,4] => [3,2,1,5,4] => [3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[2,3,4,1,5] => [4,2,3,1,5] => [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> 3
[3,1,2,4,5] => [3,1,2,4,5] => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> 1
[3,2,1,4,5] => [2,3,1,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 2
[3,2,1,5,4] => [2,3,1,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 3
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: St000148
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
St000148: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
St000148: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
=> []
=> 0
[1,2] => [1,2] => ([],2)
=> []
=> 0
[2,1] => [2,1] => ([(0,1)],2)
=> [1]
=> 1
[1,2,3] => [1,2,3] => ([],3)
=> []
=> 0
[1,3,2] => [1,3,2] => ([(1,2)],3)
=> [1]
=> 1
[2,1,3] => [2,1,3] => ([(1,2)],3)
=> [1]
=> 1
[2,3,1] => [3,1,2] => ([(0,2),(1,2)],3)
=> [1,1]
=> 2
[3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> [1,1]
=> 2
[1,2,3,4] => [1,2,3,4] => ([],4)
=> []
=> 0
[1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> [1]
=> 1
[1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> [1]
=> 1
[1,3,4,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> [1,1]
=> 2
[1,4,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 1
[1,4,3,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> [1,1]
=> 2
[2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> [1]
=> 1
[2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> [1,1]
=> 2
[2,3,1,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> [1,1]
=> 2
[2,3,4,1] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> 3
[3,1,2,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 1
[3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> [1,1]
=> 2
[3,2,4,1] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> 3
[3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> 3
[4,2,3,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> 3
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> []
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
=> [1]
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> [1]
=> 1
[1,2,4,5,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[1,2,5,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> [3]
=> 1
[1,2,5,4,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> [1]
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> [1,1]
=> 2
[1,3,4,2,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[1,3,4,5,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[1,4,2,3,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> [3]
=> 1
[1,4,3,2,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[1,4,3,5,2] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> 3
[1,4,5,2,3] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> 3
[1,5,3,4,2] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
=> [1]
=> 1
[2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> [1,1]
=> 2
[2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> [1,1]
=> 2
[2,1,4,5,3] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[2,1,5,4,3] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[2,3,1,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[2,3,1,5,4] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[2,3,4,1,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[3,1,2,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> [3]
=> 1
[3,2,1,4,5] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[3,2,1,5,4] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
Description
The number of odd parts of a partition.
Matching statistic: St000160
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
St000160: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
St000160: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
=> []
=> 0
[1,2] => [1,2] => ([],2)
=> []
=> 0
[2,1] => [2,1] => ([(0,1)],2)
=> [1]
=> 1
[1,2,3] => [1,2,3] => ([],3)
=> []
=> 0
[1,3,2] => [1,3,2] => ([(1,2)],3)
=> [1]
=> 1
[2,1,3] => [2,1,3] => ([(1,2)],3)
=> [1]
=> 1
[2,3,1] => [3,1,2] => ([(0,2),(1,2)],3)
=> [1,1]
=> 2
[3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> [1,1]
=> 2
[1,2,3,4] => [1,2,3,4] => ([],4)
=> []
=> 0
[1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> [1]
=> 1
[1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> [1]
=> 1
[1,3,4,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> [1,1]
=> 2
[1,4,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 1
[1,4,3,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> [1,1]
=> 2
[2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> [1]
=> 1
[2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> [1,1]
=> 2
[2,3,1,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> [1,1]
=> 2
[2,3,4,1] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> 3
[3,1,2,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 1
[3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> [1,1]
=> 2
[3,2,4,1] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> 3
[3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> 3
[4,2,3,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> 3
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> []
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
=> [1]
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> [1]
=> 1
[1,2,4,5,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[1,2,5,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> [3]
=> 1
[1,2,5,4,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> [1]
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> [1,1]
=> 2
[1,3,4,2,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[1,3,4,5,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[1,4,2,3,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> [3]
=> 1
[1,4,3,2,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[1,4,3,5,2] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> 3
[1,4,5,2,3] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> 3
[1,5,3,4,2] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
=> [1]
=> 1
[2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> [1,1]
=> 2
[2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> [1,1]
=> 2
[2,1,4,5,3] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[2,1,5,4,3] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[2,3,1,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[2,3,1,5,4] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[2,3,4,1,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[3,1,2,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> [3]
=> 1
[3,2,1,4,5] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[3,2,1,5,4] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
Description
The multiplicity of the smallest part of a partition.
This counts the number of occurrences of the smallest part $spt(\lambda)$ of a partition $\lambda$.
