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Your data matches 120 different statistics following compositions of up to 3 maps.
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Matching statistic: St000160
St000160: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 1
[2]
=> 1
[1,1]
=> 2
[3]
=> 1
[2,1]
=> 1
[1,1,1]
=> 3
[4]
=> 1
[3,1]
=> 1
[2,2]
=> 2
[2,1,1]
=> 2
[1,1,1,1]
=> 4
[5]
=> 1
[4,1]
=> 1
[3,2]
=> 1
[3,1,1]
=> 2
[2,2,1]
=> 1
[2,1,1,1]
=> 3
[1,1,1,1,1]
=> 5
[6]
=> 1
[5,1]
=> 1
[4,2]
=> 1
[4,1,1]
=> 2
[3,3]
=> 2
[3,2,1]
=> 1
[3,1,1,1]
=> 3
[2,2,2]
=> 3
[2,2,1,1]
=> 2
[2,1,1,1,1]
=> 4
[1,1,1,1,1,1]
=> 6
[7]
=> 1
[6,1]
=> 1
[5,2]
=> 1
[5,1,1]
=> 2
[4,3]
=> 1
[4,2,1]
=> 1
[4,1,1,1]
=> 3
[3,3,1]
=> 1
[3,2,2]
=> 2
[3,2,1,1]
=> 2
[3,1,1,1,1]
=> 4
[2,2,2,1]
=> 1
[2,2,1,1,1]
=> 3
[2,1,1,1,1,1]
=> 5
[1,1,1,1,1,1,1]
=> 7
[8]
=> 1
[7,1]
=> 1
[6,2]
=> 1
[6,1,1]
=> 2
[5,3]
=> 1
[5,2,1]
=> 1
Description
The multiplicity of the smallest part of a partition.
This counts the number of occurrences of the smallest part $spt(\lambda)$ of a partition $\lambda$.
The sum $spt(n) = \sum_{\lambda \vdash n} spt(\lambda)$ satisfies the congruences
\begin{align*}
spt(5n+4) &\equiv 0\quad \pmod{5}\\\
spt(7n+5) &\equiv 0\quad \pmod{7}\\\
spt(13n+6) &\equiv 0\quad \pmod{13},
\end{align*}
analogous to those of the counting function of partitions, see [1] and [2].
Matching statistic: St000382
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 83% ●values known / values provided: 95%●distinct values known / distinct values provided: 83%
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 83% ●values known / values provided: 95%●distinct values known / distinct values provided: 83%
Values
[1]
=> [1,0,1,0]
=> [1,1] => [1,1] => 1
[2]
=> [1,1,0,0,1,0]
=> [2,1] => [1,2] => 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,2] => [2,1] => 2
[3]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => [1,3] => 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1] => [1,1,1] => 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => [3,1] => 3
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [1,4] => 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,1,1] => [1,1,2] => 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => [2,2] => 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,2,1] => [2,1,1] => 2
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1] => 4
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,1] => [1,5] => 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,1,1] => [1,1,3] => 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,1,2] => 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,1,1] => 2
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => [1,2,1] => 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [3,1,1] => 3
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,5] => [5,1] => 5
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,1] => [1,6] => 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [4,1,1] => [1,1,4] => 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,1] => [1,1,3] => 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [2,1,2] => 2
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [2,3] => 2
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,1,1,1] => 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [3,1,1] => 3
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [3,2] => 3
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [2,2,1] => 2
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,4,1] => [4,1,1] => 4
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,6] => [6,1] => 6
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7,1] => [1,7] => 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [5,1,1] => [1,1,5] => 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [4,1,1] => [1,1,4] => 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [3,2,1] => [2,1,3] => 2
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [1,1,3] => 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => [1,1,1,2] => 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [3,1,1] => 3
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => [1,2,2] => 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [2,1,2] => 2
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [2,1,1,1] => 2
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,4,1] => [4,1,1] => 4
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [1,3,1] => 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,3,2] => [3,2,1] => 3
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,5,1] => [5,1,1] => 5
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,7] => [7,1] => 7
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8,1] => [1,8] => 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [6,1,1] => [1,1,6] => 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [5,1,1] => [1,1,5] => 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [4,2,1] => [2,1,4] => 2
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [4,1,1] => [1,1,4] => 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [3,1,1,1] => [1,1,1,3] => 1
[10,1]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,1,1] => ? => ? ∊ {1,2,9,11}
[9,1,1]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> [7,2,1] => [2,1,7] => ? ∊ {1,2,9,11}
[2,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [1,9,1] => [9,1,1] => ? ∊ {1,2,9,11}
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> ? => ? => ? ∊ {1,2,9,11}
[12]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ? => ? ∊ {1,1,1,2,3,8,9,10,12}
[11,1]
=> [1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1,1] => ? => ? ∊ {1,1,1,2,3,8,9,10,12}
[10,2]
=> [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1,1] => ? => ? ∊ {1,1,1,2,3,8,9,10,12}
[10,1,1]
=> [1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [8,2,1] => ? => ? ∊ {1,1,1,2,3,8,9,10,12}
[9,1,1,1]
=> [1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [6,3,1] => [3,1,6] => ? ∊ {1,1,1,2,3,8,9,10,12}
[3,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
=> [1,9,1] => [9,1,1] => ? ∊ {1,1,1,2,3,8,9,10,12}
[2,2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0]
=> [1,8,2] => [8,2,1] => ? ∊ {1,1,1,2,3,8,9,10,12}
[2,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
=> [1,10,1] => [10,1,1] => ? ∊ {1,1,1,2,3,8,9,10,12}
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? => ? => ? ∊ {1,1,1,2,3,8,9,10,12}
Description
The first part of an integer composition.
Matching statistic: St001933
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001933: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 78%●distinct values known / distinct values provided: 75%
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001933: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 78%●distinct values known / distinct values provided: 75%
Values
[1]
=> []
=> ?
=> ?
=> ? = 1
[2]
=> []
=> ?
=> ?
=> ? ∊ {1,2}
[1,1]
=> [1]
=> [1]
=> []
=> ? ∊ {1,2}
[3]
=> []
=> ?
=> ?
=> ? ∊ {1,1,3}
[2,1]
=> [1]
=> [1]
=> []
=> ? ∊ {1,1,3}
[1,1,1]
=> [1,1]
=> [2]
=> []
=> ? ∊ {1,1,3}
[4]
=> []
=> ?
=> ?
=> ? ∊ {2,2,4}
[3,1]
=> [1]
=> [1]
=> []
=> ? ∊ {2,2,4}
[2,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[2,1,1]
=> [1,1]
=> [2]
=> []
=> ? ∊ {2,2,4}
[1,1,1,1]
=> [1,1,1]
=> [2,1]
=> [1]
=> 1
[5]
=> []
=> ?
=> ?
=> ? ∊ {1,1,3,5}
[4,1]
=> [1]
=> [1]
=> []
=> ? ∊ {1,1,3,5}
[3,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[3,1,1]
=> [1,1]
=> [2]
=> []
=> ? ∊ {1,1,3,5}
[2,2,1]
=> [2,1]
=> [1,1,1]
=> [1,1]
=> 2
[2,1,1,1]
=> [1,1,1]
=> [2,1]
=> [1]
=> 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> []
=> ? ∊ {1,1,3,5}
[6]
=> []
=> ?
=> ?
=> ? ∊ {1,2,3,4,6}
[5,1]
=> [1]
=> [1]
=> []
=> ? ∊ {1,2,3,4,6}
[4,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[4,1,1]
=> [1,1]
=> [2]
=> []
=> ? ∊ {1,2,3,4,6}
[3,3]
=> [3]
=> [3]
=> []
=> ? ∊ {1,2,3,4,6}
[3,2,1]
=> [2,1]
=> [1,1,1]
=> [1,1]
=> 2
[3,1,1,1]
=> [1,1,1]
=> [2,1]
=> [1]
=> 1
[2,2,2]
=> [2,2]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[2,2,1,1]
=> [2,1,1]
=> [2,1,1]
=> [1,1]
=> 2
[2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> []
=> ? ∊ {1,2,3,4,6}
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [4,1]
=> [1]
=> 1
[7]
=> []
=> ?
