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Your data matches 48 different statistics following compositions of up to 3 maps.
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Matching statistic: St000907
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Mp00047: Ordered trees —to poset⟶ Posets
St000907: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000907: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[]
=> ([],1)
=> 1 = 0 + 1
[[]]
=> ([(0,1)],2)
=> 2 = 1 + 1
[[],[]]
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[[[]]]
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[[[],[]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 2 = 1 + 1
[[[[]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1 = 0 + 1
[[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1 = 0 + 1
[[],[[],[]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 1 = 0 + 1
[[],[[[]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 1 = 0 + 1
[[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1 = 0 + 1
[[[]],[[]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 1 = 0 + 1
[[[],[]],[]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 1 = 0 + 1
[[[[]]],[]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 1 = 0 + 1
[[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> 2 = 1 + 1
[[[],[[]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2 = 1 + 1
[[[[]],[]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2 = 1 + 1
[[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 3 = 2 + 1
[[[[[]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[[],[],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1 = 0 + 1
[[],[],[],[[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 1 = 0 + 1
[[],[],[[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 1 = 0 + 1
[[],[],[[],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 1 = 0 + 1
[[],[],[[[]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 1 = 0 + 1
[[],[[]],[],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 1 = 0 + 1
[[],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 1 = 0 + 1
[[],[[],[]],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 1 = 0 + 1
[[],[[[]]],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 1 = 0 + 1
[[],[[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> 1 = 0 + 1
[[],[[],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> 1 = 0 + 1
[[],[[[]],[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> 1 = 0 + 1
[[],[[[],[]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> 1 = 0 + 1
[[],[[[[]]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> 1 = 0 + 1
[[[]],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 1 = 0 + 1
[[[]],[],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 1 = 0 + 1
[[[]],[[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 1 = 0 + 1
[[[]],[[],[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> 1 = 0 + 1
[[[]],[[[]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> 1 = 0 + 1
[[[],[]],[],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 1 = 0 + 1
[[[[]]],[],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 1 = 0 + 1
[[[],[]],[[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> 1 = 0 + 1
[[[[]]],[[]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> 1 = 0 + 1
[[[],[],[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> 1 = 0 + 1
[[[],[[]]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> 1 = 0 + 1
[[[[]],[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> 1 = 0 + 1
[[[[],[]]],[]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> 1 = 0 + 1
Description
The number of maximal antichains of minimal length in a poset.
Matching statistic: St000234
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000234: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000234: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[]
=> []
=> [1] => [1] => 0
[[]]
=> [1,0]
=> [2,1] => [1,2] => 1
[[],[]]
=> [1,0,1,0]
=> [3,1,2] => [3,1,2] => 0
[[[]]]
=> [1,1,0,0]
=> [2,3,1] => [1,2,3] => 2
[[],[],[]]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => [3,4,1,2] => 0
[[],[[]]]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => [4,1,3,2] => 0
[[[]],[]]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => [1,4,2,3] => 1
[[[],[]]]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => [4,2,1,3] => 0
[[[[]]]]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => 3
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [3,4,5,1,2] => 0
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [3,5,1,4,2] => 0
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [4,1,5,3,2] => 0
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [4,5,3,1,2] => 0
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [5,1,3,4,2] => 0
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [1,4,5,2,3] => 1
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [1,5,2,4,3] => 1
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [4,2,5,1,3] => 0
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [1,2,5,3,4] => 2
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [4,5,2,1,3] => 0
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [5,2,1,4,3] => 0
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [1,5,3,2,4] => 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [5,2,3,1,4] => 0
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => 4
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [3,4,5,6,1,2] => 0
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [3,4,6,1,5,2] => 0
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [3,5,1,6,4,2] => 0
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [3,5,6,4,1,2] => 0
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [3,6,1,4,5,2] => 0
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [4,1,5,6,3,2] => 0
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [4,1,6,3,5,2] => 0
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [4,5,3,6,1,2] => 0
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [5,1,3,6,4,2] => 0
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [4,5,6,3,1,2] => 0
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [4,6,3,1,5,2] => 0
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [5,1,6,4,3,2] => 0
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [5,6,3,4,1,2] => 0
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [6,1,3,4,5,2] => 0
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [1,4,5,6,2,3] => 1
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [1,4,6,2,5,3] => 1
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [1,5,2,6,4,3] => 1
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [1,5,6,4,2,3] => 1
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [1,6,2,4,5,3] => 1
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [4,2,5,6,1,3] => 0
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [1,2,5,6,3,4] => 2
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [4,2,6,1,5,3] => 0
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [1,2,6,3,5,4] => 2
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [4,5,2,6,1,3] => 0
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [5,2,1,6,4,3] => 0
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => [1,5,3,6,2,4] => 1
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => [5,2,3,6,1,4] => 0
Description
The number of global ascents of a permutation.
