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Your data matches 21 different statistics following compositions of up to 3 maps.
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Matching statistic: St000932
(load all 23 compositions to match this statistic)
(load all 23 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000932: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000932: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> 1
[2]
=> [1,1,0,0,1,0]
=> 0
[1,1]
=> [1,0,1,1,0,0]
=> 1
[3]
=> [1,1,1,0,0,0,1,0]
=> 0
[2,1]
=> [1,0,1,0,1,0]
=> 2
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
[3,1]
=> [1,1,0,1,0,0,1,0]
=> 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 0
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 0
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 0
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 0
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 3
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 2
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> 0
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 2
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> 0
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> 0
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> 2
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 0
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> 1
Description
The number of occurrences of the pattern UDU in a Dyck path.
The number of Dyck paths with statistic value 0 are counted by the Motzkin numbers [1].
Matching statistic: St001067
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(load all 23 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001067: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001067: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> 1
[2]
=> [1,1,0,0,1,0]
=> 0
[1,1]
=> [1,0,1,1,0,0]
=> 1
[3]
=> [1,1,1,0,0,0,1,0]
=> 0
[2,1]
=> [1,0,1,0,1,0]
=> 2
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
[3,1]
=> [1,1,0,1,0,0,1,0]
=> 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 0
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 0
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 0
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 0
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 3
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 2
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> 0
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 2
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> 0
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> 0
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> 2
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 0
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> 1
Description
The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra.
Matching statistic: St001189
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001189: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001189: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[2]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 0
[1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 0
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 2
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 0
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 0
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 3
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> 2
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> 0
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> 2
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> 0
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> 0
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> 2
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 0
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 0
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> 1
Description
The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path.
This is also the number of '''twin vertices''' in the unit interval graph corresponding to the Dyck path.
A Dyck path of size $n$ determines a unit interval graph on vertex set $[n]$, see [1]. Two vertices ${i,i+1}$ are '''twins''' if they have the same closed neighbourhood in the unit interval graph.
Put differently, this is the number of indices $i$ such that the $i$-th and $(i+1)$-st up steps are consecutive, and the $i$-th and $(i+1)$-st down steps are consecutive.
Matching statistic: St000502
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000502: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000502: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> {{1,2}}
=> 1
[2]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> {{1,3},{2}}
=> 0
[1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> 0
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> {{1,2,3}}
=> 2
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> {{1,5},{2},{3},{4}}
=> 0
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 2
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,6},{2},{3},{4},{5}}
=> 0
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> {{1,5},{2,3},{4}}
=> 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 2
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> 2
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4},{5,6}}
=> 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,7},{2},{3},{4},{5},{6}}
=> 0
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> {{1,6},{2,3},{4},{5}}
=> 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> {{1,5},{2,4},{3}}
=> 0
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> {{1,5},{2},{3,4}}
=> 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> 0
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 3
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> {{1,3,4},{2},{5}}
=> 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> 0
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 2
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> {{1,2,3},{4},{5},{6}}
=> 2
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> {{1,7},{2,3},{4},{5},{6}}
=> 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> {{1,6},{2,4},{3},{5}}
=> 0
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> {{1,6},{2},{3,4},{5}}
=> 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> {{1,2,5},{3},{4}}
=> 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> {{1,4,5},{2,3}}
=> 2
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> {{1,4,5},{2},{3}}
=> 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> {{1,2,4},{3},{5}}
=> 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> 3
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> {{1,3,4},{2},{5},{6}}
=> 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 2
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> {{1},{2,3,4},{5},{6}}
=> 2
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> {{1,7},{2,4},{3},{5},{6}}
=> 0
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> {{1,7},{2},{3,4},{5},{6}}
=> 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> {{1,6},{2,5},{3},{4}}
=> 0
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> {{1,6},{2,3},{4,5}}
=> 2
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> {{1,6},{2},{3},{4,5}}
=> 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> {{1},{2,6},{3},{4},{5}}
=> 0
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> {{1,2,5},{3,4}}
=> 2
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> {{1,3,5},{2},{4}}
=> 0
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> {{1,5},{2,3,4}}
=> 2
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> {{1,4,5},{2},{3},{6}}
=> 1
Description
The number of successions of a set partitions.
This is the number of indices $i$ such that $i$ and $i+1$ belonging to the same block.
