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Your data matches 163 different statistics following compositions of up to 3 maps.
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Matching statistic: St000211
(load all 55 compositions to match this statistic)
(load all 55 compositions to match this statistic)
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%
St000211: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => {{1}}
=> 0
[1,2] => {{1},{2}}
=> 0
[2,1] => {{1,2}}
=> 1
[1,2,3] => {{1},{2},{3}}
=> 0
[1,3,2] => {{1},{2,3}}
=> 1
[2,1,3] => {{1,2},{3}}
=> 1
[2,3,1] => {{1,2,3}}
=> 2
[3,1,2] => {{1,2,3}}
=> 2
[3,2,1] => {{1,3},{2}}
=> 1
[1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[1,2,4,3] => {{1},{2},{3,4}}
=> 1
[1,3,2,4] => {{1},{2,3},{4}}
=> 1
[1,3,4,2] => {{1},{2,3,4}}
=> 2
[1,4,2,3] => {{1},{2,3,4}}
=> 2
[1,4,3,2] => {{1},{2,4},{3}}
=> 1
[2,1,3,4] => {{1,2},{3},{4}}
=> 1
[2,1,4,3] => {{1,2},{3,4}}
=> 2
[2,3,1,4] => {{1,2,3},{4}}
=> 2
[2,3,4,1] => {{1,2,3,4}}
=> 3
[2,4,1,3] => {{1,2,3,4}}
=> 3
[3,1,2,4] => {{1,2,3},{4}}
=> 2
[3,1,4,2] => {{1,2,3,4}}
=> 3
[3,2,1,4] => {{1,3},{2},{4}}
=> 1
[4,1,2,3] => {{1,2,3,4}}
=> 3
[1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 0
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 1
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 1
[1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 2
[1,2,5,3,4] => {{1},{2},{3,4,5}}
=> 2
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> 1
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 1
[1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 2
[1,3,4,2,5] => {{1},{2,3,4},{5}}
=> 2
[1,3,4,5,2] => {{1},{2,3,4,5}}
=> 3
[1,3,5,2,4] => {{1},{2,3,4,5}}
=> 3
[1,4,2,3,5] => {{1},{2,3,4},{5}}
=> 2
[1,4,2,5,3] => {{1},{2,3,4,5}}
=> 3
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> 1
[1,5,2,3,4] => {{1},{2,3,4,5}}
=> 3
[2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> 1
[2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 2
[2,1,4,3,5] => {{1,2},{3,4},{5}}
=> 2
[2,1,4,5,3] => {{1,2},{3,4,5}}
=> 3
[2,1,5,3,4] => {{1,2},{3,4,5}}
=> 3
[2,3,1,4,5] => {{1,2,3},{4},{5}}
=> 2
[2,3,1,5,4] => {{1,2,3},{4,5}}
=> 3
[2,3,4,1,5] => {{1,2,3,4},{5}}
=> 3
[2,4,1,3,5] => {{1,2,3,4},{5}}
=> 3
[3,1,2,4,5] => {{1,2,3},{4},{5}}
=> 2
[3,1,2,5,4] => {{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
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
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%
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)
=> []
=> 0
[1,2] => ([],2)
=> []
=> 0
[2,1] => ([(0,1)],2)
=> [1]
=> 1
[1,2,3] => ([],3)
=> []
=> 0
[1,3,2] => ([(1,2)],3)
=> [1]
=> 1
[2,1,3] => ([(1,2)],3)
=> [1]
=> 1
[2,3,1] => ([(0,2),(1,2)],3)
=> [1,1]
=> 2
[3,1,2] => ([(0,2),(1,2)],3)
=> [1,1]
=> 2
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[1,2,3,4] => ([],4)
=> []
=> 0
[1,2,4,3] => ([(2,3)],4)
=> [1]
=> 1
[1,3,2,4] => ([(2,3)],4)
=> [1]
=> 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> [1,1]
=> 2
[1,4,2,3] => ([(1,3),(2,3)],4)
=> [1,1]
=> 2
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 1
[2,1,3,4] => ([(2,3)],4)
=> [1]
=> 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> [1,1]
=> 2
[2,3,1,4] => ([(1,3),(2,3)],4)
=> [1,1]
=> 2
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> 3
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> 3
[3,1,2,4] => ([(1,3),(2,3)],4)
=> [1,1]
=> 2
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> 3
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> 3
[1,2,3,4,5] => ([],5)
=> []
=> 0
[1,2,3,5,4] => ([(3,4)],5)
=> [1]
=> 1
[1,2,4,3,5] => ([(3,4)],5)
=> [1]
=> 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> [3]
=> 1
[1,3,2,4,5] => ([(3,4)],5)
=> [1]
=> 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> [1,1]
=> 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> 3
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> 3
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> [3]
=> 1
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[2,1,3,4,5] => ([(3,4)],5)
=> [1]
=> 1
[2,1,3,5,4] => ([(1,4),(2,3)],5)
=> [1,1]
=> 2
[2,1,4,3,5] => ([(1,4),(2,3)],5)
=> [1,1]
=> 2
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[2,3,1,4,5] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> 3
[3,1,2,4,5] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
Description
The length of the partition.