The global ascents are the integers $i$ such that
$$C(\pi)=\{i\in [n-1] \mid \forall 1 \leq j \leq i < k \leq n: \pi(j) < \pi(k)\}.$$
Equivalently, by the pigeonhole principle,
$$C(\pi)=\{i\in [n-1] \mid \forall 1 \leq j \leq i: \pi(j) \leq i \}.$$
For $n > 1$ it can also be described as an occurrence of the mesh pattern
$$([1,2], \{(0,2),(1,0),(1,1),(2,0),(2,1) \})$$
or equivalently
$$([1,2], \{(0,1),(0,2),(1,1),(1,2),(2,0) \}),$$
see [3].
According to [2], this is also the cardinality of the connectivity set of a permutation. The permutation is connected, when the connectivity set is empty. This gives [[oeis:A003319]].
Matching statistic: St001091
Mp00047: Ordered trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St001091: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St001091: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[]
=> ([],1)
=> ([],1)
=> [1]
=> 0
[[]]
=> ([(0,1)],2)
=> ([],2)
=> [1,1]
=> 1
[[],[]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> [2,1]
=> 0
[[[]]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> [1,1,1]
=> 2
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 0
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> 0
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> 0
[[[],[]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> 1
[[[[]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> [1,1,1,1]
=> 3
[[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 0
[[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 0
[[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 0
[[],[[],[]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 0
[[],[[[]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 0
[[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 0
[[[]],[[]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> 0
[[[],[]],[]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 0
[[[[]]],[]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 0
[[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 1
[[[],[[]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> 1
[[[[]],[]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> 1
[[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> [2,1,1,1]
=> 2
[[[[[]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> [1,1,1,1,1]
=> 4
[[],[],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 0
[[],[],[],[[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 0
[[],[],[[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 0
[[],[],[[],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 0
[[],[],[[[]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 0
[[],[[]],[],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 0
[[],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> 0
[[],[[],[]],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 0
[[],[[[]]],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 0
[[],[[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 0
[[],[[],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 0
[[],[[[]],[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 0
[[],[[[],[]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 0
[[],[[[[]]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> 0
[[[]],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 0
[[[]],[],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> 0
[[[]],[[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> 0
[[[]],[[],[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> 0
[[[]],[[[]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> 0
[[[],[]],[],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 0
[[[[]]],[],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 0
[[[],[]],[[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> 0
[[[[]]],[[]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> 0
[[[],[],[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 0
[[[],[[]]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 0
[[[[]],[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 0
[[[[],[]]],[]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 0
Description
The number of parts in an integer partition whose next smaller part has the same size.
In other words, this is the number of distinct parts subtracted from the number of all parts.
