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Your data matches 232 different statistics following compositions of up to 3 maps.
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Matching statistic: St000183
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(load all 8 compositions to match this statistic)
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000183: 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
St000183: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> [1]
=> 1
[1,2] => ([],2)
=> [1,1]
=> 1
[2,1] => ([(0,1)],2)
=> [2]
=> 1
[1,2,3] => ([],3)
=> [1,1,1]
=> 1
[1,3,2] => ([(1,2)],3)
=> [2,1]
=> 1
[2,1,3] => ([(1,2)],3)
=> [2,1]
=> 1
[2,3,1] => ([(0,2),(1,2)],3)
=> [3]
=> 1
[3,1,2] => ([(0,2),(1,2)],3)
=> [3]
=> 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[1,2,3,4] => ([],4)
=> [1,1,1,1]
=> 1
[1,2,4,3] => ([(2,3)],4)
=> [2,1,1]
=> 1
[1,3,2,4] => ([(2,3)],4)
=> [2,1,1]
=> 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> 1
[1,4,2,3] => ([(1,3),(2,3)],4)
=> [3,1]
=> 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 1
[2,1,3,4] => ([(2,3)],4)
=> [2,1,1]
=> 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> [2,2]
=> 2
[2,3,1,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[3,1,2,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 1
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> 1
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[1,2,3,4,5] => ([],5)
=> [1,1,1,1,1]
=> 1
[1,2,3,5,4] => ([(3,4)],5)
=> [2,1,1,1]
=> 1
[1,2,4,3,5] => ([(3,4)],5)
=> [2,1,1,1]
=> 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 1
[1,3,2,4,5] => ([(3,4)],5)
=> [2,1,1,1]
=> 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> [2,2,1]
=> 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 1
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> 1
Description
The side length of the Durfee square of an integer partition.
Given a partition $\lambda = (\lambda_1,\ldots,\lambda_n)$, the Durfee square is the largest partition $(s^s)$ whose diagram fits inside the diagram of $\lambda$. In symbols, $s = \max\{ i \mid \lambda_i \geq i \}$.
This is also known as the Frobenius rank.
Matching statistic: St000920
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000920: 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
St000920: Dyck paths ⟶ 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]
=> [1,1,0,0]
=> 1
[2,1] => ([(0,1)],2)
=> [2]
=> [1,0,1,0]
=> 1
[1,2,3] => ([],3)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,3,2] => ([(1,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,1,3] => ([(1,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,3,1] => ([(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 1
[3,1,2] => ([(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 1
[1,2,3,4] => ([],4)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,2,4,3] => ([(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,3,2,4] => ([(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,4,2,3] => ([(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[2,1,3,4] => ([(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
[2,3,1,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 1
[3,1,2,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 1
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 1
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 1
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 1
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 1
[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]
=> 1
[1,2,3,4,5] => ([],5)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,2,3,5,4] => ([(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,2,4,3,5] => ([(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[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]
=> 1
[1,3,2,4,5] => ([(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[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]
=> 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[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]
=> 1
[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]
=> 1
[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]
=> 1
Description
The logarithmic height of a Dyck path.
