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Your data matches 31 different statistics following compositions of up to 3 maps.
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Matching statistic: St001124
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001124: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001124: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> 0
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> 0
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[3,3,3,3],[2]]
=> [2]
=> 0
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> 0
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [[4,4,4,4],[3]]
=> [3]
=> 0
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[4,4,4],[3]]
=> [3]
=> 0
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> 0
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 0
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [[4,4,4,4,4],[3]]
=> [3]
=> 0
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [[3,3,3,3,3],[2]]
=> [2]
=> 0
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [[3,3,3,3],[2,1]]
=> [2,1]
=> 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [[3,3,3,2],[2]]
=> [2]
=> 0
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> 0
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> 0
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2]
=> 0
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [[5,5,5,5,5],[4]]
=> [4]
=> 0
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [[5,5,5,5],[4]]
=> [4]
=> 0
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [[4,4,4,3],[3]]
=> [3]
=> 0
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [[4,4,4,4],[3,1]]
=> [3,1]
=> 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [[4,4,3],[3]]
=> [3]
=> 0
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [[4,4,4],[3,1]]
=> [3,1]
=> 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [[4,3,3],[2]]
=> [2]
=> 0
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> 0
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> 1
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [2]
=> 0
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> 0
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [[4,4,2],[1,1]]
=> [1,1]
=> 0
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [[5,5,5,5,5,5],[4]]
=> [4]
=> 0
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [[4,4,4,4,4,4],[3]]
=> [3]
=> 0
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [[4,4,4,4,4],[3,1]]
=> [3,1]
=> 1
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [[4,4,4,4,3],[3]]
=> [3]
=> 0
[6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [[3,3,3,3,3],[2,1]]
=> [2,1]
=> 1
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [[5,5,5],[4]]
=> [4]
=> 0
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [[3,3,3,3,2],[2]]
=> [2]
=> 0
[5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [[3,3,3,3],[2,2]]
=> [2,2]
=> 0
[5,3,1]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [[3,3,2,2],[2]]
=> [2]
=> 0
[5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [[3,3,3,2],[2,1]]
=> [2,1]
=> 1
[5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [[3,3,3,3],[2,1,1]]
=> [2,1,1]
=> 1
[5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3,1],[2]]
=> [2]
=> 0
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 0
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> 0
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [[3,3,2,2],[1,1]]
=> [1,1]
=> 0
[3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [[3,3,3,2],[1,1,1]]
=> [1,1,1]
=> 0
[3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3,1],[1,1]]
=> [1,1]
=> 0
[10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [[6,6,6,6,6,6],[5]]
=> [5]
=> 0
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [[6,6,6,6,6],[5]]
=> [5]
=> 0
[8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [[5,5,5,5,4],[4]]
=> [4]
=> 0
[8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [[5,5,5,5,5],[4,1]]
=> [4,1]
=> 1
[7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [[5,5,5,4],[4]]
=> [4]
=> 0
Description
The multiplicity of the standard representation in the Kronecker square corresponding to a partition.
The Kronecker coefficient is the multiplicity $g_{\mu,\nu}^\lambda$ of the Specht module $S^\lambda$ in $S^\mu\otimes S^\nu$:
$$ S^\mu\otimes S^\nu = \bigoplus_\lambda g_{\mu,\nu}^\lambda S^\lambda $$
This statistic records the Kronecker coefficient $g_{\lambda,\lambda}^{(n-1)1}$, for $\lambda\vdash n > 1$. For $n\leq1$ the statistic is undefined.
It follows from [3, Prop.4.1] (or, slightly easier from [3, Thm.4.2]) that this is one less than [[St000159]], the number of distinct parts of the partition.
Matching statistic: St000159
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000159: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000159: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> 1 = 0 + 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> 1 = 0 + 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[3,3,3,3],[2]]
=> [2]
=> 1 = 0 + 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> 1 = 0 + 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [[4,4,4,4],[3]]
=> [3]
=> 1 = 0 + 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[4,4,4],[3]]
=> [3]
=> 1 = 0 + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> 1 = 0 + 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> 2 = 1 + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 1 = 0 + 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [[4,4,4,4,4],[3]]
=> [3]
=> 1 = 0 + 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [[3,3,3,3,3],[2]]
=> [2]
=> 1 = 0 + 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [[3,3,3,3],[2,1]]
=> [2,1]
=> 2 = 1 + 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [[3,3,3,2],[2]]
=> [2]
=> 1 = 0 + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> 1 = 0 + 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> 1 = 0 + 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2]
=> 1 = 0 + 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [[5,5,5,5,5],[4]]
=> [4]
=> 1 = 0 + 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [[5,5,5,5],[4]]
=> [4]
=> 1 = 0 + 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [[4,4,4,3],[3]]
=> [3]
=> 1 = 0 + 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [[4,4,4,4],[3,1]]
=> [3,1]
=> 2 = 1 + 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [[4,4,3],[3]]
=> [3]
=> 1 = 0 + 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [[4,4,4],[3,1]]
=> [3,1]
=> 2 = 1 + 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [[4,3,3],[2]]
=> [2]
=> 1 = 0 + 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> 1 = 0 + 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> 2 = 1 + 1
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [2]
=> 1 = 0 + 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> 1 = 0 + 1
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [[4,4,2],[1,1]]
=> [1,1]
=> 1 = 0 + 1
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [[5,5,5,5,5,5],[4]]
=> [4]
=> 1 = 0 + 1
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [[4,4,4,4,4,4],[3]]
=> [3]
=> 1 = 0 + 1
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [[4,4,4,4,4],[3,1]]
=> [3,1]
=> 2 = 1 + 1
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [[4,4,4,4,3],[3]]
=> [3]
=> 1 = 0 + 1
[6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [[3,3,3,3,3],[2,1]]
=> [2,1]
=> 2 = 1 + 1
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [[5,5,5],[4]]
=> [4]
=> 1 = 0 + 1
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [[3,3,3,3,2],[2]]
=> [2]
=> 1 = 0 + 1
[5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [[3,3,3,3],[2,2]]
=> [2,2]
=> 1 = 0 + 1
[5,3,1]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [[3,3,2,2],[2]]
=> [2]
=> 1 = 0 + 1
[5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [[3,3,3,2],[2,1]]
=> [2,1]
=> 2 = 1 + 1
[5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [[3,3,3,3],[2,1,1]]
=> [2,1,1]
=> 2 = 1 + 1
[5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3,1],[2]]
=> [2]
=> 1 = 0 + 1
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 1 = 0 + 1
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> 1 = 0 + 1
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [[3,3,2,2],[1,1]]
=> [1,1]
=> 1 = 0 + 1
[3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [[3,3,3,2],[1,1,1]]
=> [1,1,1]
=> 1 = 0 + 1
[3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3,1],[1,1]]
=> [1,1]
=> 1 = 0 + 1
[10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [[6,6,6,6,6,6],[5]]
=> [5]
=> 1 = 0 + 1
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [[6,6,6,6,6],[5]]
=> [5]
=> 1 = 0 + 1
[8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [[5,5,5,5,4],[4]]
=> [4]
=> 1 = 0 + 1
[8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [[5,5,5,5,5],[4,1]]
=> [4,1]
=> 2 = 1 + 1
[7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [[5,5,5,4],[4]]
=> [4]
=> 1 = 0 + 1
Description
The number of distinct parts of the integer partition.
This statistic is also the number of removeable cells of the partition, and the number of valleys of the Dyck path tracing the shape of the partition.
