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Your data matches 168 different statistics following compositions of up to 3 maps.
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Matching statistic: St001805
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
St001805: Semistandard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> 0
[[2,2]]
=> 1
[[1],[2]]
=> 2
[[1,3]]
=> 0
[[2,3]]
=> 0
[[3,3]]
=> 1
[[1],[3]]
=> 2
[[2],[3]]
=> 2
[[1,1,2]]
=> 0
[[1,2,2]]
=> 0
[[2,2,2]]
=> 1
[[1,1],[2]]
=> 1
[[1,2],[2]]
=> 1
[[1,4]]
=> 0
[[2,4]]
=> 0
[[3,4]]
=> 0
[[4,4]]
=> 1
[[1],[4]]
=> 2
[[2],[4]]
=> 2
[[3],[4]]
=> 2
[[1,1,3]]
=> 0
[[1,2,3]]
=> 0
[[1,3,3]]
=> 0
[[2,2,3]]
=> 0
[[2,3,3]]
=> 0
[[3,3,3]]
=> 1
[[1,1],[3]]
=> 1
[[1,2],[3]]
=> 1
[[1,3],[2]]
=> 0
[[1,3],[3]]
=> 1
[[2,2],[3]]
=> 1
[[2,3],[3]]
=> 1
[[1],[2],[3]]
=> 3
[[1,1,1,2]]
=> 0
[[1,1,2,2]]
=> 0
[[1,2,2,2]]
=> 0
[[2,2,2,2]]
=> 1
[[1,1,1],[2]]
=> 1
[[1,1,2],[2]]
=> 1
[[1,2,2],[2]]
=> 1
[[1,1],[2,2]]
=> 2
[[1,5]]
=> 0
[[2,5]]
=> 0
[[3,5]]
=> 0
[[4,5]]
=> 0
[[5,5]]
=> 1
[[1],[5]]
=> 2
[[2],[5]]
=> 2
[[3],[5]]
=> 2
[[4],[5]]
=> 2
Description
The maximal overlap of a cylindrical tableau associated with a semistandard tableau.
A cylindrical tableau associated with a semistandard Young tableau $T$ is the skew semistandard tableau obtained by gluing two copies of $T$ such that the inner shape is a rectangle.
The overlap, recorded in this statistic, equals $\max_C\big(2\ell(T) - \ell(C)\big)$, where $\ell$ denotes the number of rows of a tableau and the maximum is taken over all cylindrical tableaux.
Matching statistic: St001767
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001767: Integer partitions ⟶ ℤResult quality: 40% ●values known / values provided: 71%●distinct values known / distinct values provided: 40%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001767: Integer partitions ⟶ ℤResult quality: 40% ●values known / values provided: 71%●distinct values known / distinct values provided: 40%
Values
[[1,2]]
=> [2]
=> []
=> ? ∊ {1,2}
[[2,2]]
=> [2]
=> []
=> ? ∊ {1,2}
[[1],[2]]
=> [1,1]
=> [1]
=> 0
[[1,3]]
=> [2]
=> []
=> ? ∊ {1,2,2}
[[2,3]]
=> [2]
=> []
=> ? ∊ {1,2,2}
[[3,3]]
=> [2]
=> []
=> ? ∊ {1,2,2}
[[1],[3]]
=> [1,1]
=> [1]
=> 0
[[2],[3]]
=> [1,1]
=> [1]
=> 0
[[1,1,2]]
=> [3]
=> []
=> ? ∊ {1,1,1}
[[1,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,1}
[[2,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,1}
[[1,1],[2]]
=> [2,1]
=> [1]
=> 0
[[1,2],[2]]
=> [2,1]
=> [1]
=> 0
[[1,4]]
=> [2]
=> []
=> ? ∊ {1,2,2,2}
[[2,4]]
=> [2]
=> []
=> ? ∊ {1,2,2,2}
[[3,4]]
=> [2]
=> []
=> ? ∊ {1,2,2,2}
[[4,4]]
=> [2]
=> []
=> ? ∊ {1,2,2,2}
[[1],[4]]
=> [1,1]
=> [1]
=> 0
[[2],[4]]
=> [1,1]
=> [1]
=> 0
[[3],[4]]
=> [1,1]
=> [1]
=> 0
[[1,1,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,3}
[[1,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,3}
[[1,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,3}
[[2,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,3}
[[2,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,3}
[[3,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,3}
[[1,1],[3]]
=> [2,1]
=> [1]
=> 0
[[1,2],[3]]
=> [2,1]
=> [1]
=> 0
[[1,3],[2]]
=> [2,1]
=> [1]
=> 0
[[1,3],[3]]
=> [2,1]
=> [1]
=> 0
[[2,2],[3]]
=> [2,1]
=> [1]
=> 0
[[2,3],[3]]
=> [2,1]
=> [1]
=> 0
[[1],[2],[3]]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,1,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,2}