=> ?
=> ? ∊ {1,1,2,5,7}
[6,1]
=> [1]
=> [1]
=> []
=> ? ∊ {1,1,2,5,7}
[5,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[5,1,1]
=> [1,1]
=> [2]
=> []
=> ? ∊ {1,1,2,5,7}
[4,3]
=> [3]
=> [3]
=> []
=> ? ∊ {1,1,2,5,7}
[4,2,1]
=> [2,1]
=> [1,1,1]
=> [1,1]
=> 2
[4,1,1,1]
=> [1,1,1]
=> [2,1]
=> [1]
=> 1
[3,3,1]
=> [3,1]
=> [3,1]
=> [1]
=> 1
[3,2,2]
=> [2,2]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[3,2,1,1]
=> [2,1,1]
=> [2,1,1]
=> [1,1]
=> 2
[3,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> []
=> ? ∊ {1,1,2,5,7}
[2,2,2,1]
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 4
[2,2,1,1,1]
=> [2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 3
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [4,1]
=> [1]
=> 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [4,2]
=> [2]
=> 1
[8]
=> []
=> ?
=> ?
=> ? ∊ {2,2,4,6,8}
[7,1]
=> [1]
=> [1]
=> []
=> ? ∊ {2,2,4,6,8}
[6,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[6,1,1]
=> [1,1]
=> [2]
=> []
=> ? ∊ {2,2,4,6,8}
[5,3]
=> [3]
=> [3]
=> []
=> ? ∊ {2,2,4,6,8}
[5,2,1]
=> [2,1]
=> [1,1,1]
=> [1,1]
=> 2
[5,1,1,1]
=> [1,1,1]
=> [2,1]
=> [1]
=> 1
[4,4]
=> [4]
=> [2,2]
=> [2]
=> 1
[4,3,1]
=> [3,1]
=> [3,1]
=> [1]
=> 1
[4,2,2]
=> [2,2]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[4,2,1,1]
=> [2,1,1]
=> [2,1,1]
=> [1,1]
=> 2
[4,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> []
=> ? ∊ {2,2,4,6,8}
[3,3,2]
=> [3,2]
=> [3,1,1]
=> [1,1]
=> 2
[3,3,1,1]
=> [3,1,1]
=> [3,2]
=> [2]
=> 1
[3,2,2,1]
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 4
[3,2,1,1,1]
=> [2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 3
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [4,1]
=> [1]
=> 1
[2,2,2,2]
=> [2,2,2]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 5
[2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 4
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [4,1,1]
=> [1,1]
=> 2
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [4,2]
=> [2]
=> 1
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [4,2,1]
=> [2,1]
=> 1
[9]
=> []
=> ?
=> ?
=> ? ∊ {1,1,2,3,3,7,9}
[8,1]
=> [1]
=> [1]
=> []
=> ? ∊ {1,1,2,3,3,7,9}
[7,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[7,1,1]
=> [1,1]
=> [2]
=> []
=> ? ∊ {1,1,2,3,3,7,9}
[6,3]
=> [3]
=> [3]
=> []
=> ? ∊ {1,1,2,3,3,7,9}
[6,2,1]
=> [2,1]
=> [1,1,1]
=> [1,1]
=> 2
[6,1,1,1]
=> [1,1,1]
=> [2,1]
=> [1]
=> 1
[5,4]
=> [4]
=> [2,2]
=> [2]
=> 1
[5,3,1]
=> [3,1]
=> [3,1]
=> [1]
=> 1
[5,2,2]
=> [2,2]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[5,2,1,1]
=> [2,1,1]
=> [2,1,1]
=> [1,1]
=> 2
[5,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> []
=> ? ∊ {1,1,2,3,3,7,9}
[4,4,1]
=> [4,1]
=> [2,2,1]
=> [2,1]
=> 1
[4,3,2]
=> [3,2]
=> [3,1,1]
=> [1,1]
=> 2
[4,3,1,1]
=> [3,1,1]
=> [3,2]
=> [2]
=> 1
[4,2,2,1]
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 4
[4,2,1,1,1]
=> [2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 3
[3,3,3]
=> [3,3]
=> [6]
=> []
=> ? ∊ {1,1,2,3,3,7,9}
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [8]
=> []
=> ? ∊ {1,1,2,3,3,7,9}
[10]
=> []
=> ?
=> ?
=> ? ∊ {1,2,2,2,3,5,8,10}
[9,1]
=> [1]
=> [1]
=> []
=> ? ∊ {1,2,2,2,3,5,8,10}
[8,1,1]
=> [1,1]
=> [2]
=> []
=> ? ∊ {1,2,2,2,3,5,8,10}
[7,3]
=> [3]
=> [3]
=> []
=> ? ∊ {1,2,2,2,3,5,8,10}
[6,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> []
=> ? ∊ {1,2,2,2,3,5,8,10}
[5,5]
=> [5]
=> [5]
=> []
=> ? ∊ {1,2,2,2,3,5,8,10}
[4,3,3]
=> [3,3]
=> [6]
=> []
=> ? ∊ {1,2,2,2,3,5,8,10}
[2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [8]
=> []
=> ? ∊ {1,2,2,2,3,5,8,10}
[11]
=> []
=> ?
=> ?
=> ? ∊ {1,1,1,1,3,4,9,11}
[10,1]
=> [1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,3,4,9,11}
[9,1,1]
=> [1,1]
=> [2]
=> []
=> ? ∊ {1,1,1,1,3,4,9,11}
[8,3]
=> [3]
=> [3]
=> []
=> ? ∊ {1,1,1,1,3,4,9,11}
[7,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> []
=> ? ∊ {1,1,1,1,3,4,9,11}
[6,5]
=> [5]
=> [5]
=> []
=> ? ∊ {1,1,1,1,3,4,9,11}
[5,3,3]
=> [3,3]
=> [6]
=> []
=> ? ∊ {1,1,1,1,3,4,9,11}
Description
The largest multiplicity of a part in an integer partition.