Matching statistic: St001640
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St001640: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St001640: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,2] => 1
[2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,3,2] => 0
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [3,1,2] => 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 0
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => 2
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => 0
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => 2
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,4,5,6,2] => 0
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => 2
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => 2
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,5,6,1,2] => 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,3,4,5,6,7,2] => 0
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,4,5,2,6] => 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 0
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => 0
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 3
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => 0
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => 2
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [3,4,5,1,2,6] => 2
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,3,4,5,6,2,7] => 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,3,4,2,6,5] => 0
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,3,4,6,2,5] => 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 2
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => 3
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [3,4,1,6,2,5] => 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [4,5,1,2,3] => 2
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [3,4,6,1,2,5] => 2
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,3,4,5,2,7,6] => 0
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,3,4,5,7,2,6] => 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,3,2,5,6,4] => 0
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,4,2,5,6] => 2
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,3,5,6,2,4] => 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [3,1,4,5,6,2] => 0
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 2
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => 0
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,3,5] => 2
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [3,1,5,6,2,4] => 1
Description
The number of ascent tops in the permutation such that all smaller elements appear before.
Matching statistic: St001484
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001484: Integer partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
St001484: Integer partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1]
=> 1
[2]
=> [1,1]
=> 0
[1,1]
=> [2]
=> 1
[3]
=> [1,1,1]
=> 0
[2,1]
=> [2,1]
=> 2
[1,1,1]
=> [3]
=> 1
[4]
=> [1,1,1,1]
=> 0
[3,1]
=> [2,1,1]
=> 1
[2,2]
=> [2,2]
=> 0
[2,1,1]
=> [3,1]
=> 2
[1,1,1,1]
=> [4]
=> 1
[5]
=> [1,1,1,1,1]
=> 0
[4,1]
=> [2,1,1,1]
=> 1
[3,2]
=> [2,2,1]
=> 1
[3,1,1]
=> [3,1,1]
=> 1
[2,2,1]
=> [3,2]
=> 2
[2,1,1,1]
=> [4,1]
=> 2
[1,1,1,1,1]
=> [5]
=> 1
[6]
=> [1,1,1,1,1,1]
=> 0
[5,1]
=> [2,1,1,1,1]
=> 1
[4,2]
=> [2,2,1,1]
=> 0
[4,1,1]
=> [3,1,1,1]
=> 1
[3,3]
=> [2,2,2]
=> 0
[3,2,1]
=> [3,2,1]
=> 3
[3,1,1,1]
=> [4,1,1]
=> 1
[2,2,2]
=> [3,3]
=> 0
[2,2,1,1]
=> [4,2]
=> 2
[2,1,1,1,1]
=> [5,1]
=> 2
[6,1]
=> [2,1,1,1,1,1]
=> 1
[5,2]
=> [2,2,1,1,1]
=> 0
[5,1,1]
=> [3,1,1,1,1]
=> 1
[4,3]
=> [2,2,2,1]
=> 1
[4,2,1]
=> [3,2,1,1]
=> 2
[4,1,1,1]
=> [4,1,1,1]
=> 1
[3,3,1]
=> [3,2,2]
=> 1
[3,2,2]
=> [3,3,1]
=> 1
[3,2,1,1]
=> [4,2,1]
=> 3
[3,1,1,1,1]
=> [5,1,1]
=> 1
[2,2,2,1]
=> [4,3]
=> 2
[2,2,1,1,1]
=> [5,2]
=> 2
[6,2]
=> [2,2,1,1,1,1]
=> 0
[6,1,1]
=> [3,1,1,1,1,1]
=> 1
[5,3]
=> [2,2,2,1,1]
=> 0
[5,2,1]
=> [3,2,1,1,1]
=> 2
[5,1,1,1]
=> [4,1,1,1,1]
=> 1
[4,4]
=> [2,2,2,2]
=> 0
[4,3,1]
=> [3,2,2,1]
=> 2
[4,2,2]
=> [3,3,1,1]
=> 0
[4,2,1,1]
=> [4,2,1,1]
=> 2
[4,1,1,1,1]
=> [5,1,1,1]
=> 1
[6,5,3,2,2,1]
=> [6,5,3,2,2,1]
=> ? = 4
[6,5,4,2,1,1]
=> [6,4,3,3,2,1]
=> ? = 4
[6,5,3,2,1,1]
=> [6,4,3,2,2,1]
=> ? = 4
[6,5,3,2,2]
=> [5,5,3,2,2,1]
=> ? = 2
[6,5,3,3,1]
=> [5,4,4,2,2,1]
=> ? = 2
Description
The number of singletons of an integer partition.
A singleton in an integer partition is a part that appear precisely once.