Matching statistic: St000318
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000318: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000318: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> 2 = 0 + 2
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> 2 = 0 + 2
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[3,3,3,3],[2]]
=> [2]
=> 2 = 0 + 2
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> 2 = 0 + 2
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [[4,4,4,4],[3]]
=> [3]
=> 2 = 0 + 2
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[4,4,4],[3]]
=> [3]
=> 2 = 0 + 2
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> 2 = 0 + 2
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> 3 = 1 + 2
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 2 = 0 + 2
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [[4,4,4,4,4],[3]]
=> [3]
=> 2 = 0 + 2
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [[3,3,3,3,3],[2]]
=> [2]
=> 2 = 0 + 2
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [[3,3,3,3],[2,1]]
=> [2,1]
=> 3 = 1 + 2
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [[3,3,3,2],[2]]
=> [2]
=> 2 = 0 + 2
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> 2 = 0 + 2
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> 2 = 0 + 2
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2]
=> 2 = 0 + 2
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [[5,5,5,5,5],[4]]
=> [4]
=> 2 = 0 + 2
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [[5,5,5,5],[4]]
=> [4]
=> 2 = 0 + 2
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [[4,4,4,3],[3]]
=> [3]
=> 2 = 0 + 2
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [[4,4,4,4],[3,1]]
=> [3,1]
=> 3 = 1 + 2
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [[4,4,3],[3]]
=> [3]
=> 2 = 0 + 2
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [[4,4,4],[3,1]]
=> [3,1]
=> 3 = 1 + 2
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [[4,3,3],[2]]
=> [2]
=> 2 = 0 + 2
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> 2 = 0 + 2
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> 3 = 1 + 2
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [2]
=> 2 = 0 + 2
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> 2 = 0 + 2
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [[4,4,2],[1,1]]
=> [1,1]
=> 2 = 0 + 2
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [[5,5,5,5,5,5],[4]]
=> [4]
=> 2 = 0 + 2
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [[4,4,4,4,4,4],[3]]
=> [3]
=> 2 = 0 + 2
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [[4,4,4,4,4],[3,1]]
=> [3,1]
=> 3 = 1 + 2
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [[4,4,4,4,3],[3]]
=> [3]
=> 2 = 0 + 2
[6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [[3,3,3,3,3],[2,1]]
=> [2,1]
=> 3 = 1 + 2
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [[5,5,5],[4]]
=> [4]
=> 2 = 0 + 2
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [[3,3,3,3,2],[2]]
=> [2]
=> 2 = 0 + 2
[5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [[3,3,3,3],[2,2]]
=> [2,2]
=> 2 = 0 + 2
[5,3,1]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [[3,3,2,2],[2]]
=> [2]
=> 2 = 0 + 2
[5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [[3,3,3,2],[2,1]]
=> [2,1]
=> 3 = 1 + 2
[5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [[3,3,3,3],[2,1,1]]
=> [2,1,1]
=> 3 = 1 + 2
[5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3,1],[2]]
=> [2]
=> 2 = 0 + 2
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 2 = 0 + 2
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> 2 = 0 + 2
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [[3,3,2,2],[1,1]]
=> [1,1]
=> 2 = 0 + 2
[3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [[3,3,3,2],[1,1,1]]
=> [1,1,1]
=> 2 = 0 + 2
[3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3,1],[1,1]]
=> [1,1]
=> 2 = 0 + 2
[10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [[6,6,6,6,6,6],[5]]
=> [5]
=> 2 = 0 + 2
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [[6,6,6,6,6],[5]]
=> [5]
=> 2 = 0 + 2
[8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [[5,5,5,5,4],[4]]
=> [4]
=> 2 = 0 + 2
[8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [[5,5,5,5,5],[4,1]]
=> [4,1]
=> 3 = 1 + 2
[7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [[5,5,5,4],[4]]
=> [4]
=> 2 = 0 + 2
Description
The number of addable cells of the Ferrers diagram of an integer partition.
Matching statistic: St001037
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
St001037: Dyck paths ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Mp00028: Dyck paths —reverse⟶ Dyck paths
St001037: Dyck paths ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Values
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 1 = 0 + 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 1 = 0 + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 1 = 0 + 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> 1 = 0 + 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 1 = 0 + 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> 2 = 1 + 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1 = 0 + 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 0 + 1
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> 2 = 1 + 1
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> 1 = 0 + 1
[6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> 2 = 1 + 1
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> 1 = 0 + 1
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> 1 = 0 + 1
[5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[5,3,1]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> 2 = 1 + 1
[5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 2 = 1 + 1
[5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> 1 = 0 + 1
[10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> ? = 0 + 1
[8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 1 + 1
[7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> 1 = 0 + 1
[7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> 1 = 0 + 1
[7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> 2 = 1 + 1
[6,4]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[6,3,1]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> 2 = 1 + 1
[6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[6,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[6,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> 2 = 1 + 1
[8,7,6,5,4,3,2]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0 + 1
[8,6,5,4,3,2,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0 + 1
[9,8,7,6,5,4,3,2]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0 + 1
[8,7,6,5,4,3]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 0 + 1
[9,7,6,5,4,3,2,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0 + 1
Description
The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path.
Matching statistic: St000386
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000386: Dyck paths ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000386: Dyck paths ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Values
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> 1 = 0 + 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> 1 = 0 + 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2 = 1 + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
=> ? = 0 + 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> 1 = 0 + 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> 2 = 1 + 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 0 + 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 0 + 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0]
=> 1 = 0 + 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> 1 = 0 + 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0]
=> 2 = 1 + 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1 = 0 + 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 0 + 1
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 0 + 1
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,1,0,0,0]
=> ? = 1 + 1
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,1,0,1,0,0,0,0]
=> ? = 0 + 1
[6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> 2 = 1 + 1
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> 1 = 0 + 1
[5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[5,3,1]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> 2 = 1 + 1
[5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 1 = 0 + 1
[3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 0 + 1
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 0 + 1
[8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 0 + 1
[8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,1,0,0,1,0,0,0]
=> ? = 1 + 1
[7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 0 + 1
[7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> 1 = 0 + 1
[7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 1 + 1
[6,4]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> 1 = 0 + 1
[6,3,1]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> 2 = 1 + 1
[6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> 2 = 1 + 1
[6,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[6,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> 2 = 1 + 1
[5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> 1 = 0 + 1
[5,4,1]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 1 = 0 + 1
[5,3,2]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[5,3,1,1]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[5,2,2,1]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[7,4,1,1]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 2 + 1
[7,2,2,1,1,1]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 2 + 1
[7,6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,1,0,0]
=> ? = 1 + 1
[7,4,3,1]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> ? = 1 + 1
[7,4,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0,1,0]
=> ? = 2 + 1
[7,4,2,1,1,1]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0,1,0]
=> ? = 2 + 1
[7,2,2,2,2,1]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0,1,0]
=> ? = 1 + 1
[7,6,3,1]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
=> ? = 1 + 1
[7,6,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0,1,0]
=> ? = 1 + 1
[7,4,3,1,1,1]
=> [1,1,0,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0,1,0]
=> ? = 2 + 1
[7,6,4,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 1
[7,6,5,4,3,1]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 0 + 1
[7,6,5,2,2,1]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0]
=> ? = 1 + 1
[7,6,5,4,1,1]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> ? = 1 + 1
[7,6,5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> ? = 0 + 1
Description
The number of factors DDU in a Dyck path.
Matching statistic: St000257
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000257: Integer partitions ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000257: Integer partitions ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Values
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 1 = 0 + 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 1 = 0 + 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [2,2,1]
=> 1 = 0 + 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1 = 0 + 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [3,3,2,1]
=> 1 = 0 + 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1]
=> 1 = 0 + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> 1 = 0 + 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 2 = 1 + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 1 = 0 + 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [3,3,2,1]
=> 1 = 0 + 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [2,2,1]
=> 1 = 0 + 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1]
=> 2 = 1 + 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [3,3,1]
=> 1 = 0 + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 1 = 0 + 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 1 = 0 + 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 1 = 0 + 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [4,4,3,2,1]
=> 1 = 0 + 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> [4,4,3,2,1]
=> 1 = 0 + 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> [4,4,2,1]
=> 1 = 0 + 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [3,3,2,1,1]
=> 2 = 1 + 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1]
=> 1 = 0 + 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1,1]
=> 2 = 1 + 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1]
=> 1 = 0 + 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 1 = 0 + 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 2 = 1 + 1
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 1 = 0 + 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 1 = 0 + 1
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1,1]
=> 1 = 0 + 1
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0]
=> [4,4,3,2,1]
=> 1 = 0 + 1
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0,0]
=> [3,3,2,1]
=> 1 = 0 + 1
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,1,1,0,0,0,0,0]
=> [3,3,2,1,1]
=> 2 = 1 + 1
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,1,1,0,0,0,0]
=> [4,4,2,1]
=> 1 = 0 + 1
[6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [2,2,1,1]
=> 2 = 1 + 1
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [4,4,3,2,1]
=> 1 = 0 + 1
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [3,3,1]
=> 1 = 0 + 1
[5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1]
=> 1 = 0 + 1
[5,3,1]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [4,4,1]
=> 1 = 0 + 1
[5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1]
=> 2 = 1 + 1
[5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1,1]
=> 2 = 1 + 1
[5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> 1 = 0 + 1
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 1 = 0 + 1
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 1 = 0 + 1
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1]
=> 1 = 0 + 1
[3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1,1]
=> 1 = 0 + 1
[3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2]
=> 1 = 0 + 1
[10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0]
=> [5,5,4,3,2,1]
=> 1 = 0 + 1
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [5,5,4,3,2,1]
=> 1 = 0 + 1
[8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,1,1,0,0,0,0]
=> [5,5,3,2,1]
=> ? = 0 + 1
[8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [4,4,3,2,1,1]
=> ? = 1 + 1
[7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,1,0,0,0]
=> [5,5,3,2,1]
=> ? = 0 + 1
[7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
=> [2,2,1]
=> 1 = 0 + 1
[7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [4,4,3,2,1,1]
=> ? = 1 + 1
[6,4]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,1,0,0]
=> [5,5,2,1]
=> ? = 0 + 1
[6,3,1]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [3,3,2,2,1]
=> 2 = 1 + 1
[6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [4,4,2,1,1]
=> 2 = 1 + 1
[6,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [3,3,2,1,1,1]
=> ? = 1 + 1
[2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> [4,3,2,2,2]
=> ? = 0 + 1
[6,2,1,1,1]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [4,4,3,2,1,1]
=> ? = 1 + 1
[6,2,2,2,1]
=> [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [4,4,3,2,2,1]
=> ? = 1 + 1
[7,2,2,1,1,1]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [4,4,3,2,2,1,1]
=> ? = 2 + 1
[6,5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2,1,1]
=> ? = 1 + 1
[7,6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [3,3,3,2,1,1]
=> ? = 1 + 1
[7,4,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [3,3,2,2,1,1,1]
=> ? = 2 + 1
[6,3,3,2,1]
=> [1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [4,4,3,3,2,1]
=> ? = 1 + 1
[7,4,2,1,1,1]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [4,4,3,3,2,1,1]
=> ? = 2 + 1
[7,2,2,2,2,1]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [4,4,3,2,2,2,1]
=> ? = 1 + 1
[7,6,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [3,3,3,2,1,1,1]
=> ? = 1 + 1
[7,4,3,1,1,1]
=> [1,1,0,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [3,3,2,2,2,1,1]
=> ? = 2 + 1
[6,6,5,3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> [7,4,4,4]
=> ? = 0 + 1
[6,6,5,4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [7,3,3,3,3]
=> ? = 0 + 1
[6,6,5,4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [7,2,2,2,2,2]
=> ? = 0 + 1
[6,6,5,4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [7,3,3,3,3,2]
=> ? = 0 + 1
[6,5,5,4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [7,6,3,3,3,2]
=> ? = 0 + 1
[6,6,5,4,3,2]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,1,1,1,1,1]
=> ? = 0 + 1
[7,5,5,4,3,2]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [6,6,1,1,1,1,1]
=> ? = 1 + 1
[7,6,4,4,3,2]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [5,5,5,1,1,1,1]
=> ? = 1 + 1
[6,5,5,4,3,1]
=> [1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [7,6,2,2,2,2,1]
=> ? = 0 + 1
[7,6,5,2,2,1]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,3,3,3,2,2,1]
=> ? = 1 + 1
Description
The number of distinct parts of a partition that occur at least twice.
