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Your data matches 144 different statistics following compositions of up to 3 maps.
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Matching statistic: St000759
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
St000759: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 2 = 1 + 1
[2]
=> 1 = 0 + 1
[1,1]
=> 2 = 1 + 1
[3]
=> 1 = 0 + 1
[2,1]
=> 3 = 2 + 1
[1,1,1]
=> 2 = 1 + 1
[4]
=> 1 = 0 + 1
[3,1]
=> 2 = 1 + 1
[2,2]
=> 1 = 0 + 1
[2,1,1]
=> 3 = 2 + 1
[1,1,1,1]
=> 2 = 1 + 1
[5]
=> 1 = 0 + 1
[4,1]
=> 2 = 1 + 1
[3,2]
=> 1 = 0 + 1
[3,1,1]
=> 2 = 1 + 1
[2,2,1]
=> 3 = 2 + 1
[2,1,1,1]
=> 3 = 2 + 1
[1,1,1,1,1]
=> 2 = 1 + 1
[6]
=> 1 = 0 + 1
[5,1]
=> 2 = 1 + 1
[4,2]
=> 1 = 0 + 1
[4,1,1]
=> 2 = 1 + 1
[3,3]
=> 1 = 0 + 1
[3,2,1]
=> 4 = 3 + 1
[3,1,1,1]
=> 2 = 1 + 1
[2,2,2]
=> 1 = 0 + 1
[2,2,1,1]
=> 3 = 2 + 1
[2,1,1,1,1]
=> 3 = 2 + 1
[1,1,1,1,1,1]
=> 2 = 1 + 1
[7]
=> 1 = 0 + 1
[6,1]
=> 2 = 1 + 1
[5,2]
=> 1 = 0 + 1
[5,1,1]
=> 2 = 1 + 1
[4,3]
=> 1 = 0 + 1
[4,2,1]
=> 3 = 2 + 1
[4,1,1,1]
=> 2 = 1 + 1
[3,3,1]
=> 2 = 1 + 1
[3,2,2]
=> 1 = 0 + 1
[3,2,1,1]
=> 4 = 3 + 1
[3,1,1,1,1]
=> 2 = 1 + 1
[2,2,2,1]
=> 3 = 2 + 1
[2,2,1,1,1]
=> 3 = 2 + 1
[2,1,1,1,1,1]
=> 3 = 2 + 1
[1,1,1,1,1,1,1]
=> 2 = 1 + 1
[8]
=> 1 = 0 + 1
[7,1]
=> 2 = 1 + 1
[6,2]
=> 1 = 0 + 1
[6,1,1]
=> 2 = 1 + 1
[5,3]
=> 1 = 0 + 1
[5,2,1]
=> 3 = 2 + 1
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
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
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%
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]
=> [1,0,1,0]
=> [1,0,1,0]
=> 2 = 1 + 1
[2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 0 + 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
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 0 + 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
[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
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 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
[1,1,1,1,1]
=> [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 = 0 + 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 2 = 1 + 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 3 = 2 + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 0 + 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 2 = 1 + 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 1 = 0 + 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> 2 = 1 + 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> 3 = 2 + 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3 = 2 + 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 4 = 3 + 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2 = 1 + 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> 2 = 1 + 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]
=> 1 = 0 + 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 1 = 0 + 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> 2 = 1 + 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> 1 = 0 + 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> 2 = 1 + 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 3 = 2 + 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> 3 = 2 + 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> 2 = 1 + 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
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: St001050
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
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%
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]
=> [1,0,1,0]
=> {{1},{2}}
=> 2 = 1 + 1
[2]
=> [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 2 = 1 + 1
[1,1]
=> [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1 = 0 + 1
[3]
=> [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 2 = 1 + 1
[2,1]
=> [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 3 = 2 + 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1 = 0 + 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 2 = 1 + 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> 3 = 2 + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 1 = 0 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 2 = 1 + 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 1 = 0 + 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> {{1,2,3,4,5},{6}}
=> 2 = 1 + 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> {{1,2,4},{3},{5}}
=> 3 = 2 + 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 3 = 2 + 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 2 = 1 + 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 1 = 0 + 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> 2 = 1 + 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> {{1},{2,3,4,5,6}}
=> 1 = 0 + 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,5,6},{7}}
=> 2 = 1 + 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> {{1,2,3,5},{4},{6}}
=> 3 = 2 + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> {{1,4},{2,3},{5}}
=> 3 = 2 + 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> {{1,3,4},{2},{5}}
=> 2 = 1 + 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 1 = 0 + 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 4 = 3 + 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> 2 = 1 + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> 1 = 0 + 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> 1 = 0 + 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> {{1},{2,3,4,6},{5}}
