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Your data matches 237 different statistics following compositions of up to 3 maps.
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Matching statistic: St001175
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Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St001175: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St001175: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> [1]
=> 0
[1,2] => ([],2)
=> [1,1]
=> 0
[2,1] => ([(0,1)],2)
=> [2]
=> 0
[1,2,3] => ([],3)
=> [1,1,1]
=> 0
[1,3,2] => ([(1,2)],3)
=> [2,1]
=> 0
[2,1,3] => ([(1,2)],3)
=> [2,1]
=> 0
[2,3,1] => ([(0,2),(1,2)],3)
=> [3]
=> 0
[3,1,2] => ([(0,2),(1,2)],3)
=> [3]
=> 0
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
[1,2,3,4] => ([],4)
=> [1,1,1,1]
=> 0
[1,2,4,3] => ([(2,3)],4)
=> [2,1,1]
=> 0
[1,3,2,4] => ([(2,3)],4)
=> [2,1,1]
=> 0
[1,3,4,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> 0
[1,4,2,3] => ([(1,3),(2,3)],4)
=> [3,1]
=> 0
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 0
[2,1,3,4] => ([(2,3)],4)
=> [2,1,1]
=> 0
[2,1,4,3] => ([(0,3),(1,2)],4)
=> [2,2]
=> 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> 0
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 0
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> 0
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 0
[3,1,2,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> 0
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> 0
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 0
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 0
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> 0
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 0
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 0
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 0
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 0
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 0
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 0
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 0
[1,2,3,4,5] => ([],5)
=> [1,1,1,1,1]
=> 0
[1,2,3,5,4] => ([(3,4)],5)
=> [2,1,1,1]
=> 0
[1,2,4,3,5] => ([(3,4)],5)
=> [2,1,1,1]
=> 0
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 0
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 0
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 0
[1,3,2,4,5] => ([(3,4)],5)
=> [2,1,1,1]
=> 0
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> [2,2,1]
=> 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 0
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 0
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> 0
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 0
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 0
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> 0
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 0
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 0
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> 0
Description
The size of a partition minus the hook length of the base cell.
This is, the number of boxes in the diagram of a partition that are neither in the first row nor in the first column.
Matching statistic: St001033
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001033: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001033: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> [1]
=> [1,0]
=> 0
[1,2] => ([],2)
=> [1,1]
=> [1,1,0,0]
=> 0
[2,1] => ([(0,1)],2)
=> [2]
=> [1,0,1,0]
=> 0
[1,2,3] => ([],3)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[1,3,2] => ([(1,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
[2,1,3] => ([(1,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
[2,3,1] => ([(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 0
[3,1,2] => ([(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 0
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 0
[1,2,3,4] => ([],4)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[1,2,4,3] => ([(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[1,3,2,4] => ([(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[1,3,4,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[1,4,2,3] => ([(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[2,1,3,4] => ([(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[2,1,4,3] => ([(0,3),(1,2)],4)
=> [2,2]
=> [1,1,1,0,0,0]
=> 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 0
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 0
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 0
[3,1,2,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 0
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 0
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 0
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 0
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 0
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 0
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 0
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 0
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 0
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,2,3,4,5] => ([],5)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[1,2,3,5,4] => ([(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0
[1,2,4,3,5] => ([(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[1,3,2,4,5] => ([(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
Description
The normalized area of the parallelogram polyomino associated with the Dyck path.
The area of the smallest parallelogram polyomino equals the semilength of the Dyck path. This statistic is therefore the area of the parallelogram polyomino minus the semilength of the Dyck path.
The area itself is equidistributed with [[St001034]] and with [[St000395]].