See Section 3.3.1 of [2].
Matching statistic: St000201
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
St000201: Binary trees ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 100%
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
St000201: Binary trees ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 100%
Values
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [.,[[[.,.],.],[.,.]]]
=> 2 = 0 + 2
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [[[.,.],.],[.,.]]
=> 2 = 0 + 2
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [.,[.,[[[.,.],.],[.,.]]]]
=> 2 = 0 + 2
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [[.,.],[.,[.,.]]]
=> 2 = 0 + 2
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [.,[.,[[[[.,.],.],.],[.,.]]]]
=> 2 = 0 + 2
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [.,[[[[.,.],.],.],[.,.]]]
=> 2 = 0 + 2
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[.,[[.,.],.]],[.,.]]
=> 2 = 0 + 2
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[.,.],[[.,.],[.,.]]]
=> 3 = 1 + 2
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[.,.],[[.,[.,.]],.]]
=> 2 = 0 + 2
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [.,[.,[.,[[[[.,.],.],.],[.,.]]]]]
=> ? = 0 + 2
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [.,[.,[.,[[[.,.],.],[.,.]]]]]
=> 2 = 0 + 2
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [.,[[.,.],[[.,.],[.,.]]]]
=> 3 = 1 + 2
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [.,[[.,[[.,.],.]],[.,.]]]
=> 2 = 0 + 2
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [.,[[.,.],[.,[.,.]]]]
=> 2 = 0 + 2
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[[[.,.],.],.],[.,.]]
=> 2 = 0 + 2
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[[[.,.],.],[.,.]],.]
=> 2 = 0 + 2
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [.,[.,[.,[[[[[.,.],.],.],.],[.,.]]]]]
=> ? = 0 + 2
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> [.,[.,[[[[[.,.],.],.],.],[.,.]]]]
=> ? = 0 + 2
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> [.,[[.,[[[.,.],.],.]],[.,.]]]
=> 2 = 0 + 2
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [.,[[.,.],[[[.,.],.],[.,.]]]]
=> 3 = 1 + 2
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [[.,[[[.,.],.],.]],[.,.]]
=> 2 = 0 + 2
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [[.,.],[[[.,.],.],[.,.]]]
=> 3 = 1 + 2
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [[.,[[[.,.],.],[.,.]]],.]
=> 2 = 0 + 2
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[[.,.],.],[.,[.,.]]]
=> 2 = 0 + 2
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[[.,.],[.,.]],[.,.]]
=> 3 = 1 + 2
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[[.,[.,.]],.],[.,.]]
=> 2 = 0 + 2
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[[.,.],[.,[.,.]]],.]
=> 2 = 0 + 2
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [[.,.],[[[.,[.,.]],.],.]]
=> 2 = 0 + 2
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0]
=> [.,[.,[.,[.,[[[[[.,.],.],.],.],[.,.]]]]]]
=> ? = 0 + 2
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0,0]
=> [.,[.,[.,[.,[[[[.,.],.],.],[.,.]]]]]]
=> ? = 0 + 2
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,1,1,0,0,0,0,0]
=> [.,[.,[[.,.],[[[.,.],.],[.,.]]]]]
=> ? = 1 + 2
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,1,1,0,0,0,0]
=> [.,[.,[[.,[[[.,.],.],.]],[.,.]]]]
=> ? = 0 + 2
[6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> 3 = 1 + 2
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [.,[[[[[.,.],.],.],.],[.,.]]]
=> 2 = 0 + 2
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [.,[.,[[.,[[.,.],.]],[.,.]]]]
=> 2 = 0 + 2
[5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [.,[[[.,.],.],[.,[.,.]]]]
=> 2 = 0 + 2
[5,3,1]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [[.,[.,[[.,.],.]]],[.,.]]
=> 2 = 0 + 2
[5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [.,[[[.,.],[.,.]],[.,.]]]
=> 3 = 1 + 2
[5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [[.,.],[.,[[.,.],[.,.]]]]
=> 3 = 1 + 2
[5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [.,[[[.,[.,.]],.],[.,.]]]
=> 2 = 0 + 2
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[.,.],[.,[.,[.,.]]]]
=> 2 = 0 + 2
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[.,[.,.]],[.,[.,.]]]
=> 2 = 0 + 2
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [.,[[[.,.],[.,[.,.]]],.]]
=> 2 = 0 + 2
[3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [[.,.],[.,[[.,[.,.]],.]]]
=> 2 = 0 + 2
[3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [[.,[.,.]],[.,[[.,.],.]]]
=> 2 = 0 + 2
[10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0]
=> [.,[.,[.,[.,[[[[[[.,.],.],.],.],.],[.,.]]]]]]
=> ? = 0 + 2
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [.,[.,[.,[[[[[[.,.],.],.],.],.],[.,.]]]]]
=> ? = 0 + 2
[8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,1,1,0,0,0,0]
=> [.,[.,[[.,[[[[.,.],.],.],.]],[.,.]]]]
=> ? = 0 + 2
[8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [.,[.,[[.,.],[[[[.,.],.],.],[.,.]]]]]
=> ? = 1 + 2
[7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,1,0,0,0]
=> [.,[[.,[[[[.,.],.],.],.]],[.,.]]]
=> ? = 0 + 2
[7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
=> [.,[.,[.,[.,[[[.,.],.],[.,.]]]]]]
=> ? = 0 + 2
[7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [.,[[.,.],[[[[.,.],.],.],[.,.]]]]
=> ? = 1 + 2
[6,4]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,1,0,0]
=> [[.,[.,[[[.,.],.],.]]],[.,.]]
=> 2 = 0 + 2
[6,3,1]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [.,[[[.,.],.],[[.,.],[.,.]]]]
=> 3 = 1 + 2
[6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [.,[[[.,.],[[.,.],.]],[.,.]]]
=> 3 = 1 + 2
[6,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,1,0,0,0]
=> [.,[[[.,[[.,.],.]],.],[.,.]]]
=> 2 = 0 + 2
[6,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [[.,.],[.,[[[.,.],.],[.,.]]]]
=> 3 = 1 + 2
[5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> [[.,[.,[[[.,.],.],[.,.]]]],.]
=> 2 = 0 + 2
[5,4,1]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [.,[.,[[.,.],[.,[.,.]]]]]
=> 2 = 0 + 2
[5,3,2]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> [[[.,[[.,.],.]],.],[.,.]]
