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Your data matches 43 different statistics following compositions of up to 3 maps.
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Matching statistic: St001449
St001449: Plane partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> 2
[[1],[1]]
=> 2
[[2]]
=> 1
[[1,1]]
=> 2
[[1],[1],[1]]
=> 2
[[2],[1]]
=> 3
[[1,1],[1]]
=> 2
[[3]]
=> 1
[[2,1]]
=> 3
[[1,1,1]]
=> 2
[[1],[1],[1],[1]]
=> 2
[[2],[1],[1]]
=> 3
[[2],[2]]
=> 1
[[1,1],[1],[1]]
=> 2
[[1,1],[1,1]]
=> 2
[[3],[1]]
=> 2
[[2,1],[1]]
=> 3
[[1,1,1],[1]]
=> 2
[[4]]
=> 1
[[3,1]]
=> 2
[[2,2]]
=> 1
[[2,1,1]]
=> 3
[[1,1,1,1]]
=> 2
[[1],[1],[1],[1],[1]]
=> 2
[[2],[1],[1],[1]]
=> 3
[[2],[2],[1]]
=> 3
[[1,1],[1],[1],[1]]
=> 2
[[1,1],[1,1],[1]]
=> 2
[[3],[1],[1]]
=> 2
[[3],[2]]
=> 1
[[2,1],[1],[1]]
=> 3
[[2,1],[2]]
=> 3
[[2,1],[1,1]]
=> 3
[[1,1,1],[1],[1]]
=> 2
[[1,1,1],[1,1]]
=> 2
[[4],[1]]
=> 2
[[3,1],[1]]
=> 2
[[2,2],[1]]
=> 3
[[2,1,1],[1]]
=> 3
[[1,1,1,1],[1]]
=> 2
[[5]]
=> 1
[[4,1]]
=> 2
[[3,2]]
=> 1
[[3,1,1]]
=> 2
[[2,2,1]]
=> 3
[[2,1,1,1]]
=> 3
[[1,1,1,1,1]]
=> 2
Description
The smallest missing nonzero part in the plane partition.
Matching statistic: St000678
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1]
=> [1,0,1,0]
=> 2
[[1],[1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 2
[[1,1]]
=> [2]
=> [1,1,0,0,1,0]
=> 2
[[1],[1],[1]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[[2],[1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 3
[[1,1],[1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 3
[[3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 2
[[2,1]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 2
[[1,1,1]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 2
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[[2],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[[2],[2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[[1,1],[1,1]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1
[[3],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 3
[[2,1],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 3
[[1,1,1],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 3
[[4]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[[3,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[[2,2]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[[2,1,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[[1,1,1,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[[2],[2],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[[3],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2
[[3],[2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 3
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2
[[2,1],[2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 3
[[2,1],[1,1]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 3
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2
[[1,1,1],[1,1]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 3
[[4],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 3
[[3,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 3
[[2,2],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 3
[[2,1,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 3
[[1,1,1,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 3
[[5]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2
[[4,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2
[[3,2]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2
[[3,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2
[[2,2,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2
[[2,1,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2
[[1,1,1,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2
Description
The number of up steps after the last double rise of a Dyck path.