Matching statistic: St000160
Mp00047: Ordered trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000160: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00037: Graphs —to partition of connected 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]
=> 1 = 0 + 1
[[]]
=> ([(0,1)],2)
=> ([],2)
=> [1,1]
=> 2 = 1 + 1
[[],[]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> [2,1]
=> 1 = 0 + 1
[[[]]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> [1,1,1]
=> 3 = 2 + 1
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 1 = 0 + 1
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> 1 = 0 + 1
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> 1 = 0 + 1
[[[],[]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> 2 = 1 + 1
[[[[]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> [1,1,1,1]
=> 4 = 3 + 1
[[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1 = 0 + 1
[[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1 = 0 + 1
[[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1 = 0 + 1
[[],[[],[]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1 = 0 + 1
[[],[[[]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 1 = 0 + 1
[[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1 = 0 + 1
[[[]],[[]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> 1 = 0 + 1
[[[],[]],[]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1 = 0 + 1
[[[[]]],[]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 1 = 0 + 1
[[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 2 = 1 + 1
[[[],[[]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> 2 = 1 + 1
[[[[]],[]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> 2 = 1 + 1
[[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> [2,1,1,1]
=> 3 = 2 + 1
[[[[[]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> [1,1,1,1,1]
=> 5 = 4 + 1
[[],[],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[],[],[[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[],[[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[],[[],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[],[[[]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[[]],[],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[[],[]],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[[[]]],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[[],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[[[]],[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[[[],[]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[[[[]]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[]],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[]],[],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[]],[[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[]],[[],[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[]],[[[]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[],[]],[],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[[]]],[],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[],[]],[[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[[]]],[[]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[],[],[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[],[[]]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[[]],[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[[],[]]],[]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
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: St000383
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[]
=> []
=> [1,0]
=> [1] => 1 = 0 + 1
[[]]
=> [1,0]
=> [1,1,0,0]
=> [2] => 2 = 1 + 1
[[],[]]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> [2,1] => 1 = 0 + 1
[[[]]]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> [3] => 3 = 2 + 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1,1] => 1 = 0 + 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,2] => 2 = 1 + 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1] => 1 = 0 + 1
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,1] => 1 = 0 + 1
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => 4 = 3 + 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => 1 = 0 + 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => 2 = 1 + 1
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => 1 = 0 + 1
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => 1 = 0 + 1
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,3] => 3 = 2 + 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,1] => 1 = 0 + 1
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2] => 2 = 1 + 1
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1,1] => 1 = 0 + 1
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1] => 1 = 0 + 1
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,1,1] => 1 = 0 + 1
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,2] => 2 = 1 + 1
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1] => 1 = 0 + 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,1] => 1 = 0 + 1
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5 = 4 + 1
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1,1] => 1 = 0 + 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [2,1,1,2] => 2 = 1 + 1
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [2,1,2,1] => 1 = 0 + 1
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,1,2,1] => 1 = 0 + 1
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,1,3] => 3 = 2 + 1
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [2,2,1,1] => 1 = 0 + 1
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [2,2,2] => 2 = 1 + 1
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [2,2,1,1] => 1 = 0 + 1
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [2,3,1] => 1 = 0 + 1
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [2,2,1,1] => 1 = 0 + 1
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,2,2] => 2 = 1 + 1
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [2,3,1] => 1 = 0 + 1
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,3,1] => 1 = 0 + 1
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,4] => 4 = 3 + 1
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,1,1,1] => 1 = 0 + 1
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,1,2] => 2 = 1 + 1
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [3,2,1] => 1 = 0 + 1
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,2,1] => 1 = 0 + 1
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,3] => 3 = 2 + 1
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [3,1,1,1] => 1 = 0 + 1
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,1,1] => 1 = 0 + 1
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [3,1,2] => 2 = 1 + 1
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [4,2] => 2 = 1 + 1
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [3,1,1,1] => 1 = 0 + 1
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [3,2,1] => 1 = 0 + 1
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [4,1,1] => 1 = 0 + 1
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,1,1] => 1 = 0 + 1
Description
The last part of an integer composition.