Matching statistic: St000326
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00135: Binary words —rotate front-to-back⟶ Binary words
Mp00278: Binary words —rowmotion⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 83%
Mp00135: Binary words —rotate front-to-back⟶ Binary words
Mp00278: Binary words —rowmotion⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 83%
Values
[1]
=> 10 => 01 => 10 => 1
[2]
=> 100 => 001 => 010 => 2
[1,1]
=> 110 => 101 => 110 => 1
[3]
=> 1000 => 0001 => 0010 => 3
[2,1]
=> 1010 => 0101 => 1010 => 1
[1,1,1]
=> 1110 => 1101 => 1110 => 1
[4]
=> 10000 => 00001 => 00010 => 4
[3,1]
=> 10010 => 00101 => 01010 => 2
[2,2]
=> 1100 => 1001 => 0110 => 2
[2,1,1]
=> 10110 => 01101 => 10110 => 1
[1,1,1,1]
=> 11110 => 11101 => 11110 => 1
[5]
=> 100000 => 000001 => 000010 => 5
[4,1]
=> 100010 => 000101 => 001010 => 3
[3,2]
=> 10100 => 01001 => 10010 => 1
[3,1,1]
=> 100110 => 001101 => 010110 => 2
[2,2,1]
=> 11010 => 10101 => 11010 => 1
[2,1,1,1]
=> 101110 => 011101 => 101110 => 1
[1,1,1,1,1]
=> 111110 => 111101 => 111110 => 1
[6]
=> 1000000 => 0000001 => 0000010 => 6
[5,1]
=> 1000010 => 0000101 => 0001010 => 4
[4,2]
=> 100100 => 001001 => 010010 => 2
[4,1,1]
=> 1000110 => 0001101 => 0010110 => 3
[3,3]
=> 11000 => 10001 => 00110 => 3
[3,2,1]
=> 101010 => 010101 => 101010 => 1
[3,1,1,1]
=> 1001110 => 0011101 => 0101110 => 2
[2,2,2]
=> 11100 => 11001 => 01110 => 2
[2,2,1,1]
=> 110110 => 101101 => 110110 => 1
[2,1,1,1,1]
=> 1011110 => 0111101 => 1011110 => 1
[1,1,1,1,1,1]
=> 1111110 => 1111101 => 1111110 => 1
[7]
=> 10000000 => 00000001 => 00000010 => 7
[6,1]
=> 10000010 => 00000101 => 00001010 => 5
[5,2]
=> 1000100 => 0001001 => 0010010 => 3
[5,1,1]
=> 10000110 => 00001101 => 00010110 => 4
[4,3]
=> 101000 => 010001 => 100010 => 1
[4,2,1]
=> 1001010 => 0010101 => 0101010 => 2
[4,1,1,1]
=> 10001110 => 00011101 => 00101110 => 3
[3,3,1]
=> 110010 => 100101 => 011010 => 2
[3,2,2]
=> 101100 => 011001 => 100110 => 1
[3,2,1,1]
=> 1010110 => 0101101 => 1010110 => 1
[3,1,1,1,1]
=> 10011110 => 00111101 => 01011110 => 2
[2,2,2,1]
=> 111010 => 110101 => 111010 => 1
[2,2,1,1,1]
=> 1101110 => 1011101 => 1101110 => 1
[2,1,1,1,1,1]
=> 10111110 => 01111101 => 10111110 => 1
[1,1,1,1,1,1,1]
=> 11111110 => 11111101 => 11111110 => 1
[8]
=> 100000000 => 000000001 => 000000010 => 8
[7,1]
=> 100000010 => 000000101 => 000001010 => 6
[6,2]
=> 10000100 => 00001001 => 00010010 => 4
[6,1,1]
=> 100000110 => 000001101 => 000010110 => 5
[5,3]
=> 1001000 => 0010001 => 0100010 => 2
[5,2,1]
=> 10001010 => 00010101 => 00101010 => 3
[4,1,1,1,1,1]
=> 1000111110 => 0001111101 => 0010111110 => ? ∊ {2,3}
[3,1,1,1,1,1,1]
=> 1001111110 => 0011111101 => 0101111110 => ? ∊ {2,3}
[9,1]
=> 10000000010 => 00000000101 => 00000001010 => ? ∊ {2,2,3,4,5,6,7,8}
[8,1,1]
=> 10000000110 => 00000001101 => 00000010110 => ? ∊ {2,2,3,4,5,6,7,8}
[7,1,1,1]
=> 10000001110 => 00000011101 => 00000101110 => ? ∊ {2,2,3,4,5,6,7,8}
[6,1,1,1,1]
=> 10000011110 => 00000111101 => 00001011110 => ? ∊ {2,2,3,4,5,6,7,8}
[5,1,1,1,1,1]
=> 10000111110 => 00001111101 => 00010111110 => ? ∊ {2,2,3,4,5,6,7,8}
[4,2,1,1,1,1]
=> 1001011110 => 0010111101 => 0101011110 => ? ∊ {2,2,3,4,5,6,7,8}
[4,1,1,1,1,1,1]
=> 10001111110 => 00011111101 => 00101111110 => ? ∊ {2,2,3,4,5,6,7,8}
[3,1,1,1,1,1,1,1]
=> 10011111110 => 00111111101 => 01011111110 => ? ∊ {2,2,3,4,5,6,7,8}
[11]
=> 100000000000 => ? => ? => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[10,1]
=> 100000000010 => ? => ? => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[9,2]
=> 10000000100 => 00000001001 => ? => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[9,1,1]
=> 100000000110 => ? => ? => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[8,3]
=> 1000001000 => ? => ? => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[8,2,1]
=> 10000001010 => 00000010101 => ? => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[8,1,1,1]
=> 100000001110 => ? => ? => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[7,3,1]
=> 1000010010 => ? => ? => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[7,2,2]
=> 1000001100 => ? => ? => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[7,2,1,1]
=> 10000010110 => 00000101101 => ? => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[7,1,1,1,1]
=> 100000011110 => ? => ? => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[6,3,1,1]
=> 1000100110 => ? => ? => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[6,2,2,1]
=> 1000011010 => ? => ? => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[6,2,1,1,1]
=> 10000101110 => 00001011101 => ? => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[6,1,1,1,1,1]
=> 100000111110 => 000001111101 => 000010111110 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[5,3,1,1,1]
=> 1001001110 => 0010011101 => 0100101110 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[5,2,1,1,1,1]
=> 10001011110 => 00010111101 => ? => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[5,1,1,1,1,1,1]
=> 100001111110 => ? => ? => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[4,3,1,1,1,1]
=> 1010011110 => ? => ? => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[4,2,2,1,1,1]
=> 1001101110 => ? => ? => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[4,2,1,1,1,1,1]
=> 10010111110 => 00101111101 => ? => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[4,1,1,1,1,1,1,1]
=> 100011111110 => ? => ? => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[3,3,1,1,1,1,1]
=> 1100111110 => ? => ? => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[3,2,2,1,1,1,1]
=> 1011011110 => ? => ? => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[3,2,1,1,1,1,1,1]
=> 10101111110 => 01011111101 => ? => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[3,1,1,1,1,1,1,1,1]
=> 100111111110 => ? => ? => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[2,2,2,1,1,1,1,1]
=> 1110111110 => 1101111101 => ? => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[2,2,1,1,1,1,1,1,1]
=> 11011111110 => 10111111101 => ? => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? => ? => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[1,1,1,1,1,1,1,1,1,1,1]
=> 111111111110 => ? => ? => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[12]
=> 1000000000000 => ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12}
[11,1]
=> 1000000000010 => ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12}
[10,2]
=> 100000000100 => 000000001001 => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12}
[10,1,1]
=> 1000000000110 => ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12}
[9,3]
=> 10000001000 => ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12}
[9,2,1]
=> 100000001010 => ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12}
[9,1,1,1]
=> 1000000001110 => ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12}
[8,4]
=> 1000010000 => 0000100001 => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12}
[8,3,1]
=> 10000010010 => ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12}
[8,2,2]
=> 10000001100 => ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12}
Description
The position of the first one in a binary word after appending a 1 at the end.
Regarding the binary word as a subset of $\{1,\dots,n,n+1\}$ that contains $n+1$, this is the minimal element of the set.