=> 2 = 1 + 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,7}}
=> 1 = 0 + 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,5,6,7},{8}}
=> 2 = 1 + 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> {{1,2,3,4,6},{5},{7}}
=> 3 = 2 + 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> {{1,2,5},{3,4},{6}}
=> 3 = 2 + 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> {{1,2,4,5},{3},{6}}
=> 2 = 1 + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> 3 = 2 + 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> {{1,4},{2},{3},{5}}
=> 4 = 3 + 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 2 = 1 + 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> {{1,3},{2},{4,5}}
=> 1 = 0 + 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> {{1,2},{3,5},{4}}
=> 2 = 1 + 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> 3 = 2 + 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> {{1},{2,3,6},{4,5}}
=> 2 = 1 + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 1 = 0 + 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> {{1},{2,3,5,6},{4}}
=> 1 = 0 + 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> {{1},{2,3,4,5,7},{6}}
=> 2 = 1 + 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,7,8}}
=> 1 = 0 + 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,5,6,7,8},{9}}
=> 2 = 1 + 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,5,7},{6},{8}}
=> 3 = 2 + 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> {{1,2,3,6},{4,5},{7}}
=> 3 = 2 + 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> {{1,2,3,5,6},{4},{7}}
=> 2 = 1 + 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> {{1,5},{2,3,4},{6}}
=> 3 = 2 + 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> {{1,2,5},{3},{4},{6}}
=> 4 = 3 + 1
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: St000439
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St000439: 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]
=> 3 = 1 + 2
[2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2 = 0 + 2
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3 = 1 + 2
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 0 + 2
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 4 = 2 + 2
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 1 + 2
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2 = 0 + 2
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 3 = 1 + 2
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 0 + 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 4 = 2 + 2
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 3 = 1 + 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 = 0 + 2
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3 = 1 + 2
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 0 + 2
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 3 = 1 + 2
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 4 = 2 + 2
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 4 = 2 + 2
[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]
=> 3 = 1 + 2
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 2 = 0 + 2
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 3 = 1 + 2
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2 = 0 + 2
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3 = 1 + 2
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2 = 0 + 2
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 5 = 3 + 2
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 3 = 1 + 2
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2 = 0 + 2
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 4 = 2 + 2
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 4 = 2 + 2
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> 3 = 1 + 2
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 2 = 0 + 2
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> 3 = 1 + 2
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> 2 = 0 + 2
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> 3 = 1 + 2
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2 = 0 + 2
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 4 = 2 + 2
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3 = 1 + 2
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3 = 1 + 2
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2 = 0 + 2
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 5 = 3 + 2
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 4 = 2 + 2
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> 4 = 2 + 2
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> 4 = 2 + 2
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 3 = 1 + 2
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 2 = 0 + 2
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> 3 = 1 + 2
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> 2 = 0 + 2
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> 3 = 1 + 2
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> 2 = 0 + 2
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> 4 = 2 + 2
Description
The position of the first down step of a Dyck path.
