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Your data matches 87 different statistics following compositions of up to 3 maps.
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Matching statistic: St000657
(load all 60 compositions to match this statistic)
(load all 60 compositions to match this statistic)
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St000657: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000657: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,1] => 1
[1,1,0,0]
=> [2] => 2
[1,0,1,0,1,0]
=> [1,1,1] => 1
[1,0,1,1,0,0]
=> [1,2] => 1
[1,1,0,0,1,0]
=> [2,1] => 1
[1,1,0,1,0,0]
=> [2,1] => 1
[1,1,1,0,0,0]
=> [3] => 3
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => 1
[1,0,1,1,0,1,0,0]
=> [1,2,1] => 1
[1,0,1,1,1,0,0,0]
=> [1,3] => 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => 1
[1,1,0,0,1,1,0,0]
=> [2,2] => 2
[1,1,0,1,0,0,1,0]
=> [2,1,1] => 1
[1,1,0,1,0,1,0,0]
=> [2,1,1] => 1
[1,1,0,1,1,0,0,0]
=> [2,2] => 2
[1,1,1,0,0,0,1,0]
=> [3,1] => 1
[1,1,1,0,0,1,0,0]
=> [3,1] => 1
[1,1,1,0,1,0,0,0]
=> [3,1] => 1
[1,1,1,1,0,0,0,0]
=> [4] => 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => 2
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => 1
Description
The smallest part of an integer composition.
Matching statistic: St000993
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,1] => [1,1]
=> [2]
=> 1
[1,1,0,0]
=> [2] => [2]
=> [1,1]
=> 2
[1,0,1,0,1,0]
=> [1,1,1] => [1,1,1]
=> [3]
=> 1
[1,0,1,1,0,0]
=> [1,2] => [2,1]
=> [2,1]
=> 1
[1,1,0,0,1,0]
=> [2,1] => [2,1]
=> [2,1]
=> 1
[1,1,0,1,0,0]
=> [2,1] => [2,1]
=> [2,1]
=> 1
[1,1,1,0,0,0]
=> [3] => [3]
=> [1,1,1]
=> 3
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,1,1,1]
=> [4]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1,1]
=> [3,1]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,1,1]
=> [3,1]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,2,1] => [2,1,1]
=> [3,1]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,3] => [3,1]
=> [2,1,1]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [2,1,1]
=> [3,1]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,2] => [2,2]
=> [2,2]
=> 2
[1,1,0,1,0,0,1,0]
=> [2,1,1] => [2,1,1]
=> [3,1]
=> 1
[1,1,0,1,0,1,0,0]
=> [2,1,1] => [2,1,1]
=> [3,1]
=> 1
[1,1,0,1,1,0,0,0]
=> [2,2] => [2,2]
=> [2,2]
=> 2
[1,1,1,0,0,0,1,0]
=> [3,1] => [3,1]
=> [2,1,1]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1] => [3,1]
=> [2,1,1]
=> 1
[1,1,1,0,1,0,0,0]
=> [3,1] => [3,1]
=> [2,1,1]
=> 1
[1,1,1,1,0,0,0,0]
=> [4] => [4]
=> [1,1,1,1]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,1,1,1,1]
=> [5]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [2,1,1,1]
=> [4,1]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,1,1,1]
=> [4,1]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [2,1,1,1]
=> [4,1]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [3,1,1]
=> [3,1,1]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [2,1,1,1]
=> [4,1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [2,2,1]
=> [3,2]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => [2,1,1,1]
=> [4,1]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [2,1,1,1]
=> [4,1]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [2,2,1]
=> [3,2]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [3,1,1]
=> [3,1,1]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [3,1,1]
=> [3,1,1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [3,1,1]
=> [3,1,1]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1]
=> [2,1,1,1]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [2,1,1,1]
=> [4,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [2,2,1]
=> [3,2]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,2,1]
=> [3,2]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [2,2,1]
=> [3,2]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [3,2]
=> [2,2,1]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => [2,1,1,1]
=> [4,1]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => [2,2,1]
=> [3,2]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => [2,1,1,1]
=> [4,1]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => [2,1,1,1]
=> [4,1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => [2,2,1]
=> [3,2]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [2,2,1]
=> [3,2]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => [2,2,1]
=> [3,2]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => [2,2,1]
=> [3,2]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => [3,2]
=> [2,2,1]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [3,1,1]
=> [3,1,1]
=> 1
Description
The multiplicity of the largest part of an integer partition.