Matching statistic: St000024
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ 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%
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] => [1,0]
=> 0
[1,2] => [1,2] => [1,0,1,0]
=> 0
[2,1] => [2,1] => [1,1,0,0]
=> 1
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 1
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 1
[2,3,1] => [3,2,1] => [1,1,1,0,0,0]
=> 2
[3,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> 2
[3,2,1] => [2,3,1] => [1,1,0,1,0,0]
=> 1
[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] => [1,0,1,0,1,1,0,0]
=> 1
[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] => [1,0,1,1,1,0,0,0]
=> 2
[1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2
[1,4,3,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 1
[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] => [1,1,0,0,1,1,0,0]
=> 2
[2,3,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2
[2,3,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 3
[2,4,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 3
[3,1,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 2
[3,1,4,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 3
[3,2,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 1
[4,1,2,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 3
[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] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[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] => [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,2,5,3,4] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,2,5,4,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> 1
[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] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,3,4,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,3,4,5,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,3,5,2,4] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,4,2,3,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,4,2,5,3] => [1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,4,3,2,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,5,2,3,4] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> 3
[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] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[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] => [1,1,0,0,1,1,1,0,0,0]
=> 3
[2,1,5,3,4] => [2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> 3
[2,3,1,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 2
[2,3,1,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[2,3,4,1,5] => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 3
[2,4,1,3,5] => [4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 3
[3,1,2,4,5] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 2
[3,1,2,5,4] => [3,1,2,5,4] => [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
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%
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)
=> []
=> 0
[1,2] => ([],2)
=> []
=> 0
[2,1] => ([(0,1)],2)
=> [1]
=> 1
[1,2,3] => ([],3)
=> []
=> 0
[1,3,2] => ([(1,2)],3)
=> [1]
=> 1
[2,1,3] => ([(1,2)],3)
=> [1]
=> 1
[2,3,1] => ([(0,2),(1,2)],3)
=> [1,1]
=> 2
[3,1,2] => ([(0,2),(1,2)],3)
=> [1,1]
=> 2
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[1,2,3,4] => ([],4)
=> []
=> 0
[1,2,4,3] => ([(2,3)],4)
=> [1]
=> 1
[1,3,2,4] => ([(2,3)],4)
=> [1]
=> 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> [1,1]
=> 2
[1,4,2,3] => ([(1,3),(2,3)],4)
=> [1,1]
=> 2
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 1
[2,1,3,4] => ([(2,3)],4)
=> [1]
=> 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> [1,1]
=> 2
[2,3,1,4] => ([(1,3),(2,3)],4)
=> [1,1]
=> 2
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> 3
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> 3
[3,1,2,4] => ([(1,3),(2,3)],4)
=> [1,1]
=> 2
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> 3
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> 3
[1,2,3,4,5] => ([],5)
=> []
=> 0
[1,2,3,5,4] => ([(3,4)],5)
=> [1]
=> 1
[1,2,4,3,5] => ([(3,4)],5)
=> [1]
=> 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> [3]
=> 1
[1,3,2,4,5] => ([(3,4)],5)
=> [1]
=> 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> [1,1]
=> 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> 3
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> 3
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> [3]
=> 1
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[2,1,3,4,5] => ([(3,4)],5)
=> [1]
=> 1
[2,1,3,5,4] => ([(1,4),(2,3)],5)
=> [1,1]
=> 2
[2,1,4,3,5] => ([(1,4),(2,3)],5)
=> [1,1]
=> 2
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[2,3,1,4,5] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> 3
[3,1,2,4,5] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
Description
The number of odd parts of a partition.