[[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,2}
[[1,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,2}
[[2,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,2}
[[1,1,1],[2]]
=> [3,1]
=> [1]
=> 0
[[1,1,2],[2]]
=> [3,1]
=> [1]
=> 0
[[1,2,2],[2]]
=> [3,1]
=> [1]
=> 0
[[1,1],[2,2]]
=> [2,2]
=> [2]
=> 1
[[1,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2}
[[2,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2}
[[3,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2}
[[4,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2}
[[5,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2}
[[1],[5]]
=> [1,1]
=> [1]
=> 0
[[2],[5]]
=> [1,1]
=> [1]
=> 0
[[3],[5]]
=> [1,1]
=> [1]
=> 0
[[4],[5]]
=> [1,1]
=> [1]
=> 0
[[1,1,4]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[1,2,4]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[1,3,4]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[1,4,4]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[2,2,4]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[2,3,4]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[2,4,4]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[3,3,4]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[3,4,4]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[4,4,4]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[1,1],[4]]
=> [2,1]
=> [1]
=> 0
[[1,2],[4]]
=> [2,1]
=> [1]
=> 0
[[1,4],[2]]
=> [2,1]
=> [1]
=> 0
[[1,3],[4]]
=> [2,1]
=> [1]
=> 0
[[1,4],[3]]
=> [2,1]
=> [1]
=> 0
[[1,4],[4]]
=> [2,1]
=> [1]
=> 0
[[2,2],[4]]
=> [2,1]
=> [1]
=> 0
[[2,3],[4]]
=> [2,1]
=> [1]
=> 0
[[2,4],[3]]
=> [2,1]
=> [1]
=> 0
[[2,4],[4]]
=> [2,1]
=> [1]
=> 0
[[3,3],[4]]
=> [2,1]
=> [1]
=> 0
[[3,4],[4]]
=> [2,1]
=> [1]
=> 0
[[1],[2],[4]]
=> [1,1,1]
=> [1,1]
=> 1
[[1],[3],[4]]
=> [1,1,1]
=> [1,1]
=> 1
[[2],[3],[4]]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1,1,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[1,1,2,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[1,1,3,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[1,2,2,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[1,2,3,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[1,3,3,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[2,2,2,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[2,2,3,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[2,3,3,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[3,3,3,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[1,1,1],[3]]
=> [3,1]
=> [1]
=> 0
[[1,1,2],[3]]
=> [3,1]
=> [1]
=> 0
[[1,1,3],[2]]
=> [3,1]
=> [1]
=> 0
[[1,1,3],[3]]
=> [3,1]
=> [1]
=> 0
[[1,2,2],[3]]
=> [3,1]
=> [1]
=> 0
[[1,2,3],[2]]
=> [3,1]
=> [1]
=> 0
[[1,2,3],[3]]
=> [3,1]
=> [1]
=> 0
[[1,3,3],[2]]
=> [3,1]
=> [1]
=> 0
[[1,3,3],[3]]
=> [3,1]
=> [1]
=> 0
[[2,2,2],[3]]
=> [3,1]
=> [1]
=> 0
[[2,2,3],[3]]
=> [3,1]
=> [1]
=> 0
[[2,3,3],[3]]
=> [3,1]
=> [1]
=> 0
[[1,1,1,1,2]]
=> [5]
=> []
=> ? ∊ {1,1,1,1,1}
[[1,1,1,2,2]]
=> [5]
=> []
=> ? ∊ {1,1,1,1,1}
[[1,1,2,2,2]]
=> [5]
=> []
=> ? ∊ {1,1,1,1,1}
Description
The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment.
Assign to each cell of the Ferrers diagram an arrow pointing north, east, south or west. Then compute for each cell the number of arrows pointing towards it, and take the minimum of those. This statistic is the maximal minimum that can be obtained by assigning arrows in any way.