Matching statistic: St000297
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [2,1] => 1 => 1
[2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 10 => 1
[1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 01 => 0
[3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 100 => 1
[2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 11 => 2
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 001 => 0
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 1000 => 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 110 => 2
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 010 => 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 101 => 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0001 => 0
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 10000 => 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 1100 => 2
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 110 => 2
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 101 => 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 011 => 0
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 1001 => 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 00001 => 0
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => 100000 => 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => 11000 => 2
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => 1100 => 2
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => 1010 => 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 0100 => 0
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 111 => 3
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 1001 => 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 0010 => 0
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 0101 => 0
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => 10001 => 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 000001 => 0
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,7] => 1000000 => 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,3,4,5,6] => 110000 => 2
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [6,3,1,2,4,5] => 11000 => 2
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,3,1,4,5] => 10100 => 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 1100 => 2
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => 1110 => 3
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 1001 => 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => 0110 => 0
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => 1010 => 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => 1101 => 2
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => 10001 => 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 0011 => 0
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => 01001 => 0
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,7,1] => 100001 => 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => 0000001 => 0
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,8] => 10000000 => 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [8,2,1,3,4,5,6,7] => 1100000 => 2
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [7,3,1,2,4,5,6] => 110000 => 2
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [7,2,3,1,4,5,6] => 101000 => 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => 11000 => 2
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [6,3,2,1,4,5] => 11100 => 3
Description
The number of leading ones in a binary word.
Matching statistic: St000326
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [1,2] => 0 => 2 = 1 + 1
[2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 10 => 1 = 0 + 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 01 => 2 = 1 + 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 110 => 1 = 0 + 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 00 => 3 = 2 + 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 011 => 2 = 1 + 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 1110 => 1 = 0 + 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 010 => 2 = 1 + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 101 => 1 = 0 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 001 => 3 = 2 + 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 0111 => 2 = 1 + 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,4,3,2,1,6] => 11110 => 1 = 0 + 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,2,1,5] => 0110 => 2 = 1 + 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 100 => 1 = 0 + 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 010 => 2 = 1 + 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 001 => 3 = 2 + 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => 0011 => 3 = 2 + 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => 01111 => 2 = 1 + 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => 111110 => 1 = 0 + 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [4,5,3,2,1,6] => 01110 => 2 = 1 + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => 1010 => 1 = 0 + 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => 0110 => 2 = 1 + 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1101 => 1 = 0 + 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 000 => 4 = 3 + 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 0101 => 2 = 1 + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 1011 => 1 = 0 + 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 0011 => 3 = 2 + 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,5,6,4,3,2] => 00111 => 3 = 2 + 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => 011111 => 2 = 1 + 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7,6,5,4,3,2,1,8] => 1111110 => 1 = 0 + 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [5,6,4,3,2,1,7] => 011110 => 2 = 1 + 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [4,3,5,2,1,6] => 10110 => 1 = 0 + 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [3,5,4,2,1,6] => 01110 => 2 = 1 + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1100 => 1 = 0 + 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 0010 => 3 = 2 + 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 0110 => 2 = 1 + 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 0101 => 2 = 1 + 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 1001 => 1 = 0 + 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 0001 => 4 = 3 + 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,5,4,6,3,2] => 01011 => 2 = 1 + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 0011 => 3 = 2 + 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,4,6,5,3,2] => 00111 => 3 = 2 + 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,6,7,5,4,3,2] => 001111 => 3 = 2 + 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,7,6,5,4,3,2] => 0111111 => 2 = 1 + 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8,7,6,5,4,3,2,1,9] => 11111110 => 1 = 0 + 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [6,7,5,4,3,2,1,8] => 0111110 => 2 = 1 + 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [5,4,6,3,2,1,7] => 101110 => 1 = 0 + 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [4,6,5,3,2,1,7] => 011110 => 2 = 1 + 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [4,3,2,5,1,6] => 11010 => 1 = 0 + 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [3,4,5,2,1,6] => 00110 => 3 = 2 + 1
Description
The position of the first one in a binary word after appending a 1 at the end.
Regarding the binary word as a subset of $\{1,\dots,n,n+1\}$ that contains $n+1$, this is the minimal element of the set.