This is the floor of the binary logarithm of the usual height increased by one:
$$
\lfloor\log_2(1+height(D))\rfloor
$$
Matching statistic: St001597
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001597: 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
St001597: Skew partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> [1]
=> [[1],[]]
=> 1
[1,2] => ([],2)
=> [1,1]
=> [[1,1],[]]
=> 1
[2,1] => ([(0,1)],2)
=> [2]
=> [[2],[]]
=> 1
[1,2,3] => ([],3)
=> [1,1,1]
=> [[1,1,1],[]]
=> 1
[1,3,2] => ([(1,2)],3)
=> [2,1]
=> [[2,1],[]]
=> 1
[2,1,3] => ([(1,2)],3)
=> [2,1]
=> [[2,1],[]]
=> 1
[2,3,1] => ([(0,2),(1,2)],3)
=> [3]
=> [[3],[]]
=> 1
[3,1,2] => ([(0,2),(1,2)],3)
=> [3]
=> [[3],[]]
=> 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> [[3],[]]
=> 1
[1,2,3,4] => ([],4)
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 1
[1,2,4,3] => ([(2,3)],4)
=> [2,1,1]
=> [[2,1,1],[]]
=> 1
[1,3,2,4] => ([(2,3)],4)
=> [2,1,1]
=> [[2,1,1],[]]
=> 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> [[3,1],[]]
=> 1
[1,4,2,3] => ([(1,3),(2,3)],4)
=> [3,1]
=> [[3,1],[]]
=> 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [[3,1],[]]
=> 1
[2,1,3,4] => ([(2,3)],4)
=> [2,1,1]
=> [[2,1,1],[]]
=> 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> [2,2]
=> [[2,2],[]]
=> 2
[2,3,1,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> [[3,1],[]]
=> 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> [[4],[]]
=> 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> [[4],[]]
=> 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[4],[]]
=> 1
[3,1,2,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> [[3,1],[]]
=> 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> [[4],[]]
=> 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [[3,1],[]]
=> 1
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[4],[]]
=> 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [[4],[]]
=> 1
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[4],[]]
=> 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> [[4],[]]
=> 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[4],[]]
=> 1
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[4],[]]
=> 1
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[4],[]]
=> 1
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[4],[]]
=> 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[4],[]]
=> 1
[1,2,3,4,5] => ([],5)
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 1
[1,2,3,5,4] => ([(3,4)],5)
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> 1
[1,2,4,3,5] => ([(3,4)],5)
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [[3,1,1],[]]
=> 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [[3,1,1],[]]
=> 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[3,1,1],[]]
=> 1
[1,3,2,4,5] => ([(3,4)],5)
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> [2,2,1]
=> [[2,2,1],[]]
=> 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [[3,1,1],[]]
=> 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [[4,1],[]]
=> 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [[4,1],[]]
=> 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [[4,1],[]]
=> 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [[3,1,1],[]]
=> 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [[4,1],[]]
=> 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[3,1,1],[]]
=> 1
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [[4,1],[]]
=> 1
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [[4,1],[]]
=> 1
Description
The Frobenius rank of a skew partition.
This is the minimal number of border strips in a border strip decomposition of the skew partition.
Matching statistic: St000480
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000480: 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
St000480: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> [1]
=> []
=> 0 = 1 - 1
[1,2] => ([],2)
=> [1,1]
=> [1]
=> 0 = 1 - 1
[2,1] => ([(0,1)],2)
=> [2]
=> []
=> 0 = 1 - 1
[1,2,3] => ([],3)
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,3,2] => ([(1,2)],3)
=> [2,1]
=> [1]
=> 0 = 1 - 1
[2,1,3] => ([(1,2)],3)
=> [2,1]
=> [1]
=> 0 = 1 - 1
[2,3,1] => ([(0,2),(1,2)],3)
=> [3]
=> []
=> 0 = 1 - 1
[3,1,2] => ([(0,2),(1,2)],3)
=> [3]
=> []
=> 0 = 1 - 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> 0 = 1 - 1
[1,2,3,4] => ([],4)
=> [1,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[1,2,4,3] => ([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,3,2,4] => ([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 0 = 1 - 1
[1,4,2,3] => ([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 0 = 1 - 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 0 = 1 - 1
[2,1,3,4] => ([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> 1 = 2 - 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 0 = 1 - 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> 0 = 1 - 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> 0 = 1 - 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> 0 = 1 - 1
[3,1,2,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 0 = 1 - 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> 0 = 1 - 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 0 = 1 - 1
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> 0 = 1 - 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> 0 = 1 - 1
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> 0 = 1 - 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> 0 = 1 - 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> 0 = 1 - 1
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> 0 = 1 - 1
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> 0 = 1 - 1
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> 0 = 1 - 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> 0 = 1 - 1
[1,2,3,4,5] => ([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0 = 1 - 1
[1,2,3,5,4] => ([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[1,2,4,3,5] => ([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,3,2,4,5] => ([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 0 = 1 - 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> 0 = 1 - 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 0 = 1 - 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> 0 = 1 - 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 0 = 1 - 1
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> 0 = 1 - 1
Description
The number of lower covers of a partition in dominance order.