Matching statistic: St000759
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000759: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000759: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1]
=> [1]
=> 2
[[1],[1]]
=> [1,1]
=> [2]
=> 1
[[2]]
=> [2]
=> [1,1]
=> 2
[[1,1]]
=> [2]
=> [1,1]
=> 2
[[1],[1],[1]]
=> [1,1,1]
=> [3]
=> 1
[[2],[1]]
=> [2,1]
=> [2,1]
=> 3
[[1,1],[1]]
=> [2,1]
=> [2,1]
=> 3
[[3]]
=> [3]
=> [1,1,1]
=> 2
[[2,1]]
=> [3]
=> [1,1,1]
=> 2
[[1,1,1]]
=> [3]
=> [1,1,1]
=> 2
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [4]
=> 1
[[2],[1],[1]]
=> [2,1,1]
=> [3,1]
=> 2
[[2],[2]]
=> [2,2]
=> [2,2]
=> 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [3,1]
=> 2
[[1,1],[1,1]]
=> [2,2]
=> [2,2]
=> 1
[[3],[1]]
=> [3,1]
=> [2,1,1]
=> 3
[[2,1],[1]]
=> [3,1]
=> [2,1,1]
=> 3
[[1,1,1],[1]]
=> [3,1]
=> [2,1,1]
=> 3
[[4]]
=> [4]
=> [1,1,1,1]
=> 2
[[3,1]]
=> [4]
=> [1,1,1,1]
=> 2
[[2,2]]
=> [4]
=> [1,1,1,1]
=> 2
[[2,1,1]]
=> [4]
=> [1,1,1,1]
=> 2
[[1,1,1,1]]
=> [4]
=> [1,1,1,1]
=> 2
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [5]
=> 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [4,1]
=> 2
[[2],[2],[1]]
=> [2,2,1]
=> [3,2]
=> 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [4,1]
=> 2
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [3,2]
=> 1
[[3],[1],[1]]
=> [3,1,1]
=> [3,1,1]
=> 2
[[3],[2]]
=> [3,2]
=> [2,2,1]
=> 3
[[2,1],[1],[1]]
=> [3,1,1]
=> [3,1,1]
=> 2
[[2,1],[2]]
=> [3,2]
=> [2,2,1]
=> 3
[[2,1],[1,1]]
=> [3,2]
=> [2,2,1]
=> 3
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [3,1,1]
=> 2
[[1,1,1],[1,1]]
=> [3,2]
=> [2,2,1]
=> 3
[[4],[1]]
=> [4,1]
=> [2,1,1,1]
=> 3
[[3,1],[1]]
=> [4,1]
=> [2,1,1,1]
=> 3
[[2,2],[1]]
=> [4,1]
=> [2,1,1,1]
=> 3
[[2,1,1],[1]]
=> [4,1]
=> [2,1,1,1]
=> 3
[[1,1,1,1],[1]]
=> [4,1]
=> [2,1,1,1]
=> 3
[[5]]
=> [5]
=> [1,1,1,1,1]
=> 2
[[4,1]]
=> [5]
=> [1,1,1,1,1]
=> 2
[[3,2]]
=> [5]
=> [1,1,1,1,1]
=> 2
[[3,1,1]]
=> [5]
=> [1,1,1,1,1]
=> 2
[[2,2,1]]
=> [5]
=> [1,1,1,1,1]
=> 2
[[2,1,1,1]]
=> [5]
=> [1,1,1,1,1]
=> 2
[[1,1,1,1,1]]
=> [5]
=> [1,1,1,1,1]
=> 2
Description
The smallest missing part in an integer partition.
In [3], this is referred to as the mex, the minimal excluded part of the partition.
For compositions, this is studied in [sec.3.2., 1].
Matching statistic: St000011
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 2
[[1],[1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[[1,1]]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[[1],[1],[1]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[[2],[1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3
[[1,1],[1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3
[[3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[[2,1]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[[1,1,1]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[[2],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[[2],[2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[[1,1],[1,1]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[[3],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[[2,1],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[[1,1,1],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[[4]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[[3,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[[2,2]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[[2,1,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[[1,1,1,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[[2],[2],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[[3],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[[3],[2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[[2,1],[2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[[2,1],[1,1]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[[1,1,1],[1,1]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[[4],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3
[[3,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3
[[2,2],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3
[[2,1,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3
[[1,1,1,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3
[[5]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2
[[4,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2
[[3,2]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2
[[3,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2
[[2,2,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2
[[2,1,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2
[[1,1,1,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2
Description
The number of touch points (or returns) of a Dyck path.