The sum $spt(n) = \sum_{\lambda \vdash n} spt(\lambda)$ satisfies the congruences
\begin{align*}
spt(5n+4) &\equiv 0\quad \pmod{5}\\\
spt(7n+5) &\equiv 0\quad \pmod{7}\\\
spt(13n+6) &\equiv 0\quad \pmod{13},
\end{align*}
analogous to those of the counting function of partitions, see [1] and [2].
Matching statistic: St000377
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000377: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000377: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1]
=> [1]
=> 0
[1,2] => [1,2] => [1,1]
=> [2]
=> 0
[2,1] => [2,1] => [2]
=> [1,1]
=> 1
[1,2,3] => [1,2,3] => [1,1,1]
=> [2,1]
=> 0
[1,3,2] => [1,3,2] => [2,1]
=> [3]
=> 1
[2,1,3] => [2,1,3] => [2,1]
=> [3]
=> 1
[2,3,1] => [3,1,2] => [3]
=> [1,1,1]
=> 2
[3,1,2] => [3,2,1] => [2,1]
=> [3]
=> 1
[3,2,1] => [2,3,1] => [3]
=> [1,1,1]
=> 2
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [3,1]
=> 0
[1,2,4,3] => [1,2,4,3] => [2,1,1]
=> [2,2]
=> 1
[1,3,2,4] => [1,3,2,4] => [2,1,1]
=> [2,2]
=> 1
[1,3,4,2] => [1,4,2,3] => [3,1]
=> [2,1,1]
=> 2
[1,4,2,3] => [1,4,3,2] => [2,1,1]
=> [2,2]
=> 1
[1,4,3,2] => [1,3,4,2] => [3,1]
=> [2,1,1]
=> 2
[2,1,3,4] => [2,1,3,4] => [2,1,1]
=> [2,2]
=> 1
[2,1,4,3] => [2,1,4,3] => [2,2]
=> [4]
=> 2
[2,3,1,4] => [3,1,2,4] => [3,1]
=> [2,1,1]
=> 2
[2,3,4,1] => [4,1,2,3] => [4]
=> [1,1,1,1]
=> 3
[3,1,2,4] => [3,2,1,4] => [2,1,1]
=> [2,2]
=> 1
[3,2,1,4] => [2,3,1,4] => [3,1]
=> [2,1,1]
=> 2
[3,2,4,1] => [2,4,1,3] => [4]
=> [1,1,1,1]
=> 3
[3,4,1,2] => [3,1,4,2] => [4]
=> [1,1,1,1]
=> 3
[4,2,3,1] => [2,3,4,1] => [4]
=> [1,1,1,1]
=> 3
[1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> [3,2]
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => [2,1,1,1]
=> [3,1,1]
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [2,1,1,1]
=> [3,1,1]
=> 1
[1,2,4,5,3] => [1,2,5,3,4] => [3,1,1]
=> [4,1]
=> 2
[1,2,5,3,4] => [1,2,5,4,3] => [2,1,1,1]
=> [3,1,1]
=> 1
[1,2,5,4,3] => [1,2,4,5,3] => [3,1,1]
=> [4,1]
=> 2
[1,3,2,4,5] => [1,3,2,4,5] => [2,1,1,1]
=> [3,1,1]
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => [2,2,1]
=> [2,2,1]
=> 2
[1,3,4,2,5] => [1,4,2,3,5] => [3,1,1]
=> [4,1]
=> 2
[1,3,4,5,2] => [1,5,2,3,4] => [4,1]
=> [2,1,1,1]
=> 3
[1,4,2,3,5] => [1,4,3,2,5] => [2,1,1,1]
=> [3,1,1]
=> 1
[1,4,3,2,5] => [1,3,4,2,5] => [3,1,1]
=> [4,1]
=> 2
[1,4,3,5,2] => [1,3,5,2,4] => [4,1]
=> [2,1,1,1]
=> 3
[1,4,5,2,3] => [1,4,2,5,3] => [4,1]
=> [2,1,1,1]
=> 3
[1,5,3,4,2] => [1,3,4,5,2] => [4,1]
=> [2,1,1,1]
=> 3
[2,1,3,4,5] => [2,1,3,4,5] => [2,1,1,1]
=> [3,1,1]
=> 1
[2,1,3,5,4] => [2,1,3,5,4] => [2,2,1]
=> [2,2,1]
=> 2
[2,1,4,3,5] => [2,1,4,3,5] => [2,2,1]
=> [2,2,1]
=> 2
[2,1,4,5,3] => [2,1,5,3,4] => [3,2]
=> [5]
=> 3
[2,1,5,4,3] => [2,1,4,5,3] => [3,2]
=> [5]
=> 3
[2,3,1,4,5] => [3,1,2,4,5] => [3,1,1]
=> [4,1]
=> 2
[2,3,1,5,4] => [3,1,2,5,4] => [3,2]
=> [5]
=> 3
[2,3,4,1,5] => [4,1,2,3,5] => [4,1]
=> [2,1,1,1]
=> 3
[3,1,2,4,5] => [3,2,1,4,5] => [2,1,1,1]
=> [3,1,1]
=> 1
[3,2,1,4,5] => [2,3,1,4,5] => [3,1,1]
=> [4,1]
=> 2
[3,2,1,5,4] => [2,3,1,5,4] => [3,2]
=> [5]
=> 3
Description
The dinv defect of an integer partition.