According to [1], Corollary 2.4, the maximum number of elements one element (apparently for $n\neq 2$) can cover is
$$
\frac{1}{2}(\sqrt{1+8n}-3)
$$
and an element which covers this number of elements is given by $(c+i,c,c-1,\dots,3,2,1)$, where $1\leq i\leq c+2$.
Matching statistic: St000660
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000660: 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
St000660: 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 - 1
[1,2] => ([],2)
=> [1,1]
=> [1,1,0,0]
=> 0 = 1 - 1
[2,1] => ([(0,1)],2)
=> [2]
=> [1,0,1,0]
=> 0 = 1 - 1
[1,2,3] => ([],3)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0 = 1 - 1
[1,3,2] => ([(1,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> 0 = 1 - 1
[2,1,3] => ([(1,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> 0 = 1 - 1
[2,3,1] => ([(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[3,1,2] => ([(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,2,3,4] => ([],4)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,2,4,3] => ([(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[1,3,2,4] => ([(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,4,2,3] => ([(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[2,1,3,4] => ([(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> [2,2]
=> [1,1,1,0,0,0]
=> 1 = 2 - 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[3,1,2,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[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 = 1 - 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[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 = 1 - 1
[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 = 1 - 1
[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 - 1
[1,2,3,4,5] => ([],5)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,2,3,5,4] => ([(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,2,4,3,5] => ([(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[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 - 1
[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 - 1
[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 - 1
[1,3,2,4,5] => ([(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 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 - 1
[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 - 1
[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 - 1
[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 - 1
[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 - 1
[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 - 1
[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 - 1
[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 - 1
[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 = 1 - 1
Description
The number of rises of length at least 3 of a Dyck path.
The number of Dyck paths without such rises are counted by the Motzkin numbers [1].
Matching statistic: St001934
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001934: Integer partitions ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001934: Integer partitions ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [1]
=> []
=> ? = 1
[1,2] => [1,1]
=> [2]
=> []
=> ? = 1
[2,1] => [2]
=> [1,1]
=> [1]
=> 1
[1,2,3] => [1,1,1]
=> [2,1]
=> [1]
=> 1
[1,3,2] => [2,1]
=> [3]
=> []
=> ? ∊ {1,1,1}
[2,1,3] => [2,1]
=> [3]
=> []
=> ? ∊ {1,1,1}
[2,3,1] => [3]
=> [1,1,1]
=> [1,1]
=> 1
[3,1,2] => [3]
=> [1,1,1]
=> [1,1]
=> 1
[3,2,1] => [2,1]
=> [3]
=> []
=> ? ∊ {1,1,1}
[1,2,3,4] => [1,1,1,1]
=> [3,1]
=> [1]
=> 1
[1,2,4,3] => [2,1,1]
=> [2,2]
=> [2]
=> 1
[1,3,2,4] => [2,1,1]
=> [2,2]
=> [2]
=> 1
[1,3,4,2] => [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[1,4,2,3] => [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[1,4,3,2] => [2,1,1]
=> [2,2]
=> [2]
=> 1
[2,1,3,4] => [2,1,1]
=> [2,2]
=> [2]
=> 1
[2,1,4,3] => [2,2]
=> [4]
=> []
=> ? ∊ {1,1,2}
[2,3,1,4] => [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[2,3,4,1] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[2,4,1,3] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[2,4,3,1] => [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[3,1,2,4] => [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[3,1,4,2] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[3,2,1,4] => [2,1,1]
=> [2,2]
=> [2]
=> 1
[3,2,4,1] => [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[3,4,1,2] => [2,2]
=> [4]
=> []
=> ? ∊ {1,1,2}
[3,4,2,1] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[4,1,2,3] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[4,1,3,2] => [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[4,2,1,3] => [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[4,2,3,1] => [2,1,1]
=> [2,2]