This is the number of points, excluding the origin, where the Dyck path has height 0.
Matching statistic: St000990
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000990: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000990: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1]
=> [1,0,1,0]
=> [2,1] => 2
[[1],[1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 1
[[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 2
[[1,1]]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 2
[[1],[1],[1]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1
[[2],[1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 3
[[1,1],[1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 3
[[3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 2
[[2,1]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 2
[[1,1,1]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 2
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1
[[2],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 2
[[2],[2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 2
[[1,1],[1,1]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 1
[[3],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 3
[[2,1],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 3
[[1,1,1],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 3
[[4]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 2
[[3,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 2
[[2,2]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 2
[[2,1,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 2
[[1,1,1,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 2
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 2
[[2],[2],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 2
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 1
[[3],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 2
[[3],[2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 3
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 2
[[2,1],[2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 3
[[2,1],[1,1]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 3
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 2
[[1,1,1],[1,1]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 3
[[4],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 3
[[3,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 3
[[2,2],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 3
[[2,1,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 3
[[1,1,1,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 3
[[5]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 2
[[4,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 2
[[3,2]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 2
[[3,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 2
[[2,2,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 2
[[2,1,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 2
[[1,1,1,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 2
Description
The first ascent of a permutation.
For a permutation $\pi$, this is the smallest index such that $\pi(i) < \pi(i+1)$.
For the first descent, see [[St000654]].
Matching statistic: St001050
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St001050: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St001050: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1]
=> [1,0,1,0]
=> {{1},{2}}
=> 2
[[1],[1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1
[[2]]
=> [2]
=> [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 2
[[1,1]]
=> [2]
=> [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 2
[[1],[1],[1]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1
[[2],[1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 3
[[1,1],[1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 3
[[3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 2
[[2,1]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 2
[[1,1,1]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 2
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 1
[[2],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 2
[[2],[2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 2
[[1,1],[1,1]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 1
[[3],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> 3
[[2,1],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> 3
[[1,1,1],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> 3
[[4]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 2
[[3,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 2
[[2,2]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 2
[[2,1,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 2
[[1,1,1,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 2
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> {{1},{2,3,4,5,6}}
=> 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> 2
[[2],[2],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> 2
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 1
[[3],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 2
[[3],[2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 3
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 2
[[2,1],[2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 3
[[2,1],[1,1]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 3
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 2
[[1,1,1],[1,1]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 3
[[4],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> {{1,2,4},{3},{5}}
=> 3
[[3,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> {{1,2,4},{3},{5}}
=> 3
[[2,2],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> {{1,2,4},{3},{5}}
=> 3
[[2,1,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> {{1,2,4},{3},{5}}
=> 3
[[1,1,1,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> {{1,2,4},{3},{5}}
=> 3
[[5]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> {{1,2,3,4,5},{6}}
=> 2
[[4,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> {{1,2,3,4,5},{6}}
=> 2
[[3,2]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> {{1,2,3,4,5},{6}}
=> 2
[[3,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> {{1,2,3,4,5},{6}}
=> 2
[[2,2,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> {{1,2,3,4,5},{6}}
=> 2
[[2,1,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> {{1,2,3,4,5},{6}}
=> 2
[[1,1,1,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> {{1,2,3,4,5},{6}}
=> 2
Description
The number of terminal closers of a set partition.
A closer of a set partition is a number that is maximal in its block. In particular, a singleton is a closer. This statistic counts the number of terminal closers. In other words, this is the number of closers such that all larger elements are also closers.
