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Your data matches 29 different statistics following compositions of up to 3 maps.
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Matching statistic: St000394
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00142: Dyck paths —promotion⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1,0]
=> [1,0]
=> 0
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 0
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 2
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 3
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 2
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 3
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 3
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 4
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 5
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [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]
=> 0
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 1
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> 2
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 2
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> 3
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 3
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> 4
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 4
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 4
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 5
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 6
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [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]
=> 0
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> 1
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> 2
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 1
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> 2
Description
The sum of the heights of the peaks of a Dyck path minus the number of peaks.
Matching statistic: St001034
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001034: Dyck paths ⟶ ℤResult quality: 95% ●values known / values provided: 96%●distinct values known / distinct values provided: 95%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001034: Dyck paths ⟶ ℤResult quality: 95% ●values known / values provided: 96%●distinct values known / distinct values provided: 95%
Values
[1]
=> []
=> ?
=> ?
=> ? = 0 - 1
[2]
=> []
=> ?
=> ?
=> ? = 0 - 1
[1,1]
=> [1]
=> []
=> []
=> 0 = 1 - 1
[3]
=> []
=> ?
=> ?
=> ? = 0 - 1
[2,1]
=> [1]
=> []
=> []
=> 0 = 1 - 1
[1,1,1]
=> [1,1]
=> [1]
=> [1,0]
=> 1 = 2 - 1
[4]
=> []
=> ?
=> ?
=> ? = 0 - 1
[3,1]
=> [1]
=> []
=> []
=> 0 = 1 - 1
[2,2]
=> [2]
=> []
=> []
=> 0 = 1 - 1
[2,1,1]
=> [1,1]
=> [1]
=> [1,0]
=> 1 = 2 - 1
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2 = 3 - 1
[5]
=> []
=> ?
=> ?
=> ? = 0 - 1
[4,1]
=> [1]
=> []
=> []
=> 0 = 1 - 1
[3,2]
=> [2]
=> []
=> []
=> 0 = 1 - 1
[3,1,1]
=> [1,1]
=> [1]
=> [1,0]
=> 1 = 2 - 1
[2,2,1]
=> [2,1]
=> [1]
=> [1,0]
=> 1 = 2 - 1
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2 = 3 - 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3 = 4 - 1
[6]
=> []
=> ?
=> ?
=> ? = 0 - 1
[5,1]
=> [1]
=> []
=> []
=> 0 = 1 - 1
[4,2]
=> [2]
=> []
=> []
=> 0 = 1 - 1
[4,1,1]
=> [1,1]
=> [1]
=> [1,0]
=> 1 = 2 - 1
[3,3]
=> [3]
=> []
=> []
=> 0 = 1 - 1
[3,2,1]
=> [2,1]
=> [1]
=> [1,0]
=> 1 = 2 - 1
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2 = 3 - 1
[2,2,2]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 2 = 3 - 1
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2 = 3 - 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3 = 4 - 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 4 = 5 - 1
[7]
=> []
=> ?
=> ?
=> ? = 0 - 1
[6,1]
=> [1]
=> []
=> []
=> 0 = 1 - 1
[5,2]
=> [2]
=> []
=> []
=> 0 = 1 - 1
[5,1,1]
=> [1,1]
=> [1]
=> [1,0]
=> 1 = 2 - 1
[4,3]
=> [3]
=> []
=> []
=> 0 = 1 - 1
[4,2,1]
=> [2,1]
=> [1]
=> [1,0]
=> 1 = 2 - 1
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2 = 3 - 1
[3,3,1]
=> [3,1]
=> [1]
=> [1,0]
=> 1 = 2 - 1
[3,2,2]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 2 = 3 - 1
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2 = 3 - 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3 = 4 - 1
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 3 = 4 - 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3 = 4 - 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 4 = 5 - 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 5 = 6 - 1
[8]
=> []
=> ?
=> ?
=> ? = 0 - 1
[7,1]
=> [1]
=> []
=> []
=> 0 = 1 - 1
[6,2]
=> [2]
=> []
=> []
=> 0 = 1 - 1
[6,1,1]
=> [1,1]
=> [1]
=> [1,0]
=> 1 = 2 - 1
[5,3]
=> [3]
=> []
=> []
=> 0 = 1 - 1
[5,2,1]
=> [2,1]
=> [1]
=> [1,0]
=> 1 = 2 - 1
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2 = 3 - 1
[4,4]
=> [4]
=> []
=> []
=> 0 = 1 - 1
[4,3,1]
=> [3,1]
=> [1]
=> [1,0]
=> 1 = 2 - 1
[4,2,2]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 2 = 3 - 1
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2 = 3 - 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3 = 4 - 1
[3,3,2]
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 2 = 3 - 1
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 2 = 3 - 1
[9]
=> []
=> ?
=> ?