Matching statistic: St000329
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000329: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 71%●distinct values known / distinct values provided: 60%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000329: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 71%●distinct values known / distinct values provided: 60%
Values
[[1,2]]
=> [2]
=> []
=> []
=> ? ∊ {1,2}
[[2,2]]
=> [2]
=> []
=> []
=> ? ∊ {1,2}
[[1],[2]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,3]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2}
[[2,3]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2}
[[3,3]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2}
[[1],[3]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2],[3]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1,2]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1}
[[1,2,2]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1}
[[2,2,2]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1}
[[1,1],[2]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,2],[2]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,4]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2,2}
[[2,4]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2,2}
[[3,4]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2,2}
[[4,4]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2,2}
[[1],[4]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2],[4]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[3],[4]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1,3]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,3}
[[1,2,3]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,3}
[[1,3,3]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,3}
[[2,2,3]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,3}
[[2,3,3]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,3}
[[3,3,3]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,3}
[[1,1],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,2],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,3],[2]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,3],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,2],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,3],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1],[2],[3]]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[[1,1,1,2]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,2}
[[1,1,2,2]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,2}
[[1,2,2,2]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,2}
[[2,2,2,2]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,2}
[[1,1,1],[2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1,2],[2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,2,2],[2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1],[2,2]]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[1,5]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2,2,2}
[[2,5]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2,2,2}
[[3,5]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2,2,2}
[[4,5]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2,2,2}
[[5,5]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2,2,2}
[[1],[5]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2],[5]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[3],[5]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[4],[5]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[1,2,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[1,3,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[1,4,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[2,2,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[2,3,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[2,4,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[3,3,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[3,4,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[4,4,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[1,1],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,2],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,4],[2]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,3],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,4],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,4],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,2],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,3],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,4],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,4],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[3,3],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[3,4],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1],[2],[4]]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[[1],[3],[4]]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[[2],[3],[4]]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[[1,1,1,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[1,1,2,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[1,1,3,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[1,2,2,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[1,2,3,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[1,3,3,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[2,2,2,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[2,2,3,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[2,3,3,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[3,3,3,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[1,1,1],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1,2],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1,3],[2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1,3],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,2,2],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,2,3],[2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,2,3],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,3,3],[2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,3,3],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,2,2],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,2,3],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,3,3],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1,1,1,2]]
=> [5]
=> []
=> []
=> ? ∊ {1,1,1,1,1}
[[1,1,1,2,2]]
=> [5]
=> []
=> []
=> ? ∊ {1,1,1,1,1}
[[1,1,2,2,2]]
=> [5]
=> []
=> []
=> ? ∊ {1,1,1,1,1}
Description
The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1.
Matching statistic: St000547
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000547: Integer partitions ⟶ ℤResult quality: 40% ●values known / values provided: 71%●distinct values known / distinct values provided: 40%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000547: Integer partitions ⟶ ℤResult quality: 40% ●values known / values provided: 71%●distinct values known / distinct values provided: 40%
Values
[[1,2]]
=> [2]
=> []
=> ?
=> ? ∊ {1,2}
[[2,2]]
=> [2]
=> []
=> ?
=> ? ∊ {1,2}
[[1],[2]]
=> [1,1]
=> [1]
=> [1]
=> 0
[[1,3]]
=> [2]
=> []
=> ?
=> ? ∊ {1,2,2}
[[2,3]]
=> [2]
=> []
=> ?
=> ? ∊ {1,2,2}
[[3,3]]
=> [2]
=> []
=> ?
=> ? ∊ {1,2,2}
[[1],[3]]
=> [1,1]
=> [1]
=> [1]
=> 0
[[2],[3]]
=> [1,1]
=> [1]
=> [1]
=> 0
[[1,1,2]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1}
[[1,2,2]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1}
[[2,2,2]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1}
[[1,1],[2]]
=> [2,1]
=> [1]
=> [1]
=> 0
[[1,2],[2]]
=> [2,1]
=> [1]
=> [1]
=> 0
[[1,4]]
=> [2]
=> []
=> ?
=> ? ∊ {1,2,2,2}
[[2,4]]
=> [2]
=> []
=> ?
=> ? ∊ {1,2,2,2}
[[3,4]]
=> [2]
=> []
=> ?
=> ? ∊ {1,2,2,2}
[[4,4]]
=> [2]
=> []
=> ?
=> ? ∊ {1,2,2,2}
[[1],[4]]
=> [1,1]
=> [1]
=> [1]
=> 0
[[2],[4]]
=> [1,1]
=> [1]
=> [1]
=> 0
[[3],[4]]
=> [1,1]
=> [1]
=> [1]
=> 0
[[1,1,3]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,3}
[[1,2,3]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,3}
[[1,3,3]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,3}
[[2,2,3]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,3}
[[2,3,3]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,3}
[[3,3,3]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,3}
[[1,1],[3]]
=> [2,1]
=> [1]
=> [1]
=> 0
[[1,2],[3]]
=> [2,1]
=> [1]
=> [1]
=> 0
[[1,3],[2]]
=> [2,1]
=> [1]
=> [1]
=> 0
[[1,3],[3]]
=> [2,1]
=> [1]
=> [1]
=> 0
[[2,2],[3]]
=> [2,1]
=> [1]
=> [1]
=> 0
[[2,3],[3]]
=> [2,1]
=> [1]
=> [1]
=> 0
[[1],[2],[3]]
=> [1,1,1]
=> [1,1]
=> [1,1]
=> 1
[[1,1,1,2]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,2}
[[1,1,2,2]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,2}
[[1,2,2,2]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,2}
[[2,2,2,2]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,2}
[[1,1,1],[2]]
=> [3,1]
=> [1]
=> [1]
=> 0
[[1,1,2],[2]]
=> [3,1]
=> [1]
=> [1]
=> 0
[[1,2,2],[2]]
=> [3,1]
=> [1]
=> [1]
=> 0
[[1,1],[2,2]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[1,5]]
=> [2]
=> []
=> ?