Matching statistic: St000445
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000445: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 100%
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000445: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0
[2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 0
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> 0
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> 2
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 0
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> 0
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 0
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 0
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> 2
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> 2
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> 0
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 2
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 3
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 2
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> 2
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> 0
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> 0
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> 2
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> 0
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> 2
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> 0
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> 2
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> 1
[6,3,2]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> ? = 1
[6,3,1,1]
=> [1,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> ? = 1
[6,2,2,1]
=> [1,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> ? = 2
[6,4,2]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 0
[6,4,1,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> ? = 1
[6,3,3]
=> [1,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> ? = 0
[6,2,2,2]
=> [1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> ? = 0
[6,2,2,1,1]
=> [1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> ? = 2
[6,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 2
[6,5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 2
[6,4,3]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 1
[6,4,1,1,1]
=> [1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> ? = 1
[6,3,3,1]
=> [1,1,1,0,1,0,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> ? = 1
[6,3,2,2]
=> [1,1,1,0,0,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> ? = 1
[6,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> ? = 1
[6,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 2
[6,5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 1
[6,4,3,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> ? = 2
[6,4,2,2]
=> [1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> ? = 0
[6,4,2,1,1]
=> [1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> ? = 2
[6,4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> ? = 1
[6,3,3,2]
=> [1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> ? = 1
[6,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> ? = 3
[6,5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> ? = 1
[6,4,3,1,1]
=> [1,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> ? = 2
[6,4,2,2,1]
=> [1,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> ? = 2
[6,4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> ? = 2
[6,3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,1,1,0,0,0,1,0,0]
=> ? = 1
[6,3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,1,0,0,0]
=> ? = 1
[6,3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> ? = 3
[6,5,3,2]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 2
[6,5,3,1,1]
=> [1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> ? = 2
[6,5,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 3
[6,4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 0
[6,4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> ? = 1
[6,4,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> ? = 2
[6,4,2,2,2]
=> [1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> ? = 0
[6,4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> ? = 2
[6,3,3,3,1]
=> [1,1,0,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> ? = 1
[6,3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> ? = 1
[6,5,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 2
[6,5,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 3
[6,4,4,2,1]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> ? = 2
[6,4,3,3,1]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> ? = 2
[6,4,3,2,2]
=> [1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> ? = 2
[6,5,4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> ? = 4
[6,5,3,2,2]
=> [1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> ? = 2
[6,5,3,3,1]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> ? = 2
Description
The number of rises of length 1 of a Dyck path.
Matching statistic: St000441
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000441: Permutations ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 100%
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000441: Permutations ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [[.,.],.]
=> [1,2] => 1
[2]
=> [1,1,0,0,1,0]
=> [[.,[.,.]],.]
=> [2,1,3] => 0
[1,1]
=> [1,0,1,1,0,0]
=> [[.,.],[.,.]]
=> [3,1,2] => 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 0
[2,1]
=> [1,0,1,0,1,0]
=> [[[.,.],.],.]
=> [1,2,3] => 2
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => 0
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => 2
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[.,[.,[.,[.,[.,.]]]]],.]
=> [5,4,3,2,1,6] => 0
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[[.,.],[.,.]],.]
=> [3,1,2,4] => 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => 2
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => 2
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[.,.],[.,[.,[.,[.,.]]]]]
=> [6,5,4,3,1,2] => 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [[.,[.,[.,[.,[.,[.,.]]]]]],.]
=> [6,5,4,3,2,1,7] => 0
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[.,[.,[.,[[.,.],.]]]],.]
=> [4,5,3,2,1,6] => 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => 0
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[.,[[.,.],[.,.]]],.]
=> [4,2,3,1,5] => 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => 0
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => 3
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 0
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => 2
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [[.,.],[.,[.,[[.,.],.]]]]
=> [5,6,4,3,1,2] => 2
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [[.,[.,[.,[.,[[.,.],.]]]]],.]
=> [5,6,4,3,2,1,7] => 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [[.,[.,[[.,[.,.]],.]]],.]
=> [4,3,5,2,1,6] => 0
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [[.,[.,[[.,.],[.,.]]]],.]
=> [5,3,4,2,1,6] => 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => 2
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => 3
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [[.,.],[.,[[.,[.,.]],.]]]
=> [5,4,6,3,1,2] => 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => 2
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [[.,.],[.,[[.,.],[.,.]]]]
=> [6,4,5,3,1,2] => 2
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [[.,[.,[.,[[.,[.,.]],.]]]],.]
=> [5,4,6,3,2,1,7] => 0
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [[.,[.,[.,[[.,.],[.,.]]]]],.]
=> [6,4,5,3,2,1,7] => ? = 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [[.,[[.,[.,[.,.]]],.]],.]
=> [4,3,2,5,1,6] => 0
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [[.,[.,[[[.,.],.],.]]],.]
=> [3,4,5,2,1,6] => 2
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [[.,[[.,.],[.,[.,.]]]],.]
=> [5,4,2,3,1,6] => 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [[.,[.,[.,[.,.]]]],[.,.]]
=> [6,4,3,2,1,5] => 0
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => 2
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[[.,[.,.]],[.,.]],.]
=> [4,2,1,3,5] => 0
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [[[.,.],[[.,.],.]],.]
=> [3,4,1,2,5] => 2
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [[.,.],[[.,[.,[.,.]]],.]]