Matching statistic: St001038
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001038: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001038: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,1] => [1,1]
=> [1,1,0,0]
=> 1
[1,1,0,0]
=> [2] => [2]
=> [1,0,1,0]
=> 2
[1,0,1,0,1,0]
=> [1,1,1] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,0,1,1,0,0]
=> [1,2] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> [2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,1,0,0,0]
=> [3] => [3]
=> [1,0,1,0,1,0]
=> 3
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,2] => [2,2]
=> [1,1,1,0,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [2,1,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [2,1,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [2,2] => [2,2]
=> [1,1,1,0,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [4] => [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
Description
The minimal height of a column in the parallelogram polyomino associated with the Dyck path.
Matching statistic: St000655
(load all 36 compositions to match this statistic)
(load all 36 compositions to match this statistic)
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000655: Dyck paths ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000655: Dyck paths ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,1] => [1,1] => [1,0,1,0]
=> 1
[1,1,0,0]
=> [2] => [2] => [1,1,0,0]
=> 2
[1,0,1,0,1,0]
=> [1,1,1] => [1,1,1] => [1,0,1,0,1,0]
=> 1
[1,0,1,1,0,0]
=> [1,2] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,1,0,0,1,0]
=> [2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[1,1,1,0,0,0]
=> [3] => [3] => [1,1,1,0,0,0]
=> 3
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,2,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [2,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [2,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [4] => [4] => [1,1,1,1,0,0,0,0]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,4] => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,5] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,2,1,3] => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,2,4] => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,0,1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,2,1,3] => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,0,1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,2,1,3] => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,0,1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,2,4] => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,3,3] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,0,1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,3,3] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,3,3] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,6] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,1,1,3] => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,1,4] => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,2,3] => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,0,1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,2,3] => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,5] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,0,1,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [1,2,1,1,3] => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,0,1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [1,2,1,4] => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,0,1,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> [1,2,1,1,3] => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,0,1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,2,1,1,3] => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,0,1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,2,1,4] => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,0,1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [1,2,2,3] => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,0,1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [1,2,2,3] => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,0,1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,2,2,3] => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,5] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,3,1,3] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,3,4] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,0,1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> [1,3,1,3] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [1,3,1,3] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,0,1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,3,4] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,0,1,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> [1,3,1,3] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,0,1,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> [1,3,1,3] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,3,1,3] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,3,4] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,4,3] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,0,1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,4,3] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,0,1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [1,4,3] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,4,3] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,1,1,1,3] => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,1,1,4] => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,2,3] => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,1,0,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,1,2,3] => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,5] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,2,1,3] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,1,0,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [2,2,1,3] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,1,0,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,2,1,3] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,3] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,1,0,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,3,3] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,1,0,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [2,3,3] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,1,0,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [2,1,1,1,3] => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
Description
The length of the minimal rise of a Dyck path.
For the length of a maximal rise, see [[St000444]].