=> ? ∊ {1,2,2,2,2}
[[2,5]]
=> [2]
=> []
=> ?
=> ? ∊ {1,2,2,2,2}
[[3,5]]
=> [2]
=> []
=> ?
=> ? ∊ {1,2,2,2,2}
[[4,5]]
=> [2]
=> []
=> ?
=> ? ∊ {1,2,2,2,2}
[[5,5]]
=> [2]
=> []
=> ?
=> ? ∊ {1,2,2,2,2}
[[1],[5]]
=> [1,1]
=> [1]
=> [1]
=> 0
[[2],[5]]
=> [1,1]
=> [1]
=> [1]
=> 0
[[3],[5]]
=> [1,1]
=> [1]
=> [1]
=> 0
[[4],[5]]
=> [1,1]
=> [1]
=> [1]
=> 0
[[1,1,4]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[1,2,4]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[1,3,4]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[1,4,4]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[2,2,4]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[2,3,4]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[2,4,4]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[3,3,4]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[3,4,4]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[4,4,4]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[1,1],[4]]
=> [2,1]
=> [1]
=> [1]
=> 0
[[1,2],[4]]
=> [2,1]
=> [1]
=> [1]
=> 0
[[1,4],[2]]
=> [2,1]
=> [1]
=> [1]
=> 0
[[1,3],[4]]
=> [2,1]
=> [1]
=> [1]
=> 0
[[1,4],[3]]
=> [2,1]
=> [1]
=> [1]
=> 0
[[1,4],[4]]
=> [2,1]
=> [1]
=> [1]
=> 0
[[2,2],[4]]
=> [2,1]
=> [1]
=> [1]
=> 0
[[2,3],[4]]
=> [2,1]
=> [1]
=> [1]
=> 0
[[2,4],[3]]
=> [2,1]
=> [1]
=> [1]
=> 0
[[2,4],[4]]
=> [2,1]
=> [1]
=> [1]
=> 0
[[3,3],[4]]
=> [2,1]
=> [1]
=> [1]
=> 0
[[3,4],[4]]
=> [2,1]
=> [1]
=> [1]
=> 0
[[1],[2],[4]]
=> [1,1,1]
=> [1,1]
=> [1,1]
=> 1
[[1],[3],[4]]
=> [1,1,1]
=> [1,1]
=> [1,1]
=> 1
[[2],[3],[4]]
=> [1,1,1]
=> [1,1]
=> [1,1]
=> 1
[[1,1,1,3]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[1,1,2,3]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[1,1,3,3]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[1,2,2,3]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[1,2,3,3]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[1,3,3,3]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[2,2,2,3]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[2,2,3,3]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[2,3,3,3]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[3,3,3,3]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[1,1,1],[3]]
=> [3,1]
=> [1]
=> [1]
=> 0
[[1,1,2],[3]]
=> [3,1]
=> [1]
=> [1]
=> 0
[[1,1,3],[2]]
=> [3,1]
=> [1]
=> [1]
=> 0
[[1,1,3],[3]]
=> [3,1]
=> [1]
=> [1]
=> 0
[[1,2,2],[3]]
=> [3,1]
=> [1]
=> [1]
=> 0
[[1,2,3],[2]]
=> [3,1]
=> [1]
=> [1]
=> 0
[[1,2,3],[3]]
=> [3,1]
=> [1]
=> [1]
=> 0
[[1,3,3],[2]]
=> [3,1]
=> [1]
=> [1]
=> 0
[[1,3,3],[3]]
=> [3,1]
=> [1]
=> [1]
=> 0
[[2,2,2],[3]]
=> [3,1]
=> [1]
=> [1]
=> 0
[[2,2,3],[3]]
=> [3,1]
=> [1]
=> [1]
=> 0
[[2,3,3],[3]]
=> [3,1]
=> [1]
=> [1]
=> 0
[[1,1,1,1,2]]
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1}
[[1,1,1,2,2]]
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1}
[[1,1,2,2,2]]
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1}
Description
The number of even non-empty partial sums of an integer partition.