Matching statistic: St001184
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001184: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001184: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[[1],[1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2
[[1,1]]
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2
[[1],[1],[1]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[[2],[1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[[1,1],[1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[[3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[[2,1]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[[1,1,1]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[[2],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[[2],[2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[[1,1],[1,1]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[[3],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 3
[[2,1],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 3
[[1,1,1],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 3
[[4]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2
[[3,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2
[[2,2]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2
[[2,1,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2
[[1,1,1,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[[2],[2],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[[3],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[[3],[2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 3
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[[2,1],[2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 3
[[2,1],[1,1]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 3
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[[1,1,1],[1,1]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 3
[[4],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3
[[3,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3
[[2,2],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3
[[2,1,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3
[[1,1,1,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3
[[5]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 2
[[4,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 2
[[3,2]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 2
[[3,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 2
[[2,2,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 2
[[2,1,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 2
[[1,1,1,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 2
Description
Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra.
Matching statistic: St001226
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001226: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001226: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 3 = 2 + 1
[[1],[1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[[1,1]]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[[1],[1],[1]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[[2],[1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 4 = 3 + 1
[[1,1],[1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 4 = 3 + 1
[[3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[[2,1]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[[1,1,1]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[[2],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[[2],[2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[[1,1],[1,1]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[[3],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 4 = 3 + 1
[[2,1],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 4 = 3 + 1
[[1,1,1],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 4 = 3 + 1
[[4]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[[3,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[[2,2]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[[2,1,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[[1,1,1,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[[1],[1],[1],[1],[1]]
=> [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]
=> 2 = 1 + 1
[[2],[1],[1],[1]]
=> [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 = 2 + 1
[[2],[2],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[[1,1],[1],[1],[1]]
=> [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 = 2 + 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[[3],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[[3],[2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[[2,1],[2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[[2,1],[1,1]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[[1,1,1],[1,1]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[[4],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[[3,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[[2,2],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[[2,1,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[[1,1,1,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[[5]]
=> [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]
=> 3 = 2 + 1
[[4,1]]
=> [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]
=> 3 = 2 + 1
[[3,2]]
=> [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]
=> 3 = 2 + 1
[[3,1,1]]
=> [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]
=> 3 = 2 + 1
[[2,2,1]]
=> [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]
=> 3 = 2 + 1
[[2,1,1,1]]
=> [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]
=> 3 = 2 + 1
[[1,1,1,1,1]]
=> [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]
=> 3 = 2 + 1
Description
The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra.
That is the number of i such that $Ext_A^1(J,e_i J)=0$.
Matching statistic: St000989
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000989: Permutations ⟶ ℤResult quality: 83% ●values known / values provided: 83%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000989: Permutations ⟶ ℤResult quality: 83% ●values known / values provided: 83%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 1 = 2 - 1
[[1],[1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 0 = 1 - 1
[[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 1 = 2 - 1
[[1,1]]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 1 = 2 - 1
[[1],[1],[1]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 0 = 1 - 1
[[2],[1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 2 = 3 - 1
[[1,1],[1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 2 = 3 - 1
[[3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 1 = 2 - 1
[[2,1]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 1 = 2 - 1
[[1,1,1]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 1 = 2 - 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => 0 = 1 - 1
[[2],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1 = 2 - 1
[[2],[2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 0 = 1 - 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1 = 2 - 1
[[1,1],[1,1]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 0 = 1 - 1
[[3],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 2 = 3 - 1
[[2,1],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 2 = 3 - 1
[[1,1,1],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 2 = 3 - 1
[[4]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 1 = 2 - 1
[[3,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 1 = 2 - 1
[[2,2]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 1 = 2 - 1
[[2,1,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 1 = 2 - 1
[[1,1,1,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 1 = 2 - 1
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? ∊ {1,2,2,2,2,2,2,2} - 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => 1 = 2 - 1
[[2],[2],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0 = 1 - 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => 1 = 2 - 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0 = 1 - 1
[[3],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 2 - 1
[[3],[2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 2 = 3 - 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 2 - 1
[[2,1],[2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 2 = 3 - 1
[[2,1],[1,1]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 2 = 3 - 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 2 - 1
[[1,1,1],[1,1]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 2 = 3 - 1
[[4],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 2 = 3 - 1
[[3,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 2 = 3 - 1
[[2,2],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 2 = 3 - 1
[[2,1,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 2 = 3 - 1
[[1,1,1,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 2 = 3 - 1
[[5]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? ∊ {1,2,2,2,2,2,2,2} - 1
[[4,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? ∊ {1,2,2,2,2,2,2,2} - 1
[[3,2]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? ∊ {1,2,2,2,2,2,2,2} - 1
[[3,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? ∊ {1,2,2,2,2,2,2,2} - 1
[[2,2,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? ∊ {1,2,2,2,2,2,2,2} - 1
[[2,1,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? ∊ {1,2,2,2,2,2,2,2} - 1
[[1,1,1,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? ∊ {1,2,2,2,2,2,2,2} - 1
Description
The number of final rises of a permutation.