=> [2]
=> 1
[4,3,1,2] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[4,3,2,1] => [2,2]
=> [4]
=> []
=> ? ∊ {1,1,2}
[1,2,3,4,5] => [1,1,1,1,1]
=> [3,2]
=> [2]
=> 1
[1,2,3,5,4] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,2,4,3,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,2,4,5,3] => [3,1,1]
=> [4,1]
=> [1]
=> 1
[1,2,5,3,4] => [3,1,1]
=> [4,1]
=> [1]
=> 1
[1,2,5,4,3] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,3,2,4,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,3,2,5,4] => [2,2,1]
=> [2,2,1]
=> [2,1]
=> 1
[1,3,4,2,5] => [3,1,1]
=> [4,1]
=> [1]
=> 1
[1,3,4,5,2] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[1,3,5,2,4] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[1,3,5,4,2] => [3,1,1]
=> [4,1]
=> [1]
=> 1
[1,4,2,3,5] => [3,1,1]
=> [4,1]
=> [1]
=> 1
[1,4,2,5,3] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[1,4,3,2,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,4,3,5,2] => [3,1,1]
=> [4,1]
=> [1]
=> 1
[1,4,5,2,3] => [2,2,1]
=> [2,2,1]
=> [2,1]
=> 1
[1,4,5,3,2] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[1,5,2,3,4] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[1,5,2,4,3] => [3,1,1]
=> [4,1]
=> [1]
=> 1
[1,5,3,2,4] => [3,1,1]
=> [4,1]
=> [1]
=> 1
[1,5,3,4,2] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,5,4,2,3] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[1,5,4,3,2] => [2,2,1]
=> [2,2,1]
=> [2,1]
=> 1
[2,1,3,4,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[2,1,4,5,3] => [3,2]
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[2,1,5,3,4] => [3,2]
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[2,3,1,5,4] => [3,2]
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[2,4,5,1,3] => [3,2]
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[2,5,4,3,1] => [3,2]
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[3,1,2,5,4] => [3,2]
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[3,4,1,5,2] => [3,2]
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[3,4,5,2,1] => [3,2]
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[3,5,1,2,4] => [3,2]
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[3,5,4,1,2] => [3,2]
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[4,1,5,2,3] => [3,2]
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[4,3,2,5,1] => [3,2]
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[4,3,5,1,2] => [3,2]
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[4,5,1,3,2] => [3,2]
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[4,5,2,1,3] => [3,2]
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[5,1,4,3,2] => [3,2]
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[5,3,2,1,4] => [3,2]
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[5,3,4,2,1] => [3,2]
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[5,4,1,2,3] => [3,2]
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[5,4,2,3,1] => [3,2]
=> [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[2,3,1,5,6,4] => [3,3]
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,1,6,4,5] => [3,3]
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,5,1,6,3] => [3,3]
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,6,1,3,5] => [3,3]
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,4,6,1,3] => [3,3]
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,6,3,1,4] => [3,3]
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,6,4,5,3,1] => [3,3]
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,6,5,3,4,1] => [3,3]
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,1,2,5,6,4] => [3,3]
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,1,2,6,4,5] => [3,3]
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,4,5,6,1,2] => [3,3]
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,4,6,5,2,1] => [3,3]
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,5,4,1,6,2] => [3,3]
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,5,6,2,4,1] => [3,3]
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,6,4,1,2,5] => [3,3]
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,6,5,2,1,4] => [3,3]
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[4,1,5,2,6,3] => [3,3]
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[4,1,6,2,3,5] => [3,3]
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[4,3,5,6,2,1] => [3,3]
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[4,3,6,5,1,2] => [3,3]
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[4,5,1,3,6,2] => [3,3]
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[4,5,2,6,3,1] => [3,3]
=> [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
Description
The number of monotone factorisations of genus zero of a permutation of given cycle type.