Matching statistic: St000659
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000659: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 71%●distinct values known / distinct values provided: 60%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000659: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 71%●distinct values known / distinct values provided: 60%
Values
[[1,2]]
=> [2]
=> []
=> []
=> ? ∊ {1,2}
[[2,2]]
=> [2]
=> []
=> []
=> ? ∊ {1,2}
[[1],[2]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,3]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2}
[[2,3]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2}
[[3,3]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2}
[[1],[3]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2],[3]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1,2]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1}
[[1,2,2]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1}
[[2,2,2]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1}
[[1,1],[2]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,2],[2]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,4]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2,2}
[[2,4]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2,2}
[[3,4]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2,2}
[[4,4]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2,2}
[[1],[4]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2],[4]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[3],[4]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1,3]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,3}
[[1,2,3]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,3}
[[1,3,3]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,3}
[[2,2,3]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,3}
[[2,3,3]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,3}
[[3,3,3]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,3}
[[1,1],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,2],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,3],[2]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,3],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,2],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,3],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1],[2],[3]]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[[1,1,1,2]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,2}
[[1,1,2,2]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,2}
[[1,2,2,2]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,2}
[[2,2,2,2]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,2}
[[1,1,1],[2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1,2],[2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,2,2],[2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1],[2,2]]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[1,5]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2,2,2}
[[2,5]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2,2,2}
[[3,5]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2,2,2}
[[4,5]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2,2,2}
[[5,5]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2,2,2}
[[1],[5]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2],[5]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[3],[5]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[4],[5]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[1,2,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[1,3,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[1,4,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[2,2,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[2,3,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[2,4,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[3,3,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[3,4,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[4,4,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[1,1],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,2],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,4],[2]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,3],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,4],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,4],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,2],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,3],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,4],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,4],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[3,3],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[3,4],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1],[2],[4]]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[[1],[3],[4]]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[[2],[3],[4]]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[[1,1,1,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[1,1,2,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[1,1,3,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[1,2,2,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[1,2,3,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[1,3,3,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[2,2,2,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[2,2,3,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[2,3,3,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[3,3,3,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[1,1,1],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1,2],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1,3],[2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1,3],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,2,2],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,2,3],[2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,2,3],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,3,3],[2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,3,3],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,2,2],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,2,3],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,3,3],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1,1,1,2]]
=> [5]
=> []
=> []
=> ? ∊ {1,1,1,1,1}
[[1,1,1,2,2]]
=> [5]
=> []
=> []
=> ? ∊ {1,1,1,1,1}
[[1,1,2,2,2]]
=> [5]
=> []
=> []
=> ? ∊ {1,1,1,1,1}
Description
The number of rises of length at least 2 of a Dyck path.
Matching statistic: St000688
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000688: Dyck paths ⟶ ℤResult quality: 40% ●values known / values provided: 71%●distinct values known / distinct values provided: 40%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000688: Dyck paths ⟶ ℤResult quality: 40% ●values known / values provided: 71%●distinct values known / distinct values provided: 40%
Values
[[1,2]]
=> [2]
=> []
=> []
=> ? ∊ {1,2}
[[2,2]]
=> [2]
=> []
=> []
=> ? ∊ {1,2}
[[1],[2]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,3]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2}
[[2,3]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2}
[[3,3]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2}
[[1],[3]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2],[3]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1,2]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1}
[[1,2,2]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1}
[[2,2,2]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1}
[[1,1],[2]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,2],[2]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,4]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2,2}
[[2,4]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2,2}
[[3,4]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2,2}
[[4,4]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2,2}
[[1],[4]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2],[4]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[3],[4]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1,3]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,3}
[[1,2,3]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,3}
[[1,3,3]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,3}
[[2,2,3]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,3}
[[2,3,3]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,3}
[[3,3,3]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,3}
[[1,1],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,2],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,3],[2]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,3],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,2],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,3],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1],[2],[3]]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[[1,1,1,2]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,2}
[[1,1,2,2]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,2}
[[1,2,2,2]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,2}
[[2,2,2,2]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,2}
[[1,1,1],[2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1,2],[2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,2,2],[2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1],[2,2]]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[1,5]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2,2,2}
[[2,5]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2,2,2}
[[3,5]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2,2,2}
[[4,5]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2,2,2}
[[5,5]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2,2,2}
[[1],[5]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2],[5]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[3],[5]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[4],[5]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[1,2,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[1,3,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[1,4,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[2,2,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[2,3,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[2,4,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[3,3,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[3,4,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[4,4,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[1,1],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,2],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,4],[2]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,3],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,4],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,4],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,2],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,3],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,4],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,4],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[3,3],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[3,4],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1],[2],[4]]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[[1],[3],[4]]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[[2],[3],[4]]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[[1,1,1,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[1,1,2,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[1,1,3,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[1,2,2,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[1,2,3,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[1,3,3,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[2,2,2,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[2,2,3,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[2,3,3,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[3,3,3,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[1,1,1],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1,2],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1,3],[2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1,3],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,2,2],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,2,3],[2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,2,3],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,3,3],[2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,3,3],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,2,2],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,2,3],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,3,3],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1,1,1,2]]
=> [5]
=> []
=> []
=> ? ∊ {1,1,1,1,1}
[[1,1,1,2,2]]
=> [5]
=> []
=> []
=> ? ∊ {1,1,1,1,1}
[[1,1,2,2,2]]
=> [5]
=> []
=> []
=> ? ∊ {1,1,1,1,1}
Description
The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path.