This is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \not\in \{0,1\}$.
Matching statistic: St000394
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00066: Permutations —inverse⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1,0]
=> 0
[1,2] => [1,2] => [1,2] => [1,0,1,0]
=> 0
[2,1] => [2,1] => [2,1] => [1,1,0,0]
=> 1
[1,2,3] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0
[1,3,2] => [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 1
[2,1,3] => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 1
[2,3,1] => [3,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> 2
[3,1,2] => [2,3,1] => [1,3,2] => [1,0,1,1,0,0]
=> 1
[3,2,1] => [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 2
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 1
[1,3,4,2] => [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2
[1,4,2,3] => [1,3,4,2] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1
[1,4,3,2] => [1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2
[2,3,1,4] => [3,1,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 2
[2,3,4,1] => [4,1,2,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 3
[3,1,2,4] => [2,3,1,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 1
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2
[3,2,4,1] => [4,2,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 3
[3,4,1,2] => [3,4,1,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 3
[4,2,3,1] => [4,2,3,1] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 3
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,2,4,5,3] => [1,2,5,3,4] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,2,5,3,4] => [1,2,4,5,3] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,3,4,2,5] => [1,4,2,3,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,3,4,5,2] => [1,5,2,3,4] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,4,2,3,5] => [1,3,4,2,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,4,3,5,2] => [1,5,3,2,4] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,4,5,2,3] => [1,4,5,2,3] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> 3
[1,5,3,4,2] => [1,5,3,4,2] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> 3
[2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 1
[2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[2,1,4,5,3] => [2,1,5,3,4] => [2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> 3
[2,1,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
[2,3,1,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 2
[2,3,1,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[2,3,4,1,5] => [4,1,2,3,5] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 3
[3,1,2,4,5] => [2,3,1,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 2
[3,2,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 3
Description
The sum of the heights of the peaks of a Dyck path minus the number of peaks.
Matching statistic: St000548
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
St000548: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
St000548: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
=> []
=> 0
[1,2] => [1,2] => ([],2)
=> []
=> 0
[2,1] => [2,1] => ([(0,1)],2)
=> [1]
=> 1
[1,2,3] => [1,2,3] => ([],3)
=> []
=> 0
[1,3,2] => [1,3,2] => ([(1,2)],3)
=> [1]
=> 1
[2,1,3] => [2,1,3] => ([(1,2)],3)
=> [1]
=> 1
[2,3,1] => [3,1,2] => ([(0,2),(1,2)],3)
=> [1,1]
=> 2
[3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> [1,1]
=> 2
[1,2,3,4] => [1,2,3,4] => ([],4)
=> []
=> 0
[1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> [1]
=> 1
[1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> [1]
=> 1
[1,3,4,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> [1,1]
=> 2
[1,4,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 1
[1,4,3,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> [1,1]
=> 2
[2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> [1]
=> 1
[2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> [1,1]
=> 2
[2,3,1,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> [1,1]
=> 2
[2,3,4,1] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> 3
[3,1,2,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 1
[3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> [1,1]
=> 2
[3,2,4,1] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> 3
[3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> 3
[4,2,3,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> 3
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> []
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
=> [1]
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> [1]
=> 1
[1,2,4,5,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[1,2,5,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> [3]
=> 1
[1,2,5,4,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> [1]
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> [1,1]
=> 2
[1,3,4,2,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[1,3,4,5,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[1,4,2,3,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> [3]
=> 1
[1,4,3,2,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[1,4,3,5,2] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> 3
[1,4,5,2,3] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> 3
[1,5,3,4,2] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
=> [1]
=> 1
[2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> [1,1]
=> 2
[2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> [1,1]
=> 2
[2,1,4,5,3] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[2,1,5,4,3] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[2,3,1,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[2,3,1,5,4] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[2,3,4,1,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[3,1,2,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> [3]
=> 1
[3,2,1,4,5] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[3,2,1,5,4] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
Description
The number of different non-empty partial sums of an integer partition.