A monotone factorisation of genus zero of a permutation $\pi\in\mathfrak S_n$ with $\ell$ cycles, including fixed points, is a tuple of $r = n - \ell$ transpositions
$$
(a_1, b_1),\dots,(a_r, b_r)
$$
with $b_1 \leq \dots \leq b_r$ and $a_i < b_i$ for all $i$, whose product, in this order, is $\pi$.
For example, the cycle $(2,3,1)$ has the two factorizations $(2,3)(1,3)$ and $(1,2)(2,3)$.
Matching statistic: St000781
(load all 20 compositions to match this statistic)
(load all 20 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
St000781: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 89%●distinct values known / distinct values provided: 50%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000781: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 89%●distinct values known / distinct values provided: 50%
Values
[1] => [1]
=> []
=> ?
=> ? = 1
[1,2] => [2]
=> []
=> ?
=> ? ∊ {1,1}
[2,1] => [1,1]
=> [1]
=> []
=> ? ∊ {1,1}
[1,2,3] => [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,1}
[1,3,2] => [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1}
[2,1,3] => [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1}
[2,3,1] => [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1}
[3,1,2] => [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1}
[3,2,1] => [1,1,1]
=> [1,1]
=> [1]
=> 1
[1,2,3,4] => [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,2,4,3] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,3,2,4] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,3,4,2] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,4,2,3] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[2,1,3,4] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[2,1,4,3] => [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[2,3,1,4] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[2,3,4,1] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[2,4,1,3] => [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[2,4,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[3,1,2,4] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[3,1,4,2] => [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[3,2,4,1] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[3,4,1,2] => [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[3,4,2,1] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[4,1,2,3] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[4,1,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[4,2,1,3] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[4,3,1,2] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[4,3,2,1] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,2,3,4,5] => [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[1,2,3,5,4] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[1,2,4,3,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[1,2,4,5,3] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[1,2,5,3,4] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[1,2,5,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,3,2,4,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[1,3,2,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[1,3,4,2,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[1,3,4,5,2] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[1,3,5,2,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,4,2,3,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[1,4,2,5,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[1,4,3,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,4,5,2,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[1,4,5,3,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,5,2,3,4] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,5,3,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,5,4,2,3] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,5,4,3,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[2,1,3,4,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[2,1,3,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[2,1,4,3,5] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[2,1,4,5,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[2,1,5,3,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[2,1,5,4,3] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,3,1,4,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[2,3,1,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[2,3,4,1,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[2,3,4,5,1] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[2,3,5,1,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[2,3,5,4,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[2,4,1,3,5] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[2,4,1,5,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[2,4,3,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[2,4,3,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[2,4,5,1,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[2,4,5,3,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[2,5,1,3,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[2,5,1,4,3] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,5,3,1,4] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[2,5,3,4,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[2,5,4,1,3] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,5,4,3,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[3,1,5,4,2] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,2,1,4,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[3,2,1,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,2,4,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[3,2,4,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[3,2,5,1,4] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,2,5,4,1] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,4,2,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[3,4,2,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[3,4,5,2,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[3,5,1,4,2] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,5,2,1,4] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,5,2,4,1] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,5,4,1,2] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,5,4,2,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[4,1,3,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[4,1,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[4,1,5,3,2] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[4,2,1,3,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
Description
The number of proper colouring schemes of a Ferrers diagram.
A colouring of a Ferrers diagram is proper if no two cells in a row or in a column have the same colour. The minimal number of colours needed is the maximum of the length and the first part of the partition, because we can restrict a latin square to the shape. We can associate to each colouring the integer partition recording how often each colour is used, see [1].
This statistic is the number of distinct such integer partitions that occur.
Matching statistic: St001432
(load all 3 compositions to match this statistic)
(load all 3 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
St001432: Integer partitions ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001432: Integer partitions ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> []
=> ?
=> ? = 1
[1,2] => [2]
=> []
=> ?