Matching statistic: St000160
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%
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)
=> []
=> 0
[1,2] => ([],2)
=> []
=> 0
[2,1] => ([(0,1)],2)
=> [1]
=> 1
[1,2,3] => ([],3)
=> []
=> 0
[1,3,2] => ([(1,2)],3)
=> [1]
=> 1
[2,1,3] => ([(1,2)],3)
=> [1]
=> 1
[2,3,1] => ([(0,2),(1,2)],3)
=> [1,1]
=> 2
[3,1,2] => ([(0,2),(1,2)],3)
=> [1,1]
=> 2
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[1,2,3,4] => ([],4)
=> []
=> 0
[1,2,4,3] => ([(2,3)],4)
=> [1]
=> 1
[1,3,2,4] => ([(2,3)],4)
=> [1]
=> 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> [1,1]
=> 2
[1,4,2,3] => ([(1,3),(2,3)],4)
=> [1,1]
=> 2
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 1
[2,1,3,4] => ([(2,3)],4)
=> [1]
=> 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> [1,1]
=> 2
[2,3,1,4] => ([(1,3),(2,3)],4)
=> [1,1]
=> 2
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> 3
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> 3
[3,1,2,4] => ([(1,3),(2,3)],4)
=> [1,1]
=> 2
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> 3
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> 3
[1,2,3,4,5] => ([],5)
=> []
=> 0
[1,2,3,5,4] => ([(3,4)],5)
=> [1]
=> 1
[1,2,4,3,5] => ([(3,4)],5)
=> [1]
=> 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> [3]
=> 1
[1,3,2,4,5] => ([(3,4)],5)
=> [1]
=> 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> [1,1]
=> 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> 3
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> 3
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> [3]
=> 1
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[2,1,3,4,5] => ([(3,4)],5)
=> [1]
=> 1
[2,1,3,5,4] => ([(1,4),(2,3)],5)
=> [1,1]
=> 2
[2,1,4,3,5] => ([(1,4),(2,3)],5)
=> [1,1]
=> 2
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[2,3,1,4,5] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> 3
[3,1,2,4,5] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[3,1,2,5,4] => ([(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
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%
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]
=> 0
[1,2] => [1,1]
=> [2]
=> 0
[2,1] => [2]
=> [1,1]
=> 1
[1,2,3] => [1,1,1]
=> [2,1]
=> 0
[1,3,2] => [2,1]
=> [3]
=> 1
[2,1,3] => [2,1]
=> [3]
=> 1
[2,3,1] => [3]
=> [1,1,1]
=> 2
[3,1,2] => [3]
=> [1,1,1]
=> 2
[3,2,1] => [2,1]
=> [3]
=> 1
[1,2,3,4] => [1,1,1,1]
=> [3,1]
=> 0
[1,2,4,3] => [2,1,1]
=> [2,2]
=> 1
[1,3,2,4] => [2,1,1]
=> [2,2]
=> 1
[1,3,4,2] => [3,1]
=> [2,1,1]
=> 2
[1,4,2,3] => [3,1]
=> [2,1,1]
=> 2
[1,4,3,2] => [2,1,1]
=> [2,2]
=> 1
[2,1,3,4] => [2,1,1]
=> [2,2]
=> 1