=> 2 = 0 + 2
[5,3,1,1]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [[[.,.],[[.,.],.]],[.,.]]
=> 3 = 1 + 2
[5,2,2,1]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [[[.,.],.],[[.,.],[.,.]]]
=> 3 = 1 + 2
[4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> [[[.,[[.,.],.]],[.,.]],.]
=> 2 = 0 + 2
[4,4,1,1]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> 3 = 1 + 2
[7,4,1,1]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [.,[[.,.],[[.,.],[[.,.],[.,.]]]]]
=> ? = 2 + 2
[7,2,2,1,1,1]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [[.,.],[[.,.],[[[.,.],.],[.,.]]]]
=> ? = 2 + 2
[7,6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [.,[[.,.],[[[.,.],.],[.,[.,.]]]]]
=> ? = 1 + 2
[7,4,3,1]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [.,[[[.,.],.],[.,[[.,.],[.,.]]]]]
=> ? = 1 + 2
[7,4,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [[.,.],[.,[[.,.],[[.,.],[.,.]]]]]
=> ? = 2 + 2
[7,4,2,1,1,1]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [[.,.],[[[.,.],.],[[.,.],[.,.]]]]
=> ? = 2 + 2
[7,2,2,2,2,1]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [[[.,.],.],[.,[[[.,.],.],[.,.]]]]
=> ? = 1 + 2
[7,6,3,1]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [.,[[[.,.],.],[[.,.],[.,[.,.]]]]]
=> ? = 1 + 2
[7,6,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [[.,.],[.,[[[.,.],.],[.,[.,.]]]]]
=> ? = 1 + 2
[7,4,3,1,1,1]
=> [1,1,0,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [[.,.],[[.,.],[.,[[.,.],[.,.]]]]]
=> ? = 2 + 2
[7,6,4,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> ? = 0 + 2
[6,6,5,3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> [[[.,[.,[.,[.,.]]]],[.,[.,.]]],.]
=> ? = 0 + 2
[6,6,5,4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [[[.,[.,[.,.]]],[.,[.,[.,.]]]],.]
=> ? = 0 + 2
[7,6,5,4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 0 + 2
[6,6,5,4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [[[.,[.,.]],[.,[.,[.,[.,.]]]]],.]
=> ? = 0 + 2
[6,6,5,4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [[[[.,[.,.]],.],[.,[.,[.,.]]]],.]
=> ? = 0 + 2
[6,5,5,4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [[[[[.,[.,.]],.],[.,[.,.]]],.],.]
=> ? = 0 + 2
[6,6,5,4,3,2]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [[[.,.],[.,[.,[.,[.,[.,.]]]]]],.]
=> ? = 0 + 2
[7,5,5,4,3,2]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [[[.,.],[.,[.,[.,[.,.]]]]],[.,.]]
=> ? = 1 + 2
[7,6,4,4,3,2]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [[[.,.],[.,[.,[.,.]]]],[.,[.,.]]]
=> ? = 1 + 2
[7,6,5,4,3,1]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 0 + 2
[6,5,5,4,3,1]
=> [1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [[[[[.,.],.],[.,[.,[.,.]]]],.],.]
=> ? = 0 + 2
[7,6,4,3,2,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> ? = 0 + 2
[7,6,5,2,2,1]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [[[.,.],.],[[.,.],[.,[.,[.,.]]]]]
=> ? = 1 + 2
[7,6,5,4,1,1]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [[.,.],[[.,.],[.,[.,[.,[.,.]]]]]]
=> ? = 1 + 2
[7,6,5,4,3]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]]
=> ? = 0 + 2
[7,6,5,4,1]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]]
=> ? = 0 + 2
[7,6,5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [.,[[[.,.],.],[.,[.,[.,[.,.]]]]]]
=> ? = 0 + 2
[8,7,6,5,4,3]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [.,[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ? = 0 + 2
Description
The number of leaf nodes in a binary tree.
Equivalently, the number of cherries [1] in the complete binary tree.
The number of binary trees of size $n$, at least $1$, with exactly one leaf node for is $2^{n-1}$, see [2].
The number of binary tree of size $n$, at least $3$, with exactly two leaf nodes is $n(n+1)2^{n-2}$, see [3].
Matching statistic: St000196
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
St000196: Binary trees ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 100%
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
St000196: Binary trees ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 100%
Values
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [.,[[[.,.],.],[.,.]]]
=> 1 = 0 + 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [[[.,.],.],[.,.]]
=> 1 = 0 + 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [.,[.,[[[.,.],.],[.,.]]]]
=> 1 = 0 + 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [[.,.],[.,[.,.]]]
=> 1 = 0 + 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [.,[.,[[[[.,.],.],.],[.,.]]]]
=> 1 = 0 + 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [.,[[[[.,.],.],.],[.,.]]]
=> 1 = 0 + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[.,[[.,.],.]],[.,.]]
=> 1 = 0 + 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[.,.],[[.,.],[.,.]]]
=> 2 = 1 + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[.,.],[[.,[.,.]],.]]
=> 1 = 0 + 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [.,[.,[.,[[[[.,.],.],.],[.,.]]]]]
=> ? = 0 + 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [.,[.,[.,[[[.,.],.],[.,.]]]]]
=> 1 = 0 + 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [.,[[.,.],[[.,.],[.,.]]]]
=> 2 = 1 + 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [.,[[.,[[.,.],.]],[.,.]]]
=> 1 = 0 + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [.,[[.,.],[.,[.,.]]]]
=> 1 = 0 + 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[[[.,.],.],.],[.,.]]
=> 1 = 0 + 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[[[.,.],.],[.,.]],.]
=> 1 = 0 + 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [.,[.,[.,[[[[[.,.],.],.],.],[.,.]]]]]
=> ? = 0 + 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> [.,[.,[[[[[.,.],.],.],.],[.,.]]]]
=> ? = 0 + 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> [.,[[.,[[[.,.],.],.]],[.,.]]]
=> 1 = 0 + 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [.,[[.,.],[[[.,.],.],[.,.]]]]
=> 2 = 1 + 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [[.,[[[.,.],.],.]],[.,.]]
=> 1 = 0 + 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [[.,.],[[[.,.],.],[.,.]]]
=> 2 = 1 + 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [[.,[[[.,.],.],[.,.]]],.]
=> 1 = 0 + 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[[.,.],.],[.,[.,.]]]
=> 1 = 0 + 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[[.,.],[.,.]],[.,.]]
=> 2 = 1 + 1
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[[.,[.,.]],.],[.,.]]
=> 1 = 0 + 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[[.,.],[.,[.,.]]],.]
=> 1 = 0 + 1
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [[.,.],[[[.,[.,.]],.],.]]
=> 1 = 0 + 1
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0]
=> [.,[.,[.,[.,[[[[[.,.],.],.],.],[.,.]]]]]]
=> ? = 0 + 1
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0,0]
=> [.,[.,[.,[.,[[[[.,.],.],.],[.,.]]]]]]
=> ? = 0 + 1
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,1,1,0,0,0,0,0]
=> [.,[.,[[.,.],[[[.,.],.],[.,.]]]]]
=> ? = 1 + 1
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,1,1,0,0,0,0]
=> [.,[.,[[.,[[[.,.],.],.]],[.,.]]]]
=> ? = 0 + 1
[6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> 2 = 1 + 1
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [.,[[[[[.,.],.],.],.],[.,.]]]
=> 1 = 0 + 1
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [.,[.,[[.,[[.,.],.]],[.,.]]]]
=> 1 = 0 + 1
[5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [.,[[[.,.],.],[.,[.,.]]]]
=> 1 = 0 + 1
[5,3,1]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [[.,[.,[[.,.],.]]],[.,.]]
=> 1 = 0 + 1
[5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [.,[[[.,.],[.,.]],[.,.]]]
=> 2 = 1 + 1
[5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [[.,.],[.,[[.,.],[.,.]]]]
=> 2 = 1 + 1
[5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [.,[[[.,[.,.]],.],[.,.]]]
=> 1 = 0 + 1
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[.,.],[.,[.,[.,.]]]]
=> 1 = 0 + 1
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[.,[.,.]],[.,[.,.]]]
=> 1 = 0 + 1
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [.,[[[.,.],[.,[.,.]]],.]]
=> 1 = 0 + 1
[3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [[.,.],[.,[[.,[.,.]],.]]]
=> 1 = 0 + 1
[3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [[.,[.,.]],[.,[[.,.],.]]]
=> 1 = 0 + 1
[10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0]
=> [.,[.,[.,[.,[[[[[[.,.],.],.],.],.],[.,.]]]]]]
=> ? = 0 + 1
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [.,[.,[.,[[[[[[.,.],.],.],.],.],[.,.]]]]]