=> [5,4,3,6,1,2] => 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => 1
[6,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [[.,[.,[[.,.],[[.,.],.]]]],.]
=> [5,6,3,4,2,1,7] => ? = 2
[6,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [[.,[[.,.],[.,[.,[.,.]]]]],.]
=> [6,5,4,2,3,1,7] => ? = 1
[6,4,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> [[.,[[.,[.,[[.,.],.]]],.]],.]
=> [4,5,3,2,6,1,7] => ? = 1
[6,3,2]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [[.,[.,[[[.,[.,.]],.],.]]],.]
=> [4,3,5,6,2,1,7] => ? = 1
[6,3,1,1]
=> [1,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> [[.,[.,[[[.,.],[.,.]],.]]],.]
=> [5,3,4,6,2,1,7] => ? = 1
[6,2,1,1,1]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [[.,[[.,.],[.,[[.,.],.]]]],.]
=> [5,6,4,2,3,1,7] => ? = 2
[6,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [[[.,.],[.,[.,[.,[.,.]]]]],.]
=> [6,5,4,3,1,2,7] => ? = 1
[6,5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [[[.,[.,[.,[[.,.],.]]]],.],.]
=> [4,5,3,2,1,6,7] => ? = 2
[6,4,2]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [[.,[[.,[[.,[.,.]],.]],.]],.]
=> [4,3,5,2,6,1,7] => ? = 0
[6,4,1,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0,1,0]
=> [[.,[[.,[[.,.],[.,.]]],.]],.]
=> [5,3,4,2,6,1,7] => ? = 1
[6,3,2,1]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [[.,[.,[[[[.,.],.],.],.]]],.]
=> [3,4,5,6,2,1,7] => ? = 3
[6,3,1,1,1]
=> [1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> [[.,[[.,.],[[.,[.,.]],.]]],.]
=> [5,4,6,2,3,1,7] => ? = 1
[6,2,2,2]
=> [1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> [[.,[[.,[.,.]],[.,[.,.]]]],.]
=> [6,5,3,2,4,1,7] => ? = 0
[6,2,2,1,1]
=> [1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> [[.,[[.,.],[[.,.],[.,.]]]],.]
=> [6,4,5,2,3,1,7] => ? = 2
[6,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [[[.,.],[.,[.,[[.,.],.]]]],.]
=> [5,6,4,3,1,2,7] => ? = 2
[6,5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [[[.,[.,[[.,[.,.]],.]]],.],.]
=> [4,3,5,2,1,6,7] => ? = 1
[6,5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0,1,0]
=> [[[.,[.,[[.,.],[.,.]]]],.],.]
=> [5,3,4,2,1,6,7] => ? = 2
[6,4,3]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [[.,[[[.,[.,[.,.]]],.],.]],.]
=> [4,3,2,5,6,1,7] => ? = 1
[6,4,2,1]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [[.,[[.,[[[.,.],.],.]],.]],.]
=> [3,4,5,2,6,1,7] => ? = 2
[6,4,1,1,1]
=> [1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [[.,[[[.,.],[.,[.,.]]],.]],.]
=> [5,4,2,3,6,1,7] => ? = 1
[6,3,2,2]
=> [1,1,1,0,0,1,1,0,1,0,0,0,1,0]
=> [[.,[[.,[.,.]],[[.,.],.]]],.]
=> [5,6,3,2,4,1,7] => ? = 1
[6,3,2,1,1]
=> [1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [[.,[[.,.],[[[.,.],.],.]]],.]
=> [4,5,6,2,3,1,7] => ? = 3
[6,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [[[.,.],[.,[[.,[.,.]],.]]],.]
=> [5,4,6,3,1,2,7] => ? = 1
[6,2,2,2,1]
=> [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [[.,[[[.,.],.],[.,[.,.]]]],.]
=> [6,5,2,3,4,1,7] => ? = 2
[6,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [[[.,.],[.,[[.,.],[.,.]]]],.]
=> [6,4,5,3,1,2,7] => ? = 2
[6,5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [[[.,[[.,[.,[.,.]]],.]],.],.]
=> [4,3,2,5,1,6,7] => ? = 1
[6,5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [[[.,[.,[[[.,.],.],.]]],.],.]
=> [3,4,5,2,1,6,7] => ? = 3
[6,5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [[[.,[[.,.],[.,[.,.]]]],.],.]
=> [5,4,2,3,1,6,7] => ? = 2
[6,4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [[[.,[.,[.,[.,.]]]],[.,.]],.]
=> [6,4,3,2,1,5,7] => ? = 0
[6,4,3,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [[.,[[[.,[[.,.],.]],.],.]],.]
=> [3,4,2,5,6,1,7] => ? = 2
[6,4,2,2]
=> [1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [[.,[[[.,[.,.]],[.,.]],.]],.]