Matching statistic: St001483
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001483: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 66%●distinct values known / distinct values provided: 50%
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001483: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 66%●distinct values known / distinct values provided: 50%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[2]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 2
[1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 5
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 4
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 3
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [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]
=> ? ∊ {1,7}
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> 5
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> 3
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> 4
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> 2
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? ∊ {1,7}
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [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]
=> ? ∊ {1,1,6,8}
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,6,8}
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> 4
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> 5
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 2
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> 3
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> 4
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 4
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [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]
=> ? ∊ {1,1,6,8}
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? ∊ {1,1,6,8}
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,5,6,7,9}
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,5,6,7,9}
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,5,6,7,9}
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,5,6,7,9}
[3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [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]
=> ? ∊ {1,1,1,1,5,6,7,9}
[2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,5,6,7,9}
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,5,6,7,9}
[1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? ∊ {1,1,1,1,5,6,7,9}
[10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,3,4,5,6,6,7,8,10}
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,3,4,5,6,6,7,8,10}
[8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,3,4,5,6,6,7,8,10}
[8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,3,4,5,6,6,7,8,10}
[7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,3,4,5,6,6,7,8,10}
[7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,3,4,5,6,6,7,8,10}
[7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,3,4,5,6,6,7,8,10}
[4,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [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]
=> ? ∊ {1,1,1,1,1,1,3,4,5,6,6,7,8,10}
[3,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,3,4,5,6,6,7,8,10}
[3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,3,4,5,6,6,7,8,10}
[2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,3,4,5,6,6,7,8,10}
[2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,3,4,5,6,6,7,8,10}
[2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,3,4,5,6,6,7,8,10}
[1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? ∊ {1,1,1,1,1,1,3,4,5,6,6,7,8,10}
[11]
=> [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,4,4,5,5,5,6,6,7,7,8,9,11}
[10,1]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,4,4,5,5,5,6,6,7,7,8,9,11}
[9,2]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,4,4,5,5,5,6,6,7,7,8,9,11}
[9,1,1]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,4,4,5,5,5,6,6,7,7,8,9,11}
[8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,4,4,5,5,5,6,6,7,7,8,9,11}
[8,2,1]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,4,4,5,5,5,6,6,7,7,8,9,11}
[8,1,1,1]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,4,4,5,5,5,6,6,7,7,8,9,11}
[7,4]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,4,4,5,5,5,6,6,7,7,8,9,11}
[7,3,1]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,4,4,5,5,5,6,6,7,7,8,9,11}
[7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,4,4,5,5,5,6,6,7,7,8,9,11}
[7,2,1,1]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,4,4,5,5,5,6,6,7,7,8,9,11}
[7,1,1,1,1]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,4,4,5,5,5,6,6,7,7,8,9,11}
[5,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,4,4,5,5,5,6,6,7,7,8,9,11}
[4,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,4,4,5,5,5,6,6,7,7,8,9,11}
[4,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,4,4,5,5,5,6,6,7,7,8,9,11}
[3,3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,4,4,5,5,5,6,6,7,7,8,9,11}
[3,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,4,4,5,5,5,6,6,7,7,8,9,11}
[3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,4,4,5,5,5,6,6,7,7,8,9,11}
[3,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,4,4,5,5,5,6,6,7,7,8,9,11}
[2,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,4,4,5,5,5,6,6,7,7,8,9,11}
[2,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,4,4,5,5,5,6,6,7,7,8,9,11}
[2,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,4,4,5,5,5,6,6,7,7,8,9,11}
Description
The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module.
Matching statistic: St000686
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000686: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 66%●distinct values known / distinct values provided: 50%
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000686: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 66%●distinct values known / distinct values provided: 50%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
[2]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 4 = 3 + 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 1 + 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [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]
=> 7 = 6 + 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 3 = 2 + 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3 = 2 + 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2 = 1 + 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 2 = 1 + 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 2 = 1 + 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [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]
=> ? ∊ {1,7} + 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> 4 = 3 + 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> 5 = 4 + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2 = 1 + 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 4 = 3 + 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3 = 2 + 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [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 = 2 + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> 2 = 1 + 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? ∊ {1,7} + 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [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]
=> ? ∊ {1,1,6,8} + 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,6,8} + 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 3 = 2 + 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> 4 = 3 + 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> 5 = 4 + 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,6,8} + 1
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? ∊ {1,1,6,8} + 1
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,5,6,7,9} + 1
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,5,6,7,9} + 1
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,5,6,7,9} + 1
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,5,6,7,9} + 1
[3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,5,6,7,9} + 1
[2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,5,6,7,9} + 1
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,5,6,7,9} + 1
[1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? ∊ {1,1,1,1,5,6,7,9} + 1
[10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,3,4,5,6,6,7,8,10} + 1
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,3,4,5,6,6,7,8,10} + 1
[8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,3,4,5,6,6,7,8,10} + 1
[8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,3,4,5,6,6,7,8,10} + 1
[7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,3,4,5,6,6,7,8,10} + 1
[7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,3,4,5,6,6,7,8,10} + 1
[7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,3,4,5,6,6,7,8,10} + 1
[4,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,3,4,5,6,6,7,8,10} + 1
[3,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,3,4,5,6,6,7,8,10} + 1
[3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,3,4,5,6,6,7,8,10} + 1
[2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,3,4,5,6,6,7,8,10} + 1
[2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,3,4,5,6,6,7,8,10} + 1
[2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,3,4,5,6,6,7,8,10} + 1
[1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? ∊ {1,1,1,1,1,1,3,4,5,6,6,7,8,10} + 1
[11]
=> [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,4,4,5,5,5,6,6,7,7,8,9,11} + 1
[10,1]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,4,4,5,5,5,6,6,7,7,8,9,11} + 1
[9,2]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,4,4,5,5,5,6,6,7,7,8,9,11} + 1
[9,1,1]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,4,4,5,5,5,6,6,7,7,8,9,11} + 1
[8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,4,4,5,5,5,6,6,7,7,8,9,11} + 1
[8,2,1]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,4,4,5,5,5,6,6,7,7,8,9,11} + 1
[8,1,1,1]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,4,4,5,5,5,6,6,7,7,8,9,11} + 1
[7,4]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,4,4,5,5,5,6,6,7,7,8,9,11} + 1
[7,3,1]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,4,4,5,5,5,6,6,7,7,8,9,11} + 1
[7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,4,4,5,5,5,6,6,7,7,8,9,11} + 1
[7,2,1,1]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,4,4,5,5,5,6,6,7,7,8,9,11} + 1
[7,1,1,1,1]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,4,4,5,5,5,6,6,7,7,8,9,11} + 1
[5,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,4,4,5,5,5,6,6,7,7,8,9,11} + 1
[4,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,4,4,5,5,5,6,6,7,7,8,9,11} + 1
[4,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,4,4,5,5,5,6,6,7,7,8,9,11} + 1
[3,3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,4,4,5,5,5,6,6,7,7,8,9,11} + 1
[3,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,4,4,5,5,5,6,6,7,7,8,9,11} + 1
[3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,4,4,5,5,5,6,6,7,7,8,9,11} + 1
[3,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,4,4,5,5,5,6,6,7,7,8,9,11} + 1
[2,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,4,4,5,5,5,6,6,7,7,8,9,11} + 1
[2,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,4,4,5,5,5,6,6,7,7,8,9,11} + 1
[2,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,4,4,5,5,5,6,6,7,7,8,9,11} + 1
Description
The finitistic dominant dimension of a Dyck path.
To every LNakayama algebra there is a corresponding Dyck path, see also [[St000684]]. We associate the finitistic dominant dimension of the algebra to the corresponding Dyck path.