[2,1,4,3] => [2,2]
=> [4]
=> 2
[2,3,1,4] => [3,1]
=> [2,1,1]
=> 2
[2,3,4,1] => [4]
=> [1,1,1,1]
=> 3
[2,4,1,3] => [4]
=> [1,1,1,1]
=> 3
[3,1,2,4] => [3,1]
=> [2,1,1]
=> 2
[3,1,4,2] => [4]
=> [1,1,1,1]
=> 3
[3,2,1,4] => [2,1,1]
=> [2,2]
=> 1
[4,1,2,3] => [4]
=> [1,1,1,1]
=> 3
[1,2,3,4,5] => [1,1,1,1,1]
=> [3,2]
=> 0
[1,2,3,5,4] => [2,1,1,1]
=> [3,1,1]
=> 1
[1,2,4,3,5] => [2,1,1,1]
=> [3,1,1]
=> 1
[1,2,4,5,3] => [3,1,1]
=> [4,1]
=> 2
[1,2,5,3,4] => [3,1,1]
=> [4,1]
=> 2
[1,2,5,4,3] => [2,1,1,1]
=> [3,1,1]
=> 1
[1,3,2,4,5] => [2,1,1,1]
=> [3,1,1]
=> 1
[1,3,2,5,4] => [2,2,1]
=> [2,2,1]
=> 2
[1,3,4,2,5] => [3,1,1]
=> [4,1]
=> 2
[1,3,4,5,2] => [4,1]
=> [2,1,1,1]
=> 3
[1,3,5,2,4] => [4,1]
=> [2,1,1,1]
=> 3
[1,4,2,3,5] => [3,1,1]
=> [4,1]
=> 2
[1,4,2,5,3] => [4,1]
=> [2,1,1,1]
=> 3
[1,4,3,2,5] => [2,1,1,1]
=> [3,1,1]
=> 1
[1,5,2,3,4] => [4,1]
=> [2,1,1,1]
=> 3
[2,1,3,4,5] => [2,1,1,1]
=> [3,1,1]
=> 1
[2,1,3,5,4] => [2,2,1]
=> [2,2,1]
=> 2
[2,1,4,3,5] => [2,2,1]
=> [2,2,1]
=> 2
[2,1,4,5,3] => [3,2]
=> [5]
=> 3
[2,1,5,3,4] => [3,2]
=> [5]
=> 3
[2,3,1,4,5] => [3,1,1]
=> [4,1]
=> 2
[2,3,1,5,4] => [3,2]
=> [5]
=> 3
[2,3,4,1,5] => [4,1]
=> [2,1,1,1]
=> 3
[2,4,1,3,5] => [4,1]
=> [2,1,1,1]
=> 3
[3,1,2,4,5] => [3,1,1]
=> [4,1]
=> 2
[3,1,2,5,4] => [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: St000548
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%
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)
=> []
=> 0
[1,2] => ([],2)
=> []
=> 0
[2,1] => ([(0,1)],2)
=> [1]
=> 1
[1,2,3] => ([],3)
=> []
=> 0
[1,3,2] => ([(1,2)],3)
=> [1]
=> 1
[2,1,3] => ([(1,2)],3)
=> [1]
=> 1
[2,3,1] => ([(0,2),(1,2)],3)
=> [1,1]
=> 2
[3,1,2] => ([(0,2),(1,2)],3)
=> [1,1]
=> 2
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[1,2,3,4] => ([],4)
=> []
=> 0
[1,2,4,3] => ([(2,3)],4)
=> [1]
=> 1
[1,3,2,4] => ([(2,3)],4)
=> [1]
=> 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> [1,1]
=> 2
[1,4,2,3] => ([(1,3),(2,3)],4)
=> [1,1]
=> 2
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 1
[2,1,3,4] => ([(2,3)],4)
=> [1]
=> 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> [1,1]
=> 2
[2,3,1,4] => ([(1,3),(2,3)],4)
=> [1,1]
=> 2
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> 3
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> 3
[3,1,2,4] => ([(1,3),(2,3)],4)
=> [1,1]
=> 2
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> 3
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> 3
[1,2,3,4,5] => ([],5)
=> []
=> 0
[1,2,3,5,4] => ([(3,4)],5)
=> [1]
=> 1
[1,2,4,3,5] => ([(3,4)],5)
=> [1]
=> 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> [3]
=> 1
[1,3,2,4,5] => ([(3,4)],5)
=> [1]