Matching statistic: St000475
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00047: Ordered trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000475: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000475: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[]
=> ([],1)
=> ([],1)
=> [1]
=> 1 = 0 + 1
[[]]
=> ([(0,1)],2)
=> ([],2)
=> [1,1]
=> 2 = 1 + 1
[[],[]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> [2,1]
=> 1 = 0 + 1
[[[]]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> [1,1,1]
=> 3 = 2 + 1
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 1 = 0 + 1
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> 1 = 0 + 1
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> 1 = 0 + 1
[[[],[]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> 2 = 1 + 1
[[[[]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> [1,1,1,1]
=> 4 = 3 + 1
[[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1 = 0 + 1
[[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1 = 0 + 1
[[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1 = 0 + 1
[[],[[],[]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1 = 0 + 1
[[],[[[]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 1 = 0 + 1
[[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1 = 0 + 1
[[[]],[[]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> 1 = 0 + 1
[[[],[]],[]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1 = 0 + 1
[[[[]]],[]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 1 = 0 + 1
[[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 2 = 1 + 1
[[[],[[]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> 2 = 1 + 1
[[[[]],[]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> 2 = 1 + 1
[[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> [2,1,1,1]
=> 3 = 2 + 1
[[[[[]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> [1,1,1,1,1]
=> 5 = 4 + 1
[[],[],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[],[],[[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[],[[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[],[[],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[],[[[]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[[]],[],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[[],[]],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[[[]]],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[[],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[[[]],[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[[[],[]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[[[[]]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[]],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[]],[],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[]],[[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[]],[[],[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[]],[[[]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[],[]],[],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[[]]],[],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[],[]],[[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[[]]],[[]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[],[],[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[],[[]]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[[]],[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[[],[]]],[]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
Description
The number of parts equal to 1 in a partition.
Matching statistic: St000974
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000974: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000974: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[]
=> []
=> [1,0]
=> [[]]
=> 1 = 0 + 1
[[]]
=> [1,0]
=> [1,1,0,0]
=> [[[]]]
=> 2 = 1 + 1
[[],[]]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> [[[],[]]]
=> 1 = 0 + 1
[[[]]]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> [[[[]]]]
=> 3 = 2 + 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [[[],[],[]]]
=> 1 = 0 + 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [[[],[[]]]]
=> 1 = 0 + 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [[[[]],[]]]
=> 1 = 0 + 1
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [[[[],[]]]]
=> 2 = 1 + 1
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [[[[[]]]]]
=> 4 = 3 + 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[[],[],[],[]]]
=> 1 = 0 + 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [[[],[],[[]]]]
=> 1 = 0 + 1
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [[[],[[]],[]]]
=> 1 = 0 + 1
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [[[],[[],[]]]]
=> 1 = 0 + 1
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [[[],[[[]]]]]
=> 1 = 0 + 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[[[]],[],[]]]
=> 1 = 0 + 1
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[[[]],[[]]]]
=> 1 = 0 + 1
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[[[],[]],[]]]
=> 1 = 0 + 1
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[[[[]]],[]]]
=> 1 = 0 + 1
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[[[],[],[]]]]
=> 2 = 1 + 1
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [[[[],[[]]]]]
=> 2 = 1 + 1
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[[[[]],[]]]]
=> 2 = 1 + 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[[[[],[]]]]]
=> 3 = 2 + 1
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[[[[[]]]]]]
=> 5 = 4 + 1
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [[[],[],[],[],[]]]
=> 1 = 0 + 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [[[],[],[],[[]]]]
=> 1 = 0 + 1
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [[[],[],[[]],[]]]
=> 1 = 0 + 1
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [[[],[],[[],[]]]]
=> 1 = 0 + 1
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [[[],[],[[[]]]]]
=> 1 = 0 + 1
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [[[],[[]],[],[]]]
=> 1 = 0 + 1
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [[[],[[]],[[]]]]
=> 1 = 0 + 1
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [[[],[[],[]],[]]]
=> 1 = 0 + 1
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [[[],[[[]]],[]]]
=> 1 = 0 + 1
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [[[],[[],[],[]]]]
=> 1 = 0 + 1
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [[[],[[],[[]]]]]
=> 1 = 0 + 1
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [[[],[[[]],[]]]]
=> 1 = 0 + 1
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [[[],[[[],[]]]]]
=> 1 = 0 + 1
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [[[],[[[[]]]]]]
=> 1 = 0 + 1
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [[[[]],[],[],[]]]
=> 1 = 0 + 1
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [[[[]],[],[[]]]]
=> 1 = 0 + 1
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [[[[]],[[]],[]]]
=> 1 = 0 + 1
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [[[[]],[[],[]]]]
=> 1 = 0 + 1
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [[[[]],[[[]]]]]
=> 1 = 0 + 1
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [[[[],[]],[],[]]]
=> 1 = 0 + 1
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [[[[[]]],[],[]]]
=> 1 = 0 + 1
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [[[[],[]],[[]]]]
=> 1 = 0 + 1
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [[[[[]]],[[]]]]
=> 1 = 0 + 1
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [[[[],[],[]],[]]]
=> 1 = 0 + 1
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [[[[],[[]]],[]]]
=> 1 = 0 + 1
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [[[[[]],[]],[]]]
=> 1 = 0 + 1
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [[[[[],[]]],[]]]
=> 1 = 0 + 1
Description
The length of the trunk of an ordered tree.