The global dimension is given by [[St000684]] and the dominant dimension is given by [[St000685]]. To every Dyck path there is an LNakayama algebra associated as described in [[St000684]].
Dyck paths for which the global dimension and the dominant dimension of the the LNakayama algebra coincide and both dimensions at least $2$ correspond to the LNakayama algebras that are higher Auslander algebras in the sense of [1].
Matching statistic: St000970
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000970: Dyck paths ⟶ ℤResult quality: 40% ●values known / values provided: 71%●distinct values known / distinct values provided: 40%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000970: Dyck paths ⟶ ℤResult quality: 40% ●values known / values provided: 71%●distinct values known / distinct values provided: 40%
Values
[[1,2]]
=> [2]
=> []
=> []
=> ? ∊ {1,2}
[[2,2]]
=> [2]
=> []
=> []
=> ? ∊ {1,2}
[[1],[2]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,3]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2}
[[2,3]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2}
[[3,3]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2}
[[1],[3]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2],[3]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1,2]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1}
[[1,2,2]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1}
[[2,2,2]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1}
[[1,1],[2]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,2],[2]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,4]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2,2}
[[2,4]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2,2}
[[3,4]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2,2}
[[4,4]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2,2}
[[1],[4]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2],[4]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[3],[4]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1,3]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,3}
[[1,2,3]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,3}
[[1,3,3]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,3}
[[2,2,3]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,3}
[[2,3,3]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,3}
[[3,3,3]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,3}
[[1,1],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,2],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,3],[2]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,3],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,2],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,3],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1],[2],[3]]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[[1,1,1,2]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,2}
[[1,1,2,2]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,2}
[[1,2,2,2]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,2}
[[2,2,2,2]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,2}
[[1,1,1],[2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1,2],[2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,2,2],[2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1],[2,2]]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[1,5]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2,2,2}
[[2,5]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2,2,2}
[[3,5]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2,2,2}
[[4,5]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2,2,2}
[[5,5]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2,2,2}
[[1],[5]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2],[5]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[3],[5]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[4],[5]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[1,2,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[1,3,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[1,4,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[2,2,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[2,3,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[2,4,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[3,3,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[3,4,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[4,4,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[1,1],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,2],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,4],[2]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,3],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,4],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,4],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,2],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,3],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,4],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,4],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[3,3],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[3,4],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1],[2],[4]]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[[1],[3],[4]]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[[2],[3],[4]]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[[1,1,1,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[1,1,2,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[1,1,3,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[1,2,2,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[1,2,3,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[1,3,3,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[2,2,2,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[2,2,3,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[2,3,3,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[3,3,3,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[1,1,1],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1,2],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1,3],[2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1,3],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,2,2],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,2,3],[2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,2,3],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,3,3],[2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,3,3],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,2,2],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,2,3],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,3,3],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1,1,1,2]]
=> [5]
=> []
=> []
=> ? ∊ {1,1,1,1,1}
[[1,1,1,2,2]]
=> [5]
=> []
=> []
=> ? ∊ {1,1,1,1,1}
[[1,1,2,2,2]]
=> [5]
=> []
=> []
=> ? ∊ {1,1,1,1,1}
Description
Number of peaks minus the dominant dimension of the corresponding LNakayama algebra.