=> ? = 0 + 1
[8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,1,1,0,0,0,0]
=> [.,[.,[[.,[[[[.,.],.],.],.]],[.,.]]]]
=> ? = 0 + 1
[8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [.,[.,[[.,.],[[[[.,.],.],.],[.,.]]]]]
=> ? = 1 + 1
[7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,1,0,0,0]
=> [.,[[.,[[[[.,.],.],.],.]],[.,.]]]
=> ? = 0 + 1
[7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
=> [.,[.,[.,[.,[[[.,.],.],[.,.]]]]]]
=> ? = 0 + 1
[7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [.,[[.,.],[[[[.,.],.],.],[.,.]]]]
=> ? = 1 + 1
[6,4]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,1,0,0]
=> [[.,[.,[[[.,.],.],.]]],[.,.]]
=> 1 = 0 + 1
[6,3,1]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [.,[[[.,.],.],[[.,.],[.,.]]]]
=> 2 = 1 + 1
[6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [.,[[[.,.],[[.,.],.]],[.,.]]]
=> 2 = 1 + 1
[6,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,1,0,0,0]
=> [.,[[[.,[[.,.],.]],.],[.,.]]]
=> 1 = 0 + 1
[6,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [[.,.],[.,[[[.,.],.],[.,.]]]]
=> 2 = 1 + 1
[5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> [[.,[.,[[[.,.],.],[.,.]]]],.]
=> 1 = 0 + 1
[5,4,1]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [.,[.,[[.,.],[.,[.,.]]]]]
=> 1 = 0 + 1
[5,3,2]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> [[[.,[[.,.],.]],.],[.,.]]
=> 1 = 0 + 1
[5,3,1,1]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [[[.,.],[[.,.],.]],[.,.]]
=> 2 = 1 + 1
[5,2,2,1]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [[[.,.],.],[[.,.],[.,.]]]
=> 2 = 1 + 1
[4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> [[[.,[[.,.],.]],[.,.]],.]
=> 1 = 0 + 1
[4,4,1,1]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> 2 = 1 + 1
[7,4,1,1]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [.,[[.,.],[[.,.],[[.,.],[.,.]]]]]
=> ? = 2 + 1
[7,2,2,1,1,1]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [[.,.],[[.,.],[[[.,.],.],[.,.]]]]
=> ? = 2 + 1
[7,6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [.,[[.,.],[[[.,.],.],[.,[.,.]]]]]
=> ? = 1 + 1
[7,4,3,1]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [.,[[[.,.],.],[.,[[.,.],[.,.]]]]]
=> ? = 1 + 1
[7,4,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [[.,.],[.,[[.,.],[[.,.],[.,.]]]]]
=> ? = 2 + 1
[7,4,2,1,1,1]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [[.,.],[[[.,.],.],[[.,.],[.,.]]]]
=> ? = 2 + 1
[7,2,2,2,2,1]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [[[.,.],.],[.,[[[.,.],.],[.,.]]]]
=> ? = 1 + 1
[7,6,3,1]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [.,[[[.,.],.],[[.,.],[.,[.,.]]]]]
=> ? = 1 + 1
[7,6,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [[.,.],[.,[[[.,.],.],[.,[.,.]]]]]
=> ? = 1 + 1
[7,4,3,1,1,1]
=> [1,1,0,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [[.,.],[[.,.],[.,[[.,.],[.,.]]]]]
=> ? = 2 + 1
[7,6,4,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> ? = 0 + 1
[6,6,5,3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> [[[.,[.,[.,[.,.]]]],[.,[.,.]]],.]
=> ? = 0 + 1
[6,6,5,4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [[[.,[.,[.,.]]],[.,[.,[.,.]]]],.]
=> ? = 0 + 1
[7,6,5,4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 0 + 1
[6,6,5,4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [[[.,[.,.]],[.,[.,[.,[.,.]]]]],.]
=> ? = 0 + 1
[6,6,5,4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [[[[.,[.,.]],.],[.,[.,[.,.]]]],.]
=> ? = 0 + 1
[6,5,5,4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [[[[[.,[.,.]],.],[.,[.,.]]],.],.]
=> ? = 0 + 1
[7,6,5,4,3,2]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ? = 0 + 1
[6,6,5,4,3,2]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [[[.,.],[.,[.,[.,[.,[.,.]]]]]],.]
=> ? = 0 + 1
[7,5,5,4,3,2]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [[[.,.],[.,[.,[.,[.,.]]]]],[.,.]]
=> ? = 1 + 1
[7,6,4,4,3,2]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [[[.,.],[.,[.,[.,.]]]],[.,[.,.]]]
=> ? = 1 + 1
[7,6,5,4,3,1]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 0 + 1
[6,5,5,4,3,1]
=> [1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [[[[[.,.],.],[.,[.,[.,.]]]],.],.]
=> ? = 0 + 1
[7,6,4,3,2,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> ? = 0 + 1
[7,5,4,3,2,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> ? = 0 + 1
[7,6,5,2,2,1]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [[[.,.],.],[[.,.],[.,[.,[.,.]]]]]
=> ? = 1 + 1
[7,6,5,4,1,1]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [[.,.],[[.,.],[.,[.,[.,[.,.]]]]]]
=> ? = 1 + 1
[7,6,5,4,3]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]]
=> ? = 0 + 1
[7,6,5,4,1]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]]
=> ? = 0 + 1
[7,6,5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [.,[[[.,.],.],[.,[.,[.,[.,.]]]]]]
=> ? = 0 + 1
[8,7,6,5,4,3,2]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ? = 0 + 1
[8,6,5,4,3,2,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [[[[[[[[.,.],.],.],.],.],.],.],[.,.]]
=> ? = 0 + 1
[9,8,7,6,5,4,3,2]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ? = 0 + 1
[8,7,6,5,4,3]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [.,[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ? = 0 + 1
[9,7,6,5,4,3,2,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [[[[[[[[[.,.],.],.],.],.],.],.],.],[.,.]]
=> ? = 0 + 1
Description
The number of occurrences of the contiguous pattern {{{[[.,.],[.,.]]}}} in a binary tree.
Equivalently, this is the number of branches in the tree, i.e. the number of nodes with two children. Binary trees avoiding this pattern are counted by $2^{n-2}$.