This is the length of the path from the root to the first vertex which has not exactly one child.
Matching statistic: St001933
Mp00047: Ordered trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St001933: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St001933: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[]
=> ([],1)
=> ([],1)
=> [1]
=> 1 = 0 + 1
[[]]
=> ([(0,1)],2)
=> ([],2)
=> [1,1]
=> 2 = 1 + 1
[[],[]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> [2,1]
=> 1 = 0 + 1
[[[]]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> [1,1,1]
=> 3 = 2 + 1
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 1 = 0 + 1
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> 1 = 0 + 1
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> 1 = 0 + 1
[[[],[]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> 2 = 1 + 1
[[[[]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> [1,1,1,1]
=> 4 = 3 + 1
[[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1 = 0 + 1
[[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1 = 0 + 1
[[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1 = 0 + 1
[[],[[],[]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1 = 0 + 1
[[],[[[]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 1 = 0 + 1
[[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1 = 0 + 1
[[[]],[[]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> 1 = 0 + 1
[[[],[]],[]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1 = 0 + 1
[[[[]]],[]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 1 = 0 + 1
[[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 2 = 1 + 1
[[[],[[]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> 2 = 1 + 1
[[[[]],[]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> 2 = 1 + 1
[[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> [2,1,1,1]
=> 3 = 2 + 1
[[[[[]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> [1,1,1,1,1]
=> 5 = 4 + 1
[[],[],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[],[],[[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[],[[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[],[[],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[],[[[]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[[]],[],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[[],[]],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[[[]]],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[[],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[[[]],[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[[[],[]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[],[[[[]]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[]],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[]],[],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[]],[[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[]],[[],[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[]],[[[]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[],[]],[],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[[]]],[],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[],[]],[[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[[]]],[[]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[],[],[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[],[[]]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[[]],[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
[[[[],[]]],[]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 0 + 1
Description
The largest multiplicity of a part in an integer partition.
Matching statistic: St000221
(load all 25 compositions to match this statistic)
(load all 25 compositions to match this statistic)
Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000221: Permutations ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000221: Permutations ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[]
=> .
=> ? => ? = 0
[[]]
=> [.,.]
=> [1] => 1
[[],[]]
=> [[.,.],.]
=> [1,2] => 2
[[[]]]
=> [.,[.,.]]
=> [2,1] => 0
[[],[],[]]
=> [[[.,.],.],.]
=> [1,2,3] => 3
[[],[[]]]
=> [[.,.],[.,.]]
=> [3,1,2] => 0
[[[]],[]]
=> [[.,[.,.]],.]
=> [2,1,3] => 1
[[[],[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => 0
[[[[]]]]
=> [.,[.,[.,.]]]
=> [3,2,1] => 0
[[],[],[],[]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => 4
[[],[],[[]]]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => 0
[[],[[]],[]]
=> [[[.,.],[.,.]],.]
=> [3,1,2,4] => 1
[[],[[],[]]]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => 0
[[],[[[]]]]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 0
[[[]],[],[]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => 2
[[[]],[[]]]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 0
[[[],[]],[]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => 1
[[[[]]],[]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 1
[[[],[],[]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 0
[[[],[[]]]]
=> [.,[[.,.],[.,.]]]
=> [4,2,3,1] => 0
[[[[]],[]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 0
[[[[],[]]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 0
[[[[[]]]]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 0
[[],[],[],[],[]]
=> [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => 5
[[],[],[],[[]]]
=> [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 0
[[],[],[[]],[]]
=> [[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => 1
[[],[],[[],[]]]
=> [[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => 0
[[],[],[[[]]]]
=> [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => 0
[[],[[]],[],[]]
=> [[[[.,.],[.,.]],.],.]