Matching statistic: St000297
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00280: Binary words —path rowmotion⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 67%
Mp00280: Binary words —path rowmotion⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 67%
Values
[1]
=> 10 => 11 => 00 => 0 = 1 - 1
[2]
=> 100 => 011 => 100 => 1 = 2 - 1
[1,1]
=> 110 => 111 => 000 => 0 = 1 - 1
[3]
=> 1000 => 0011 => 1100 => 2 = 3 - 1
[2,1]
=> 1010 => 1101 => 0010 => 0 = 1 - 1
[1,1,1]
=> 1110 => 1111 => 0000 => 0 = 1 - 1
[4]
=> 10000 => 00011 => 11100 => 3 = 4 - 1
[3,1]
=> 10010 => 01101 => 10010 => 1 = 2 - 1
[2,2]
=> 1100 => 0111 => 1000 => 1 = 2 - 1
[2,1,1]
=> 10110 => 11011 => 00100 => 0 = 1 - 1
[1,1,1,1]
=> 11110 => 11111 => 00000 => 0 = 1 - 1
[5]
=> 100000 => 000011 => 111100 => 4 = 5 - 1
[4,1]
=> 100010 => 001101 => 110010 => 2 = 3 - 1
[3,2]
=> 10100 => 11001 => 00110 => 0 = 1 - 1
[3,1,1]
=> 100110 => 011011 => 100100 => 1 = 2 - 1
[2,2,1]
=> 11010 => 11101 => 00010 => 0 = 1 - 1
[2,1,1,1]
=> 101110 => 110111 => 001000 => 0 = 1 - 1
[1,1,1,1,1]
=> 111110 => 111111 => 000000 => 0 = 1 - 1
[6]
=> 1000000 => 0000011 => 1111100 => 5 = 6 - 1
[5,1]
=> 1000010 => 0001101 => 1110010 => 3 = 4 - 1
[4,2]
=> 100100 => 011001 => 100110 => 1 = 2 - 1
[4,1,1]
=> 1000110 => 0011011 => 1100100 => 2 = 3 - 1
[3,3]
=> 11000 => 00111 => 11000 => 2 = 3 - 1
[3,2,1]
=> 101010 => 110101 => 001010 => 0 = 1 - 1
[3,1,1,1]
=> 1001110 => 0110111 => 1001000 => 1 = 2 - 1
[2,2,2]
=> 11100 => 01111 => 10000 => 1 = 2 - 1
[2,2,1,1]
=> 110110 => 111011 => 000100 => 0 = 1 - 1
[2,1,1,1,1]
=> 1011110 => 1101111 => 0010000 => 0 = 1 - 1
[1,1,1,1,1,1]
=> 1111110 => 1111111 => 0000000 => 0 = 1 - 1
[7]
=> 10000000 => 00000011 => 11111100 => 6 = 7 - 1
[6,1]
=> 10000010 => 00001101 => 11110010 => 4 = 5 - 1
[5,2]
=> 1000100 => 0011001 => 1100110 => 2 = 3 - 1
[5,1,1]
=> 10000110 => 00011011 => 11100100 => 3 = 4 - 1
[4,3]
=> 101000 => 110001 => 001110 => 0 = 1 - 1
[4,2,1]
=> 1001010 => 0110101 => 1001010 => 1 = 2 - 1
[4,1,1,1]
=> 10001110 => 00110111 => 11001000 => 2 = 3 - 1
[3,3,1]
=> 110010 => 011101 => 100010 => 1 = 2 - 1
[3,2,2]
=> 101100 => 110011 => 001100 => 0 = 1 - 1
[3,2,1,1]
=> 1010110 => 1101011 => 0010100 => 0 = 1 - 1
[3,1,1,1,1]
=> 10011110 => 01101111 => 10010000 => 1 = 2 - 1
[2,2,2,1]
=> 111010 => 111101 => 000010 => 0 = 1 - 1
[2,2,1,1,1]
=> 1101110 => 1110111 => 0001000 => 0 = 1 - 1
[2,1,1,1,1,1]
=> 10111110 => 11011111 => 00100000 => 0 = 1 - 1
[1,1,1,1,1,1,1]
=> 11111110 => 11111111 => 00000000 => 0 = 1 - 1
[8]
=> 100000000 => 000000011 => 111111100 => 7 = 8 - 1
[7,1]
=> 100000010 => 000001101 => 111110010 => 5 = 6 - 1
[6,2]
=> 10000100 => 00011001 => 11100110 => 3 = 4 - 1
[6,1,1]
=> 100000110 => 000011011 => 111100100 => 4 = 5 - 1
[5,3]
=> 1001000 => 0110001 => 1001110 => 1 = 2 - 1
[5,2,1]
=> 10001010 => 00110101 => 11001010 => 2 = 3 - 1
[9]
=> 1000000000 => 0000000011 => 1111111100 => ? ∊ {2,3,4,6,7,9} - 1
[8,1]
=> 1000000010 => 0000001101 => 1111110010 => ? ∊ {2,3,4,6,7,9} - 1
[7,1,1]
=> 1000000110 => 0000011011 => 1111100100 => ? ∊ {2,3,4,6,7,9} - 1
[5,1,1,1,1]
=> 1000011110 => 0001101111 => 1110010000 => ? ∊ {2,3,4,6,7,9} - 1
[4,1,1,1,1,1]
=> 1000111110 => 0011011111 => 1100100000 => ? ∊ {2,3,4,6,7,9} - 1
[3,1,1,1,1,1,1]
=> 1001111110 => 0110111111 => 1001000000 => ? ∊ {2,3,4,6,7,9} - 1
[10]
=> 10000000000 => 00000000011 => 11111111100 => ? ∊ {1,1,2,2,3,3,4,5,5,6,6,7,8,10} - 1
[9,1]
=> 10000000010 => 00000001101 => 11111110010 => ? ∊ {1,1,2,2,3,3,4,5,5,6,6,7,8,10} - 1
[8,2]
=> 1000000100 => 0000011001 => 1111100110 => ? ∊ {1,1,2,2,3,3,4,5,5,6,6,7,8,10} - 1
[8,1,1]
=> 10000000110 => 00000011011 => 11111100100 => ? ∊ {1,1,2,2,3,3,4,5,5,6,6,7,8,10} - 1
[7,2,1]
=> 1000001010 => 0000110101 => 1111001010 => ? ∊ {1,1,2,2,3,3,4,5,5,6,6,7,8,10} - 1
[7,1,1,1]
=> 10000001110 => 00000110111 => 11111001000 => ? ∊ {1,1,2,2,3,3,4,5,5,6,6,7,8,10} - 1
[6,1,1,1,1]
=> 10000011110 => 00001101111 => 11110010000 => ? ∊ {1,1,2,2,3,3,4,5,5,6,6,7,8,10} - 1
[5,2,1,1,1]
=> 1000101110 => 0011010111 => 1100101000 => ? ∊ {1,1,2,2,3,3,4,5,5,6,6,7,8,10} - 1
[5,1,1,1,1,1]
=> 10000111110 => 00011011111 => 11100100000 => ? ∊ {1,1,2,2,3,3,4,5,5,6,6,7,8,10} - 1
[4,2,1,1,1,1]
=> 1001011110 => 0110101111 => 1001010000 => ? ∊ {1,1,2,2,3,3,4,5,5,6,6,7,8,10} - 1
[4,1,1,1,1,1,1]
=> 10001111110 => 00110111111 => 11001000000 => ? ∊ {1,1,2,2,3,3,4,5,5,6,6,7,8,10} - 1
[3,1,1,1,1,1,1,1]
=> 10011111110 => 01101111111 => 10010000000 => ? ∊ {1,1,2,2,3,3,4,5,5,6,6,7,8,10} - 1
[2,1,1,1,1,1,1,1,1]
=> 10111111110 => 11011111111 => 00100000000 => ? ∊ {1,1,2,2,3,3,4,5,5,6,6,7,8,10} - 1
[1,1,1,1,1,1,1,1,1,1]
=> 11111111110 => 11111111111 => 00000000000 => ? ∊ {1,1,2,2,3,3,4,5,5,6,6,7,8,10} - 1
[11]
=> 100000000000 => ? => ? => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} - 1
[10,1]
=> 100000000010 => ? => ? => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} - 1
[9,2]
=> 10000000100 => 00000011001 => 11111100110 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} - 1
[9,1,1]
=> 100000000110 => ? => ? => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} - 1
[8,3]
=> 1000001000 => ? => ? => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} - 1
[8,2,1]
=> 10000001010 => 00000110101 => 11111001010 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} - 1
[8,1,1,1]
=> 100000001110 => ? => ? => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} - 1
[7,3,1]
=> 1000010010 => ? => ? => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} - 1
[7,2,2]
=> 1000001100 => ? => ? => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} - 1
[7,2,1,1]
=> 10000010110 => 00001101011 => 11110010100 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} - 1
[7,1,1,1,1]
=> 100000011110 => ? => ? => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} - 1
[6,3,1,1]
=> 1000100110 => ? => ? => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} - 1
[6,2,2,1]
=> 1000011010 => ? => ? => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} - 1
[6,2,1,1,1]
=> 10000101110 => 00011010111 => 11100101000 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} - 1
[6,1,1,1,1,1]
=> 100000111110 => ? => ? => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} - 1
[5,3,1,1,1]
=> 1001001110 => 0110010111 => 1001101000 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} - 1
[5,2,2,1,1]
=> 1000110110 => 0011011011 => 1100100100 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} - 1
[5,2,1,1,1,1]
=> 10001011110 => 00110101111 => ? => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} - 1
[5,1,1,1,1,1,1]
=> 100001111110 => ? => ? => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} - 1
[4,3,1,1,1,1]
=> 1010011110 => ? => ? => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} - 1
[4,2,2,1,1,1]
=> 1001101110 => ? => ? => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} - 1
[4,2,1,1,1,1,1]
=> 10010111110 => 01101011111 => 10010100000 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} - 1
[4,1,1,1,1,1,1,1]
=> 100011111110 => ? => ? => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} - 1
[3,3,1,1,1,1,1]
=> 1100111110 => ? => ? => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} - 1
[3,2,2,1,1,1,1]
=> 1011011110 => ? => ? => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} - 1
[3,2,1,1,1,1,1,1]
=> 10101111110 => 11010111111 => 00101000000 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} - 1
[3,1,1,1,1,1,1,1,1]
=> 100111111110 => ? => ? => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} - 1
[2,2,1,1,1,1,1,1,1]
=> 11011111110 => 11101111111 => 00010000000 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} - 1
[2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? => ? => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} - 1
[1,1,1,1,1,1,1,1,1,1,1]
=> 111111111110 => ? => ? => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} - 1
Description
The number of leading ones in a binary word.