Matching statistic: St001026
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001026: Dyck paths ⟶ ℤResult quality: 40% ●values known / values provided: 71%●distinct values known / distinct values provided: 40%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001026: Dyck paths ⟶ ℤResult quality: 40% ●values known / values provided: 71%●distinct values known / distinct values provided: 40%
Values
[[1,2]]
=> [2]
=> []
=> []
=> ? ∊ {1,2}
[[2,2]]
=> [2]
=> []
=> []
=> ? ∊ {1,2}
[[1],[2]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,3]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2}
[[2,3]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2}
[[3,3]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2}
[[1],[3]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2],[3]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1,2]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1}
[[1,2,2]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1}
[[2,2,2]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1}
[[1,1],[2]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,2],[2]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,4]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2,2}
[[2,4]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2,2}
[[3,4]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2,2}
[[4,4]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2,2}
[[1],[4]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2],[4]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[3],[4]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1,3]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,3}
[[1,2,3]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,3}
[[1,3,3]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,3}
[[2,2,3]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,3}
[[2,3,3]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,3}
[[3,3,3]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,3}
[[1,1],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,2],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,3],[2]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,3],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,2],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,3],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1],[2],[3]]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[[1,1,1,2]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,2}
[[1,1,2,2]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,2}
[[1,2,2,2]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,2}
[[2,2,2,2]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,2}
[[1,1,1],[2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1,2],[2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,2,2],[2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1],[2,2]]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[1,5]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2,2,2}
[[2,5]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2,2,2}
[[3,5]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2,2,2}
[[4,5]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2,2,2}
[[5,5]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2,2,2}
[[1],[5]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2],[5]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[3],[5]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[4],[5]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[1,2,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[1,3,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[1,4,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[2,2,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[2,3,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[2,4,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[3,3,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[3,4,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[4,4,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[1,1],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,2],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,4],[2]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,3],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,4],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,4],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,2],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,3],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,4],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,4],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[3,3],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[3,4],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1],[2],[4]]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[[1],[3],[4]]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[[2],[3],[4]]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[[1,1,1,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[1,1,2,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[1,1,3,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[1,2,2,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[1,2,3,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[1,3,3,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[2,2,2,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[2,2,3,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[2,3,3,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[3,3,3,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[1,1,1],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1,2],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1,3],[2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1,3],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,2,2],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,2,3],[2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,2,3],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,3,3],[2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,3,3],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,2,2],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,2,3],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,3,3],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1,1,1,2]]
=> [5]
=> []
=> []
=> ? ∊ {1,1,1,1,1}
[[1,1,1,2,2]]
=> [5]
=> []
=> []
=> ? ∊ {1,1,1,1,1}
[[1,1,2,2,2]]
=> [5]
=> []
=> []
=> ? ∊ {1,1,1,1,1}
Description
The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001031
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001031: Dyck paths ⟶ ℤResult quality: 40% ●values known / values provided: 71%●distinct values known / distinct values provided: 40%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001031: Dyck paths ⟶ ℤResult quality: 40% ●values known / values provided: 71%●distinct values known / distinct values provided: 40%
Values
[[1,2]]
=> [2]
=> []
=> []
=> ? ∊ {1,2}
[[2,2]]
=> [2]
=> []
=> []
=> ? ∊ {1,2}
[[1],[2]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,3]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2}
[[2,3]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2}
[[3,3]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2}
[[1],[3]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2],[3]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1,2]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1}
[[1,2,2]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1}
[[2,2,2]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1}
[[1,1],[2]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,2],[2]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,4]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2,2}
[[2,4]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2,2}
[[3,4]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2,2}
[[4,4]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2,2}
[[1],[4]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2],[4]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[3],[4]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1,3]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,3}
[[1,2,3]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,3}
[[1,3,3]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,3}
[[2,2,3]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,3}
[[2,3,3]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,3}
[[3,3,3]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,3}
[[1,1],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,2],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,3],[2]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,3],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,2],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,3],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1],[2],[3]]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[[1,1,1,2]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,2}
[[1,1,2,2]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,2}
[[1,2,2,2]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,2}
[[2,2,2,2]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,2}
[[1,1,1],[2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1,2],[2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,2,2],[2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1],[2,2]]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[1,5]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2,2,2}
[[2,5]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2,2,2}
[[3,5]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2,2,2}
[[4,5]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2,2,2}
[[5,5]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2,2,2}
[[1],[5]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2],[5]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[3],[5]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[4],[5]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[1,2,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[1,3,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[1,4,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[2,2,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[2,3,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[2,4,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[3,3,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[3,4,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[4,4,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[1,1],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,2],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,4],[2]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,3],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,4],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,4],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,2],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,3],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,4],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,4],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[3,3],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[3,4],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1],[2],[4]]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[[1],[3],[4]]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[[2],[3],[4]]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[[1,1,1,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[1,1,2,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[1,1,3,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[1,2,2,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[1,2,3,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[1,3,3,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[2,2,2,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[2,2,3,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[2,3,3,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[3,3,3,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[1,1,1],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1,2],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1,3],[2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1,3],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,2,2],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,2,3],[2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,2,3],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,3,3],[2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,3,3],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,2,2],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,2,3],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,3,3],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1,1,1,2]]
=> [5]
=> []
=> []
=> ? ∊ {1,1,1,1,1}
[[1,1,1,2,2]]
=> [5]
=> []
=> []
=> ? ∊ {1,1,1,1,1}
[[1,1,2,2,2]]
=> [5]
=> []
=> []
=> ? ∊ {1,1,1,1,1}
Description
The height of the bicoloured Motzkin path associated with the Dyck path.