Matching statistic: St001596
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001596: Skew partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001596: Skew partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> [1]
=> [[1],[]]
=> 0
[1,2] => ([],2)
=> [1,1]
=> [[1,1],[]]
=> 0
[2,1] => ([(0,1)],2)
=> [2]
=> [[2],[]]
=> 0
[1,2,3] => ([],3)
=> [1,1,1]
=> [[1,1,1],[]]
=> 0
[1,3,2] => ([(1,2)],3)
=> [2,1]
=> [[2,1],[]]
=> 0
[2,1,3] => ([(1,2)],3)
=> [2,1]
=> [[2,1],[]]
=> 0
[2,3,1] => ([(0,2),(1,2)],3)
=> [3]
=> [[3],[]]
=> 0
[3,1,2] => ([(0,2),(1,2)],3)
=> [3]
=> [[3],[]]
=> 0
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> [[3],[]]
=> 0
[1,2,3,4] => ([],4)
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 0
[1,2,4,3] => ([(2,3)],4)
=> [2,1,1]
=> [[2,1,1],[]]
=> 0
[1,3,2,4] => ([(2,3)],4)
=> [2,1,1]
=> [[2,1,1],[]]
=> 0
[1,3,4,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> [[3,1],[]]
=> 0
[1,4,2,3] => ([(1,3),(2,3)],4)
=> [3,1]
=> [[3,1],[]]
=> 0
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [[3,1],[]]
=> 0
[2,1,3,4] => ([(2,3)],4)
=> [2,1,1]
=> [[2,1,1],[]]
=> 0
[2,1,4,3] => ([(0,3),(1,2)],4)
=> [2,2]
=> [[2,2],[]]
=> 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> [[3,1],[]]
=> 0
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> [[4],[]]
=> 0
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> [[4],[]]
=> 0
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[4],[]]
=> 0
[3,1,2,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> [[3,1],[]]
=> 0
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> [[4],[]]
=> 0
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [[3,1],[]]
=> 0
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[4],[]]
=> 0
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [[4],[]]
=> 0
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[4],[]]
=> 0
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> [[4],[]]
=> 0
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[4],[]]
=> 0
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[4],[]]
=> 0
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[4],[]]
=> 0
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[4],[]]
=> 0
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[4],[]]
=> 0
[1,2,3,4,5] => ([],5)
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 0
[1,2,3,5,4] => ([(3,4)],5)
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> 0
[1,2,4,3,5] => ([(3,4)],5)
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> 0
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [[3,1,1],[]]
=> 0
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [[3,1,1],[]]
=> 0
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[3,1,1],[]]
=> 0
[1,3,2,4,5] => ([(3,4)],5)
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> 0
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> [2,2,1]
=> [[2,2,1],[]]
=> 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [[3,1,1],[]]
=> 0
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [[4,1],[]]
=> 0
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [[4,1],[]]
=> 0
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [[4,1],[]]
=> 0
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [[3,1,1],[]]
=> 0
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [[4,1],[]]
=> 0
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[3,1,1],[]]
=> 0
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [[4,1],[]]
=> 0
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [[4,1],[]]
=> 0
Description
The number of two-by-two squares inside a skew partition.
This is, the number of cells $(i,j)$ in a skew partition for which the box $(i+1,j+1)$ is also a cell inside the skew partition.