Matching statistic: St000439
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 75%
Mp00142: Dyck paths —promotion⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 75%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> 4 = 1 + 3
[2]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 5 = 2 + 3
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 4 = 1 + 3
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 6 = 3 + 3
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4 = 1 + 3
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 4 = 1 + 3
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 7 = 4 + 3
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 5 = 2 + 3
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 5 = 2 + 3
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 4 = 1 + 3
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 4 = 1 + 3
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 8 = 5 + 3
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 6 = 3 + 3
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 5 = 2 + 3
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 4 = 1 + 3
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 4 = 1 + 3
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> 4 = 1 + 3
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> 4 = 1 + 3
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> 9 = 6 + 3
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 7 = 4 + 3
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 6 = 3 + 3
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 5 = 2 + 3
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 6 = 3 + 3
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 4 = 1 + 3
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 4 = 1 + 3
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 5 = 2 + 3
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> 4 = 1 + 3
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> 4 = 1 + 3
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 + 3
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> 10 = 7 + 3
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> 8 = 5 + 3
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> 7 = 4 + 3
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> 6 = 3 + 3
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 6 = 3 + 3
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 5 = 2 + 3
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 4 = 1 + 3
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> 5 = 2 + 3
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 5 = 2 + 3
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 4 = 1 + 3
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> 4 = 1 + 3
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> 4 = 1 + 3
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> 4 = 1 + 3
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? ∊ {1,1} + 3
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1} + 3
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> 11 = 8 + 3
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> 9 = 6 + 3
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> 8 = 5 + 3
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> 7 = 4 + 3
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> 7 = 4 + 3
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> 6 = 3 + 3
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> 5 = 2 + 3
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> 7 = 4 + 3
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 5 = 2 + 3
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,1,0,0,0,0]
=> ? ∊ {1,1,2,3} + 3
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,0,1,1,0,0,0,0,0]
=> ? ∊ {1,1,2,3} + 3
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? ∊ {1,1,2,3} + 3
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {1,1,2,3} + 3
[4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,1,0,0,0]
=> ? ∊ {1,1,1,1,2,2,3} + 3
[3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,1,0,1,0,0,0,0]
=> ? ∊ {1,1,1,1,2,2,3} + 3
[3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,2,2,3} + 3
[2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,0,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,2,2,3} + 3
[2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,2,2,3} + 3
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,2,2,3} + 3
[1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,2,2,3} + 3
[10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,4,4,5,6,6,10} + 3
[8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,4,4,5,6,6,10} + 3
[7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,4,4,5,6,6,10} + 3
[7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,4,4,5,6,6,10} + 3
[7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,4,4,5,6,6,10} + 3
[5,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,4,4,5,6,6,10} + 3
[4,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,1,0,0,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,4,4,5,6,6,10} + 3
[4,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,0,0,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,4,4,5,6,6,10} + 3
[3,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,4,4,5,6,6,10} + 3
[3,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,1,0,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,4,4,5,6,6,10} + 3
[3,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,4,4,5,6,6,10} + 3
[3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,4,4,5,6,6,10} + 3
[2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,4,4,5,6,6,10} + 3
[2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,4,4,5,6,6,10} + 3
[2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,0,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,4,4,5,6,6,10} + 3
[2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,1,0,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,4,4,5,6,6,10} + 3
[2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,1,0,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,4,4,5,6,6,10} + 3
[1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,1,0,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,4,4,5,6,6,10} + 3
[11]
=> [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,4,5,5,5,6,6,7,7,9,11} + 3
[10,1]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,4,5,5,5,6,6,7,7,9,11} + 3
[9,1,1]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,4,5,5,5,6,6,7,7,9,11} + 3
[8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,4,5,5,5,6,6,7,7,9,11} + 3
[8,2,1]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,4,5,5,5,6,6,7,7,9,11} + 3
[8,1,1,1]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,4,5,5,5,6,6,7,7,9,11} + 3
[7,4]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,4,5,5,5,6,6,7,7,9,11} + 3
[7,3,1]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,4,5,5,5,6,6,7,7,9,11} + 3
[7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,4,5,5,5,6,6,7,7,9,11} + 3
[7,1,1,1,1]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,4,5,5,5,6,6,7,7,9,11} + 3
[5,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,1,0,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,4,5,5,5,6,6,7,7,9,11} + 3
[5,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,0,0,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,4,5,5,5,6,6,7,7,9,11} + 3
[4,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,1,0,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,4,5,5,5,6,6,7,7,9,11} + 3
[4,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,1,0,0,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,4,5,5,5,6,6,7,7,9,11} + 3
[4,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,4,5,5,5,6,6,7,7,9,11} + 3
[4,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,4,5,5,5,6,6,7,7,9,11} + 3
[3,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,4,5,5,5,6,6,7,7,9,11} + 3
[3,3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,4,5,5,5,6,6,7,7,9,11} + 3
Description
The position of the first down step of a Dyck path.