=> 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> [1,1]
=> 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> 3
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> 3
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> [3]
=> 1
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[2,1,3,4,5] => ([(3,4)],5)
=> [1]
=> 1
[2,1,3,5,4] => ([(1,4),(2,3)],5)
=> [1,1]
=> 2
[2,1,4,3,5] => ([(1,4),(2,3)],5)
=> [1,1]
=> 2
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[2,3,1,4,5] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> 3
[3,1,2,4,5] => ([(2,4),(3,4)],5)
=> [1,1]
=> 2
[3,1,2,5,4] => ([(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: St001176
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
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%
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]
=> 0
[1,2] => [1,1]
=> [2]
=> 0
[2,1] => [2]
=> [1,1]
=> 1
[1,2,3] => [1,1,1]
=> [3]
=> 0
[1,3,2] => [2,1]
=> [2,1]
=> 1
[2,1,3] => [2,1]
=> [2,1]
=> 1
[2,3,1] => [3]
=> [1,1,1]
=> 2
[3,1,2] => [3]
=> [1,1,1]
=> 2
[3,2,1] => [2,1]
=> [2,1]
=> 1
[1,2,3,4] => [1,1,1,1]
=> [4]
=> 0
[1,2,4,3] => [2,1,1]
=> [3,1]
=> 1
[1,3,2,4] => [2,1,1]
=> [3,1]
=> 1
[1,3,4,2] => [3,1]
=> [2,1,1]
=> 2
[1,4,2,3] => [3,1]
=> [2,1,1]
=> 2
[1,4,3,2] => [2,1,1]
=> [3,1]
=> 1
[2,1,3,4] => [2,1,1]
=> [3,1]
=> 1
[2,1,4,3] => [2,2]
=> [2,2]
=> 2
[2,3,1,4] => [3,1]
=> [2,1,1]
=> 2
[2,3,4,1] => [4]
=> [1,1,1,1]
=> 3
[2,4,1,3] => [4]
=> [1,1,1,1]
=> 3
[3,1,2,4] => [3,1]
=> [2,1,1]
=> 2
[3,1,4,2] => [4]
=> [1,1,1,1]
=> 3
[3,2,1,4] => [2,1,1]
=> [3,1]
=> 1
[4,1,2,3] => [4]
=> [1,1,1,1]
=> 3
[1,2,3,4,5] => [1,1,1,1,1]
=> [5]
=> 0
[1,2,3,5,4] => [2,1,1,1]
=> [4,1]
=> 1
[1,2,4,3,5] => [2,1,1,1]
=> [4,1]
=> 1
[1,2,4,5,3] => [3,1,1]
=> [3,1,1]
=> 2
[1,2,5,3,4] => [3,1,1]
=> [3,1,1]
=> 2
[1,2,5,4,3] => [2,1,1,1]
=> [4,1]
=> 1
[1,3,2,4,5] => [2,1,1,1]
=> [4,1]
=> 1
[1,3,2,5,4] => [2,2,1]
=> [3,2]
=> 2
[1,3,4,2,5] => [3,1,1]
=> [3,1,1]
=> 2
[1,3,4,5,2] => [4,1]
=> [2,1,1,1]
=> 3
[1,3,5,2,4] => [4,1]
=> [2,1,1,1]
=> 3
[1,4,2,3,5] => [3,1,1]
=> [3,1,1]
=> 2
[1,4,2,5,3] => [4,1]
=> [2,1,1,1]
=> 3
[1,4,3,2,5] => [2,1,1,1]
=> [4,1]
=> 1
[1,5,2,3,4] => [4,1]
=> [2,1,1,1]
=> 3
[2,1,3,4,5] => [2,1,1,1]
=> [4,1]
=> 1
[2,1,3,5,4] => [2,2,1]
=> [3,2]
=> 2
[2,1,4,3,5] => [2,2,1]
=> [3,2]
=> 2
[2,1,4,5,3] => [3,2]
=> [2,2,1]
=> 3
[2,1,5,3,4] => [3,2]
=> [2,2,1]
=> 3
[2,3,1,4,5] => [3,1,1]
=> [3,1,1]
=> 2
[2,3,1,5,4] => [3,2]
=> [2,2,1]
=> 3
[2,3,4,1,5] => [4,1]
=> [2,1,1,1]
=> 3
[2,4,1,3,5] => [4,1]
=> [2,1,1,1]
=> 3
[3,1,2,4,5] => [3,1,1]
=> [3,1,1]
=> 2
[3,1,2,5,4] => [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.