Matching statistic: St000284
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000284: Integer partitions ⟶ ℤResult quality: 12% ●values known / values provided: 76%●distinct values known / distinct values provided: 12%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000284: Integer partitions ⟶ ℤResult quality: 12% ●values known / values provided: 76%●distinct values known / distinct values provided: 12%
Values
[1,0,1,0]
=> [2,1] => [1,1]
=> [1]
=> ? ∊ {1,2}
[1,1,0,0]
=> [1,2] => [2]
=> []
=> ? ∊ {1,2}
[1,0,1,0,1,0]
=> [2,3,1] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,3}
[1,0,1,1,0,0]
=> [2,1,3] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,3}
[1,1,0,0,1,0]
=> [1,3,2] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,3}
[1,1,0,1,0,0]
=> [3,1,2] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,3}
[1,1,1,0,0,0]
=> [1,2,3] => [3]
=> []
=> ? ∊ {1,1,1,1,3}
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [2,2]
=> [2]
=> 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [2,2]
=> [2]
=> 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [2,2]
=> [2]
=> 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,2]
=> [2]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [3,2]
=> [2]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [3,2]
=> [2]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [3,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [3,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [3,2]
=> [2]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [3,2]
=> [2]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [3,2]
=> [2]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [3,2]
=> [2]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [3,2]
=> [2]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [3,2]
=> [2]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [3,2]
=> [2]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [3,2]
=> [2]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [3,2]
=> [2]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [3,2]
=> [2]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [3,2]
=> [2]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [3,2]
=> [2]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,2]
=> [2]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [3,2]
=> [2]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [3,2]
=> [2]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => [3,2]
=> [2]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => [3,2]
=> [2]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,3,5] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => [3,2]
=> [2]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => [3,2]
=> [2]
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> [4,5,1,2,3] => [3,2]
=> [2]
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [4,1,2,3,5] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,1] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,1,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,3,4,1,6,5] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,3,4,6,1,5] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,3,4,1,5,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,1,5,6,4] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,3,1,5,4,6] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,3,5,1,6,4] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,3,5,6,1,4] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4,6] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,3,1,4,6,5] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,6,4,5] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,3,6,1,4,5] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,3,1,4,5,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,5,6,3] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,1,4,5,3,6] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5] => [3,3]
=> [3]
=> 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,6,3,5] => [3,3]
=> [3]
=> 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,5,6] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,4,1,5,6,3] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [2,4,1,5,3,6] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [2,4,5,1,6,3] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,4,5,6,1,3] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [2,4,5,1,3,6] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,6,5] => [3,3]
=> [3]
=> 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,4,5,6,2] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,4,5,2,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,4,2,5,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,4,5,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,1,2,4,5,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,2,4,5,6,3] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,2,4,5,3,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,2,4,3,5,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,4,2,3,5,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [4,1,2,3,5,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,2,3,5,6,4] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
Description
The Plancherel distribution on integer partitions.
This is defined as the distribution induced by the RSK shape of the uniform distribution on permutations. In other words, this is the size of the preimage of the map 'Robinson-Schensted tableau shape' from permutations to integer partitions.
Equivalently, this is given by the square of the number of standard Young tableaux of the given shape.
Matching statistic: St000704
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000704: Integer partitions ⟶ ℤResult quality: 12% ●values known / values provided: 76%●distinct values known / distinct values provided: 12%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000704: Integer partitions ⟶ ℤResult quality: 12% ●values known / values provided: 76%●distinct values known / distinct values provided: 12%
Values
[1,0,1,0]
=> [2,1] => [1,1]
=> [1]
=> ? ∊ {1,2}
[1,1,0,0]
=> [1,2] => [2]
=> []
=> ? ∊ {1,2}
[1,0,1,0,1,0]
=> [2,3,1] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,3}
[1,0,1,1,0,0]
=> [2,1,3] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,3}
[1,1,0,0,1,0]
=> [1,3,2] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,3}
[1,1,0,1,0,0]
=> [3,1,2] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,3}
[1,1,1,0,0,0]
=> [1,2,3] => [3]
=> []
=> ? ∊ {1,1,1,1,3}
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [2,2]
=> [2]
=> 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [2,2]
=> [2]
=> 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [2,2]
=> [2]
=> 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,2]
=> [2]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [3,2]
=> [2]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [3,2]
=> [2]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [3,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [3,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [3,2]
=> [2]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [3,2]
=> [2]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [3,2]
=> [2]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [3,2]
=> [2]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [3,2]
=> [2]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [3,2]
=> [2]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [3,2]
=> [2]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [3,2]
=> [2]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [3,2]
=> [2]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [3,2]
=> [2]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [3,2]
=> [2]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [3,2]
=> [2]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,2]
=> [2]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [3,2]
=> [2]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [3,2]
=> [2]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => [3,2]
=> [2]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => [3,2]
=> [2]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,3,5] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => [3,2]
=> [2]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => [3,2]
=> [2]
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> [4,5,1,2,3] => [3,2]