Matching statistic: St000731
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000731: Permutations ⟶ ℤResult quality: 58% ●values known / values provided: 58%●distinct values known / distinct values provided: 83%
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000731: Permutations ⟶ ℤResult quality: 58% ●values known / values provided: 58%●distinct values known / distinct values provided: 83%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> [2,1] => 0 = 1 - 1
[2]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1 = 2 - 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0 = 1 - 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 2 = 3 - 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0 = 1 - 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0 = 1 - 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 3 = 4 - 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1 = 2 - 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1 = 2 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 0 = 1 - 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0 = 1 - 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => 4 = 5 - 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => 2 = 3 - 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 0 = 1 - 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 1 = 2 - 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 0 = 1 - 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 0 = 1 - 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => 0 = 1 - 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,1] => 5 = 6 - 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,1] => 3 = 4 - 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,2,3,5,1] => 1 = 2 - 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => 2 = 3 - 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 2 = 3 - 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 0 = 1 - 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => 1 = 2 - 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 1 = 2 - 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 0 = 1 - 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [3,2,1,4,5,6] => 0 = 1 - 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => 0 = 1 - 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,1] => 6 = 7 - 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,7,1] => 4 = 5 - 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [4,2,3,5,6,1] => 2 = 3 - 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [2,4,3,5,6,1] => 3 = 4 - 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [5,2,3,4,1] => 0 = 1 - 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => 1 = 2 - 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => 2 = 3 - 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 1 = 2 - 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => 0 = 1 - 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 0 = 1 - 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [2,4,3,1,5,6] => 1 = 2 - 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 0 = 1 - 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,4,3,2,5,6] => 0 = 1 - 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [3,2,1,4,5,6,7] => 0 = 1 - 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,8,7] => 0 = 1 - 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,9,1] => 7 = 8 - 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,7,8,1] => 5 = 6 - 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [4,2,3,5,6,7,1] => 3 = 4 - 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [2,4,3,5,6,7,1] => ? ∊ {2,5} - 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [5,2,3,4,6,1] => 1 = 2 - 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [3,2,5,4,6,1] => 2 = 3 - 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [2,3,5,4,6,1] => 3 = 4 - 1
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [2,4,3,1,5,6,7] => ? ∊ {2,5} - 1
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [2,4,3,5,6,7,8,1] => ? ∊ {1,1,1,3,4,5,6} - 1
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> [3,2,5,4,6,7,1] => ? ∊ {1,1,1,3,4,5,6} - 1
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [2,3,5,4,6,7,1] => ? ∊ {1,1,1,3,4,5,6} - 1
[4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [2,3,5,4,1,6,7] => ? ∊ {1,1,1,3,4,5,6} - 1
[3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,5,4,3,6,7] => ? ∊ {1,1,1,3,4,5,6} - 1
[3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [2,4,3,1,5,6,7,8] => ? ∊ {1,1,1,3,4,5,6} - 1
[2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,4,3,2,5,6,7,8] => ? ∊ {1,1,1,3,4,5,6} - 1
[8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,4,3,5,6,7,8,9,1] => ? ∊ {1,1,1,1,1,2,3,3,4,4,4,5,5,6,7} - 1
[7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
=> [3,2,5,4,6,7,8,1] => ? ∊ {1,1,1,1,1,2,3,3,4,4,4,5,5,6,7} - 1
[7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [2,3,5,4,6,7,8,1] => ? ∊ {1,1,1,1,1,2,3,3,4,4,4,5,5,6,7} - 1
[6,3,1]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> [4,2,3,6,5,7,1] => ? ∊ {1,1,1,1,1,2,3,3,4,4,4,5,5,6,7} - 1
[6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [2,5,3,4,6,7,1] => ? ∊ {1,1,1,1,1,2,3,3,4,4,4,5,5,6,7} - 1
[6,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [3,2,4,6,5,7,1] => ? ∊ {1,1,1,1,1,2,3,3,4,4,4,5,5,6,7} - 1
[6,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [2,3,4,6,5,7,1] => ? ∊ {1,1,1,1,1,2,3,3,4,4,4,5,5,6,7} - 1
[5,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [2,3,4,6,5,1,7] => ? ∊ {1,1,1,1,1,2,3,3,4,4,4,5,5,6,7} - 1
[4,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0,1,0]
=> [2,3,1,6,5,4,7] => ? ∊ {1,1,1,1,1,2,3,3,4,4,4,5,5,6,7} - 1
[4,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,5,4,1,6,7,8] => ? ∊ {1,1,1,1,1,2,3,3,4,4,4,5,5,6,7} - 1
[3,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,6,5,4,7] => ? ∊ {1,1,1,1,1,2,3,3,4,4,4,5,5,6,7} - 1
[3,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,1,5,4,3,6,7,8] => ? ∊ {1,1,1,1,1,2,3,3,4,4,4,5,5,6,7} - 1
[3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [2,4,3,1,5,6,7,8,9] => ? ∊ {1,1,1,1,1,2,3,3,4,4,4,5,5,6,7} - 1
[2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,2,5,4,3,6,7,8] => ? ∊ {1,1,1,1,1,2,3,3,4,4,4,5,5,6,7} - 1
[2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,4,3,2,5,6,7,8,9] => ? ∊ {1,1,1,1,1,2,3,3,4,4,4,5,5,6,7} - 1
[11]
=> [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} - 1
[10,1]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,7,8,9,10,11,1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} - 1
[9,2]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [4,2,3,5,6,7,8,9,10,1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} - 1
[9,1,1]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,4,3,5,6,7,8,9,10,1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} - 1
[8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [5,2,3,4,6,7,8,9,1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} - 1
[8,2,1]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [3,2,5,4,6,7,8,9,1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} - 1
[8,1,1,1]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [2,3,5,4,6,7,8,9,1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} - 1
[7,4]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> [6,2,3,4,5,7,8,1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} - 1
[7,3,1]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,1,0,0]
=> [4,2,3,6,5,7,8,1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} - 1
[7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [2,5,3,4,6,7,8,1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} - 1
[7,2,1,1]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0]
=> [3,2,4,6,5,7,8,1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} - 1
[7,1,1,1,1]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [2,3,4,6,5,7,8,1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} - 1
[6,5]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [7,2,3,4,5,6,1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} - 1
[6,4,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> [5,2,3,4,7,6,1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} - 1
[6,3,2]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [3,2,6,4,5,7,1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} - 1
[6,3,1,1]
=> [1,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,1,0,0,0]
=> [4,2,3,5,7,6,1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} - 1
[6,2,2,1]
=> [1,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [2,4,3,6,5,7,1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} - 1
[6,2,1,1,1]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [3,2,4,5,7,6,1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} - 1
[6,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,5,7,6,1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} - 1
[5,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> [2,3,4,1,7,6,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} - 1
[5,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [2,3,4,6,5,1,7,8] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} - 1
[4,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> [2,1,4,6,5,3,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} - 1
[4,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,7,6,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} - 1
[4,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [2,3,1,6,5,4,7,8] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} - 1
[4,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [2,3,5,4,1,6,7,8,9] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} - 1
[3,3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [1,3,5,4,2,6,7,8] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} - 1
Description
The number of double exceedences of a permutation.
A double exceedence is an index $\sigma(i)$ such that $i < \sigma(i) < \sigma(\sigma(i))$.
Matching statistic: St000383
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 67%
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 67%
Values
[1]
=> [[1]]
=> [1] => [1] => 1
[2]
=> [[1,2]]
=> [2] => [1] => 1
[1,1]
=> [[1],[2]]
=> [1,1] => [2] => 2
[3]
=> [[1,2,3]]
=> [3] => [1] => 1
[2,1]
=> [[1,2],[3]]
=> [2,1] => [1,1] => 1
[1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => [3] => 3
[4]
=> [[1,2,3,4]]
=> [4] => [1] => 1
[3,1]
=> [[1,2,3],[4]]
=> [3,1] => [1,1] => 1
[2,2]
=> [[1,2],[3,4]]
=> [2,2] => [2] => 2
[2,1,1]
=> [[1,2],[3],[4]]
=> [2,1,1] => [1,2] => 2
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => [4] => 4
[5]
=> [[1,2,3,4,5]]
=> [5] => [1] => 1
[4,1]
=> [[1,2,3,4],[5]]
=> [4,1] => [1,1] => 1
[3,2]
=> [[1,2,3],[4,5]]
=> [3,2] => [1,1] => 1
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [3,1,1] => [1,2] => 2
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [2,2,1] => [2,1] => 1
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1] => [1,3] => 3
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => [5] => 5
[6]
=> [[1,2,3,4,5,6]]
=> [6] => [1] => 1
[5,1]
=> [[1,2,3,4,5],[6]]
=> [5,1] => [1,1] => 1
[4,2]
=> [[1,2,3,4],[5,6]]
=> [4,2] => [1,1] => 1
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [4,1,1] => [1,2] => 2
[3,3]
=> [[1,2,3],[4,5,6]]
=> [3,3] => [2] => 2
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [3,2,1] => [1,1,1] => 1
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [3,1,1,1] => [1,3] => 3
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [2,2,2] => [3] => 3
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [2,2,1,1] => [2,2] => 2
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [2,1,1,1,1] => [1,4] => 4
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1] => [6] => 6
[7]
=> [[1,2,3,4,5,6,7]]
=> [7] => [1] => 1
[6,1]
=> [[1,2,3,4,5,6],[7]]
=> [6,1] => [1,1] => 1
[5,2]
=> [[1,2,3,4,5],[6,7]]
=> [5,2] => [1,1] => 1
[5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [5,1,1] => [1,2] => 2