=> [5,3,2,4,6,1,7] => ? = 0
[6,4,2,1,1]
=> [1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [[.,[[[.,.],[[.,.],.]],.]],.]
=> [4,5,2,3,6,1,7] => ? = 2
[6,4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [[[.,.],[[.,[.,[.,.]]],.]],.]
=> [5,4,3,6,1,2,7] => ? = 1
[6,3,3,2]
=> [1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> [[.,[[[.,[.,.]],.],[.,.]]],.]
=> [6,3,2,4,5,1,7] => ? = 1
[6,3,3,1,1]
=> [1,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> [[.,[[[.,.],[.,.]],[.,.]]],.]
=> [6,4,2,3,5,1,7] => ? = 1
[6,3,2,2,1]
=> [1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> [[.,[[[.,.],.],[[.,.],.]]],.]
=> [5,6,2,3,4,1,7] => ? = 3
[6,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [[[.,.],[.,[[[.,.],.],.]]],.]
=> [4,5,6,3,1,2,7] => ? = 3
[6,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [[[.,[.,.]],[.,[.,[.,.]]]],.]
=> [6,5,4,2,1,3,7] => ? = 0
[6,5,3,1]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [[[.,[[.,[[.,.],.]],.]],.],.]
=> [3,4,2,5,1,6,7] => ? = 2
[6,5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0,1,0]
=> [[[.,[[.,[.,.]],[.,.]]],.],.]
=> [5,3,2,4,1,6,7] => ? = 1
[6,5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> [[[.,[[.,.],[[.,.],.]]],.],.]
=> [4,5,2,3,1,6,7] => ? = 3
[6,5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [[[[.,.],[.,[.,[.,.]]]],.],.]
=> [5,4,3,1,2,6,7] => ? = 2
[6,4,4,1]
=> [1,1,1,0,1,0,0,0,1,1,0,0,1,0]
=> [[[.,[.,[[.,.],.]]],[.,.]],.]
=> [6,3,4,2,1,5,7] => ? = 1
[6,4,3,2]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> [[.,[[[[.,[.,.]],.],.],.]],.]
=> [3,2,4,5,6,1,7] => ? = 2
[6,4,3,1,1]
=> [1,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> [[.,[[[[.,.],[.,.]],.],.]],.]
=> [4,2,3,5,6,1,7] => ? = 2
[6,4,2,2,1]
=> [1,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [[.,[[[[.,.],.],[.,.]],.]],.]
=> [5,2,3,4,6,1,7] => ? = 2
[6,4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [[[.,.],[[.,[[.,.],.]],.]],.]
=> [4,5,3,6,1,2,7] => ? = 2
[6,3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [[[.,[.,[.,.]]],[.,[.,.]]],.]
=> [6,5,3,2,1,4,7] => ? = 0
[6,3,3,2,1]
=> [1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [[.,[[[[.,.],.],.],[.,.]]],.]
=> [6,2,3,4,5,1,7] => ? = 3
Description
The number of successions of a permutation.
A succession of a permutation $\pi$ is an index $i$ such that $\pi(i)+1 = \pi(i+1)$. Successions are also known as ''small ascents'' or ''1-rises''.
Matching statistic: St001061
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St001061: Permutations ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 86%
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St001061: Permutations ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 86%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> [2,1] => 1
[2]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => 0
[1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => 2
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => 0
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 2
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 2
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => 0
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,1,3,5,6,4] => 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => 0
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 0
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 3
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 0
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 2
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,6,5] => 2
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [2,1,3,4,6,7,5] => ? = 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [2,3,1,5,6,4] => 0
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,3,2,5,6,4] => 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => 2
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,2,5,1] => 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 3
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> [2,3,1,4,6,5] => 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 2
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,3,2,4,6,5] => 2
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [2,3,1,4,6,7,5] => ? = 0
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,3,2,4,6,7,5] => 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,3,5,4,6,2] => 0
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> [3,2,1,5,6,4] => 2
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,5,4,6,3] => 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => 0
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,3,5,2,1] => 2
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,2,1] => 0