=> [2]
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [4,1,2,3,5] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,1] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,1,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,3,4,1,6,5] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,3,4,6,1,5] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,3,4,1,5,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,1,5,6,4] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,3,1,5,4,6] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,3,5,1,6,4] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,3,5,6,1,4] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4,6] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,3,1,4,6,5] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,6,4,5] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,3,6,1,4,5] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,3,1,4,5,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,5,6,3] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,1,4,5,3,6] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5] => [3,3]
=> [3]
=> 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,6,3,5] => [3,3]
=> [3]
=> 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,5,6] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,4,1,5,6,3] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [2,4,1,5,3,6] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [2,4,5,1,6,3] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,4,5,6,1,3] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [2,4,5,1,3,6] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,6,5] => [3,3]
=> [3]
=> 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,4,5,6,2] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,4,5,2,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,4,2,5,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,4,5,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,1,2,4,5,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,2,4,5,6,3] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,2,4,5,3,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,2,4,3,5,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,4,2,3,5,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [4,1,2,3,5,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,2,3,5,6,4] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
Description
The number of semistandard tableaux on a given integer partition with minimal maximal entry.
This is, for an integer partition $\lambda = (\lambda_1 > \cdots > \lambda_k > 0)$, the number of [[SemistandardTableaux|semistandard tableaux]] of shape $\lambda$ with maximal entry $k$.
Equivalently, this is the evaluation $s_\lambda(1,\ldots,1)$ of the Schur function $s_\lambda$ in $k$ variables, or, explicitly,
$$ \prod_{(i,j) \in L} \frac{k + j - i}{ \operatorname{hook}(i,j) }$$
where the product is over all cells $(i,j) \in L$ and $\operatorname{hook}(i,j)$ is the hook length of a cell.
See [Theorem 6.3, 1] for details.
Matching statistic: St000706
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000706: Integer partitions ⟶ ℤResult quality: 12% ●values known / values provided: 76%●distinct values known / distinct values provided: 12%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000706: Integer partitions ⟶ ℤResult quality: 12% ●values known / values provided: 76%●distinct values known / distinct values provided: 12%
Values
[1,0,1,0]
=> [2,1] => [1,1]
=> [1]
=> ? ∊ {1,2}
[1,1,0,0]
=> [1,2] => [2]
=> []
=> ? ∊ {1,2}
[1,0,1,0,1,0]
=> [2,3,1] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,3}
[1,0,1,1,0,0]
=> [2,1,3] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,3}
[1,1,0,0,1,0]
=> [1,3,2] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,3}
[1,1,0,1,0,0]
=> [3,1,2] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,3}
[1,1,1,0,0,0]
=> [1,2,3] => [3]
=> []
=> ? ∊ {1,1,1,1,3}
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [2,2]
=> [2]
=> 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [2,2]
=> [2]
=> 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [2,2]
=> [2]
=> 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,2]
=> [2]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [3,2]
=> [2]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [3,2]
=> [2]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [3,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [3,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [3,2]
=> [2]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [3,2]
=> [2]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [3,2]
=> [2]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [3,2]
=> [2]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [3,2]
=> [2]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [3,2]
=> [2]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [3,2]
=> [2]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [3,2]
=> [2]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [3,2]
=> [2]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [3,2]
=> [2]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [3,2]
=> [2]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [3,2]
=> [2]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,2]
=> [2]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [3,2]
=> [2]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [3,2]
=> [2]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => [3,2]
=> [2]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => [3,2]
=> [2]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,3,5] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => [3,2]
=> [2]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => [3,2]
=> [2]
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> [4,5,1,2,3] => [3,2]
=> [2]
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [4,1,2,3,5] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,1] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,1,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,3,4,1,6,5] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,3,4,6,1,5] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,3,4,1,5,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,1,5,6,4] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,3,1,5,4,6] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,3,5,1,6,4] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,3,5,6,1,4] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4,6] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,3,1,4,6,5] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,6,4,5] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,3,6,1,4,5] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,3,1,4,5,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,5,6,3] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,1,4,5,3,6] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5] => [3,3]
=> [3]
=> 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,6,3,5] => [3,3]
=> [3]
=> 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,5,6] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,4,1,5,6,3] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [2,4,1,5,3,6] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [2,4,5,1,6,3] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,4,5,6,1,3] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [2,4,5,1,3,6] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,6,5] => [3,3]
=> [3]
=> 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,4,5,6,2] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,4,5,2,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,4,2,5,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,4,5,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,1,2,4,5,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,2,4,5,6,3] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,2,4,5,3,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,2,4,3,5,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,4,2,3,5,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [4,1,2,3,5,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,2,3,5,6,4] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
Description
The product of the factorials of the multiplicities of an integer partition.