Matching statistic: St000382
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [1,2] => [2] => 2 = 1 + 1
[2]
=> [1,1,0,0,1,0]
=> [2,1,3] => [1,2] => 1 = 0 + 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => [2,1] => 2 = 1 + 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [1,1,2] => 1 = 0 + 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => [3] => 3 = 2 + 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [2,1,1] => 2 = 1 + 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => [1,1,1,2] => 1 = 0 + 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,2] => 2 = 1 + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,2,1] => 1 = 0 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [3,1] => 3 = 2 + 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [2,1,1,1] => 2 = 1 + 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,4,3,2,1,6] => [1,1,1,1,2] => 1 = 0 + 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,2,1,5] => [2,1,2] => 2 = 1 + 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [1,3] => 1 = 0 + 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,2] => 2 = 1 + 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [3,1] => 3 = 2 + 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [3,1,1] => 3 = 2 + 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => [2,1,1,1,1] => 2 = 1 + 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => [1,1,1,1,1,2] => 1 = 0 + 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [4,5,3,2,1,6] => [2,1,1,2] => 2 = 1 + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => [1,2,2] => 1 = 0 + 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [2,1,2] => 2 = 1 + 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => [1,1,2,1] => 1 = 0 + 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [4] => 4 = 3 + 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [2,2,1] => 2 = 1 + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [1,2,1,1] => 1 = 0 + 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [3,1,1] => 3 = 2 + 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,5,6,4,3,2] => [3,1,1,1] => 3 = 2 + 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => [2,1,1,1,1,1] => 2 = 1 + 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7,6,5,4,3,2,1,8] => [1,1,1,1,1,1,2] => 1 = 0 + 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [5,6,4,3,2,1,7] => [2,1,1,1,2] => 2 = 1 + 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [4,3,5,2,1,6] => [1,2,1,2] => 1 = 0 + 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [3,5,4,2,1,6] => [2,1,1,2] => 2 = 1 + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [1,1,3] => 1 = 0 + 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [3,2] => 3 = 2 + 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [2,1,2] => 2 = 1 + 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,2,1] => 2 = 1 + 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,3,1] => 1 = 0 + 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [4,1] => 4 = 3 + 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,5,4,6,3,2] => [2,2,1,1] => 2 = 1 + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [3,1,1] => 3 = 2 + 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,4,6,5,3,2] => [3,1,1,1] => 3 = 2 + 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,6,7,5,4,3,2] => [3,1,1,1,1] => 3 = 2 + 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,7,6,5,4,3,2] => [2,1,1,1,1,1,1] => 2 = 1 + 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8,7,6,5,4,3,2,1,9] => [1,1,1,1,1,1,1,2] => 1 = 0 + 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [6,7,5,4,3,2,1,8] => [2,1,1,1,1,2] => 2 = 1 + 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [5,4,6,3,2,1,7] => [1,2,1,1,2] => 1 = 0 + 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [4,6,5,3,2,1,7] => [2,1,1,1,2] => 2 = 1 + 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [4,3,2,5,1,6] => [1,1,2,2] => 1 = 0 + 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [3,4,5,2,1,6] => [3,1,2] => 3 = 2 + 1
[1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,11,10,9,8,7,6,5,4,3,2] => [2,1,1,1,1,1,1,1,1,1] => ? = 1 + 1
Description
The first part of an integer composition.
Matching statistic: St000745
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [2,1] => [[1],[2]]
=> 2 = 1 + 1
[2]
=> [1,1,0,0,1,0]
=> [3,1,2] => [[1,3],[2]]
=> 2 = 1 + 1
[1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => [[1,2],[3]]
=> 1 = 0 + 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [[1,3,4],[2]]