=> ? ∊ {1,1}
[2,1] => [1,1]
=> [1]
=> []
=> ? ∊ {1,1}
[1,2,3] => [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,1}
[1,3,2] => [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1}
[2,1,3] => [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1}
[2,3,1] => [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1}
[3,1,2] => [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1}
[3,2,1] => [1,1,1]
=> [1,1]
=> [1]
=> 1
[1,2,3,4] => [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,2,4,3] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,3,2,4] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,3,4,2] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,4,2,3] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[2,1,3,4] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[2,1,4,3] => [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[2,3,1,4] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[2,3,4,1] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[2,4,1,3] => [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[2,4,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[3,1,2,4] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[3,1,4,2] => [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[3,2,4,1] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[3,4,1,2] => [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[3,4,2,1] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[4,1,2,3] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[4,1,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[4,2,1,3] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[4,3,1,2] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[4,3,2,1] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,2,3,4,5] => [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[1,2,3,5,4] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[1,2,4,3,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[1,2,4,5,3] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[1,2,5,3,4] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[1,2,5,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,3,2,4,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[1,3,2,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[1,3,4,2,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[1,3,4,5,2] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[1,3,5,2,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,4,2,3,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[1,4,2,5,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[1,4,3,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,4,5,2,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[1,4,5,3,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,5,2,3,4] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,5,3,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,5,4,2,3] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,5,4,3,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[2,1,3,4,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[2,1,3,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[2,1,4,3,5] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[2,1,4,5,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[2,1,5,3,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[2,1,5,4,3] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,3,1,4,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[2,3,1,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[2,3,4,1,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[2,3,4,5,1] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[2,3,5,1,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[2,3,5,4,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[2,4,1,3,5] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[2,4,1,5,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[2,4,3,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[2,4,3,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[2,4,5,1,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[2,4,5,3,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[2,5,1,3,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[2,5,1,4,3] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,5,3,1,4] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[2,5,3,4,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[2,5,4,1,3] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,5,4,3,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[3,1,5,4,2] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,2,1,4,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[3,2,1,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,2,4,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[3,2,4,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[3,2,5,1,4] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,2,5,4,1] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,4,2,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[3,4,2,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[3,4,5,2,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[3,5,1,4,2] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,5,2,1,4] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,5,2,4,1] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,5,4,1,2] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,5,4,2,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[4,1,3,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[4,1,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[4,1,5,3,2] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[4,2,1,3,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
Description
The order dimension of the partition.
Given a partition $\lambda$, let $I(\lambda)$ be the principal order ideal in the Young lattice generated by $\lambda$. The order dimension of a partition is defined as the order dimension of the poset $I(\lambda)$.
Matching statistic: St001780
(load all 2 compositions to match this statistic)
(load all 2 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
St001780: Integer partitions ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001780: Integer partitions ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> []
=> ?
=> ? = 1
[1,2] => [2]
=> []
=> ?
=> ? ∊ {1,1}
[2,1] => [1,1]
=> [1]
=> []
=> ? ∊ {1,1}
[1,2,3] => [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,1}
[1,3,2] => [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1}
[2,1,3] => [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1}
[2,3,1] => [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1}
[3,1,2] => [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1}
[3,2,1] => [1,1,1]
=> [1,1]
=> [1]
=> 1
[1,2,3,4] => [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,2,4,3] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,3,2,4] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,3,4,2] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,4,2,3] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[2,1,3,4] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[2,1,4,3] => [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[2,3,1,4] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[2,3,4,1] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[2,4,1,3] => [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[2,4,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[3,1,2,4] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[3,1,4,2] => [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[3,2,4,1] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[3,4,1,2] => [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[3,4,2,1] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[4,1,2,3] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[4,1,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[4,2,1,3] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[4,3,1,2] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[4,3,2,1] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,2,3,4,5] => [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[1,2,3,5,4] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[1,2,4,3,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[1,2,4,5,3] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[1,2,5,3,4] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[1,2,5,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,3,2,4,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[1,3,2,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[1,3,4,2,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[1,3,4,5,2] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[1,3,5,2,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,4,2,3,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[1,4,2,5,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[1,4,3,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,4,5,2,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[1,4,5,3,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,5,2,3,4] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,5,3,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,5,4,2,3] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,5,4,3,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[2,1,3,4,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[2,1,3,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[2,1,4,3,5] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[2,1,4,5,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[2,1,5,3,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[2,1,5,4,3] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,3,1,4,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[2,3,1,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[2,3,4,1,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[2,3,4,5,1] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[2,3,5,1,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[2,3,5,4,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[2,4,1,3,5] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[2,4,1,5,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[2,4,3,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[2,4,3,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[2,4,5,1,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[2,4,5,3,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[2,5,1,3,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[2,5,1,4,3] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,5,3,1,4] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[2,5,3,4,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[2,5,4,1,3] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,5,4,3,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[3,1,5,4,2] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,2,1,4,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[3,2,1,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,2,4,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[3,2,4,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[3,2,5,1,4] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,2,5,4,1] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,4,2,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[3,4,2,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[3,4,5,2,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[3,5,1,4,2] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,5,2,1,4] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,5,2,4,1] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,5,4,1,2] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,5,4,2,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[4,1,3,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[4,1,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[4,1,5,3,2] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[4,2,1,3,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
Description
The order of promotion on the set of standard tableaux of given shape.