=> ? = 0 - 1
[10]
=> []
=> ?
=> ?
=> ? = 0 - 1
[11]
=> []
=> ?
=> ?
=> ? = 0 - 1
Description
The area of the parallelogram polyomino associated with the Dyck path.
The (bivariate) generating function is given in [1].
Matching statistic: St000293
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00135: Binary words —rotate front-to-back⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 84% ●values known / values provided: 94%●distinct values known / distinct values provided: 84%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00135: Binary words —rotate front-to-back⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 84% ●values known / values provided: 94%●distinct values known / distinct values provided: 84%
Values
[1]
=> []
=> => ? => ? = 0 - 1
[2]
=> []
=> => ? => ? = 0 - 1
[1,1]
=> [1]
=> 10 => 01 => 0 = 1 - 1
[3]
=> []
=> => ? => ? = 0 - 1
[2,1]
=> [1]
=> 10 => 01 => 0 = 1 - 1
[1,1,1]
=> [1,1]
=> 110 => 101 => 1 = 2 - 1
[4]
=> []
=> => ? => ? = 0 - 1
[3,1]
=> [1]
=> 10 => 01 => 0 = 1 - 1
[2,2]
=> [2]
=> 100 => 001 => 0 = 1 - 1
[2,1,1]
=> [1,1]
=> 110 => 101 => 1 = 2 - 1
[1,1,1,1]
=> [1,1,1]
=> 1110 => 1101 => 2 = 3 - 1
[5]
=> []
=> => ? => ? = 0 - 1
[4,1]
=> [1]
=> 10 => 01 => 0 = 1 - 1
[3,2]
=> [2]
=> 100 => 001 => 0 = 1 - 1
[3,1,1]
=> [1,1]
=> 110 => 101 => 1 = 2 - 1
[2,2,1]
=> [2,1]
=> 1010 => 0101 => 1 = 2 - 1
[2,1,1,1]
=> [1,1,1]
=> 1110 => 1101 => 2 = 3 - 1
[1,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 11101 => 3 = 4 - 1
[6]
=> []
=> => ? => ? = 0 - 1
[5,1]
=> [1]
=> 10 => 01 => 0 = 1 - 1
[4,2]
=> [2]
=> 100 => 001 => 0 = 1 - 1
[4,1,1]
=> [1,1]
=> 110 => 101 => 1 = 2 - 1
[3,3]
=> [3]
=> 1000 => 0001 => 0 = 1 - 1
[3,2,1]
=> [2,1]
=> 1010 => 0101 => 1 = 2 - 1
[3,1,1,1]
=> [1,1,1]
=> 1110 => 1101 => 2 = 3 - 1
[2,2,2]
=> [2,2]
=> 1100 => 1001 => 2 = 3 - 1
[2,2,1,1]
=> [2,1,1]
=> 10110 => 01101 => 2 = 3 - 1
[2,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 11101 => 3 = 4 - 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 111101 => 4 = 5 - 1
[7]
=> []
=> => ? => ? = 0 - 1
[6,1]
=> [1]
=> 10 => 01 => 0 = 1 - 1
[5,2]
=> [2]
=> 100 => 001 => 0 = 1 - 1
[5,1,1]
=> [1,1]
=> 110 => 101 => 1 = 2 - 1
[4,3]
=> [3]
=> 1000 => 0001 => 0 = 1 - 1
[4,2,1]
=> [2,1]
=> 1010 => 0101 => 1 = 2 - 1
[4,1,1,1]
=> [1,1,1]
=> 1110 => 1101 => 2 = 3 - 1
[3,3,1]
=> [3,1]
=> 10010 => 00101 => 1 = 2 - 1
[3,2,2]
=> [2,2]
=> 1100 => 1001 => 2 = 3 - 1
[3,2,1,1]
=> [2,1,1]
=> 10110 => 01101 => 2 = 3 - 1
[3,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 11101 => 3 = 4 - 1
[2,2,2,1]
=> [2,2,1]
=> 11010 => 10101 => 3 = 4 - 1
[2,2,1,1,1]
=> [2,1,1,1]
=> 101110 => 011101 => 3 = 4 - 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 111101 => 4 = 5 - 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 1111110 => 1111101 => 5 = 6 - 1
[8]
=> []
=> => ? => ? = 0 - 1
[7,1]
=> [1]
=> 10 => 01 => 0 = 1 - 1
[6,2]
=> [2]
=> 100 => 001 => 0 = 1 - 1
[6,1,1]
=> [1,1]
=> 110 => 101 => 1 = 2 - 1
[5,3]
=> [3]