For a permutation $\pi$ of length $n$, this is the maximal $k$ such that
$$\pi(n-k) \leq \pi(n-k+1) \leq \cdots \leq \pi(n-1) \leq \pi(n).$$
Equivalently, this is $n-1$ minus the position of the last descent [[St000653]].
Matching statistic: St001640
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001640: Permutations ⟶ ℤResult quality: 83% ●values known / values provided: 83%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001640: Permutations ⟶ ℤResult quality: 83% ●values known / values provided: 83%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 1 = 2 - 1
[[1],[1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 0 = 1 - 1
[[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 1 = 2 - 1
[[1,1]]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 1 = 2 - 1
[[1],[1],[1]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 0 = 1 - 1
[[2],[1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 2 = 3 - 1
[[1,1],[1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 2 = 3 - 1
[[3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 1 = 2 - 1
[[2,1]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 1 = 2 - 1
[[1,1,1]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 1 = 2 - 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => 0 = 1 - 1
[[2],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1 = 2 - 1
[[2],[2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 0 = 1 - 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1 = 2 - 1
[[1,1],[1,1]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 0 = 1 - 1
[[3],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 2 = 3 - 1
[[2,1],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 2 = 3 - 1
[[1,1,1],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 2 = 3 - 1
[[4]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 1 = 2 - 1
[[3,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 1 = 2 - 1
[[2,2]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 1 = 2 - 1
[[2,1,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 1 = 2 - 1
[[1,1,1,1]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 1 = 2 - 1
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? ∊ {1,1,1,2,2,2,2,2} - 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => 1 = 2 - 1
[[2],[2],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1 = 2 - 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => 1 = 2 - 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1 = 2 - 1
[[3],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 2 - 1
[[3],[2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 2 = 3 - 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 2 - 1
[[2,1],[2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 2 = 3 - 1
[[2,1],[1,1]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 2 = 3 - 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 2 - 1
[[1,1,1],[1,1]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 2 = 3 - 1
[[4],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 2 = 3 - 1
[[3,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 2 = 3 - 1
[[2,2],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 2 = 3 - 1
[[2,1,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 2 = 3 - 1
[[1,1,1,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 2 = 3 - 1
[[5]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? ∊ {1,1,1,2,2,2,2,2} - 1
[[4,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? ∊ {1,1,1,2,2,2,2,2} - 1
[[3,2]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? ∊ {1,1,1,2,2,2,2,2} - 1
[[3,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? ∊ {1,1,1,2,2,2,2,2} - 1
[[2,2,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? ∊ {1,1,1,2,2,2,2,2} - 1
[[2,1,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? ∊ {1,1,1,2,2,2,2,2} - 1
[[1,1,1,1,1]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? ∊ {1,1,1,2,2,2,2,2} - 1
Description
The number of ascent tops in the permutation such that all smaller elements appear before.
The following 33 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000439The position of the first down step of a Dyck path. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001180Number of indecomposable injective modules with projective dimension at most 1. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000284The Plancherel distribution on integer partitions. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001128The exponens consonantiae of a partition. St001568The smallest positive integer that does not appear twice in the partition.
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