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => 2
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,3,4,2,6,5] => 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => 1
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => 1
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> [3,2,1,4,6,7,5] => ? = 2
[6,3,1]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,0,1,0,0]
=> [3,2,4,1,6,7,5] => ? = 1
[6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [2,3,4,1,6,7,5] => ? = 0
[6,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,3,6,7,5] => ? = 2
[6,4,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [2,1,4,6,5,7,3] => ? = 1
[6,3,2]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> [3,4,2,1,6,7,5] => ? = 1
[6,3,1,1]
=> [1,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [2,4,3,1,6,7,5] => ? = 1
[6,2,1,1,1]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> [2,1,3,6,5,7,4] => ? = 2
[6,5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [2,1,3,6,7,5,4] => ? = 2
[6,4,2]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [2,4,3,6,5,7,1] => ? = 0
[6,3,3]
=> [1,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [2,3,4,6,5,7,1] => ? = 0
[6,3,2,1]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,0,0]
=> [4,3,2,1,6,7,5] => ? = 3
[6,3,1,1,1]
=> [1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,1,0,0,1,0,0]
=> [2,3,1,6,5,7,4] => ? = 1
[6,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [2,1,3,5,7,6,4] => ? = 2
[6,5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,1,0,1,0,0,0]
=> [2,3,1,6,7,5,4] => ? = 1
[6,4,2,1]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> [4,3,2,6,5,7,1] => ? = 2
[6,3,3,1]
=> [1,1,1,0,1,0,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> [3,4,2,6,5,7,1] => ? = 1
[6,3,2,2]
=> [1,1,1,0,0,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [3,2,4,6,5,7,1] => ? = 1
[6,3,2,1,1]
=> [1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [3,2,1,6,5,7,4] => ? = 3
[6,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,1,0,0,0]
=> [2,3,1,5,7,6,4] => ? = 1
[6,5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> [3,2,1,6,7,5,4] => ? = 3
[6,4,3,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,1,0,0]
=> [4,3,6,5,2,7,1] => ? = 2
[6,4,2,2]
=> [1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [3,4,6,5,2,7,1] => ? = 0
[6,4,2,1,1]
=> [1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> [3,2,6,5,4,7,1] => ? = 2
[6,3,3,2]
=> [1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [2,4,6,5,3,7,1] => ? = 1
[6,3,3,1,1]
=> [1,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,4,7,1] => ? = 1
[6,3,2,2,1]
=> [1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> [2,1,6,5,4,7,3] => ? = 3
[6,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [3,2,1,5,7,6,4] => ? = 3
[6,5,3,1]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [3,2,6,5,7,4,1] => ? = 2
[6,5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [2,3,6,5,7,4,1] => ? = 1
[6,5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [2,1,6,5,7,4,3] => ? = 3
[6,4,4,1]
=> [1,1,1,0,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [3,2,5,6,7,4,1] => ? = 1
[6,4,3,2]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> [4,6,5,3,2,7,1] => ? = 2
[6,4,3,1,1]
=> [1,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [3,6,5,4,2,7,1] => ? = 2
[6,4,2,2,1]
=> [1,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [2,6,5,4,3,7,1] => ? = 2
[6,4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,7,6,1] => ? = 2
[6,3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [2,3,5,6,7,4,1] => ? = 0
[6,3,3,2,1]
=> [1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,6,5,4,3,7,2] => ? = 3
[6,3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [2,3,5,4,7,6,1] => ? = 1
[6,3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> [2,1,5,6,7,4,3] => ? = 1
[6,3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> [2,1,5,4,7,6,3] => ? = 3
[6,5,4,1]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,6,7,5,4,3] => ? = 3
[6,5,3,2]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [3,6,5,4,7,2,1] => ? = 2
[6,5,3,1,1]
=> [1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [2,6,5,4,7,3,1] => ? = 2
[6,5,2,2,1]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,6,5,4,7,3,2] => ? = 3
[6,5,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [2,1,5,7,6,4,3] => ? = 3
[6,4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [3,5,6,4,7,2,1] => ? = 0
[6,4,4,1,1]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [2,5,6,4,7,3,1] => ? = 1
Description
The number of indices that are both descents and recoils of a permutation.