Matching statistic: St001035
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001035: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 71%●distinct values known / distinct values provided: 60%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001035: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 71%●distinct values known / distinct values provided: 60%
Values
[[1,2]]
=> [2]
=> []
=> []
=> ? ∊ {1,2}
[[2,2]]
=> [2]
=> []
=> []
=> ? ∊ {1,2}
[[1],[2]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,3]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2}
[[2,3]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2}
[[3,3]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2}
[[1],[3]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2],[3]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1,2]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1}
[[1,2,2]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1}
[[2,2,2]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1}
[[1,1],[2]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,2],[2]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,4]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2,2}
[[2,4]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2,2}
[[3,4]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2,2}
[[4,4]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2,2}
[[1],[4]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2],[4]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[3],[4]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1,3]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,3}
[[1,2,3]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,3}
[[1,3,3]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,3}
[[2,2,3]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,3}
[[2,3,3]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,3}
[[3,3,3]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,3}
[[1,1],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,2],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,3],[2]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,3],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,2],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,3],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1],[2],[3]]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[[1,1,1,2]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,2}
[[1,1,2,2]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,2}
[[1,2,2,2]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,2}
[[2,2,2,2]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,2}
[[1,1,1],[2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1,2],[2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,2,2],[2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1],[2,2]]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[[1,5]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2,2,2}
[[2,5]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2,2,2}
[[3,5]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2,2,2}
[[4,5]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2,2,2}
[[5,5]]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2,2,2}
[[1],[5]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2],[5]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[3],[5]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[4],[5]]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[1,2,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[1,3,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[1,4,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[2,2,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[2,3,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[2,4,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[3,3,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[3,4,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[4,4,4]]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,3,3,3}
[[1,1],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,2],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,4],[2]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,3],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,4],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,4],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,2],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,3],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,4],[3]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,4],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[3,3],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[3,4],[4]]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1],[2],[4]]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[[1],[3],[4]]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[[2],[3],[4]]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[[1,1,1,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[1,1,2,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[1,1,3,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[1,2,2,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[1,2,3,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[1,3,3,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[2,2,2,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[2,2,3,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[2,3,3,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[3,3,3,3]]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2}
[[1,1,1],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1,2],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1,3],[2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1,3],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,2,2],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,2,3],[2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,2,3],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,3,3],[2]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,3,3],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,2,2],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,2,3],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[2,3,3],[3]]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[[1,1,1,1,2]]
=> [5]
=> []
=> []
=> ? ∊ {1,1,1,1,1}
[[1,1,1,2,2]]
=> [5]
=> []
=> []
=> ? ∊ {1,1,1,1,1}
[[1,1,2,2,2]]
=> [5]
=> []
=> []
=> ? ∊ {1,1,1,1,1}
Description
The convexity degree of the parallelogram polyomino associated with the Dyck path.
A parallelogram polyomino is $k$-convex if $k$ is the maximal number of turns an axis-parallel path must take to connect two cells of the polyomino.
For example, any rotation of a Ferrers shape has convexity degree at most one.
The (bivariate) generating function is given in Theorem 2 of [1].
The following 158 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. 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). St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. 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. St000934The 2-degree of an integer partition. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000442The maximal area to the right of an up step of a Dyck path. St000658The number of rises of length 2 of a Dyck path. St000683The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps. St000693The modular (standard) major index of a standard tableau. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000929The constant term of the character polynomial of an integer partition. St000941The number of characters of the symmetric group whose value on the partition is even. St000984The number of boxes below precisely one peak. St001128The exponens consonantiae of a partition. St001139The number of occurrences of hills of size 2 in a Dyck path. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001204Call 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. St001480The number of simple summands of the module J^2/J^3. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001498The normalised height of a Nakayama algebra with magnitude 1. St000137The Grundy value of an integer partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000225Difference between largest and smallest parts in a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000618The number of self-evacuating tableaux of given shape. St000667The greatest common divisor of the parts of the partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000944The 3-degree of an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001175The size of a partition minus the hook length of the base cell. St001176The size of a partition minus its first part. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001248Sum of the even parts of a partition. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001279The sum of the parts of an integer partition that are at least two. St001280The number of parts of an integer partition that are at least two. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001360The number of covering relations in Young's lattice below a partition. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001383The BG-rank of an integer partition. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001389The number of partitions of the same length below the given integer partition. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001432The order dimension of the partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001525The number of symmetric hooks on the diagonal of a partition. St001527The cyclic permutation representation number of an integer partition. St001541The Gini index of an integer partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St001571The Cartan determinant of the integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001657The number of twos in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St001933The largest multiplicity of a part in an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St000455The second largest eigenvalue of a graph if it is integral. St000284The Plancherel distribution on integer partitions. St000478Another weight of a partition according to Alladi. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000567The sum of the products of all pairs of parts. St000681The Grundy value of Chomp on Ferrers diagrams. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. 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. St001100The 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 trees. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001645The pebbling number of a connected graph. St000379The number of Hamiltonian cycles in a graph. St000454The largest eigenvalue of a graph if it is integral. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. 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. St000260The radius of a connected graph. St000456The monochromatic index of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000259The diameter of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000668The least common multiple of the parts of the partition. St000706The product of the factorials of the multiplicities of an integer partition. 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. 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. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The 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 vertex labelled trees. St001568The smallest positive integer that does not appear twice in the partition.
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