Matching statistic: St001251
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00251: Graphs —clique sizes⟶ Integer partitions
St001251: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00251: Graphs —clique sizes⟶ Integer partitions
St001251: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> [1]
=> 0
[1,2] => ([],2)
=> [1,1]
=> 0
[2,1] => ([(0,1)],2)
=> [2]
=> 1
[1,2,3] => ([],3)
=> [1,1,1]
=> 0
[1,3,2] => ([(1,2)],3)
=> [2,1]
=> 1
[2,1,3] => ([(1,2)],3)
=> [2,1]
=> 1
[2,3,1] => ([(0,2),(1,2)],3)
=> [2,2]
=> 2
[3,1,2] => ([(0,2),(1,2)],3)
=> [2,2]
=> 2
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[1,2,3,4] => ([],4)
=> [1,1,1,1]
=> 0
[1,2,4,3] => ([(2,3)],4)
=> [2,1,1]
=> 1
[1,3,2,4] => ([(2,3)],4)
=> [2,1,1]
=> 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> [2,2,1]
=> 2
[1,4,2,3] => ([(1,3),(2,3)],4)
=> [2,2,1]
=> 2
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 1
[2,1,3,4] => ([(2,3)],4)
=> [2,1,1]
=> 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> [2,2]
=> 2
[2,3,1,4] => ([(1,3),(2,3)],4)
=> [2,2,1]
=> 2
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> 3
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> 3
[3,1,2,4] => ([(1,3),(2,3)],4)
=> [2,2,1]
=> 2
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> 3
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> 3
[1,2,3,4,5] => ([],5)
=> [1,1,1,1,1]
=> 0
[1,2,3,5,4] => ([(3,4)],5)
=> [2,1,1,1]
=> 1
[1,2,4,3,5] => ([(3,4)],5)
=> [2,1,1,1]
=> 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> [2,2,1,1]
=> 2
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> [2,2,1,1]
=> 2
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 1
[1,3,2,4,5] => ([(3,4)],5)
=> [2,1,1,1]
=> 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> [2,2,1]
=> 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> [2,2,1,1]
=> 2
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> 3
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [2,2,2,1]
=> 3
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> [2,2,1,1]
=> 2
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> [2,2,2,1]
=> 3
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 1
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> 3
[2,1,3,4,5] => ([(3,4)],5)
=> [2,1,1,1]
=> 1
[2,1,3,5,4] => ([(1,4),(2,3)],5)
=> [2,2,1]
=> 2
[2,1,4,3,5] => ([(1,4),(2,3)],5)
=> [2,2,1]
=> 2
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> [2,2,2]
=> 3
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> [2,2,2]
=> 3
[2,3,1,4,5] => ([(2,4),(3,4)],5)
=> [2,2,1,1]
=> 2
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> [2,2,2]
=> 3
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> 3
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> [2,2,2,1]
=> 3
[3,1,2,4,5] => ([(2,4),(3,4)],5)
=> [2,2,1,1]
=> 2
[3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> [2,2,2]
=> 3
Description
The number of parts of a partition that are not congruent 1 modulo 3.
Matching statistic: St001280
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00251: Graphs —clique sizes⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00251: Graphs —clique sizes⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> [1]
=> 0
[1,2] => ([],2)
=> [1,1]
=> 0
[2,1] => ([(0,1)],2)
=> [2]
=> 1
[1,2,3] => ([],3)
=> [1,1,1]
=> 0
[1,3,2] => ([(1,2)],3)
=> [2,1]
=> 1
[2,1,3] => ([(1,2)],3)
=> [2,1]
=> 1
[2,3,1] => ([(0,2),(1,2)],3)
=> [2,2]
=> 2
[3,1,2] => ([(0,2),(1,2)],3)
=> [2,2]
=> 2
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[1,2,3,4] => ([],4)
=> [1,1,1,1]
=> 0
[1,2,4,3] => ([(2,3)],4)
=> [2,1,1]
=> 1
[1,3,2,4] => ([(2,3)],4)
=> [2,1,1]
=> 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> [2,2,1]
=> 2
[1,4,2,3] => ([(1,3),(2,3)],4)
=> [2,2,1]
=> 2
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 1
[2,1,3,4] => ([(2,3)],4)
=> [2,1,1]
=> 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> [2,2]
=> 2
[2,3,1,4] => ([(1,3),(2,3)],4)
=> [2,2,1]
=> 2
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> 3
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> 3
[3,1,2,4] => ([(1,3),(2,3)],4)
=> [2,2,1]
=> 2
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> 3
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> 3
[1,2,3,4,5] => ([],5)
=> [1,1,1,1,1]
=> 0
[1,2,3,5,4] => ([(3,4)],5)
=> [2,1,1,1]
=> 1
[1,2,4,3,5] => ([(3,4)],5)
=> [2,1,1,1]
=> 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> [2,2,1,1]
=> 2
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> [2,2,1,1]
=> 2
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 1
[1,3,2,4,5] => ([(3,4)],5)
=> [2,1,1,1]
=> 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> [2,2,1]
=> 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> [2,2,1,1]
=> 2
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> 3
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [2,2,2,1]
=> 3
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> [2,2,1,1]
=> 2
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> [2,2,2,1]
=> 3
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 1
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> 3
[2,1,3,4,5] => ([(3,4)],5)
=> [2,1,1,1]
=> 1
[2,1,3,5,4] => ([(1,4),(2,3)],5)
=> [2,2,1]
=> 2
[2,1,4,3,5] => ([(1,4),(2,3)],5)
=> [2,2,1]
=> 2
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> [2,2,2]
=> 3
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> [2,2,2]
=> 3
[2,3,1,4,5] => ([(2,4),(3,4)],5)
=> [2,2,1,1]
=> 2
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> [2,2,2]
=> 3
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> 3
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> [2,2,2,1]
=> 3
[3,1,2,4,5] => ([(2,4),(3,4)],5)
=> [2,2,1,1]
=> 2
[3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> [2,2,2]
=> 3
Description
The number of parts of an integer partition that are at least two.