=> 1000 => 0001 => 0 = 1 - 1
[5,2,1]
=> [2,1]
=> 1010 => 0101 => 1 = 2 - 1
[5,1,1,1]
=> [1,1,1]
=> 1110 => 1101 => 2 = 3 - 1
[4,4]
=> [4]
=> 10000 => 00001 => 0 = 1 - 1
[4,3,1]
=> [3,1]
=> 10010 => 00101 => 1 = 2 - 1
[4,2,2]
=> [2,2]
=> 1100 => 1001 => 2 = 3 - 1
[4,2,1,1]
=> [2,1,1]
=> 10110 => 01101 => 2 = 3 - 1
[4,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 11101 => 3 = 4 - 1
[3,3,2]
=> [3,2]
=> 10100 => 01001 => 2 = 3 - 1
[3,3,1,1]
=> [3,1,1]
=> 100110 => 001101 => 2 = 3 - 1
[9]
=> []
=> => ? => ? = 0 - 1
[10]
=> []
=> => ? => ? = 0 - 1
[11]
=> []
=> => ? => ? = 0 - 1
[2,2,2,2,2,2,2,2,2]
=> [2,2,2,2,2,2,2,2]
=> 1111111100 => 1111111001 => ? = 15 - 1
[3,3,3,3,3,3,3,3]
=> [3,3,3,3,3,3,3]
=> 1111111000 => ? => ? = 19 - 1
[2,2,2,2,2,2,2,2,2,2]
=> [2,2,2,2,2,2,2,2,2]
=> 11111111100 => ? => ? = 17 - 1
[6,6,6,6,6]
=> [6,6,6,6]
=> 1111000000 => 1110000001 => ? = 19 - 1
[7,7,7,7]
=> [7,7,7]
=> 1110000000 => ? => ? = 15 - 1
[8,8,8]
=> [8,8]
=> 1100000000 => 1000000001 => ? = 9 - 1
Description
The number of inversions of a binary word.
Matching statistic: St001176
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 74% ●values known / values provided: 93%●distinct values known / distinct values provided: 74%
St001176: Integer partitions ⟶ ℤResult quality: 74% ●values known / values provided: 93%●distinct values known / distinct values provided: 74%
Values
[1]
=> []
=> ? = 0 - 1
[2]
=> []
=> ? = 0 - 1
[1,1]
=> [1]
=> 0 = 1 - 1
[3]
=> []
=> ? = 0 - 1
[2,1]
=> [1]
=> 0 = 1 - 1
[1,1,1]
=> [1,1]
=> 1 = 2 - 1
[4]
=> []
=> ? = 0 - 1
[3,1]
=> [1]
=> 0 = 1 - 1
[2,2]
=> [2]
=> 0 = 1 - 1
[2,1,1]
=> [1,1]
=> 1 = 2 - 1
[1,1,1,1]
=> [1,1,1]
=> 2 = 3 - 1
[5]
=> []
=> ? = 0 - 1
[4,1]
=> [1]
=> 0 = 1 - 1
[3,2]
=> [2]
=> 0 = 1 - 1
[3,1,1]
=> [1,1]
=> 1 = 2 - 1
[2,2,1]
=> [2,1]
=> 1 = 2 - 1
[2,1,1,1]
=> [1,1,1]
=> 2 = 3 - 1
[1,1,1,1,1]
=> [1,1,1,1]
=> 3 = 4 - 1
[6]
=> []
=> ? = 0 - 1
[5,1]
=> [1]
=> 0 = 1 - 1
[4,2]
=> [2]
=> 0 = 1 - 1
[4,1,1]
=> [1,1]
=> 1 = 2 - 1
[3,3]
=> [3]
=> 0 = 1 - 1
[3,2,1]
=> [2,1]
=> 1 = 2 - 1
[3,1,1,1]
=> [1,1,1]
=> 2 = 3 - 1
[2,2,2]
=> [2,2]
=> 2 = 3 - 1
[2,2,1,1]
=> [2,1,1]
=> 2 = 3 - 1
[2,1,1,1,1]
=> [1,1,1,1]
=> 3 = 4 - 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 4 = 5 - 1
[7]
=> []
=> ? = 0 - 1
[6,1]
=> [1]
=> 0 = 1 - 1
[5,2]
=> [2]
=> 0 = 1 - 1
[5,1,1]
=> [1,1]
=> 1 = 2 - 1
[4,3]
=> [3]
=> 0 = 1 - 1
[4,2,1]
=> [2,1]
=> 1 = 2 - 1
[4,1,1,1]
=> [1,1,1]
=> 2 = 3 - 1
[3,3,1]
=> [3,1]
=> 1 = 2 - 1
[3,2,2]
=> [2,2]
=> 2 = 3 - 1
[3,2,1,1]
=> [2,1,1]
=> 2 = 3 - 1
[3,1,1,1,1]
=> [1,1,1,1]
=> 3 = 4 - 1
[2,2,2,1]
=> [2,2,1]
=> 3 = 4 - 1
[2,2,1,1,1]
=> [2,1,1,1]
=> 3 = 4 - 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> 4 = 5 - 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 5 = 6 - 1
[8]
=> []
=> ? = 0 - 1
[7,1]
=> [1]
=> 0 = 1 - 1
[6,2]
=> [2]
=> 0 = 1 - 1
[6,1,1]
=> [1,1]
=> 1 = 2 - 1
[5,3]
=> [3]
=> 0 = 1 - 1
[5,2,1]
=> [2,1]
=> 1 = 2 - 1
[5,1,1,1]
=> [1,1,1]
=> 2 = 3 - 1
[4,4]
=> [4]
=> 0 = 1 - 1
[4,3,1]