Matching statistic: St001657
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001657: Integer partitions ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 100%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001657: Integer partitions ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 100%
Values
[1]
=> 10 => [1,2] => [2,1]
=> 1
[2]
=> 100 => [1,3] => [3,1]
=> 0
[1,1]
=> 110 => [1,1,2] => [2,1,1]
=> 1
[3]
=> 1000 => [1,4] => [4,1]
=> 0
[2,1]
=> 1010 => [1,2,2] => [2,2,1]
=> 2
[1,1,1]
=> 1110 => [1,1,1,2] => [2,1,1,1]
=> 1
[4]
=> 10000 => [1,5] => [5,1]
=> 0
[3,1]
=> 10010 => [1,3,2] => [3,2,1]
=> 1
[2,2]
=> 1100 => [1,1,3] => [3,1,1]
=> 0
[2,1,1]
=> 10110 => [1,2,1,2] => [2,2,1,1]
=> 2
[1,1,1,1]
=> 11110 => [1,1,1,1,2] => [2,1,1,1,1]
=> 1
[5]
=> 100000 => [1,6] => [6,1]
=> 0
[4,1]
=> 100010 => [1,4,2] => [4,2,1]
=> 1
[3,2]
=> 10100 => [1,2,3] => [3,2,1]
=> 1
[3,1,1]
=> 100110 => [1,3,1,2] => [3,2,1,1]
=> 1
[2,2,1]
=> 11010 => [1,1,2,2] => [2,2,1,1]
=> 2
[2,1,1,1]
=> 101110 => [1,2,1,1,2] => [2,2,1,1,1]
=> 2
[1,1,1,1,1]
=> 111110 => [1,1,1,1,1,2] => [2,1,1,1,1,1]
=> 1
[6]
=> 1000000 => [1,7] => [7,1]
=> 0
[5,1]
=> 1000010 => [1,5,2] => [5,2,1]
=> 1
[4,2]
=> 100100 => [1,3,3] => [3,3,1]
=> 0
[4,1,1]
=> 1000110 => [1,4,1,2] => [4,2,1,1]
=> 1
[3,3]
=> 11000 => [1,1,4] => [4,1,1]
=> 0
[3,2,1]
=> 101010 => [1,2,2,2] => [2,2,2,1]
=> 3
[3,1,1,1]
=> 1001110 => [1,3,1,1,2] => [3,2,1,1,1]
=> 1
[2,2,2]
=> 11100 => [1,1,1,3] => [3,1,1,1]
=> 0
[2,2,1,1]
=> 110110 => [1,1,2,1,2] => [2,2,1,1,1]
=> 2
[2,1,1,1,1]
=> 1011110 => [1,2,1,1,1,2] => [2,2,1,1,1,1]
=> 2
[6,1]
=> 10000010 => [1,6,2] => [6,2,1]
=> 1
[5,2]
=> 1000100 => [1,4,3] => [4,3,1]
=> 0
[5,1,1]
=> 10000110 => [1,5,1,2] => [5,2,1,1]
=> 1
[4,3]
=> 101000 => [1,2,4] => [4,2,1]
=> 1
[4,2,1]
=> 1001010 => [1,3,2,2] => [3,2,2,1]
=> 2
[4,1,1,1]
=> 10001110 => [1,4,1,1,2] => [4,2,1,1,1]
=> 1
[3,3,1]
=> 110010 => [1,1,3,2] => [3,2,1,1]
=> 1
[3,2,2]
=> 101100 => [1,2,1,3] => [3,2,1,1]
=> 1
[3,2,1,1]
=> 1010110 => [1,2,2,1,2] => [2,2,2,1,1]
=> 3
[3,1,1,1,1]
=> 10011110 => [1,3,1,1,1,2] => [3,2,1,1,1,1]
=> 1
[2,2,2,1]
=> 111010 => [1,1,1,2,2] => [2,2,1,1,1]
=> 2
[2,2,1,1,1]
=> 1101110 => [1,1,2,1,1,2] => [2,2,1,1,1,1]
=> 2
[6,2]
=> 10000100 => [1,5,3] => [5,3,1]
=> 0
[6,1,1]
=> 100000110 => [1,6,1,2] => [6,2,1,1]
=> 1
[5,3]
=> 1001000 => [1,3,4] => [4,3,1]
=> 0
[5,2,1]
=> 10001010 => [1,4,2,2] => [4,2,2,1]
=> 2
[5,1,1,1]
=> 100001110 => [1,5,1,1,2] => [5,2,1,1,1]
=> 1
[4,4]
=> 110000 => [1,1,5] => [5,1,1]
=> 0
[4,3,1]
=> 1010010 => [1,2,3,2] => [3,2,2,1]
=> 2
[4,2,2]
=> 1001100 => [1,3,1,3] => [3,3,1,1]
=> 0
[4,2,1,1]
=> 10010110 => [1,3,2,1,2] => [3,2,2,1,1]
=> 2
[4,1,1,1,1]
=> 100011110 => [1,4,1,1,1,2] => [4,2,1,1,1,1]
=> 1
[6,3,1,1]
=> 1000100110 => ? => ?
=> ? = 1
[6,2,2,1]
=> 1000011010 => ? => ?
=> ? = 2
[6,2,1,1,1]
=> 10000101110 => [1,5,2,1,1,2] => ?
=> ? = 2
[6,1,1,1,1,1]
=> 100000111110 => [1,6,1,1,1,1,2] => ?
=> ? = 1
[5,3,1,1,1]
=> 1001001110 => [1,3,3,1,1,2] => ?
=> ? = 1
[5,2,2,1,1]
=> 1000110110 => [1,4,1,2,1,2] => ?
=> ? = 2
[6,4,1,1]
=> 1001000110 => [1,3,4,1,2] => ?
=> ? = 1
[6,3,2,1]
=> 1000101010 => [1,4,2,2,2] => ?
=> ? = 3
[6,3,1,1,1]
=> 10001001110 => ? => ?
=> ? = 1
[6,2,2,2]
=> 1000011100 => [1,5,1,1,3] => ?