Matching statistic: St000068
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St000068: Posets ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 100%
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St000068: Posets ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 100%
Values
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> ([(0,2),(0,3),(1,5),(2,6),(3,5),(3,6),(5,4),(6,4)],7)
=> 2 = 0 + 2
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[3,3,3,3],[2]]
=> ([(0,3),(0,4),(1,7),(2,8),(3,2),(3,9),(4,7),(4,9),(6,5),(7,6),(8,5),(9,6),(9,8)],10)
=> ? = 0 + 2
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [[4,4,4,4],[3]]
=> ([(0,3),(0,4),(1,10),(2,7),(3,2),(3,12),(4,5),(4,12),(5,10),(5,11),(7,8),(8,6),(9,6),(10,9),(11,8),(11,9),(12,7),(12,11)],13)
=> ? = 0 + 2
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[4,4,4],[3]]
=> ([(0,2),(0,3),(1,7),(2,8),(3,4),(3,8),(4,6),(4,7),(6,5),(7,5),(8,6)],9)
=> ? = 0 + 2
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> ([(0,4),(1,2),(1,3),(2,5),(3,4),(3,5)],6)
=> 2 = 0 + 2
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> ([(0,4),(1,2),(1,3),(2,5),(3,4),(3,5)],6)
=> 2 = 0 + 2
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [[4,4,4,4,4],[3]]
=> ([(0,4),(0,5),(1,12),(2,3),(2,16),(3,9),(4,6),(4,14),(5,2),(5,14),(6,12),(6,15),(8,10),(9,11),(10,7),(11,7),(12,8),(13,10),(13,11),(14,15),(14,16),(15,8),(15,13),(16,9),(16,13)],17)
=> ? = 0 + 2
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [[3,3,3,3,3],[2]]
=> ([(0,4),(0,5),(1,10),(2,3),(2,12),(3,9),(4,2),(4,11),(5,10),(5,11),(7,8),(8,6),(9,6),(10,7),(11,7),(11,12),(12,8),(12,9)],13)
=> ? = 0 + 2
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [[3,3,3,3],[2,1]]
=> ([(0,7),(1,3),(1,8),(2,7),(2,8),(3,6),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 1 + 2
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [[3,3,3,2],[2]]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,8),(4,7),(4,8),(7,5),(8,5),(8,6)],9)
=> ? = 0 + 2
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> ([(0,3),(1,2),(1,4),(2,5),(3,4),(4,5)],6)
=> 2 = 0 + 2
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2 = 0 + 2
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> 2 = 0 + 2
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [[5,5,5,5,5],[4]]
=> ([(0,5),(0,6),(1,14),(2,4),(2,19),(3,7),(3,20),(4,10),(5,2),(5,16),(6,3),(6,16),(7,14),(7,18),(9,12),(10,11),(11,13),(12,8),(13,8),(14,9),(15,11),(15,17),(16,19),(16,20),(17,12),(17,13),(18,9),(18,17),(19,10),(19,15),(20,15),(20,18)],21)
=> ? = 0 + 2
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [[5,5,5,5],[4]]
=> ([(0,4),(0,5),(1,11),(2,12),(3,6),(3,15),(4,2),(4,13),(5,3),(5,13),(6,11),(6,14),(8,9),(9,7),(10,7),(11,10),(12,8),(13,12),(13,15),(14,9),(14,10),(15,8),(15,14)],16)
=> ? = 0 + 2
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [[4,4,4,3],[3]]
=> ([(0,3),(0,4),(1,9),(2,8),(3,2),(3,11),(4,5),(4,11),(5,9),(5,10),(8,7),(9,6),(10,6),(10,7),(11,8),(11,10)],12)
=> ? = 0 + 2
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [[4,4,4,4],[3,1]]
=> ([(0,9),(1,4),(1,11),(2,3),(2,11),(3,8),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7),(11,8),(11,10)],12)
=> ? = 1 + 2
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [[4,4,3],[3]]
=> ([(0,7),(1,3),(1,4),(2,6),(2,7),(3,5),(4,2),(4,5),(5,6)],8)
=> ? = 0 + 2
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [[4,4,4],[3,1]]
=> ([(0,7),(1,6),(2,3),(2,7),(3,5),(3,6),(5,4),(6,4),(7,5)],8)
=> ? = 1 + 2
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [[4,3,3],[2]]
=> ([(0,4),(0,7),(1,2),(1,3),(2,5),(3,5),(3,7),(5,6),(7,6)],8)
=> ? = 0 + 2
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2 = 0 + 2
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 3 = 1 + 2
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> 2 = 0 + 2
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> 2 = 0 + 2
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [[4,4,2],[1,1]]
=> ([(0,7),(1,3),(1,4),(2,6),(3,5),(3,7),(4,2),(4,5),(5,6)],8)
=> ? = 0 + 2
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [[5,5,5,5,5,5],[4]]
=> ([(0,6),(0,7),(1,16),(2,4),(2,24),(3,8),(3,25),(4,5),(4,22),(5,10),(6,2),(6,21),(7,3),(7,21),(8,16),(8,23),(10,12),(11,13),(12,14),(13,15),(14,9),(15,9),(16,11),(17,12),(17,18),(18,14),(18,15),(19,13),(19,18),(20,17),(20,19),(21,24),(21,25),(22,10),(22,17),(23,11),(23,19),(24,20),(24,22),(25,20),(25,23)],26)
=> ? = 0 + 2
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [[4,4,4,4,4,4],[3]]
=> ([(0,5),(0,6),(1,13),(2,4),(2,18),(3,14),(4,3),(4,20),(5,2),(5,15),(6,7),(6,15),(7,13),(7,19),(9,10),(10,11),(11,8),(12,8),(13,9),(14,12),(15,18),(15,19),(16,11),(16,12),(17,10),(17,16),(18,17),(18,20),(19,9),(19,17),(20,14),(20,16)],21)
=> ? = 0 + 2
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [[4,4,4,4,4],[3,1]]
=> ([(0,11),(1,4),(1,15),(2,5),(2,15),(3,10),(4,3),(4,12),(5,11),(5,14),(7,8),(8,6),(9,6),(10,9),(11,7),(12,10),(12,13),(13,8),(13,9),(14,7),(14,13),(15,12),(15,14)],16)
=> ? = 1 + 2
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [[4,4,4,4,3],[3]]
=> ([(0,4),(0,5),(1,11),(2,3),(2,15),(3,10),(4,6),(4,12),(5,2),(5,12),(6,11),(6,14),(9,7),(10,8),(11,9),(12,14),(12,15),(13,7),(13,8),(14,9),(14,13),(15,10),(15,13)],16)
=> ? = 0 + 2
[6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [[3,3,3,3,3],[2,1]]
=> ([(0,9),(1,4),(1,11),(2,9),(2,11),(3,8),(4,3),(4,10),(6,7),(7,5),(8,5),(9,6),(10,7),(10,8),(11,6),(11,10)],12)
=> ? = 1 + 2
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [[5,5,5],[4]]
=> ([(0,3),(0,4),(1,9),(2,5),(2,7),(3,10),(4,2),(4,10),(5,8),(5,9),(7,8),(8,6),(9,6),(10,7)],11)
=> ? = 0 + 2
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [[3,3,3,3,2],[2]]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,6),(4,2),(4,10),(5,9),(5,10),(8,7),(9,8),(10,8),(10,11),(11,6),(11,7)],12)
=> ? = 0 + 2
[5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [[3,3,3,3],[2,2]]
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(3,6),(4,6),(5,7),(6,7)],8)
=> ? = 0 + 2
[5,3,1]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [[3,3,2,2],[2]]
=> ([(0,7),(1,3),(1,4),(2,6),(3,5),(3,7),(4,2),(4,5),(5,6)],8)
=> ? = 0 + 2
[5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [[3,3,3,2],[2,1]]
=> ([(0,6),(1,3),(1,7),(2,6),(2,7),(3,5),(6,4),(7,4),(7,5)],8)
=> ? = 1 + 2
[5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [[3,3,3,3],[2,1,1]]
=> ([(0,7),(1,6),(2,3),(2,7),(3,5),(3,6),(5,4),(6,4),(7,5)],8)
=> ? = 1 + 2
[5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3,1],[2]]
=> ([(0,7),(1,3),(1,4),(3,5),(3,7),(4,2),(4,5),(5,6),(7,6)],8)
=> ? = 0 + 2
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2 = 0 + 2
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> 2 = 0 + 2
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [[3,3,2,2],[1,1]]
=> ([(0,4),(0,7),(1,2),(1,3),(2,5),(3,5),(3,7),(4,6),(7,6)],8)
=> ? = 0 + 2
[3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [[3,3,3,2],[1,1,1]]
=> ([(0,7),(1,3),(1,4),(2,6),(2,7),(3,5),(4,2),(4,5),(5,6)],8)
=> ? = 0 + 2
[3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3,1],[1,1]]
=> ([(0,4),(0,7),(1,2),(1,3),(2,5),(3,5),(3,7),(5,6),(7,6)],8)
=> ? = 0 + 2
[10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [[6,6,6,6,6,6],[5]]
=> ([(0,7),(0,8),(1,17),(2,5),(2,25),(3,6),(3,24),(4,18),(5,9),(5,29),(6,4),(6,30),(7,2),(7,26),(8,3),(8,26),(9,17),(9,28),(11,13),(12,14),(13,15),(14,16),(15,10),(16,10),(17,11),(18,12),(19,23),(19,27),(20,14),(20,21),(21,15),(21,16),(22,13),(22,21),(23,20),(23,22),(24,19),(24,30),(25,19),(25,29),(26,24),(26,25),(27,12),(27,20),(28,11),(28,22),(29,23),(29,28),(30,18),(30,27)],31)
=> ? = 0 + 2
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [[6,6,6,6,6],[5]]
=> ([(0,6),(0,7),(1,16),(2,5),(2,23),(3,4),(3,24),(4,11),(5,8),(5,20),(6,2),(6,21),(7,3),(7,21),(8,16),(8,22),(10,15),(11,12),(12,13),(13,14),(14,9),(15,9),(16,10),(17,13),(17,18),(18,14),(18,15),(19,12),(19,17),(20,17),(20,22),(21,23),(21,24),(22,10),(22,18),(23,19),(23,20),(24,11),(24,19)],25)
=> ? = 0 + 2
[8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [[5,5,5,5,4],[4]]
=> ([(0,5),(0,6),(1,13),(2,4),(2,18),(3,7),(3,19),(4,10),(5,2),(5,14),(6,3),(6,14),(7,13),(7,17),(10,11),(11,8),(12,9),(13,12),(14,18),(14,19),(15,11),(15,16),(16,8),(16,9),(17,12),(17,16),(18,10),(18,15),(19,15),(19,17)],20)