Matching statistic: St000345
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000345: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000345: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> [1]
=> []
=> 1 = 0 + 1
[1,2] => ([],2)
=> [1,1]
=> [1]
=> 1 = 0 + 1
[2,1] => ([(0,1)],2)
=> [2]
=> []
=> 1 = 0 + 1
[1,2,3] => ([],3)
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,3,2] => ([(1,2)],3)
=> [2,1]
=> [1]
=> 1 = 0 + 1
[2,1,3] => ([(1,2)],3)
=> [2,1]
=> [1]
=> 1 = 0 + 1
[2,3,1] => ([(0,2),(1,2)],3)
=> [3]
=> []
=> 1 = 0 + 1
[3,1,2] => ([(0,2),(1,2)],3)
=> [3]
=> []
=> 1 = 0 + 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> 1 = 0 + 1
[1,2,3,4] => ([],4)
=> [1,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,2,4,3] => ([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,3,2,4] => ([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1 = 0 + 1
[1,4,2,3] => ([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1 = 0 + 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1 = 0 + 1
[2,1,3,4] => ([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> 2 = 1 + 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1 = 0 + 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> 1 = 0 + 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> 1 = 0 + 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> 1 = 0 + 1
[3,1,2,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1 = 0 + 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> 1 = 0 + 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1 = 0 + 1
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> 1 = 0 + 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> 1 = 0 + 1
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> 1 = 0 + 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> 1 = 0 + 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> 1 = 0 + 1
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> 1 = 0 + 1
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> 1 = 0 + 1
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> 1 = 0 + 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> 1 = 0 + 1
[1,2,3,4,5] => ([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1 = 0 + 1
[1,2,3,5,4] => ([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,2,4,3,5] => ([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,3,2,4,5] => ([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 2 = 1 + 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1 = 0 + 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> 1 = 0 + 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1 = 0 + 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> 1 = 0 + 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1 = 0 + 1
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> 1 = 0 + 1
Description
The number of refinements of a partition.
A partition $\lambda$ refines a partition $\mu$ if the parts of $\mu$ can be subdivided to obtain the parts of $\lambda$.
Matching statistic: St000935
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000935: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000935: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> [1]
=> []
=> 1 = 0 + 1
[1,2] => ([],2)
=> [1,1]
=> [1]
=> 1 = 0 + 1
[2,1] => ([(0,1)],2)
=> [2]
=> []
=> 1 = 0 + 1
[1,2,3] => ([],3)
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,3,2] => ([(1,2)],3)
=> [2,1]
=> [1]
=> 1 = 0 + 1
[2,1,3] => ([(1,2)],3)
=> [2,1]
=> [1]
=> 1 = 0 + 1
[2,3,1] => ([(0,2),(1,2)],3)
=> [3]
=> []
=> 1 = 0 + 1
[3,1,2] => ([(0,2),(1,2)],3)
=> [3]
=> []
=> 1 = 0 + 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> 1 = 0 + 1
[1,2,3,4] => ([],4)
=> [1,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,2,4,3] => ([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,3,2,4] => ([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1 = 0 + 1
[1,4,2,3] => ([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1 = 0 + 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1 = 0 + 1
[2,1,3,4] => ([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> 2 = 1 + 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1 = 0 + 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> 1 = 0 + 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> 1 = 0 + 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> 1 = 0 + 1
[3,1,2,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1 = 0 + 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> 1 = 0 + 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1 = 0 + 1
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> 1 = 0 + 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> 1 = 0 + 1
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> 1 = 0 + 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> 1 = 0 + 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> 1 = 0 + 1
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> 1 = 0 + 1
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> 1 = 0 + 1
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> 1 = 0 + 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> 1 = 0 + 1
[1,2,3,4,5] => ([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1 = 0 + 1
[1,2,3,5,4] => ([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,2,4,3,5] => ([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,3,2,4,5] => ([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 2 = 1 + 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1 = 0 + 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> 1 = 0 + 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1 = 0 + 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> 1 = 0 + 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1 = 0 + 1
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> 1 = 0 + 1
Description
The number of ordered refinements of an integer partition.
This is, for an integer partition $\mu = (\mu_1,\ldots,\mu_n)$ the number of integer partition $\lambda = (\lambda_1,\ldots,\lambda_m)$ such that there are indices $1 = a_0 < \ldots < a_n = m$ with $\mu_j = \lambda_{a_{j-1}} + \ldots + \lambda_{a_j-1}$.
Matching statistic: St000175
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000175: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 89%●distinct values known / distinct values provided: 67%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000175: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 89%●distinct values known / distinct values provided: 67%
Values
[1] => [1]
=> []
=> ?
=> ? = 0
[1,2] => [2]
=> []
=> ?