Matching statistic: St001899
(load all 2 compositions to match this statistic)
(load all 2 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
St001899: Integer partitions ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001899: Integer partitions ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> []
=> ?
=> ? = 1
[1,2] => [2]
=> []
=> ?
=> ? ∊ {1,1}
[2,1] => [1,1]
=> [1]
=> []
=> ? ∊ {1,1}
[1,2,3] => [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,1}
[1,3,2] => [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1}
[2,1,3] => [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1}
[2,3,1] => [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1}
[3,1,2] => [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1}
[3,2,1] => [1,1,1]
=> [1,1]
=> [1]
=> 1
[1,2,3,4] => [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,2,4,3] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,3,2,4] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,3,4,2] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,4,2,3] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[2,1,3,4] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[2,1,4,3] => [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[2,3,1,4] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[2,3,4,1] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[2,4,1,3] => [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[2,4,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[3,1,2,4] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[3,1,4,2] => [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[3,2,4,1] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[3,4,1,2] => [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[3,4,2,1] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[4,1,2,3] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[4,1,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[4,2,1,3] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[4,3,1,2] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[4,3,2,1] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,2,3,4,5] => [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[1,2,3,5,4] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[1,2,4,3,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[1,2,4,5,3] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[1,2,5,3,4] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[1,2,5,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,3,2,4,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[1,3,2,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[1,3,4,2,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[1,3,4,5,2] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[1,3,5,2,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,4,2,3,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[1,4,2,5,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[1,4,3,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,4,5,2,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[1,4,5,3,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,5,2,3,4] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,5,3,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,5,4,2,3] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,5,4,3,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[2,1,3,4,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[2,1,3,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[2,1,4,3,5] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[2,1,4,5,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[2,1,5,3,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[2,1,5,4,3] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,3,1,4,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[2,3,1,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[2,3,4,1,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[2,3,4,5,1] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[2,3,5,1,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[2,3,5,4,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[2,4,1,3,5] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[2,4,1,5,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[2,4,3,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[2,4,3,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[2,4,5,1,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[2,4,5,3,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[2,5,1,3,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
[2,5,1,4,3] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,5,3,1,4] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[2,5,3,4,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[2,5,4,1,3] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,5,4,3,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[3,1,5,4,2] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,2,1,4,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[3,2,1,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,2,4,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[3,2,4,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[3,2,5,1,4] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,2,5,4,1] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,4,2,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[3,4,2,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[3,4,5,2,1] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[3,5,1,4,2] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,5,2,1,4] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,5,2,4,1] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,5,4,1,2] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,5,4,2,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[4,1,3,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[4,1,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[4,1,5,3,2] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[4,2,1,3,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
Description
The total number of irreducible representations contained in the higher Lie character for an integer partition.