=> ? = 0 + 2
[8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [[5,5,5,5,5],[4,1]]
=> ([(0,13),(1,4),(1,19),(2,5),(2,19),(3,9),(4,3),(4,17),(5,6),(5,18),(6,13),(6,16),(8,11),(9,10),(10,12),(11,7),(12,7),(13,8),(14,10),(14,15),(15,11),(15,12),(16,8),(16,15),(17,9),(17,14),(18,14),(18,16),(19,17),(19,18)],20)
=> ? = 1 + 2
[7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [[5,5,5,4],[4]]
=> ([(0,4),(0,5),(1,10),(2,11),(3,6),(3,14),(4,2),(4,12),(5,3),(5,12),(6,10),(6,13),(9,7),(10,8),(11,9),(12,11),(12,14),(13,7),(13,8),(14,9),(14,13)],15)
=> ? = 0 + 2
[7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [[3,3,3,3,3,3],[2]]
=> ([(0,5),(0,6),(1,11),(2,4),(2,13),(3,12),(4,3),(4,15),(5,2),(5,14),(6,11),(6,14),(8,9),(9,10),(10,7),(11,8),(12,7),(13,9),(13,15),(14,8),(14,13),(15,10),(15,12)],16)
=> ? = 0 + 2
[7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [[5,5,5,5],[4,1]]
=> ([(0,11),(1,3),(1,14),(2,4),(2,14),(3,10),(4,5),(4,13),(5,11),(5,12),(7,8),(8,6),(9,6),(10,7),(11,9),(12,8),(12,9),(13,7),(13,12),(14,10),(14,13)],15)
=> ? = 1 + 2
[6,4]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [[4,4,3,3],[3]]
=> ([(0,3),(0,4),(1,7),(2,8),(3,2),(3,10),(4,5),(4,10),(5,7),(5,9),(8,6),(9,6),(10,8),(10,9)],11)
=> ? = 0 + 2
[6,3,1]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [[4,4,4,4],[3,2]]
=> ([(0,3),(0,4),(1,8),(2,8),(2,10),(3,9),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 1 + 2
[6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [[4,4,4,3],[3,1]]
=> ([(0,8),(1,4),(1,10),(2,3),(2,10),(3,7),(4,8),(4,9),(7,6),(8,5),(9,5),(9,6),(10,7),(10,9)],11)
=> ? = 1 + 2
[6,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [[4,4,4,2],[3]]
=> ([(0,3),(0,4),(1,9),(2,6),(3,2),(3,10),(4,5),(4,10),(5,8),(5,9),(8,7),(9,7),(10,6),(10,8)],11)
=> ? = 0 + 2
[6,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [[4,4,4,4],[3,1,1]]
=> ([(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 1 + 2
[5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [[4,3,3,3],[2]]
=> ([(0,4),(0,5),(1,2),(1,8),(3,9),(4,3),(4,10),(5,8),(5,10),(7,6),(8,7),(9,6),(10,7),(10,9)],11)
=> ? = 0 + 2
[5,4,1]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [[2,2,2,2,2],[1,1]]
=> ([(0,3),(1,4),(1,7),(2,6),(3,7),(4,2),(4,5),(5,6),(7,5)],8)
=> ? = 0 + 2
[5,3,2]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [[4,4,2],[3]]
=> ([(0,6),(1,3),(1,4),(2,6),(3,5),(4,2),(4,5)],7)
=> 2 = 0 + 2
[5,3,1,1]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [[4,4,3],[3,1]]
=> ([(0,4),(1,5),(2,3),(2,4),(3,5),(3,6),(4,6)],7)
=> 3 = 1 + 2
[5,2,2,1]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [[4,4,4],[3,2]]
=> ([(0,4),(1,4),(1,6),(2,3),(3,6),(4,5),(6,5)],7)
=> 3 = 1 + 2
[4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [[4,3,2],[2]]
=> ([(0,4),(0,6),(1,2),(1,3),(2,5),(3,5),(3,6)],7)
=> 2 = 0 + 2
[4,4,1,1]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [[4,3,3],[2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(2,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
[4,2,2,2]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [[4,3,2],[1,1]]
=> ([(0,6),(1,3),(1,4),(3,5),(3,6),(4,2),(4,5)],7)
=> 2 = 0 + 2
[4,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [[4,4,4,1],[2]]
=> ([(0,4),(0,5),(1,3),(1,10),(3,9),(4,2),(4,8),(5,8),(5,10),(7,6),(8,7),(9,6),(10,7),(10,9)],11)
=> ? = 0 + 2
[3,3,3,1]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> [[4,4,3],[2,2]]
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(3,6),(4,6)],7)
=> 2 = 0 + 2
[3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [[4,4,2],[2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(2,5),(3,6),(5,6)],7)
=> 3 = 1 + 2
[3,3,2,1,1]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1],[2]]
=> ([(0,4),(0,6),(1,2),(1,3),(3,6),(4,5),(6,5)],7)
=> 2 = 0 + 2
[2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [[4,4,4,2],[1,1,1]]
=> ([(0,3),(0,4),(1,7),(2,8),(3,2),(3,10),(4,5),(4,10),(5,7),(5,9),(8,6),(9,6),(10,8),(10,9)],11)
=> ? = 0 + 2
[2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [[4,4,4,1],[1,1]]
=> ([(0,4),(0,5),(1,2),(1,8),(3,9),(4,3),(4,10),(5,8),(5,10),(7,6),(8,7),(9,6),(10,7),(10,9)],11)
=> ? = 0 + 2
[6,5]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [[3,3,3,3,3],[2,2]]
=> ([(0,3),(1,4),(1,5),(2,8),(3,9),(4,2),(4,10),(5,9),(5,10),(7,6),(8,6),(9,7),(10,7),(10,8)],11)
=> ? = 0 + 2
[6,3,2]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [[3,3,3,3,3],[2,1,1]]
=> ([(0,9),(1,4),(1,9),(2,3),(2,10),(3,8),(4,7),(4,10),(6,5),(7,6),(8,5),(9,7),(10,6),(10,8)],11)
=> ? = 1 + 2
[6,2,1,1,1]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [[5,5,5],[4,1]]
=> ([(0,9),(1,8),(2,3),(2,9),(3,4),(3,7),(4,6),(4,8),(6,5),(7,6),(8,5),(9,7)],10)
=> ? = 1 + 2
[5,4,2]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> [[3,3,3,2],[2,2]]
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(3,6),(4,6)],7)
=> 2 = 0 + 2
[5,4,1,1]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [[3,3,3,3],[2,2,1]]
=> ([(0,4),(1,4),(1,6),(2,3),(3,6),(4,5),(6,5)],7)
=> 3 = 1 + 2
[5,3,3]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [[3,3,2,2],[2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(2,5),(3,6),(5,6)],7)
=> 3 = 1 + 2
[5,3,2,1]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [[5,5],[4]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> 2 = 0 + 2
[5,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2,1],[2]]
=> ([(0,6),(1,3),(1,4),(3,5),(3,6),(4,2),(4,5)],7)
=> 2 = 0 + 2
[5,2,2,2]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [[3,3,3,2],[2,1,1]]
=> ([(0,4),(1,5),(2,3),(2,4),(3,5),(3,6),(4,6)],7)
=> 3 = 1 + 2
[5,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(2,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
[4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [[3,2,2,2],[1,1]]
=> ([(0,4),(0,6),(1,2),(1,3),(3,6),(4,5),(6,5)],7)
=> 2 = 0 + 2
[4,4,2,1]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [[5,4],[3]]
=> ([(0,4),(1,3),(1,5),(2,5),(4,2)],6)
=> 2 = 0 + 2
[4,3,3,1]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [[5,3],[2]]
=> ([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
=> 2 = 0 + 2
[3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [[3,3,2,2],[1,1,1]]
=> ([(0,6),(1,3),(1,4),(2,6),(3,5),(4,2),(4,5)],7)
=> 2 = 0 + 2
[3,3,3,1,1]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> ([(0,4),(0,6),(1,2),(1,3),(2,5),(3,5),(3,6)],7)
=> 2 = 0 + 2
[6,3,1,1,1]
=> [1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> [[4,4,4,4],[3,2,1]]
=> ([(0,8),(1,7),(2,7),(2,9),(3,8),(3,9),(5,4),(6,4),(7,5),(8,6),(9,5),(9,6)],10)
=> 4 = 2 + 2
[5,4,3]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,2],[1,1,1]]
=> ([(0,3),(0,6),(1,4),(2,6),(3,5),(4,2),(6,5)],7)
=> 2 = 0 + 2
[5,4,2,1]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [[4,4,4],[3,3]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> 2 = 0 + 2
[5,4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1]]
=> ([(0,3),(1,4),(1,6),(3,6),(4,2),(4,5),(6,5)],7)
=> 2 = 0 + 2
[5,3,3,1]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> [[4,4,3],[3,2]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> 3 = 1 + 2
[5,3,2,2]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> [[4,4,2],[3,1]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 3 = 1 + 2
[5,3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [[4,4,1],[3]]
=> ([(0,5),(1,2),(1,4),(3,5),(4,3)],6)
=> 2 = 0 + 2
[4,4,3,1]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> [[4,3,3],[2,2]]
=> ([(0,4),(1,2),(1,3),(3,5),(4,5)],6)
=> 2 = 0 + 2
[4,4,2,2]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [[4,3,2],[2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
=> 3 = 1 + 2
[4,4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> ([(0,4),(0,5),(1,2),(1,3),(3,5)],6)
=> 2 = 0 + 2
[4,3,3,2]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> [[4,2,2],[1,1]]
=> ([(0,5),(1,3),(1,4),(3,5),(4,2)],6)
=> 2 = 0 + 2
[5,4,3,1]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [[3,3,3,3],[2,2,2]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> 2 = 0 + 2
[5,4,2,2]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [[3,3,3,2],[2,2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> 3 = 1 + 2
[5,4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3,1],[2,2]]
=> ([(0,4),(1,2),(1,3),(3,5),(4,5)],6)
=> 2 = 0 + 2
Description
The number of minimal elements in a poset.