Matching statistic: St000813
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000813: Integer partitions ⟶ ℤResult quality: 12% ●values known / values provided: 76%●distinct values known / distinct values provided: 12%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000813: Integer partitions ⟶ ℤResult quality: 12% ●values known / values provided: 76%●distinct values known / distinct values provided: 12%
Values
[1,0,1,0]
=> [2,1] => [1,1]
=> [1]
=> ? ∊ {1,2}
[1,1,0,0]
=> [1,2] => [2]
=> []
=> ? ∊ {1,2}
[1,0,1,0,1,0]
=> [2,3,1] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,3}
[1,0,1,1,0,0]
=> [2,1,3] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,3}
[1,1,0,0,1,0]
=> [1,3,2] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,3}
[1,1,0,1,0,0]
=> [3,1,2] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,3}
[1,1,1,0,0,0]
=> [1,2,3] => [3]
=> []
=> ? ∊ {1,1,1,1,3}
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [2,2]
=> [2]
=> 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [2,2]
=> [2]
=> 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [2,2]
=> [2]
=> 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,2]
=> [2]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [3,2]
=> [2]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [3,2]
=> [2]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [3,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [3,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [3,2]
=> [2]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [3,2]
=> [2]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [3,2]
=> [2]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [3,2]
=> [2]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [3,2]
=> [2]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [3,2]
=> [2]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [3,2]
=> [2]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [3,2]
=> [2]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [3,2]
=> [2]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [3,2]
=> [2]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [3,2]
=> [2]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [3,2]
=> [2]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,2]
=> [2]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [3,2]
=> [2]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [3,2]
=> [2]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => [3,2]
=> [2]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => [3,2]
=> [2]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,3,5] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => [3,2]
=> [2]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => [3,2]
=> [2]
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> [4,5,1,2,3] => [3,2]
=> [2]
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [4,1,2,3,5] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,1] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,1,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,3,4,1,6,5] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,3,4,6,1,5] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,3,4,1,5,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,1,5,6,4] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,3,1,5,4,6] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,3,5,1,6,4] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,3,5,6,1,4] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4,6] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,3,1,4,6,5] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,6,4,5] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,3,6,1,4,5] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,3,1,4,5,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,5,6,3] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,1,4,5,3,6] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5] => [3,3]
=> [3]
=> 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,6,3,5] => [3,3]
=> [3]
=> 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,5,6] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,4,1,5,6,3] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [2,4,1,5,3,6] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [2,4,5,1,6,3] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,4,5,6,1,3] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [2,4,5,1,3,6] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,6,5] => [3,3]
=> [3]
=> 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,4,5,6,2] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,4,5,2,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,4,2,5,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,4,5,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,1,2,4,5,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,2,4,5,6,3] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,2,4,5,3,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,2,4,3,5,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,4,2,3,5,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [4,1,2,3,5,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,2,3,5,6,4] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
Description
The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition.