=> 2 = 1 + 1
[2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => [[1],[2],[3]]
=> 3 = 2 + 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [[1,2,3],[4]]
=> 1 = 0 + 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [[1,3,4,5],[2]]
=> 2 = 1 + 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [[1,4],[2],[3]]
=> 3 = 2 + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [[1,2],[3,4]]
=> 1 = 0 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [[1,3],[2],[4]]
=> 2 = 1 + 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [[1,2,3,4],[5]]
=> 1 = 0 + 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => [[1,3,4,5,6],[2]]
=> 2 = 1 + 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [[1,4,5],[2],[3]]
=> 3 = 2 + 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [[1,4],[2],[3]]
=> 3 = 2 + 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [[1,3],[2],[4]]
=> 2 = 1 + 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [[1,2],[3],[4]]
=> 1 = 0 + 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [[1,3,4],[2],[5]]
=> 2 = 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] => [[1,2,3,4,5],[6]]
=> 1 = 0 + 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => [[1,3,4,5,6,7],[2]]
=> 2 = 1 + 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => [[1,4,5,6],[2],[3]]
=> 3 = 2 + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [[1,4,5],[2],[3]]
=> 3 = 2 + 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [[1,3,5],[2],[4]]
=> 2 = 1 + 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [[1,2,5],[3,4]]
=> 1 = 0 + 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [[1],[2],[3],[4]]
=> 4 = 3 + 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [[1,3,4],[2],[5]]
=> 2 = 1 + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [[1,2,3],[4,5]]
=> 1 = 0 + 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [[1,2,4],[3],[5]]
=> 1 = 0 + 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => [[1,3,4,5],[2],[6]]
=> 2 = 1 + 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [[1,2,3,4,5,6],[7]]
=> 1 = 0 + 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,7] => [[1,3,4,5,6,7,8],[2]]
=> 2 = 1 + 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,3,4,5,6] => [[1,4,5,6,7],[2],[3]]
=> 3 = 2 + 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [6,3,1,2,4,5] => [[1,4,5,6],[2],[3]]
=> 3 = 2 + 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,3,1,4,5] => [[1,3,5,6],[2],[4]]
=> 2 = 1 + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [[1,4,5],[2],[3]]
=> 3 = 2 + 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [[1,5],[2],[3],[4]]
=> 4 = 3 + 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [[1,3,4],[2],[5]]
=> 2 = 1 + 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [[1,2],[3,5],[4]]
=> 1 = 0 + 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [[1,3],[2,5],[4]]
=> 2 = 1 + 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [[1,4],[2],[3],[5]]
=> 3 = 2 + 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => [[1,3,4,5],[2],[6]]
=> 2 = 1 + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [[1,2,3],[4],[5]]
=> 1 = 0 + 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => [[1,2,4,5],[3],[6]]
=> 1 = 0 + 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,7,1] => [[1,3,4,5,6],[2],[7]]
=> 2 = 1 + 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => [[1,2,3,4,5,6,7],[8]]
=> 1 = 0 + 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,8] => [[1,3,4,5,6,7,8,9],[2]]
=> 2 = 1 + 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [8,2,1,3,4,5,6,7] => [[1,4,5,6,7,8],[2],[3]]
=> 3 = 2 + 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [7,3,1,2,4,5,6] => [[1,4,5,6,7],[2],[3]]
=> 3 = 2 + 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [7,2,3,1,4,5,6] => [[1,3,5,6,7],[2],[4]]
=> 2 = 1 + 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [[1,4,5,6],[2],[3]]
=> 3 = 2 + 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [6,3,2,1,4,5] => [[1,5,6],[2],[3],[4]]
=> 4 = 3 + 1
[10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [11,1,2,3,4,5,6,7,8,9,10] => [[1,3,4,5,6,7,8,9,10,11],[2]]
=> ? ∊ {0,1,1} + 1
[2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,9,10,1] => [[1,3,4,5,6,7,8,9],[2],[10]]
=> ? ∊ {0,1,1} + 1
[1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => [[1,2,3,4,5,6,7,8,9,10],[11]]
=> ? ∊ {0,1,1} + 1
Description
The index of the last row whose first entry is the row number in a standard Young tableau.