=> ? ∊ {0,0}
[2,1] => [1,1]
=> [1]
=> []
=> ? ∊ {0,0}
[1,2,3] => [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,0}
[1,3,2] => [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0}
[2,1,3] => [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0}
[2,3,1] => [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0}
[3,1,2] => [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0}
[3,2,1] => [1,1,1]
=> [1,1]
=> [1]
=> 0
[1,2,3,4] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,2,4,3] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,3,2,4] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,3,4,2] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,4,2,3] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[2,1,3,4] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[2,1,4,3] => [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[2,3,1,4] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[2,3,4,1] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[2,4,1,3] => [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[2,4,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[3,1,2,4] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[3,1,4,2] => [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,4,1] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[3,4,1,2] => [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[3,4,2,1] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,1,2,3] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[4,1,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,2,1,3] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,3,1,2] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,3,2,1] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,4,5] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,2,3,5,4] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,2,4,3,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,2,4,5,3] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,2,5,3,4] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,2,5,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,3,2,4,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,3,2,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,3,4,2,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,3,4,5,2] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,3,5,2,4] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,4,2,3,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,4,2,5,3] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,4,3,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,4,5,2,3] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,4,5,3,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,2,3,4] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,3,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,4,2,3] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,4,3,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,1,3,4,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,1,3,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,1,4,3,5] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,1,4,5,3] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,1,5,3,4] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,1,5,4,3] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[2,3,1,4,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,3,1,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,3,4,1,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,3,4,5,1] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,3,5,1,4] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,3,5,4,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,4,1,3,5] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,4,1,5,3] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,4,3,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,4,3,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,4,5,1,3] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,4,5,3,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,5,1,3,4] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,5,1,4,3] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[2,5,3,1,4] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,5,3,4,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,5,4,1,3] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[2,5,4,3,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[3,1,5,4,2] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,2,1,4,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,1,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,2,4,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,4,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,5,1,4] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,2,5,4,1] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,4,2,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,4,2,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,4,5,2,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,5,1,4,2] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,5,2,1,4] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,5,2,4,1] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,5,4,1,2] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,5,4,2,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[4,1,3,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[4,1,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[4,1,5,3,2] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[4,2,1,3,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
Description
Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape.
Given a partition $\lambda$ with $r$ parts, the number of semi-standard Young-tableaux of shape $k\lambda$ and boxes with values in $[r]$ grows as a polynomial in $k$. This follows by setting $q=1$ in (7.105) on page 375 of [1], which yields the polynomial
$$p(k) = \prod_{i < j}\frac{k(\lambda_j-\lambda_i)+j-i}{j-i}.$$
The statistic of the degree of this polynomial.
For example, the partition $(3, 2, 1, 1, 1)$ gives
$$p(k) = \frac{-1}{36} (k - 3) (2k - 3) (k - 2)^2 (k - 1)^3$$
which has degree 7 in $k$. Thus, $[3, 2, 1, 1, 1] \mapsto 7$.
This is the same as the number of unordered pairs of different parts, which follows from:
$$\deg p(k)=\sum_{i < j}\begin{cases}1& \lambda_j \neq \lambda_i\\0&\lambda_i=\lambda_j\end{cases}=\sum_{\stackrel{i < j}{\lambda_j \neq \lambda_i}} 1$$
Matching statistic: St000205
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000205: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 89%●distinct values known / distinct values provided: 33%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000205: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 89%●distinct values known / distinct values provided: 33%
Values
[1] => [1]
=> []
=> ?
=> ? = 0
[1,2] => [2]
=> []
=> ?