This is also the sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to monomial symmetric functions.
Matching statistic: St000901
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000901: Integer partitions ⟶ ℤResult quality: 12% ●values known / values provided: 76%●distinct values known / distinct values provided: 12%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000901: Integer partitions ⟶ ℤResult quality: 12% ●values known / values provided: 76%●distinct values known / distinct values provided: 12%
Values
[1,0,1,0]
=> [2,1] => [1,1]
=> [1]
=> ? ∊ {1,2}
[1,1,0,0]
=> [1,2] => [2]
=> []
=> ? ∊ {1,2}
[1,0,1,0,1,0]
=> [2,3,1] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,3}
[1,0,1,1,0,0]
=> [2,1,3] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,3}
[1,1,0,0,1,0]
=> [1,3,2] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,3}
[1,1,0,1,0,0]
=> [3,1,2] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,3}
[1,1,1,0,0,0]
=> [1,2,3] => [3]
=> []
=> ? ∊ {1,1,1,1,3}
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [2,2]
=> [2]
=> 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [2,2]
=> [2]
=> 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [2,2]
=> [2]
=> 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,2]
=> [2]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [3,2]
=> [2]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [3,2]
=> [2]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [3,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [3,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [3,2]
=> [2]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [3,2]
=> [2]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [3,2]
=> [2]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [3,2]
=> [2]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [3,2]
=> [2]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [3,2]
=> [2]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [3,2]
=> [2]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [3,2]
=> [2]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [3,2]
=> [2]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [3,2]
=> [2]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [3,2]
=> [2]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [3,2]
=> [2]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,2]
=> [2]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [3,2]
=> [2]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [3,2]
=> [2]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => [3,2]
=> [2]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => [3,2]
=> [2]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,3,5] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => [3,2]
=> [2]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => [3,2]
=> [2]
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> [4,5,1,2,3] => [3,2]
=> [2]
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [4,1,2,3,5] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,1] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,1,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,3,4,1,6,5] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,3,4,6,1,5] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,3,4,1,5,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,1,5,6,4] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,3,1,5,4,6] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,3,5,1,6,4] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,3,5,6,1,4] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4,6] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,3,1,4,6,5] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,6,4,5] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,3,6,1,4,5] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,3,1,4,5,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,5,6,3] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,1,4,5,3,6] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5] => [3,3]
=> [3]
=> 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,6,3,5] => [3,3]
=> [3]
=> 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,5,6] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,4,1,5,6,3] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [2,4,1,5,3,6] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [2,4,5,1,6,3] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,4,5,6,1,3] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [2,4,5,1,3,6] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,6,5] => [3,3]
=> [3]
=> 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,4,5,6,2] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,4,5,2,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,4,2,5,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,4,5,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,1,2,4,5,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,2,4,5,6,3] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,2,4,5,3,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,2,4,3,5,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,4,2,3,5,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [4,1,2,3,5,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,2,3,5,6,4] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
Description
The cube of the number of standard Young tableaux with shape given by the partition.