=> [3,1]
=> 1 = 2 - 1
[4,2,2]
=> [2,2]
=> 2 = 3 - 1
[4,2,1,1]
=> [2,1,1]
=> 2 = 3 - 1
[4,1,1,1,1]
=> [1,1,1,1]
=> 3 = 4 - 1
[3,3,2]
=> [3,2]
=> 2 = 3 - 1
[3,3,1,1]
=> [3,1,1]
=> 2 = 3 - 1
[9]
=> []
=> ? = 0 - 1
[10]
=> []
=> ? = 0 - 1
[11]
=> []
=> ? = 0 - 1
[3,3,3,3,3,3,3,3]
=> [3,3,3,3,3,3,3]
=> ? = 19 - 1
[2,2,2,2,2,2,2,2,2,2]
=> [2,2,2,2,2,2,2,2,2]
=> ? = 17 - 1
[3,3,3,3,3,3,3]
=> [3,3,3,3,3,3]
=> ? = 16 - 1
[4,4,4,4,4,4]
=> [4,4,4,4,4]
=> ? = 17 - 1
[6,6,6,6]
=> [6,6,6]
=> ? = 13 - 1
[6,6,6,6,6]
=> [6,6,6,6]
=> ? = 19 - 1
[5,5,5,5,5,5]
=> [5,5,5,5,5]
=> ? = 21 - 1
[7,7,7,7]
=> [7,7,7]
=> ? = 15 - 1
[5,5,5,5,5]
=> [5,5,5,5]
=> ? = 16 - 1
Description
The size of a partition minus its first part.
This is the number of boxes in its diagram that are not in the first row.
Matching statistic: St000228
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 58% ●values known / values provided: 91%●distinct values known / distinct values provided: 58%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 58% ●values known / values provided: 91%●distinct values known / distinct values provided: 58%
Values
[1]
=> []
=> ?
=> ? = 0 - 1
[2]
=> []
=> ?
=> ? = 0 - 1
[1,1]
=> [1]
=> []
=> 0 = 1 - 1
[3]
=> []
=> ?
=> ? = 0 - 1
[2,1]
=> [1]
=> []
=> 0 = 1 - 1
[1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[4]
=> []
=> ?
=> ? = 0 - 1
[3,1]
=> [1]
=> []
=> 0 = 1 - 1
[2,2]
=> [2]
=> []
=> 0 = 1 - 1
[2,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2 = 3 - 1
[5]
=> []
=> ?
=> ? = 0 - 1
[4,1]
=> [1]
=> []
=> 0 = 1 - 1
[3,2]
=> [2]
=> []
=> 0 = 1 - 1
[3,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[2,2,1]
=> [2,1]
=> [1]
=> 1 = 2 - 1
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3 = 4 - 1
[6]
=> []
=> ?
=> ? = 0 - 1
[5,1]
=> [1]
=> []
=> 0 = 1 - 1
[4,2]
=> [2]
=> []
=> 0 = 1 - 1
[4,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[3,3]
=> [3]
=> []
=> 0 = 1 - 1
[3,2,1]
=> [2,1]
=> [1]
=> 1 = 2 - 1
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2 = 3 - 1
[2,2,2]
=> [2,2]
=> [2]
=> 2 = 3 - 1
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 2 = 3 - 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3 = 4 - 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 4 = 5 - 1
[7]
=> []
=> ?
=> ? = 0 - 1
[6,1]
=> [1]
=> []
=> 0 = 1 - 1
[5,2]
=> [2]
=> []
=> 0 = 1 - 1
[5,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[4,3]
=> [3]
=> []
=> 0 = 1 - 1
[4,2,1]
=> [2,1]
=> [1]
=> 1 = 2 - 1
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2 = 3 - 1
[3,3,1]
=> [3,1]
=> [1]
=> 1 = 2 - 1
[3,2,2]
=> [2,2]
=> [2]
=> 2 = 3 - 1
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 2 = 3 - 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3 = 4 - 1
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 3 = 4 - 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 3 = 4 - 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 4 = 5 - 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 5 = 6 - 1
[8]
=> []
=> ?