The following 222 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000003The number of standard Young tableaux of the partition. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000321The number of integer partitions of n that are dominated by an integer partition. St000326The position of the first one in a binary word after appending a 1 at the end. St000345The number of refinements of a partition. St000628The balance of a binary word. St000655The length of the minimal rise of a Dyck path. St000847The number of standard Young tableaux whose descent set is the binary word. St000913The number of ways to refine the partition into singletons. St000935The number of ordered refinements of an integer partition. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001256Number of simple reflexive modules that are 2-stable reflexive. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001722The number of minimal chains with small intervals between a binary word and the top element. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001481The minimal height of a peak of a Dyck path. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001128The exponens consonantiae of a partition. St000481The number of upper covers of a partition in dominance order. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000914The sum of the values of the Möbius function of a poset. St001890The maximum magnitude of the Möbius function of a poset. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000667The greatest common divisor of the parts of the partition. St001389The number of partitions of the same length below the given integer partition. St001571The Cartan determinant of the integer partition. St000456The monochromatic index of a connected graph. St001651The Frankl number of a lattice. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000031The number of cycles in the cycle decomposition of a permutation. St001330The hat guessing number of a graph. St001095The number of non-isomorphic posets with precisely one further covering relation. St000553The number of blocks of a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St000916The packing number of a graph. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. 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. 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. St001496The number of graphs with the same Laplacian spectrum as the given graph. St000068The number of minimal elements in a poset. St000842The breadth of a permutation. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. 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. St001518The number of graphs with the same ordinary spectrum as the given graph. St000287The number of connected components of a graph. St001765The number of connected components of the friends and strangers graph. St001162The minimum jump of a permutation. 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 $eA$. St001271The competition number of a graph. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000897The number of different multiplicities of parts of an integer partition. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000284The Plancherel distribution on integer partitions. St000706The product of the factorials of the multiplicities of an integer partition. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000993The multiplicity of the largest part of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St001964The interval resolution global dimension of a poset. St000759The smallest missing part in an integer partition. St000455The second largest eigenvalue of a graph if it is integral. 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 $K[x]/(x^n)$. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001371The length of the longest Yamanouchi prefix of a binary word. 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. 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. St000929The constant term of the character polynomial of an integer partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000908The length of the shortest maximal antichain in a poset. St001301The first Betti number of the order complex associated with the poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001396Number of triples of incomparable elements in a finite poset. St001820The size of the image of the pop stack sorting operator. St001846The number of elements which do not have a complement in the lattice. 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. 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. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. 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. St000181The number of connected components of the Hasse diagram for the poset. 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. 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. St000069The number of maximal elements of a poset. St001621The number of atoms of a lattice. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St000256The number of parts from which one can substract 2 and still get an integer partition. St000629The defect of a binary word. 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. 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. 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". St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St001316The domatic number of a graph. St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St000323The minimal crossing number of a graph. St000351The determinant of the adjacency matrix of a graph. St000368The Altshuler-Steinberg determinant of a graph. St000370The genus 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. St001307The number of induced stars on four vertices in a graph. St001309The number of four-cliques 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. St001325The minimal number of occurrences of the comparability-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. St001334The minimal number of occurrences of the 3-colorable 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. St001793The difference between the clique number and the chromatic number 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. St000273The domination number of a graph. St000544The cop number of a graph. St001829The common independence number 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. St000379The number of Hamiltonian cycles in a graph. St000699The toughness times the least common multiple of 1,. St001281The normalized isoperimetric number of a graph. St001322The size of a minimal independent dominating set in a graph. St000264The girth of a graph, which is not a tree. 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 $c_{n−1}$. St001344The neighbouring number of a permutation. St001768The number of reduced words of a signed permutation. St001549The number of restricted non-inversions between exceedances. St001811The Castelnuovo-Mumford regularity of a permutation. St000322The skewness of a graph. St000286The number of connected components of the complement of a graph. 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. St001578The minimal number of edges to add or remove to make a graph a line graph. 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. St001871The number of triconnected components of a graph. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St001570The minimal number of edges to add to make a graph Hamiltonian.
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