=> [3,1,2,4,5] => 2
[[],[[]],[[]]]
=> [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 0
[[],[[],[]],[]]
=> [[[.,.],[[.,.],.]],.]
=> [3,4,1,2,5] => 1
[[],[[[]]],[]]
=> [[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => 1
[[],[[],[],[]]]
=> [[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => 0
[[],[[],[[]]]]
=> [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => 0
[[],[[[]],[]]]
=> [[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => 0
[[],[[[],[]]]]
=> [[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => 0
[[],[[[[]]]]]
=> [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => 0
[[[]],[],[],[]]
=> [[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => 3
[[[]],[],[[]]]
=> [[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => 0
[[[]],[[]],[]]
=> [[[.,[.,.]],[.,.]],.]
=> [4,2,1,3,5] => 1
[[[]],[[],[]]]
=> [[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => 0
[[[]],[[[]]]]
=> [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 0
[[[],[]],[],[]]
=> [[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => 2
[[[[]]],[],[]]
=> [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => 2
[[[],[]],[[]]]
=> [[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => 0
[[[[]]],[[]]]
=> [[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => 0
[[[],[],[]],[]]
=> [[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => 1
[[[],[[]]],[]]
=> [[.,[[.,.],[.,.]]],.]
=> [4,2,3,1,5] => 1
[[[[]],[]],[]]
=> [[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => 1
[[[[],[]]],[]]
=> [[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => 1
[[[[[]]]],[]]
=> [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => 1
Description
The number of strong fixed points of a permutation.
$i$ is called a strong fixed point of $\pi$ if
1. $j < i$ implies $\pi_j < \pi_i$, and
2. $j > i$ implies $\pi_j > \pi_i$
This can be described as an occurrence of the mesh pattern ([1], {(0,1),(1,0)}), i.e., the upper left and the lower right quadrants are shaded, see [3].
The generating function for the joint-distribution (RLmin, LRmax, strong fixed points) has a continued fraction expression as given in [4, Lemma 3.2], for LRmax see [[St000314]].
Matching statistic: St001107
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001107: Dyck paths ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001107: Dyck paths ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[]
=> []
=> [1,0]
=> ? = 0 + 1
[[]]
=> [1,0]
=> [1,1,0,0]
=> 2 = 1 + 1
[[],[]]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[[[]]]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> 3 = 2 + 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1 = 0 + 1
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1 = 0 + 1
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 0 + 1
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2 = 1 + 1
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2 = 1 + 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 3 = 2 + 1
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 4 + 1
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> 1 = 0 + 1
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> 1 = 0 + 1
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 1 = 0 + 1
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 1 = 0 + 1
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> 1 = 0 + 1
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 1 = 0 + 1
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> 1 = 0 + 1
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 1 = 0 + 1
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 1 = 0 + 1
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 1 = 0 + 1
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> 1 = 0 + 1
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> 1 = 0 + 1
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1 = 0 + 1
Description
The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path.
In other words, this is the lowest height of a valley of a Dyck path, or its semilength in case of the unique path without valleys.
The following 38 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000297The number of leading ones in a binary word. St000315The number of isolated vertices of a graph. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St000461The rix statistic of a permutation. St000674The number of hills of a Dyck path. St001342The number of vertices in the center of a graph. St001368The number of vertices of maximal degree in a graph. St000326The position of the first one in a binary word after appending a 1 at the end. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000225Difference between largest and smallest parts in a partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St001175The size of a partition minus the hook length of the base cell. St001586The number of odd parts smaller than the largest even part in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St000455The second largest eigenvalue of a graph if it is integral. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000929The constant term of the character polynomial of an integer partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001651The Frankl number of a lattice. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000056The decomposition (or block) number of a permutation. St000654The first descent of a permutation. St001826The maximal number of leaves on a vertex of a graph. St001672The restrained domination number of a graph. St000990The first ascent of a permutation. St001479The number of bridges of a graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid.
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