Matching statistic: St001051
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00091: Set partitions —rotate increasing⟶ Set partitions
St001051: Set partitions ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 100%
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00091: Set partitions —rotate increasing⟶ Set partitions
St001051: Set partitions ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> {{1},{2}}
=> {{1},{2}}
=> 2 = 1 + 1
[2]
=> [1,1,0,0,1,0]
=> {{1,2},{3}}
=> {{1},{2,3}}
=> 2 = 1 + 1
[1,1]
=> [1,0,1,1,0,0]
=> {{1},{2,3}}
=> {{1,3},{2}}
=> 1 = 0 + 1
[3]
=> [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> {{1},{2,3,4}}
=> 2 = 1 + 1
[2,1]
=> [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> {{1},{2},{3}}
=> 3 = 2 + 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> {{1,3,4},{2}}
=> 1 = 0 + 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> {{1},{2,3,4,5}}
=> 2 = 1 + 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> {{1},{2,4},{3}}
=> 3 = 2 + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> {{1,4},{2,3}}
=> 1 = 0 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> {{1,3},{2},{4}}
=> 2 = 1 + 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> {{1,3,4,5},{2}}
=> 1 = 0 + 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> {{1,2,3,4,5},{6}}
=> {{1},{2,3,4,5,6}}
=> 2 = 1 + 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> {{1,2,4},{3},{5}}
=> {{1},{2,3,5},{4}}
=> 3 = 2 + 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> {{1},{2,3},{4}}
=> 3 = 2 + 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> {{1},{2},{3,4}}
=> 2 = 1 + 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> {{1,4},{2},{3}}
=> 1 = 0 + 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> {{1,3,4},{2},{5}}
=> 2 = 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}}
=> {{1,3,4,5,6},{2}}
=> 1 = 0 + 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,5,6},{7}}
=> {{1},{2,3,4,5,6,7}}
=> 2 = 1 + 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> {{1,2,3,5},{4},{6}}
=> {{1},{2,3,4,6},{5}}
=> 3 = 2 + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> {{1,4},{2,3},{5}}
=> {{1},{2,5},{3,4}}
=> 3 = 2 + 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> {{1,3,4},{2},{5}}
=> {{1},{2,4,5},{3}}
=> 2 = 1 + 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> {{1,5},{2,3,4}}
=> 1 = 0 + 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 4 = 3 + 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> {{1,3},{2},{4,5}}
=> 2 = 1 + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> {{1,4,5},{2,3}}
=> 1 = 0 + 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> {{1,3,5},{2},{4}}
=> 1 = 0 + 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> {{1},{2,3,4,6},{5}}
=> {{1,3,4,5},{2},{6}}
=> 2 = 1 + 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,7}}
=> {{1,3,4,5,6,7},{2}}
=> 1 = 0 + 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,5,6,7},{8}}
=> {{1},{2,3,4,5,6,7,8}}
=> 2 = 1 + 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> {{1,2,3,4,6},{5},{7}}
=> {{1},{2,3,4,5,7},{6}}
=> 3 = 2 + 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> {{1,2,5},{3,4},{6}}
=> {{1},{2,3,6},{4,5}}
=> 3 = 2 + 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> {{1,2,4,5},{3},{6}}
=> {{1},{2,3,5,6},{4}}
=> 2 = 1 + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> {{1},{2,3,4},{5}}
=> 3 = 2 + 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> {{1,4},{2},{3},{5}}
=> {{1},{2,5},{3},{4}}
=> 4 = 3 + 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> {{1},{2},{3,4,5}}
=> 2 = 1 + 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> {{1,3},{2},{4,5}}
=> {{1,5},{2,4},{3}}
=> 1 = 0 + 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> {{1,2},{3,5},{4}}
=> {{1,4},{2,3},{5}}
=> 2 = 1 + 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> {{1,3},{2},{4},{5}}
=> 3 = 2 + 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> {{1},{2,3,6},{4,5}}
=> {{1,3,4},{2},{5,6}}
=> 2 = 1 + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> {{1,4,5},{2},{3}}
=> 1 = 0 + 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> {{1},{2,3,5,6},{4}}
=> {{1,3,4,6},{2},{5}}
=> 1 = 0 + 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> {{1},{2,3,4,5,7},{6}}
=> {{1,3,4,5,6},{2},{7}}
=> 2 = 1 + 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,7,8}}
=> {{1,3,4,5,6,7,8},{2}}
=> 1 = 0 + 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,5,6,7,8},{9}}
=> {{1},{2,3,4,5,6,7,8,9}}
=> 2 = 1 + 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,5,7},{6},{8}}
=> {{1},{2,3,4,5,6,8},{7}}
=> 3 = 2 + 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> {{1,2,3,6},{4,5},{7}}
=> {{1},{2,3,4,7},{5,6}}
=> 3 = 2 + 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> {{1,2,3,5,6},{4},{7}}
=> {{1},{2,3,4,6,7},{5}}
=> 2 = 1 + 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> {{1,5},{2,3,4},{6}}
=> {{1},{2,6},{3,4,5}}
=> 3 = 2 + 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> {{1,2,5},{3},{4},{6}}
=> {{1},{2,3,6},{4},{5}}
=> 4 = 3 + 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,8},{7}}
=> {{1,3,4,5,6,7},{2},{8}}
=> ? ∊ {0,2} + 1