[4,3]
=> [[1,2,3,4],[5,6,7]]
=> [4,3] => [1,1] => 1
[4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [4,2,1] => [1,1,1] => 1
[4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [4,1,1,1] => [1,3] => 3
[3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [3,3,1] => [2,1] => 1
[3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [3,2,2] => [1,2] => 2
[3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [3,2,1,1] => [1,1,2] => 2
[3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [3,1,1,1,1] => [1,4] => 4
[2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [2,2,2,1] => [3,1] => 1
[2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [2,2,1,1,1] => [2,3] => 3
[2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [2,1,1,1,1,1] => [1,5] => 5
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [1,1,1,1,1,1,1] => [7] => 7
[8]
=> [[1,2,3,4,5,6,7,8]]
=> [8] => [1] => 1
[7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [7,1] => [1,1] => 1
[6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [6,2] => [1,1] => 1
[6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> [6,1,1] => [1,2] => 2
[5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [5,3] => [1,1] => 1
[5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> [5,2,1] => [1,1,1] => 1
[1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [1,1,1,1,1,1,1,1] => [8] => ? = 8
[1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> [1,1,1,1,1,1,1,1,1] => [9] => ? = 9
[2,2,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10]]
=> [2,2,1,1,1,1,1,1] => [2,6] => ? ∊ {6,10}
[1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> [1,1,1,1,1,1,1,1,1,1] => [10] => ? ∊ {6,10}
[11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[10,1]
=> [[1,2,3,4,5,6,7,8,9,10],[11]]
=> ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[9,2]
=> [[1,2,3,4,5,6,7,8,9],[10,11]]
=> ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[9,1,1]
=> [[1,2,3,4,5,6,7,8,9],[10],[11]]
=> ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[8,3]
=> [[1,2,3,4,5,6,7,8],[9,10,11]]
=> ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[8,2,1]
=> [[1,2,3,4,5,6,7,8],[9,10],[11]]
=> ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[8,1,1,1]
=> [[1,2,3,4,5,6,7,8],[9],[10],[11]]
=> ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[7,4]
=> [[1,2,3,4,5,6,7],[8,9,10,11]]
=> ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[7,3,1]
=> [[1,2,3,4,5,6,7],[8,9,10],[11]]
=> ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[7,2,2]
=> [[1,2,3,4,5,6,7],[8,9],[10,11]]
=> ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[7,2,1,1]
=> [[1,2,3,4,5,6,7],[8,9],[10],[11]]
=> ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[7,1,1,1,1]
=> [[1,2,3,4,5,6,7],[8],[9],[10],[11]]
=> ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[6,5]
=> [[1,2,3,4,5,6],[7,8,9,10,11]]
=> ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[6,4,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11]]
=> ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[6,3,2]
=> [[1,2,3,4,5,6],[7,8,9],[10,11]]
=> ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[6,3,1,1]
=> [[1,2,3,4,5,6],[7,8,9],[10],[11]]
=> ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[6,2,2,1]
=> [[1,2,3,4,5,6],[7,8],[9,10],[11]]
=> ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[6,2,1,1,1]
=> [[1,2,3,4,5,6],[7,8],[9],[10],[11]]
=> ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[6,1,1,1,1,1]
=> [[1,2,3,4,5,6],[7],[8],[9],[10],[11]]
=> ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[5,5,1]
=> [[1,2,3,4,5],[6,7,8,9,10],[11]]
=> ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[5,4,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11]]
=> [5,4,2] => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[5,4,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10],[11]]
=> [5,4,1,1] => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[5,3,3]
=> [[1,2,3,4,5],[6,7,8],[9,10,11]]
=> [5,3,3] => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[5,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11]]
=> [5,3,2,1] => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[5,3,1,1,1]
=> [[1,2,3,4,5],[6,7,8],[9],[10],[11]]
=> [5,3,1,1,1] => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[5,2,2,2]
=> [[1,2,3,4,5],[6,7],[8,9],[10,11]]
=> [5,2,2,2] => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[5,2,2,1,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10],[11]]
=> [5,2,2,1,1] => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[5,2,1,1,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9],[10],[11]]
=> ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[5,1,1,1,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8],[9],[10],[11]]
=> ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[4,4,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11]]
=> [4,4,2,1] => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[4,4,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11]]
=> [4,4,1,1,1] => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[4,3,3,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11]]
=> [4,3,3,1] => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[4,3,2,2]
=> [[1,2,3,4],[5,6,7],[8,9],[10,11]]
=> [4,3,2,2] => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[4,3,2,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10],[11]]
=> [4,3,2,1,1] => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[4,3,1,1,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9],[10],[11]]
=> ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[4,2,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9,10],[11]]
=> [4,2,2,2,1] => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[4,2,2,1,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9],[10],[11]]
=> ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[4,2,1,1,1,1,1]
=> [[1,2,3,4],[5,6],[7],[8],[9],[10],[11]]
=> ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[4,1,1,1,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8],[9],[10],[11]]
=> ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[3,3,3,1,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10],[11]]
=> [3,3,3,1,1] => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[3,3,2,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11]]
=> [3,3,2,2,1] => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[3,3,2,1,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10],[11]]
=> ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[3,3,1,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10],[11]]
=> ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[3,2,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11]]
=> ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[3,2,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10],[11]]
=> ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
[3,2,2,1,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9],[10],[11]]
=> ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11}
Description
The last part of an integer composition.
The following 110 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000648The number of 2-excedences of a permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000654The first descent of a permutation. St000717The number of ordinal summands of a poset. St000740The last entry of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St000873The aix statistic of a permutation. St000907The number of maximal antichains of minimal length in a poset. St000678The number of up steps after the last double rise of a Dyck path. St000485The length of the longest cycle of a permutation. St000993The multiplicity of the largest part of an integer partition. St000989The number of final rises of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St001571The Cartan determinant of the integer partition. St000990The first ascent of a permutation. St000542The number of left-to-right-minima of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000996The number of exclusive left-to-right maxima of a permutation. St000054The first entry of the permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000203The number of external nodes of a binary tree. St000843The decomposition number of a perfect matching. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St000061The number of nodes on the left branch of a binary tree. St000314The number of left-to-right-maxima of a permutation. St000991The number of right-to-left minima of a permutation. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St000025The number of initial rises of a Dyck path. St000338The number of pixed points of a permutation. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St000015The number of peaks of a Dyck path. St000051The size of the left subtree of a binary tree. St000056The decomposition (or block) number of a permutation. St000084The number of subtrees. St000133The "bounce" of a permutation. St000352The Elizalde-Pak rank of a permutation. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000838The number of terminal right-hand endpoints when the vertices are written in order. St001050The number of terminal closers of a set partition. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001481The minimal height of a peak of a Dyck path. St000039The number of crossings of a permutation. St000117The number of centered tunnels of a Dyck path. St000221The number of strong fixed points of a permutation. St000241The number of cyclical small excedances. St000331The number of upper interactions of a Dyck path. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000734The last entry in the first row of a standard tableau. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000895The number of ones on the main diagonal of an alternating sign matrix. St000971The smallest closer of a set partition. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001010Number of indecomposable injective modules with projective dimension g-1 when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001346The number of parking functions that give the same permutation. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001530The depth of a Dyck path. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St000756The sum of the positions of the left to right maxima of a permutation. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001060The distinguishing index of a graph. St001061The number of indices that are both descents and recoils of a permutation. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000456The monochromatic index of a connected graph. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001134The largest label in the subtree rooted at the sister of 1 in the leaf labelled binary unordered tree associated with the perfect matching. St000153The number of adjacent cycles of a permutation. St000007The number of saliances of the permutation. St000193The row of the unique '1' in the first column of the alternating sign matrix. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000461The rix statistic of a permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001557The number of inversions of the second entry of a permutation. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St000738The first entry in the last row of a standard tableau. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000910The number of maximal chains of minimal length in a poset. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000782The indicator function of whether a given perfect matching is an L & P matching. St001438The number of missing boxes of a skew partition. St001868The number of alignments of type NE of a signed permutation. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000259The diameter of a connected graph. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$.
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