=> ? ∊ {0,0}
[2,1] => [1,1]
=> [1]
=> []
=> ? ∊ {0,0}
[1,2,3] => [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,0}
[1,3,2] => [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0}
[2,1,3] => [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0}
[2,3,1] => [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0}
[3,1,2] => [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0}
[3,2,1] => [1,1,1]
=> [1,1]
=> [1]
=> 0
[1,2,3,4] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,2,4,3] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,3,2,4] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,3,4,2] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,4,2,3] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[2,1,3,4] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[2,1,4,3] => [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[2,3,1,4] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[2,3,4,1] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[2,4,1,3] => [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[2,4,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[3,1,2,4] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[3,1,4,2] => [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,4,1] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[3,4,1,2] => [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[3,4,2,1] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,1,2,3] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[4,1,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,2,1,3] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,3,1,2] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,3,2,1] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,4,5] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,2,3,5,4] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,2,4,3,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,2,4,5,3] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,2,5,3,4] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,2,5,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,3,2,4,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,3,2,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,3,4,2,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,3,4,5,2] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,3,5,2,4] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,4,2,3,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,4,2,5,3] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,4,3,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,4,5,2,3] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,4,5,3,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,2,3,4] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,3,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,4,2,3] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,4,3,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,1,3,4,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,1,3,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,1,4,3,5] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,1,4,5,3] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,1,5,3,4] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,1,5,4,3] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[2,3,1,4,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,3,1,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,3,4,1,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,3,4,5,1] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,3,5,1,4] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,3,5,4,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,4,1,3,5] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,4,1,5,3] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,4,3,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,4,3,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,4,5,1,3] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,4,5,3,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,5,1,3,4] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,5,1,4,3] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[2,5,3,1,4] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,5,3,4,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,5,4,1,3] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[2,5,4,3,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[3,1,5,4,2] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,2,1,4,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,1,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,2,4,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,4,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,5,1,4] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,2,5,4,1] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,4,2,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,4,2,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,4,5,2,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,5,1,4,2] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,5,2,1,4] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,5,2,4,1] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,5,4,1,2] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,5,4,2,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[4,1,3,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[4,1,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[4,1,5,3,2] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[4,2,1,3,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
Description
Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight.
Given $\lambda$ count how many ''integer partitions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is non-integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has at least one non-integral vertex.
Matching statistic: St000206
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000206: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 89%●distinct values known / distinct values provided: 33%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000206: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 89%●distinct values known / distinct values provided: 33%
Values
[1] => [1]
=> []
=> ?
=> ? = 0
[1,2] => [2]
=> []
=> ?
=> ? ∊ {0,0}
[2,1] => [1,1]
=> [1]
=> []
=> ? ∊ {0,0}
[1,2,3] => [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,0}
[1,3,2] => [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0}
[2,1,3] => [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0}
[2,3,1] => [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0}
[3,1,2] => [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0}
[3,2,1] => [1,1,1]
=> [1,1]
=> [1]
=> 0
[1,2,3,4] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,2,4,3] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,3,2,4] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,3,4,2] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,4,2,3] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[2,1,3,4] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[2,1,4,3] => [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[2,3,1,4] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[2,3,4,1] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[2,4,1,3] => [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[2,4,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[3,1,2,4] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[3,1,4,2] => [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,4,1] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[3,4,1,2] => [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[3,4,2,1] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,1,2,3] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[4,1,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,2,1,3] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,3,1,2] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,3,2,1] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,4,5] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,2,3,5,4] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,2,4,3,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,2,4,5,3] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,2,5,3,4] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,2,5,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,3,2,4,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,3,2,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,3,4,2,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,3,4,5,2] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,3,5,2,4] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,4,2,3,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,4,2,5,3] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,4,3,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,4,5,2,3] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,4,5,3,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,2,3,4] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,3,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,4,2,3] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,4,3,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,1,3,4,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,1,3,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,1,4,3,5] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,1,4,5,3] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,1,5,3,4] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,1,5,4,3] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[2,3,1,4,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,3,1,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,3,4,1,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,3,4,5,1] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,3,5,1,4] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,3,5,4,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,4,1,3,5] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,4,1,5,3] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,4,3,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,4,3,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,4,5,1,3] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,4,5,3,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,5,1,3,4] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,5,1,4,3] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[2,5,3,1,4] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,5,3,4,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,5,4,1,3] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[2,5,4,3,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[3,1,5,4,2] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,2,1,4,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,1,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,2,4,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,4,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,5,1,4] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,2,5,4,1] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,4,2,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,4,2,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,4,5,2,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,5,1,4,2] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,5,2,1,4] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,5,2,4,1] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,5,4,1,2] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,5,4,2,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[4,1,3,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[4,1,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[4,1,5,3,2] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[4,2,1,3,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
Description
Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight.