The following 153 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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. St000053The number of valleys of the Dyck path. St000074The number of special entries. St000147The largest part of an integer partition. St000157The number of descents of a standard tableau. St000228The size of a partition. St000245The number of ascents of a permutation. St000293The number of inversions of a binary word. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000369The dinv deficit of a Dyck path. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000441The number of successions of a permutation. St000459The hook length of the base cell of a partition. St000672The number of minimal elements in Bruhat order not less than the permutation. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St000992The alternating sum of the parts of an integer partition. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000013The height of a Dyck path. St000026The position of the first return of a Dyck path. St000093The cardinality of a maximal independent set of vertices of a graph. St000381The largest part of an integer composition. St000676The number of odd rises of a Dyck path. St000738The first entry in the last row of a standard tableau. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000808The number of up steps of the associated bargraph. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. 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:
St001389The number of partitions of the same length below the given integer partition. St001814The number of partitions interlacing the given partition. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000925The number of topologically connected components of a set partition. St001062The maximal size of a block of a set partition. St000442The maximal area to the right of an up step of a Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000502The number of successions of a set partitions. St000728The dimension of a set partition. St000444The length of the maximal rise of a Dyck path. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer 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. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000460The hook length of the last cell along the main diagonal of an integer partition. St000667The greatest common divisor of the parts of the partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001571The Cartan determinant of the integer partition. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St000809The reduced reflection length of the permutation. St000632The jump number of the poset. St000996The number of exclusive left-to-right maxima of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St000058The order of a permutation. St001726The number of visible inversions of a permutation. St000957The number of Bruhat lower covers of a permutation. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a 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. St000477The weight of a partition according to Alladi. St000494The number of inversions of distance at most 3 of a permutation. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000831The number of indices that are either descents or recoils. St000933The number of multipartitions of sizes given by an integer partition. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St000478Another weight of a partition according to Alladi. St000539The number of odd inversions of a permutation. St000795The mad of a permutation. St000306The bounce count of a Dyck path. St001489The maximum of the number of descents and the number of inverse descents. St000470The number of runs in a permutation. St000354The number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000028The number of stack-sorts needed to sort a permutation. St000240The number of indices that are not small excedances. St001480The number of simple summands of the module J^2/J^3. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000702The number of weak deficiencies of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. St000454The largest eigenvalue of a graph if it is integral. St000740The last entry of a permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St000485The length of the longest cycle of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000029The depth 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. 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. St000021The number of descents of a permutation. St000120The number of left tunnels of a Dyck path. St000168The number of internal nodes of an ordered tree. St000224The sorting index of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000331The number of upper interactions of a Dyck path. St000332The positive inversions of an alternating sign matrix. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. 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 $eA$. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001298The number of repeated entries in the Lehmer code of a permutation. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001869The maximum cut size of a graph. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St000015The number of peaks of a Dyck path. St000325The width of the tree associated to a permutation. St000991The number of right-to-left minima of a permutation. St001235The global dimension of the corresponding Comp-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. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St000216The absolute length of a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000039The number of crossings of a permutation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000213The number of weak exceedances (also weak excedences) of a permutation. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000264The girth of a graph, which is not a tree. St001861The number of Bruhat lower covers of a permutation. St001894The depth of a signed permutation. St001769The reflection length of a signed permutation. St001864The number of excedances of a signed permutation. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001866The nesting alignments of a signed permutation. St001823The Stasinski-Voll length of a signed permutation. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001877Number of indecomposable injective modules with projective dimension 2.
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