=> ? = 0 - 1
[7,1]
=> [1]
=> []
=> 0 = 1 - 1
[6,2]
=> [2]
=> []
=> 0 = 1 - 1
[6,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[5,3]
=> [3]
=> []
=> 0 = 1 - 1
[5,2,1]
=> [2,1]
=> [1]
=> 1 = 2 - 1
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2 = 3 - 1
[4,4]
=> [4]
=> []
=> 0 = 1 - 1
[4,3,1]
=> [3,1]
=> [1]
=> 1 = 2 - 1
[4,2,2]
=> [2,2]
=> [2]
=> 2 = 3 - 1
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> 2 = 3 - 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3 = 4 - 1
[3,3,2]
=> [3,2]
=> [2]
=> 2 = 3 - 1
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> 2 = 3 - 1
[9]
=> []
=> ?
=> ? = 0 - 1
[10]
=> []
=> ?
=> ? = 0 - 1
[11]
=> []
=> ?
=> ? = 0 - 1
[2,2,2,2,2,2,2,2]
=> [2,2,2,2,2,2,2]
=> [2,2,2,2,2,2]
=> ? = 13 - 1
[3,3,3,3,3,2]
=> [3,3,3,3,2]
=> [3,3,3,2]
=> ? = 12 - 1
[3,2,2,2,2,2,2,2]
=> [2,2,2,2,2,2,2]
=> [2,2,2,2,2,2]
=> ? = 13 - 1
[2,2,2,2,2,2,2,2,2]
=> [2,2,2,2,2,2,2,2]
=> [2,2,2,2,2,2,2]
=> ? = 15 - 1
[3,3,3,3,3,3]
=> [3,3,3,3,3]
=> [3,3,3,3]
=> ? = 13 - 1
[3,3,3,3,3,3,3,3]
=> [3,3,3,3,3,3,3]
=> [3,3,3,3,3,3]
=> ? = 19 - 1
[2,2,2,2,2,2,2,2,2,2]
=> [2,2,2,2,2,2,2,2,2]
=> [2,2,2,2,2,2,2,2]
=> ? = 17 - 1
[3,3,3,3,3,3,3]
=> [3,3,3,3,3,3]
=> [3,3,3,3,3]
=> ? = 16 - 1
[4,3,3,3,3,3]
=> [3,3,3,3,3]
=> [3,3,3,3]
=> ? = 13 - 1
[4,4,4,4,4]
=> [4,4,4,4]
=> [4,4,4]
=> ? = 13 - 1
[4,4,4,4,4,4]
=> [4,4,4,4,4]
=> [4,4,4,4]
=> ? = 17 - 1
[6,6,6,6]
=> [6,6,6]
=> [6,6]
=> ? = 13 - 1
[6,6,6,6,6]
=> [6,6,6,6]
=> [6,6,6]
=> ? = 19 - 1
[5,5,5,5,5,5]
=> [5,5,5,5,5]
=> [5,5,5,5]
=> ? = 21 - 1
[7,7,7,7]
=> [7,7,7]
=> [7,7]
=> ? = 15 - 1
[5,5,5,5,5]
=> [5,5,5,5]
=> [5,5,5]
=> ? = 16 - 1
[5,4,4,4,4]
=> [4,4,4,4]
=> [4,4,4]
=> ? = 13 - 1
Description
The size of a partition.
This statistic is the constant statistic of the level sets.
Matching statistic: St000507
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 63% ●values known / values provided: 88%●distinct values known / distinct values provided: 63%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 63% ●values known / values provided: 88%●distinct values known / distinct values provided: 63%
Values
[1]
=> []
=> []
=> []
=> 0
[2]
=> []
=> []
=> []
=> 0
[1,1]
=> [1]
=> [1]
=> [[1]]
=> 1
[3]
=> []
=> []
=> []
=> 0
[2,1]
=> [1]
=> [1]
=> [[1]]
=> 1
[1,1,1]
=> [1,1]
=> [2]
=> [[1,2]]
=> 2
[4]
=> []
=> []
=> []
=> 0
[3,1]
=> [1]
=> [1]
=> [[1]]
=> 1
[2,2]
=> [2]
=> [1,1]
=> [[1],[2]]
=> 1
[2,1,1]
=> [1,1]
=> [2]
=> [[1,2]]
=> 2
[1,1,1,1]
=> [1,1,1]
=> [3]
=> [[1,2,3]]
=> 3
[5]
=> []
=> []
=> []
=> 0
[4,1]
=> [1]
=> [1]
=> [[1]]
=> 1
[3,2]
=> [2]
=> [1,1]
=> [[1],[2]]
=> 1
[3,1,1]
=> [1,1]
=> [2]
=> [[1,2]]
=> 2
[2,2,1]
=> [2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2
[2,1,1,1]
=> [1,1,1]
=> [3]
=> [[1,2,3]]
=> 3
[1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> [[1,2,3,4]]
=> 4
[6]
=> []
=> []
=> []
=> 0
[5,1]
=> [1]
=> [1]
=> [[1]]
=> 1
[4,2]
=> [2]
=> [1,1]
=> [[1],[2]]
=> 1
[4,1,1]
=> [1,1]
=> [2]
=> [[1,2]]
=> 2
[3,3]
=> [3]
=> [1,1,1]
=> [[1],[2],[3]]
=> 1
[3,2,1]
=> [2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2
[3,1,1,1]
=> [1,1,1]
=> [3]
=> [[1,2,3]]
=> 3
[2,2,2]
=> [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 3
[2,2,1,1]
=> [2,1,1]
=> [3,1]
=> [[1,2,3],[4]]
=> 3
[2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> [[1,2,3,4]]
=> 4
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> [[1,2,3,4,5]]
=> 5
[7]
=> []
=> []
=> []
=> 0
[6,1]
=> [1]
=> [1]
=> [[1]]
=> 1
[5,2]
=> [2]
=> [1,1]
=> [[1],[2]]
=> 1
[5,1,1]
=> [1,1]
=> [2]
=> [[1,2]]
=> 2
[4,3]