Given $\lambda$ count how many ''integer compositions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is non-integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has at least one non-integral vertex.
See also [[St000205]].
Each value in this statistic is greater than or equal to corresponding value in [[St000205]].
Matching statistic: St000225
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000225: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 89%●distinct values known / distinct values provided: 67%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000225: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 89%●distinct values known / distinct values provided: 67%
Values
[1] => [1]
=> []
=> ?
=> ? = 0
[1,2] => [2]
=> []
=> ?
=> ? ∊ {0,0}
[2,1] => [1,1]
=> [1]
=> []
=> ? ∊ {0,0}
[1,2,3] => [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,0}
[1,3,2] => [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0}
[2,1,3] => [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0}
[2,3,1] => [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0}
[3,1,2] => [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0}
[3,2,1] => [1,1,1]
=> [1,1]
=> [1]
=> 0
[1,2,3,4] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,2,4,3] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,3,2,4] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,3,4,2] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,4,2,3] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[2,1,3,4] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[2,1,4,3] => [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[2,3,1,4] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[2,3,4,1] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[2,4,1,3] => [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[2,4,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[3,1,2,4] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[3,1,4,2] => [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,4,1] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[3,4,1,2] => [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[3,4,2,1] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,1,2,3] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[4,1,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,2,1,3] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,3,1,2] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,3,2,1] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,4,5] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,2,3,5,4] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,2,4,3,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,2,4,5,3] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,2,5,3,4] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,2,5,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,3,2,4,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,3,2,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,3,4,2,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,3,4,5,2] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,3,5,2,4] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,4,2,3,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,4,2,5,3] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,4,3,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,4,5,2,3] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,4,5,3,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,2,3,4] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,3,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,4,2,3] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,4,3,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,1,3,4,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,1,3,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,1,4,3,5] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,1,4,5,3] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,1,5,3,4] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,1,5,4,3] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[2,3,1,4,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,3,1,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,3,4,1,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,3,4,5,1] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,3,5,1,4] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,3,5,4,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,4,1,3,5] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,4,1,5,3] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,4,3,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,4,3,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,4,5,1,3] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,4,5,3,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,5,1,3,4] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,5,1,4,3] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[2,5,3,1,4] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,5,3,4,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,5,4,1,3] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[2,5,4,3,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[3,1,5,4,2] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,2,1,4,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,1,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,2,4,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,4,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,5,1,4] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,2,5,4,1] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,4,2,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,4,2,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,4,5,2,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,5,1,4,2] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,5,2,1,4] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,5,2,4,1] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,5,4,1,2] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,5,4,2,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[4,1,3,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[4,1,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[4,1,5,3,2] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[4,2,1,3,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
Description
Difference between largest and smallest parts in a partition.
Matching statistic: St000749
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000749: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 89%●distinct values known / distinct values provided: 67%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000749: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 89%●distinct values known / distinct values provided: 67%
Values
[1] => [1]
=> []
=> ?
=> ? = 0
[1,2] => [2]
=> []
=> ?