Matching statistic: St000987
Mp00066: Permutations —inverse⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000987: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000987: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => ([],1)
=> 0
[1,2] => [1,2] => [1,2] => ([],2)
=> 0
[2,1] => [2,1] => [2,1] => ([(0,1)],2)
=> 1
[1,2,3] => [1,2,3] => [1,2,3] => ([],3)
=> 0
[1,3,2] => [1,3,2] => [1,3,2] => ([(1,2)],3)
=> 1
[2,1,3] => [2,1,3] => [2,1,3] => ([(1,2)],3)
=> 1
[2,3,1] => [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[3,1,2] => [2,3,1] => [1,3,2] => ([(1,2)],3)
=> 1
[3,2,1] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> 1
[1,3,4,2] => [1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[1,4,2,3] => [1,3,4,2] => [1,2,4,3] => ([(2,3)],4)
=> 1
[1,4,3,2] => [1,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2
[2,3,1,4] => [3,1,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> 2
[2,3,4,1] => [4,1,2,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 3
[3,1,2,4] => [2,3,1,4] => [1,3,2,4] => ([(2,3)],4)
=> 1
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 2
[3,2,4,1] => [4,2,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[3,4,1,2] => [3,4,1,2] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 3
[4,2,3,1] => [4,2,3,1] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> 1
[1,2,4,5,3] => [1,2,5,3,4] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 2
[1,2,5,3,4] => [1,2,4,5,3] => [1,2,3,5,4] => ([(3,4)],5)
=> 1
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 2
[1,3,4,2,5] => [1,4,2,3,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 2
[1,3,4,5,2] => [1,5,2,3,4] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 3
[1,4,2,3,5] => [1,3,4,2,5] => [1,2,4,3,5] => ([(3,4)],5)
=> 1
[1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> 2
[1,4,3,5,2] => [1,5,3,2,4] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,4,5,2,3] => [1,4,5,2,3] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 3
[1,5,3,4,2] => [1,5,3,4,2] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
=> 1
[2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 2
[2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> 2
[2,1,4,5,3] => [2,1,5,3,4] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> 3
[2,1,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> 3
[2,3,1,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> 2
[2,3,1,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> 3
[2,3,4,1,5] => [4,1,2,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> 3
[3,1,2,4,5] => [2,3,1,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> 1
[3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> 2
[3,2,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> 3
Description
The number of positive eigenvalues of the Laplacian matrix of the graph.
This is the number of vertices minus the number of connected components of the graph.
Matching statistic: St001176
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1]
=> [1]
=> 0
[1,2] => [1,2] => [1,1]
=> [2]
=> 0
[2,1] => [2,1] => [2]
=> [1,1]
=> 1
[1,2,3] => [1,2,3] => [1,1,1]
=> [3]
=> 0
[1,3,2] => [1,3,2] => [2,1]
=> [2,1]
=> 1
[2,1,3] => [2,1,3] => [2,1]
=> [2,1]
=> 1
[2,3,1] => [3,1,2] => [3]
=> [1,1,1]
=> 2
[3,1,2] => [3,2,1] => [2,1]
=> [2,1]
=> 1
[3,2,1] => [2,3,1] => [3]
=> [1,1,1]
=> 2
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [4]
=> 0
[1,2,4,3] => [1,2,4,3] => [2,1,1]
=> [3,1]
=> 1
[1,3,2,4] => [1,3,2,4] => [2,1,1]
=> [3,1]
=> 1
[1,3,4,2] => [1,4,2,3] => [3,1]
=> [2,1,1]
=> 2
[1,4,2,3] => [1,4,3,2] => [2,1,1]
=> [3,1]
=> 1
[1,4,3,2] => [1,3,4,2] => [3,1]
=> [2,1,1]
=> 2
[2,1,3,4] => [2,1,3,4] => [2,1,1]
=> [3,1]
=> 1
[2,1,4,3] => [2,1,4,3] => [2,2]
=> [2,2]
=> 2
[2,3,1,4] => [3,1,2,4] => [3,1]
=> [2,1,1]
=> 2
[2,3,4,1] => [4,1,2,3] => [4]
=> [1,1,1,1]
=> 3
[3,1,2,4] => [3,2,1,4] => [2,1,1]
=> [3,1]
=> 1
[3,2,1,4] => [2,3,1,4] => [3,1]
=> [2,1,1]
=> 2
[3,2,4,1] => [2,4,1,3] => [4]
=> [1,1,1,1]
=> 3
[3,4,1,2] => [3,1,4,2] => [4]
=> [1,1,1,1]
=> 3
[4,2,3,1] => [2,3,4,1] => [4]
=> [1,1,1,1]
=> 3
[1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> [5]
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => [2,1,1,1]