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,7,8,9}}
=> {{1,3,4,5,6,7,8,9},{2}}
=> ? ∊ {0,2} + 1
[3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> {{1},{2,3,4,5,8},{6,7}}
=> {{1,3,4,5,6},{2},{7,8}}
=> ? ∊ {0,0,1,2} + 1
[2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> {{1},{2,3,4,5,7,8},{6}}
=> {{1,3,4,5,6,8},{2},{7}}
=> ? ∊ {0,0,1,2} + 1
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,7,9},{8}}
=> {{1,3,4,5,6,7,8},{2},{9}}
=> ? ∊ {0,0,1,2} + 1
[1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,7,8,9,10}}
=> {{1,3,4,5,6,7,8,9,10},{2}}
=> ? ∊ {0,0,1,2} + 1
[4,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> {{1},{2,3,4,8},{5,6,7}}
=> {{1,3,4,5},{2},{6,7,8}}
=> ? ∊ {0,0,0,1,1,2,2} + 1
[3,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> {{1},{2,3,4,5,8},{6},{7}}
=> {{1,3,4,5,6},{2},{7},{8}}
=> ? ∊ {0,0,0,1,1,2,2} + 1
[3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,9},{7,8}}
=> {{1,3,4,5,6,7},{2},{8,9}}
=> ? ∊ {0,0,0,1,1,2,2} + 1
[2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> {{1},{2,3,4,6,7,8},{5}}
=> {{1,3,4,5,7,8},{2},{6}}
=> ? ∊ {0,0,0,1,1,2,2} + 1
[2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,8,9},{7}}
=> {{1,3,4,5,6,7,9},{2},{8}}
=> ? ∊ {0,0,0,1,1,2,2} + 1
[2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,7,8,10},{9}}
=> {{1,3,4,5,6,7,8,9},{2},{10}}
=> ? ∊ {0,0,0,1,1,2,2} + 1
[1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,7,8,9,10,11}}
=> {{1,3,4,5,6,7,8,9,10,11},{2}}
=> ? ∊ {0,0,0,1,1,2,2} + 1
Description
The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition.
The bijection between set partitions of $\{1,\dots,n\}$ into $k$ blocks and trees with $n+1-k$ leaves is described in Theorem 1 of [1].
Matching statistic: St001801
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St001801: Permutations ⟶ ℤResult quality: 80% ●values known / values provided: 87%●distinct values known / distinct values provided: 80%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St001801: Permutations ⟶ ℤResult quality: 80% ●values known / values provided: 87%●distinct values known / distinct values provided: 80%
Values
[1]
=> []
=> []
=> [] => ? = 1
[2]
=> []
=> []
=> [] => ? = 1
[1,1]
=> [1]
=> [1,0]
=> [1] => 0
[3]
=> []
=> []
=> [] => ? = 2
[2,1]
=> [1]
=> [1,0]
=> [1] => 0
[1,1,1]
=> [1,1]
=> [1,1,0,0]
=> [2,1] => 1
[4]
=> []
=> []
=> [] => ? = 2
[3,1]
=> [1]
=> [1,0]
=> [1] => 0
[2,2]
=> [2]
=> [1,0,1,0]
=> [1,2] => 0
[2,1,1]
=> [1,1]
=> [1,1,0,0]
=> [2,1] => 1
[1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1
[5]
=> []
=> []
=> [] => ? = 2
[4,1]
=> [1]
=> [1,0]
=> [1] => 0
[3,2]
=> [2]
=> [1,0,1,0]
=> [1,2] => 0
[3,1,1]
=> [1,1]
=> [1,1,0,0]
=> [2,1] => 1
[2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 2
[6]
=> []
=> []
=> [] => ? = 3
[5,1]
=> [1]
=> [1,0]
=> [1] => 0
[4,2]
=> [2]
=> [1,0,1,0]
=> [1,2] => 0
[4,1,1]
=> [1,1]
=> [1,1,0,0]
=> [2,1] => 1
[3,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1
[2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 2
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 2
[7]
=> []
=> []
=> [] => ? = 2
[6,1]
=> [1]
=> [1,0]
=> [1] => 0
[5,2]
=> [2]
=> [1,0,1,0]
=> [1,2] => 0
[5,1,1]
=> [1,1]
=> [1,1,0,0]
=> [2,1] => 1
[4,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1
[3,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
[3,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 2
[2,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 2
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 2
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => 3
[8]
=> []
=> []
=> [] => ? ∊ {0,3}
[7,1]
=> [1]
=> [1,0]
=> [1] => 0
[6,2]
=> [2]
=> [1,0,1,0]
=> [1,2] => 0
[6,1,1]
=> [1,1]
=> [1,1,0,0]
=> [2,1] => 1
[5,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[5,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[5,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1
[4,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[4,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
[4,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 2
[3,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 0
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 1
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,1] => ? ∊ {0,3}
[9]
=> []
=> []
=> [] => ? ∊ {0,0,3}
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,1] => ? ∊ {0,0,3}
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,1] => ? ∊ {0,0,3}
[10]
=> []
=> []
=> [] => ? ∊ {0,0,0,0,3,4}
[3,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,1] => ? ∊ {0,0,0,0,3,4}
[2,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,7,1] => ? ∊ {0,0,0,0,3,4}
[2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,8,2] => ? ∊ {0,0,0,0,3,4}
[2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,1] => ? ∊ {0,0,0,0,3,4}
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,9,1] => ? ∊ {0,0,0,0,3,4}
Description
Half the number of preimage-image pairs of different parity in a permutation.