Matching statistic: St001128
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001128: Integer partitions ⟶ ℤResult quality: 12% ●values known / values provided: 76%●distinct values known / distinct values provided: 12%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001128: Integer partitions ⟶ ℤResult quality: 12% ●values known / values provided: 76%●distinct values known / distinct values provided: 12%
Values
[1,0,1,0]
=> [2,1] => [1,1]
=> [1]
=> ? ∊ {1,2}
[1,1,0,0]
=> [1,2] => [2]
=> []
=> ? ∊ {1,2}
[1,0,1,0,1,0]
=> [2,3,1] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,3}
[1,0,1,1,0,0]
=> [2,1,3] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,3}
[1,1,0,0,1,0]
=> [1,3,2] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,3}
[1,1,0,1,0,0]
=> [3,1,2] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,3}
[1,1,1,0,0,0]
=> [1,2,3] => [3]
=> []
=> ? ∊ {1,1,1,1,3}
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [2,2]
=> [2]
=> 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [2,2]
=> [2]
=> 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [2,2]
=> [2]
=> 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,4}
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,2]
=> [2]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [3,2]
=> [2]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [3,2]
=> [2]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [3,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [3,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [3,2]
=> [2]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [3,2]
=> [2]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [3,2]
=> [2]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [3,2]
=> [2]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [3,2]
=> [2]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [3,2]
=> [2]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [3,2]
=> [2]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [3,2]
=> [2]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [3,2]
=> [2]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [3,2]
=> [2]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [3,2]
=> [2]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [3,2]
=> [2]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,2]
=> [2]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [3,2]
=> [2]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [3,2]
=> [2]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => [3,2]
=> [2]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => [3,2]
=> [2]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,3,5] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => [3,2]
=> [2]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => [3,2]
=> [2]
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> [4,5,1,2,3] => [3,2]
=> [2]
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [4,1,2,3,5] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [5]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,5}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,1] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,1,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,3,4,1,6,5] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,3,4,6,1,5] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,3,4,1,5,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,1,5,6,4] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,3,1,5,4,6] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,3,5,1,6,4] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,3,5,6,1,4] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4,6] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,3,1,4,6,5] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,6,4,5] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,3,6,1,4,5] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,3,1,4,5,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,5,6,3] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,1,4,5,3,6] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5] => [3,3]
=> [3]
=> 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,6,3,5] => [3,3]
=> [3]
=> 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,5,6] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,4,1,5,6,3] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [2,4,1,5,3,6] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [2,4,5,1,6,3] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,4,5,6,1,3] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [2,4,5,1,3,6] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,6,5] => [3,3]
=> [3]
=> 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,4,5,6,2] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,4,5,2,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,4,2,5,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,4,5,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,1,2,4,5,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,2,4,5,6,3] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,2,4,5,3,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,2,4,3,5,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,4,2,3,5,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [4,1,2,3,5,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,2,3,5,6,4] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,6}
Description
The exponens consonantiae of a partition.
This is the quotient of the least common multiple and the greatest common divior of the parts of the partiton. See [1, Caput sextum, §19-§22].
The following 77 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001568The smallest positive integer that does not appear twice in the partition. St000700The protection number of an ordered tree. St000667The greatest common divisor of the parts of the partition. St000781The number of proper colouring schemes of a Ferrers diagram. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001933The largest multiplicity of a part in an integer partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001075The minimal size of a block of a set partition. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St001119The length of a shortest maximal path in a graph. St001316The domatic number of a graph. St000908The length of the shortest maximal antichain in a poset. St000487The length of the shortest cycle of a permutation. St000210Minimum over maximum difference of elements in cycles. St001571The Cartan determinant of the integer partition. St000326The position of the first one in a binary word after appending a 1 at the end. St000627The exponent of a binary word. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001256Number of simple reflexive modules that are 2-stable reflexive. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001481The minimal height of a peak of a Dyck path. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St000260The radius of a connected graph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001722The number of minimal chains with small intervals between a binary word and the top element. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St000618The number of self-evacuating tableaux of given shape. St001280The number of parts of an integer partition that are at least two. St001432The order dimension of the partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000640The rank of the largest boolean interval in a poset. St000914The sum of the values of the Möbius function of a poset. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001890The maximum magnitude of the Möbius function of a poset. St001162The minimum jump of a permutation. St001330The hat guessing number of a graph. St000456The monochromatic index of a connected graph. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St000310The minimal degree of a vertex of a graph. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001344The neighbouring number of a permutation. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St001877Number of indecomposable injective modules with projective dimension 2. St000633The size of the automorphism group of a poset. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001820The size of the image of the pop stack sorting operator. St001846The number of elements which do not have a complement in the lattice. St001096The size of the overlap set of a permutation. St000741The Colin de Verdière graph invariant. St000750The number of occurrences of the pattern 4213 in a permutation. St001884The number of borders of a binary word. St000782The indicator function of whether a given perfect matching is an L & P matching. St000902 The minimal number of repetitions of an integer composition. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001130The number of two successive successions in a permutation.
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