=> [4,1]
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [2,1,1,1]
=> [4,1]
=> 1
[1,2,4,5,3] => [1,2,5,3,4] => [3,1,1]
=> [3,1,1]
=> 2
[1,2,5,3,4] => [1,2,5,4,3] => [2,1,1,1]
=> [4,1]
=> 1
[1,2,5,4,3] => [1,2,4,5,3] => [3,1,1]
=> [3,1,1]
=> 2
[1,3,2,4,5] => [1,3,2,4,5] => [2,1,1,1]
=> [4,1]
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => [2,2,1]
=> [3,2]
=> 2
[1,3,4,2,5] => [1,4,2,3,5] => [3,1,1]
=> [3,1,1]
=> 2
[1,3,4,5,2] => [1,5,2,3,4] => [4,1]
=> [2,1,1,1]
=> 3
[1,4,2,3,5] => [1,4,3,2,5] => [2,1,1,1]
=> [4,1]
=> 1
[1,4,3,2,5] => [1,3,4,2,5] => [3,1,1]
=> [3,1,1]
=> 2
[1,4,3,5,2] => [1,3,5,2,4] => [4,1]
=> [2,1,1,1]
=> 3
[1,4,5,2,3] => [1,4,2,5,3] => [4,1]
=> [2,1,1,1]
=> 3
[1,5,3,4,2] => [1,3,4,5,2] => [4,1]
=> [2,1,1,1]
=> 3
[2,1,3,4,5] => [2,1,3,4,5] => [2,1,1,1]
=> [4,1]
=> 1
[2,1,3,5,4] => [2,1,3,5,4] => [2,2,1]
=> [3,2]
=> 2
[2,1,4,3,5] => [2,1,4,3,5] => [2,2,1]
=> [3,2]
=> 2
[2,1,4,5,3] => [2,1,5,3,4] => [3,2]
=> [2,2,1]
=> 3
[2,1,5,4,3] => [2,1,4,5,3] => [3,2]
=> [2,2,1]
=> 3
[2,3,1,4,5] => [3,1,2,4,5] => [3,1,1]
=> [3,1,1]
=> 2
[2,3,1,5,4] => [3,1,2,5,4] => [3,2]
=> [2,2,1]
=> 3
[2,3,4,1,5] => [4,1,2,3,5] => [4,1]
=> [2,1,1,1]
=> 3
[3,1,2,4,5] => [3,2,1,4,5] => [2,1,1,1]
=> [4,1]
=> 1
[3,2,1,4,5] => [2,3,1,4,5] => [3,1,1]
=> [3,1,1]
=> 2
[3,2,1,5,4] => [2,3,1,5,4] => [3,2]
=> [2,2,1]
=> 3
Description
The size of a partition minus its first part.
This is the number of boxes in its diagram that are not in the first row.
The following 55 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001251The number of parts of a partition that are not congruent 1 modulo 3. St001280The number of parts of an integer partition that are at least two. St001613The binary logarithm of the size of the center of a lattice. St001617The dimension of the space of valuations of a lattice. St000105The number of blocks in the set partition. St000507The number of ascents of a standard tableau. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St000728The dimension of a set partition. St000925The number of topologically connected components of a set partition. St001062The maximal size of a block of a set partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001933The largest multiplicity of a part in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St000996The number of exclusive left-to-right maxima of a permutation. St000957The number of Bruhat lower covers of a permutation. St001726The number of visible inversions of a permutation. St000809The reduced reflection length of the permutation. St000993The multiplicity of the largest part of an integer partition. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St000831The number of indices that are either descents or recoils. St000539The number of odd inversions of a permutation. St000795The mad of a permutation. St000141The maximum drop size of a permutation. St000019The cardinality of the support of a permutation. St000740The last entry of a permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000029The depth of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000030The sum of the descent differences of a permutations. St000155The number of exceedances (also excedences) of a permutation. St000238The number of indices that are not small weak excedances. St001869The maximum cut size of a graph. St000443The number of long tunnels of a Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000216The absolute length of a permutation. St001480The number of simple summands of the module J^2/J^3. St000454The largest eigenvalue of a graph if it is integral. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St001875The number of simple modules with projective dimension at most 1. St000264The girth of a graph, which is not a tree. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001861The number of Bruhat lower covers of a permutation. St001894The depth of a signed permutation. St001769The reflection length of a signed permutation. St001866The nesting alignments of a signed permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001877Number of indecomposable injective modules with projective dimension 2.
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