Matching statistic: St000023
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000023: Permutations ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 67%
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000023: Permutations ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 67%
Values
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => 1 = 0 + 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 1 = 0 + 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [3,4,2,1,5,6] => 1 = 0 + 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1 = 0 + 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [4,5,3,2,1,6,7] => ? = 0 + 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [4,5,3,2,1,6] => 1 = 0 + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => 1 = 0 + 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 2 = 1 + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 1 = 0 + 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [4,5,3,2,1,6,7,8] => ? = 0 + 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [3,4,2,1,5,6,7] => ? = 0 + 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,1,6] => 2 = 1 + 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [4,5,2,1,3,6] => 1 = 0 + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => 1 = 0 + 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => 1 = 0 + 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => 1 = 0 + 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [5,6,4,3,2,1,7,8,9] => ? = 0 + 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> [5,6,4,3,2,1,7,8] => ? = 0 + 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> [5,6,3,2,1,4,7] => ? = 0 + 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [4,5,3,2,6,1,7] => ? = 1 + 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [5,6,3,2,1,4] => 1 = 0 + 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [4,5,3,2,6,1] => 2 = 1 + 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [6,3,4,2,1,5] => 1 = 0 + 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 1 = 0 + 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => 2 = 1 + 1
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => 1 = 0 + 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 1 = 0 + 1
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [5,4,2,3,6,1] => 1 = 0 + 1
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0]
=> [5,6,4,3,2,1,7,8,9,10] => ? = 0 + 1
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0,0]
=> [4,5,3,2,1,6,7,8,9] => ? = 0 + 1
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,1,1,0,0,0,0,0]
=> [4,5,3,2,6,1,7,8] => ? = 1 + 1
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,1,1,0,0,0,0]
=> [5,6,3,2,1,4,7,8] => ? = 0 + 1
[6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,1,6,7] => ? = 1 + 1
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [5,6,4,3,2,1,7] => ? = 0 + 1
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [4,5,2,1,3,6,7] => ? = 0 + 1
[5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [3,4,5,2,1,6] => 1 = 0 + 1
[5,3,1]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [5,6,2,1,3,4] => 1 = 0 + 1
[5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [4,5,2,3,1,6] => 2 = 1 + 1
[5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => 2 = 1 + 1
[5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [4,5,3,1,2,6] => 1 = 0 + 1
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1 = 0 + 1
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 1 = 0 + 1
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [5,2,3,4,1,6] => 1 = 0 + 1
[3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => 1 = 0 + 1
[3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [4,3,5,6,1,2] => 1 = 0 + 1
[10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0]
=> [6,7,5,4,3,2,1,8,9,10,11] => ? = 0 + 1
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [6,7,5,4,3,2,1,8,9,10] => ? = 0 + 1
[8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,1,1,0,0,0,0]
=> [6,7,4,3,2,1,5,8,9] => ? = 0 + 1
[8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [5,6,4,3,2,7,1,8,9] => ? = 1 + 1
[7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,1,0,0,0]
=> [6,7,4,3,2,1,5,8] => ? = 0 + 1
[7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
=> [3,4,2,1,5,6,7,8] => ? = 0 + 1
[7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [5,6,4,3,2,7,1,8] => ? = 1 + 1
[6,4]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,1,0,0]
=> [6,7,3,2,1,4,5] => ? = 0 + 1
[6,3,1]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [4,5,3,6,2,1,7] => ? = 1 + 1
[6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [5,6,3,2,4,1,7] => ? = 1 + 1
[6,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,1,0,0,0]
=> [5,6,4,2,1,3,7] => ? = 0 + 1
[6,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [4,5,3,2,6,7,1] => ? = 1 + 1
[5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> [7,3,4,2,1,5,6] => ? = 0 + 1
[5,4,1]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [2,3,4,1,5,6] => 1 = 0 + 1
[5,3,2]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> [5,6,4,2,1,3] => 1 = 0 + 1
[5,3,1,1]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [5,6,3,2,4,1] => 2 = 1 + 1
[5,2,2,1]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [4,5,3,6,2,1] => 2 = 1 + 1
[4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> [6,4,5,2,1,3] => 1 = 0 + 1
[4,4,1,1]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [6,3,4,2,5,1] => 2 = 1 + 1
[4,2,2,2]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [6,4,2,3,5,1] => 1 = 0 + 1
[4,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> [5,4,6,3,1,2,7] => ? = 0 + 1
[3,3,3,1]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [5,3,4,6,2,1] => 1 = 0 + 1
[3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,6,2,3,1] => 2 = 1 + 1
[3,3,2,1,1]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> [5,4,6,3,1,2] => 1 = 0 + 1
[2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [5,4,2,3,6,7,1] => ? = 0 + 1
[2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> [5,4,3,6,7,1,2] => ? = 0 + 1
[6,5]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [3,4,5,2,1,6,7] => ? = 0 + 1
[6,3,2]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,6,1,7] => ? = 1 + 1
[6,2,1,1,1]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [5,6,4,3,2,7,1] => ? = 1 + 1
[5,4,2]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> [4,5,6,2,1,3] => 1 = 0 + 1
[5,4,1,1]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [3,4,5,2,6,1] => 2 = 1 + 1
[5,3,3]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [5,6,2,3,1,4] => 2 = 1 + 1
[5,3,2,1]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,2,1] => 1 = 0 + 1
[5,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [5,6,3,1,2,4] => 1 = 0 + 1
[5,2,2,2]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [4,5,2,3,6,1] => 2 = 1 + 1
[5,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [4,5,3,6,1,2] => 2 = 1 + 1
[4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,1,5] => 1 = 0 + 1
[4,4,2,1]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,2,1] => 1 = 0 + 1
[6,5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,2,1,7] => ? = 0 + 1
[6,3,1,1,1]
=> [1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [4,5,3,6,2,7,1] => ? = 2 + 1
[7,4,1,1]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [4,5,3,6,2,7,1,8] => ? = 2 + 1
[6,5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [3,4,5,2,6,1,7] => ? = 1 + 1
[6,3,2,1,1]
=> [1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,6,7,1] => ? = 1 + 1
[6,2,2,2,1]
=> [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [5,6,4,3,7,2,1] => ? = 1 + 1
[7,2,2,1,1,1]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [5,6,4,3,7,2,8,1] => ? = 2 + 1
[6,5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [2,3,4,1,5,6,7] => ? = 0 + 1
[6,5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,2,7,1] => ? = 1 + 1
[6,3,2,2,1]
=> [1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [4,5,3,6,7,2,1] => ? = 1 + 1
[7,6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [4,5,6,3,2,7,1,8] => ? = 1 + 1
[7,4,3,1]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [4,5,3,6,7,2,1,8] => ? = 1 + 1
[7,4,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [4,5,3,6,2,7,8,1] => ? = 2 + 1
[6,5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,2,1,7] => ? = 0 + 1
[6,5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [3,4,5,2,6,7,1] => ? = 1 + 1
[6,3,3,2,1]
=> [1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [5,6,4,7,3,2,1] => ? = 1 + 1
[7,4,2,1,1,1]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [5,6,4,7,3,2,8,1] => ? = 2 + 1
Description
The number of inner peaks of a permutation.
The number of peaks including the boundary is [[St000092]].
The following 21 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000779The tier of a permutation. St000099The number of valleys of a permutation, including the boundary. St000659The number of rises of length at least 2 of a Dyck path. St000035The number of left outer peaks of a permutation. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St000054The first entry of the permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000891The number of distinct diagonal sums of a permutation matrix. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St001044The number of pairs whose larger element is at most one more than half the size of the perfect matching. St000052The number of valleys of a Dyck path not on the x-axis. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000366The number of double descents of a permutation. St000534The number of 2-rises of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000669The number of permutations obtained by switching ascents or descents of size 2. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St000022The number of fixed points of a permutation. St000405The number of occurrences of the pattern 1324 in a permutation. St000842The breadth of a permutation.
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