=> [3]
=> [1,1,1]
=> [[1],[2],[3]]
=> 1
[4,2,1]
=> [2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2
[4,1,1,1]
=> [1,1,1]
=> [3]
=> [[1,2,3]]
=> 3
[3,3,1]
=> [3,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[3,2,2]
=> [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 3
[3,2,1,1]
=> [2,1,1]
=> [3,1]
=> [[1,2,3],[4]]
=> 3
[3,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> [[1,2,3,4]]
=> 4
[2,2,2,1]
=> [2,2,1]
=> [3,2]
=> [[1,2,3],[4,5]]
=> 4
[2,2,1,1,1]
=> [2,1,1,1]
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> [[1,2,3,4,5]]
=> 5
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [6]
=> [[1,2,3,4,5,6]]
=> 6
[8]
=> []
=> []
=> []
=> 0
[7,1]
=> [1]
=> [1]
=> [[1]]
=> 1
[6,2]
=> [2]
=> [1,1]
=> [[1],[2]]
=> 1
[6,1,1]
=> [1,1]
=> [2]
=> [[1,2]]
=> 2
[5,3]
=> [3]
=> [1,1,1]
=> [[1],[2],[3]]
=> 1
[5,2,1]
=> [2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2
[2,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [6,5]
=> [[1,2,3,4,5,6],[7,8,9,10,11]]
=> ? = 10
[3,3,2,2,2,2]
=> [3,2,2,2,2]
=> [5,5,1]
=> [[1,2,3,4,5],[6,7,8,9,10],[11]]
=> ? = 9
[3,3,3,3,3]
=> [3,3,3,3]
=> [4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12]]
=> ? = 10
[3,3,3,2,2,2]
=> [3,3,2,2,2]
=> [5,5,2]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12]]
=> ? = 10
[5,5,5,1]
=> [5,5,1]
=> [3,2,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11]]
=> ? = 7
[4,4,4,4]
=> [4,4,4]
=> [3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12]]
=> ? = 9
[4,3,3,3,3]
=> [3,3,3,3]
=> [4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12]]
=> ? = 10
[3,3,3,3,3,1]
=> [3,3,3,3,1]
=> [5,4,4]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12,13]]
=> ? = 11
[3,3,3,3,2,2]
=> [3,3,3,2,2]
=> [5,5,3]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12,13]]
=> ? = 11
[3,3,2,2,2,2,2]
=> [3,2,2,2,2,2]
=> [6,6,1]
=> [[1,2,3,4,5,6],[7,8,9,10,11,12],[13]]
=> ? = 11
[2,2,2,2,2,2,2,2]
=> [2,2,2,2,2,2,2]
=> [7,7]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> ? = 13
[6,6,5]
=> [6,5]
=> [2,2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11]]
=> ? = 6
[5,4,4,4]
=> [4,4,4]
=> [3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12]]
=> ? = 9
[4,4,4,4,1]
=> [4,4,4,1]
=> [4,3,3,3]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11,12,13]]
=> ? = 10
[4,4,3,3,3]
=> [4,3,3,3]
=> [4,4,4,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13]]
=> ? = 10
[3,3,3,3,3,2]
=> [3,3,3,3,2]
=> [5,5,4]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12,13,14]]
=> ? = 12
[3,2,2,2,2,2,2,2]
=> [2,2,2,2,2,2,2]
=> [7,7]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> ? = 13
[4,4,4,4,2]
=> [4,4,4,2]
=> [4,4,3,3]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13,14]]
=> ? = 11
[2,2,2,2,2,2,2,2,2]
=> [2,2,2,2,2,2,2,2]
=> [8,8]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12,13,14,15,16]]
=> ? = 15
[3,3,3,3,3,3]
=> [3,3,3,3,3]
=> [5,5,5]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12,13,14,15]]
=> ? = 13
[3,3,3,3,3,3,3,3]
=> [3,3,3,3,3,3,3]
=> [7,7,7]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14],[15,16,17,18,19,20,21]]
=> ? = 19
[2,2,2,2,2,2,2,2,2,2]
=> [2,2,2,2,2,2,2,2,2]
=> [9,9]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12,13,14,15,16,17,18]]
=> ? = 17
[3,3,3,3,3,3,3]
=> [3,3,3,3,3,3]
=> [6,6,6]
=> [[1,2,3,4,5,6],[7,8,9,10,11,12],[13,14,15,16,17,18]]
=> ? = 16
[4,3,3,3,3,3]
=> [3,3,3,3,3]
=> [5,5,5]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12,13,14,15]]
=> ? = 13
[4,4,4,4,4]
=> [4,4,4,4]