=> ? = 0
[6,2,2,1,1]
=> 10000110110 => ? => ?
=> ? = 2
[6,2,1,1,1,1]
=> 100001011110 => [1,5,2,1,1,1,2] => ?
=> ? = 2
[5,4,1,1,1]
=> 1010001110 => [1,2,4,1,1,2] => ?
=> ? = 2
[5,3,2,1,1]
=> 1001010110 => [1,3,2,2,1,2] => ?
=> ? = 3
[5,2,2,2,1]
=> 1000111010 => [1,4,1,1,2,2] => ?
=> ? = 2
[6,5,1,1]
=> 1010000110 => ? => ?
=> ? = 2
[6,4,2,1]
=> 1001001010 => [1,3,3,2,2] => ?
=> ? = 2
[6,4,1,1,1]
=> 10010001110 => ? => ?
=> ? = 1
[6,3,3,1]
=> 1000110010 => [1,4,1,3,2] => ?
=> ? = 1
[6,3,2,2]
=> 1000101100 => [1,4,2,1,3] => ?
=> ? = 1
[6,3,2,1,1]
=> 10001010110 => ? => ?
=> ? = 3
[6,3,1,1,1,1]
=> 100010011110 => [1,4,3,1,1,1,2] => ?
=> ? = 1
[6,2,2,2,1]
=> 10000111010 => ? => ?
=> ? = 2
[6,2,2,1,1,1]
=> 100001101110 => [1,5,1,2,1,1,2] => ?
=> ? = 2
[5,4,2,1,1]
=> 1010010110 => [1,2,3,2,1,2] => ?
=> ? = 3
[5,3,3,1,1]
=> 1001100110 => [1,3,1,3,1,2] => ?
=> ? = 1
[5,3,2,2,1]
=> 1001011010 => [1,3,2,1,2,2] => ?
=> ? = 3
[6,5,2,1]
=> 1010001010 => [1,2,4,2,2] => ?
=> ? = 3
[6,5,1,1,1]
=> 10100001110 => ? => ?
=> ? = 2
[6,4,3,1]
=> 1001010010 => [1,3,2,3,2] => ?
=> ? = 2
[6,4,2,2]
=> 1001001100 => [1,3,3,1,3] => ?
=> ? = 0
[6,4,2,1,1]
=> 10010010110 => ? => ?
=> ? = 2
[6,4,1,1,1,1]
=> 100100011110 => [1,3,4,1,1,1,2] => ?
=> ? = 1
[6,3,3,2]
=> 1000110100 => [1,4,1,2,3] => ?
=> ? = 1
[6,3,3,1,1]
=> 10001100110 => ? => ?
=> ? = 1
[6,3,2,2,1]
=> 10001011010 => ? => ?
=> ? = 3
[6,3,2,1,1,1]
=> 100010101110 => [1,4,2,2,1,1,2] => ?
=> ? = 3
[6,2,2,2,2]
=> 10000111100 => ? => ?
=> ? = 0
[5,4,3,1,1]
=> 1010100110 => [1,2,2,3,1,2] => ?
=> ? = 3
[5,4,2,2,1]
=> 1010011010 => [1,2,3,1,2,2] => ?
=> ? = 3
[5,3,3,2,1]
=> 1001101010 => [1,3,1,2,2,2] => ?
=> ? = 3
[6,5,3,1]
=> 1010010010 => [1,2,3,3,2] => ?
=> ? = 2
[6,5,2,2]
=> 1010001100 => [1,2,4,1,3] => ?
=> ? = 1
[6,5,2,1,1]
=> 10100010110 => ? => ?
=> ? = 3
[6,5,1,1,1,1]
=> 101000011110 => [1,2,5,1,1,1,2] => ?
=> ? = 2
[6,4,4,1]
=> 1001100010 => [1,3,1,4,2] => ?
=> ? = 1
[6,4,3,2]
=> 1001010100 => [1,3,2,2,3] => ?
=> ? = 2
[6,4,3,1,1]
=> 10010100110 => ? => ?
=> ? = 2
[6,4,2,2,1]
=> 10010011010 => ? => ?
=> ? = 2
[6,4,2,1,1,1]
=> 100100101110 => [1,3,3,2,1,1,2] => ?
=> ? = 2
Description
The number of twos in an integer partition.
The total number of twos in all partitions of $n$ is equal to the total number of singletons [[St001484]] in all partitions of $n-1$, see [1].
The following 11 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000214The number of adjacencies of a permutation. St000247The number of singleton blocks of a set partition. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St000931The number of occurrences of the pattern UUU in a Dyck path. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St000658The number of rises of length 2 of a Dyck path. St001139The number of occurrences of hills of size 2 in a Dyck path. St000731The number of double exceedences of a permutation. St001948The number of augmented double ascents of a permutation.
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