The following 134 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000678The number of up steps after the last double rise of a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St000007The number of saliances of the permutation. St000383The last part of an integer composition. St000546The number of global descents of a permutation. St000237The number of small exceedances. St000675The number of centered multitunnels of a Dyck path. St000054The first entry of the permutation. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St000025The number of initial rises of a Dyck path. St000234The number of global ascents of a permutation. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St000069The number of maximal elements of a poset. St000717The number of ordinal summands of a poset. St000971The smallest closer of a set partition. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001316The domatic number of a graph. St000214The number of adjacencies of a permutation. St000843The decomposition number of a perfect matching. St000068The number of minimal elements in a poset. St000549The number of odd partial sums of an integer partition. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St001114The number of odd descents of a permutation. St000989The number of final rises of a permutation. St000654The first descent of a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St001733The number of weak left to right maxima of a Dyck path. St000740The last entry of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St000542The number of left-to-right-minima of a permutation. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000990The first ascent of a permutation. St000335The difference of lower and upper interactions. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St000051The size of the left subtree of a binary tree. St000133The "bounce" of a permutation. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. 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. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001274The number of indecomposable injective modules with projective dimension equal to two. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000056The decomposition (or block) number of a permutation. St000061The number of nodes on the left branch of a binary tree. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000084The number of subtrees. St000286The number of connected components of the complement of a graph. St000314The number of left-to-right-maxima of a permutation. St000991The number of right-to-left minima of a permutation. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001481The minimal height of a peak of a Dyck path. St000738The first entry in the last row of a standard tableau. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St000137The Grundy value of an integer partition. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000352The Elizalde-Pak rank of a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000681The Grundy value of Chomp on Ferrers diagrams. St000993The multiplicity of the largest part of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St000454The largest eigenvalue of a graph if it is integral. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000731The number of double exceedences of a permutation. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000996The number of exclusive left-to-right maxima of a permutation. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000665The number of rafts of a permutation. St001115The number of even descents of a permutation. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001948The number of augmented double ascents of a permutation. St000338The number of pixed points of a permutation. St000732The number of double deficiencies of a permutation. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St000193The row of the unique '1' in the first column of the alternating sign matrix. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000741The Colin de Verdière graph invariant. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001713The difference of the first and last value in the first row of the Gelfand-Tsetlin pattern. St001498The normalised height of a Nakayama algebra with magnitude 1. St000782The indicator function of whether a given perfect matching is an L & P matching. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. 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 $eA$. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001423The number of distinct cubes in a binary word. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001730The number of times the path corresponding to a binary word crosses the base line. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000455The second largest eigenvalue of a graph if it is integral. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001267The length of the Lyndon factorization of the binary word. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001462The number of factors of a standard tableaux under concatenation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001904The length of the initial strictly increasing segment of a parking function. St001937The size of the center of a parking function.
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