=> [4,4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16]]
=> ? = 13
[4,4,4,4,4,4]
=> [4,4,4,4,4]
=> [5,5,5,5]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12,13,14,15],[16,17,18,19,20]]
=> ? = 17
[5,5,5,5]
=> [5,5,5]
=> [3,3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12],[13,14,15]]
=> ? = 11
[6,6,6,6]
=> [6,6,6]
=> [3,3,3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12],[13,14,15],[16,17,18]]
=> ? = 13
[6,6,6,6,6]
=> [6,6,6,6]
=> [4,4,4,4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16],[17,18,19,20],[21,22,23,24]]
=> ? = 19
[5,5,5,5,5,5]
=> [5,5,5,5,5]
=> [5,5,5,5,5]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12,13,14,15],[16,17,18,19,20],[21,22,23,24,25]]
=> ? = 21
[7,7,7,7]
=> [7,7,7]
=> [3,3,3,3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12],[13,14,15],[16,17,18],[19,20,21]]
=> ? = 15
[7,7,7]
=> [7,7]
=> [2,2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14]]
=> ? = 8
[5,5,5,5,5]
=> [5,5,5,5]
=> [4,4,4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16],[17,18,19,20]]
=> ? = 16
[8,8,8]
=> [8,8]
=> [2,2,2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14],[15,16]]
=> ? = 9
[6,5,5,5]
=> [5,5,5]
=> [3,3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12],[13,14,15]]
=> ? = 11
[5,4,4,4,4]
=> [4,4,4,4]
=> [4,4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16]]
=> ? = 13
Description
The number of ascents of a standard tableau.
Entry $i$ of a standard Young tableau is an '''ascent''' if $i+1$ appears to the right or above $i$ in the tableau (with respect to the English notation for tableaux).
Matching statistic: St000738
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000738: Standard tableaux ⟶ ℤResult quality: 58% ●values known / values provided: 85%●distinct values known / distinct values provided: 58%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000738: Standard tableaux ⟶ ℤResult quality: 58% ●values known / values provided: 85%●distinct values known / distinct values provided: 58%
Values
[1]
=> []
=> []
=> ? = 0
[2]
=> []
=> []
=> ? = 0
[1,1]
=> [1]
=> [[1]]
=> 1
[3]
=> []
=> []
=> ? = 0
[2,1]
=> [1]
=> [[1]]
=> 1
[1,1,1]
=> [1,1]
=> [[1],[2]]
=> 2
[4]
=> []
=> []
=> ? = 0
[3,1]
=> [1]
=> [[1]]
=> 1
[2,2]
=> [2]
=> [[1,2]]
=> 1
[2,1,1]
=> [1,1]
=> [[1],[2]]
=> 2
[1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3
[5]
=> []
=> []
=> ? = 0
[4,1]
=> [1]
=> [[1]]
=> 1
[3,2]
=> [2]
=> [[1,2]]
=> 1
[3,1,1]
=> [1,1]
=> [[1],[2]]
=> 2
[2,2,1]
=> [2,1]
=> [[1,3],[2]]
=> 2
[2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3
[1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4
[6]
=> []
=> []
=> ? = 0
[5,1]
=> [1]
=> [[1]]
=> 1
[4,2]
=> [2]
=> [[1,2]]
=> 1
[4,1,1]
=> [1,1]
=> [[1],[2]]
=> 2
[3,3]
=> [3]
=> [[1,2,3]]
=> 1
[3,2,1]
=> [2,1]
=> [[1,3],[2]]
=> 2
[3,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3
[2,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 3
[2,2,1,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 3
[2,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 5
[7]
=> []
=> []
=> ? = 0
[6,1]
=> [1]
=> [[1]]
=> 1
[5,2]
=> [2]
=> [[1,2]]
=> 1
[5,1,1]
=> [1,1]
=> [[1],[2]]
=> 2
[4,3]
=> [3]
=> [[1,2,3]]
=> 1
[4,2,1]
=> [2,1]
=> [[1,3],[2]]
=> 2
[4,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3
[3,3,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 2
[3,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 3
[3,2,1,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 3
[3,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4