=> ? ∊ {0,0}
[2,1] => [1,1]
=> [1]
=> []
=> ? ∊ {0,0}
[1,2,3] => [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,0}
[1,3,2] => [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0}
[2,1,3] => [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0}
[2,3,1] => [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0}
[3,1,2] => [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0}
[3,2,1] => [1,1,1]
=> [1,1]
=> [1]
=> 0
[1,2,3,4] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,2,4,3] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,3,2,4] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,3,4,2] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,4,2,3] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[2,1,3,4] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[2,1,4,3] => [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[2,3,1,4] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[2,3,4,1] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[2,4,1,3] => [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[2,4,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[3,1,2,4] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[3,1,4,2] => [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,4,1] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[3,4,1,2] => [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[3,4,2,1] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,1,2,3] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[4,1,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,2,1,3] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,3,1,2] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,3,2,1] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,4,5] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,2,3,5,4] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,2,4,3,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,2,4,5,3] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,2,5,3,4] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,2,5,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,3,2,4,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,3,2,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,3,4,2,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,3,4,5,2] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,3,5,2,4] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,4,2,3,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,4,2,5,3] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,4,3,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,4,5,2,3] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,4,5,3,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,2,3,4] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,3,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,4,2,3] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,4,3,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,1,3,4,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,1,3,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,1,4,3,5] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,1,4,5,3] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,1,5,3,4] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,1,5,4,3] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[2,3,1,4,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,3,1,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,3,4,1,5] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,3,4,5,1] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,3,5,1,4] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,3,5,4,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,4,1,3,5] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,4,1,5,3] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,4,3,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,4,3,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,4,5,1,3] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,4,5,3,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,5,1,3,4] => [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1}
[2,5,1,4,3] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[2,5,3,1,4] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,5,3,4,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,5,4,1,3] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[2,5,4,3,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[3,1,5,4,2] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,2,1,4,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,1,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,2,4,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,4,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,5,1,4] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,2,5,4,1] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,4,2,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,4,2,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,4,5,2,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,5,1,4,2] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,5,2,1,4] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,5,2,4,1] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,5,4,1,2] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,5,4,2,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[4,1,3,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[4,1,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[4,1,5,3,2] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[4,2,1,3,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
Description
The 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.
For example, restricting $S_{(6,3)}$ to $\mathfrak S_8$ yields $$S_{(5,3)}\oplus S_{(6,2)}$$ of degrees (number of standard Young tableaux) 28 and 20, none of which are odd. Restricting to $\mathfrak S_7$ yields $$S_{(4,3)}\oplus 2S_{(5,2)}\oplus S_{(6,1)}$$ of degrees 14, 14 and 6. However, restricting to $\mathfrak S_6$ yields
$$S_{(3,3)}\oplus 3S_{(4,2)}\oplus 3S_{(5,1)}\oplus S_6$$ of degrees 5,9,5 and 1. Therefore, the statistic on the partition $(6,3)$ gives 3.
This is related to $2$-saturations of Welter's game, see [1, Corollary 1.2].
The following 227 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000944The 3-degree of an integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St000929The constant term of the character polynomial of an integer partition. St000149The number of cells of the partition whose leg is zero and arm is odd. St000256The number of parts from which one can substract 2 and still get an integer partition. St000290The major index of a binary word. St000291The number of descents of a binary word. St000293The number of inversions of a binary word. St000296The length of the symmetric border of a binary word. St000347The inversion sum of a binary word. St000629The defect of a binary word. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000921The number of internal inversions of a binary word. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. 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. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001214The aft of an integer partition. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001371The length of the longest Yamanouchi prefix of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001485The modular major index of a binary word. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St000478Another weight of a partition according to Alladi. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St000379The number of Hamiltonian cycles in a graph. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000934The 2-degree of an integer 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. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001651The Frankl number of a lattice. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St001141The number of occurrences of hills of size 3 in a Dyck path. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001095The number of non-isomorphic posets with precisely one further covering relation. St000455The second largest eigenvalue of a graph if it is integral. St001490The number of connected components of a skew partition. St000352The Elizalde-Pak rank of a permutation. St000007The number of saliances of the permutation. St000699The toughness times the least common multiple of 1,. St001541The Gini index of an integer partition. St001657The number of twos in an integer partition. St000031The number of cycles in the cycle decomposition of a permutation. St001330The hat guessing number of a graph. St000914The sum of the values of the Möbius function of a poset. St001890The maximum magnitude of the Möbius function of a poset. St000323The minimal crossing number of a graph. St000370The genus of a graph. St000447The number of pairs of vertices of a graph with distance 3. St000449The number of pairs of vertices of a graph with distance 4. St000552The number of cut vertices of a graph. St001309The number of four-cliques in a graph. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001793The difference between the clique number and the chromatic number of a graph. St001877Number of indecomposable injective modules with projective dimension 2. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001875The number of simple modules with projective dimension at most 1. St001307The number of induced stars on four vertices in a graph. St000068The number of minimal elements in a poset. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St000322The skewness of a graph. St001578The minimal number of edges to add or remove to make a graph a line graph. St001871The number of triconnected components of a graph. St000842The breadth of a permutation. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. 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 $eA$. St000627The exponent of a binary word. St001271The competition number of a graph. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000897The number of different multiplicities of parts of an integer partition. St000567The sum of the products of all pairs of parts. 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. 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. 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. St001964The interval resolution global dimension of a poset. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000759The smallest missing part in an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. 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. St001730The number of times the path corresponding to a binary word crosses the base line. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St000475The number of parts equal to 1 in a partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001301The first Betti number of the order complex associated with the poset. St000908The length of the shortest maximal antichain in a poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001396Number of triples of incomparable elements in a finite poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001846The number of elements which do not have a complement in the lattice. St001820The size of the image of the pop stack sorting operator. St000162The number of nontrivial cycles in the cycle decomposition 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. St000570The Edelman-Greene number of a permutation. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St000542The number of left-to-right-minima of a permutation. St001520The number of strict 3-descents. St001568The smallest positive integer that does not appear twice in the partition. St000259The diameter of a connected graph. St000260The radius 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. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St000181The number of connected components of the Hasse diagram for the poset. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001845The number of join irreducibles minus the rank of a lattice. St001613The binary logarithm of the size of the center of a lattice. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001881The number of factors of a lattice as a Cartesian product of lattices. St000069The number of maximal elements of a poset. St001621The number of atoms of a lattice. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St001249Sum of the odd parts of a partition. St001383The BG-rank of an integer partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001868The number of alignments of type NE of a signed permutation. St000022The number of fixed points of a permutation. St000731The number of double exceedences of a permutation. St001862The number of crossings of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St000632The jump number of the poset. St001397Number of pairs of incomparable elements in a finite poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001510The number of self-evacuating linear extensions of a finite poset. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001779The order of promotion on the set of linear extensions of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St000351The determinant of the adjacency matrix of a graph. St000368The Altshuler-Steinberg determinant of a graph. St000403The Szeged index minus the Wiener index of a graph. St000671The maximin edge-connectivity for choosing a subgraph. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001119The length of a shortest maximal path in a graph. St001305The number of induced cycles on four vertices in a graph. St001310The number of induced diamond graphs in a graph. St001323The independence gap of a graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001357The maximal degree of a regular spanning subgraph of a graph. St001395The number of strictly unfriendly partitions of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001797The number of overfull subgraphs of a graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St001316The domatic number of a graph. St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001496The number of graphs with the same Laplacian spectrum as the given graph. St000273The domination number of a graph. St000544The cop number of a graph. St000553The number of blocks of a graph. St000916The packing number of a graph. St001739The number of graphs with the same edge polytope as the given graph. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St001829The common independence number of a graph. St001281The normalized isoperimetric number of a graph. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St000264The girth of a graph, which is not a tree. St001549The number of restricted non-inversions between exceedances. St001811The Castelnuovo-Mumford regularity of a permutation. St000805The number of peaks of the associated bargraph. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St001162The minimum jump of a permutation. St001344The neighbouring number of a permutation. St001768The number of reduced words of a signed permutation. St000095The number of triangles of a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000274The number of perfect matchings of a graph. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000310The minimal degree of a vertex of a graph. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St001518The number of graphs with the same ordinary spectrum as the given graph. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St001765The number of connected components of the friends and strangers graph.
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