[2,2,2,1]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 4
[2,2,1,1,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 4
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 5
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> 6
[8]
=> []
=> []
=> ? = 0
[7,1]
=> [1]
=> [[1]]
=> 1
[6,2]
=> [2]
=> [[1,2]]
=> 1
[6,1,1]
=> [1,1]
=> [[1],[2]]
=> 2
[5,3]
=> [3]
=> [[1,2,3]]
=> 1
[5,2,1]
=> [2,1]
=> [[1,3],[2]]
=> 2
[5,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3
[4,4]
=> [4]
=> [[1,2,3,4]]
=> 1
[4,3,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 2
[4,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 3
[4,2,1,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 3
[4,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4
[3,3,2]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 3
[3,3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3
[9]
=> []
=> []
=> ? = 0
[10]
=> []
=> []
=> ? = 0
[11]
=> []
=> []
=> ? = 0
[2,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? = 10
[3,3,2,2,2,2]
=> [3,2,2,2,2]
=> [[1,2,11],[3,4],[5,6],[7,8],[9,10]]
=> ? = 9
[3,3,3,3,3]
=> [3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12]]
=> ? = 10
[3,3,3,2,2,2]
=> [3,3,2,2,2]
=> [[1,2,9],[3,4,12],[5,6],[7,8],[10,11]]
=> ? = 10
[5,5,5,1]
=> [5,5,1]
=> [[1,3,4,5,6],[2,8,9,10,11],[7]]
=> ? = 7
[4,4,4,4]
=> [4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12]]
=> ? = 9
[4,3,3,3,3]
=> [3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12]]
=> ? = 10
[3,3,3,3,3,1]
=> [3,3,3,3,1]
=> [[1,3,4],[2,6,7],[5,9,10],[8,12,13],[11]]
=> ? = 11
[3,3,3,3,2,2]
=> [3,3,3,2,2]
=> [[1,2,7],[3,4,10],[5,6,13],[8,9],[11,12]]
=> ? = 11
[3,3,2,2,2,2,2]
=> [3,2,2,2,2,2]
=> [[1,2,13],[3,4],[5,6],[7,8],[9,10],[11,12]]
=> ? = 11
[2,2,2,2,2,2,2,2]
=> [2,2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14]]
=> ? = 13
[6,6,5]
=> [6,5]
=> [[1,2,3,4,5,11],[6,7,8,9,10]]
=> ? = 6
[5,4,4,4]
=> [4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12]]
=> ? = 9
[4,4,4,4,1]
=> [4,4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6,11,12,13],[10]]
=> ? = 10
[4,4,3,3,3]
=> [4,3,3,3]
=> [[1,2,3,13],[4,5,6],[7,8,9],[10,11,12]]
=> ? = 10
[3,3,3,3,3,2]
=> [3,3,3,3,2]
=> [[1,2,5],[3,4,8],[6,7,11],[9,10,14],[12,13]]
=> ? = 12
[3,2,2,2,2,2,2,2]
=> [2,2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14]]
=> ? = 13
[4,4,4,4,2]
=> [4,4,4,2]
=> [[1,2,5,6],[3,4,9,10],[7,8,13,14],[11,12]]
=> ? = 11
[2,2,2,2,2,2,2,2,2]
=> [2,2,2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14],[15,16]]
=> ? = 15
[3,3,3,3,3,3]
=> [3,3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12],[13,14,15]]
=> ? = 13
[3,3,3,3,3,3,3,3]
=> [3,3,3,3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12],[13,14,15],[16,17,18],[19,20,21]]
=> ? = 19
[2,2,2,2,2,2,2,2,2,2]
=> [2,2,2,2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14],[15,16],[17,18]]
=> ? = 17
[3,3,3,3,3,3,3]
=> [3,3,3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12],[13,14,15],[16,17,18]]
=> ? = 16
[4,3,3,3,3,3]
=> [3,3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12],[13,14,15]]
=> ? = 13
[4,4,4,4,4]
=> [4,4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16]]
=> ? = 13
[4,4,4,4,4,4]
=> [4,4,4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16],[17,18,19,20]]
=> ? = 17
[5,5,5,5]
=> [5,5,5]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12,13,14,15]]
=> ? = 11
[6,6,6,6]
=> [6,6,6]
=> [[1,2,3,4,5,6],[7,8,9,10,11,12],[13,14,15,16,17,18]]
=> ? = 13
[6,6,6,6,6]
=> [6,6,6,6]
=> [[1,2,3,4,5,6],[7,8,9,10,11,12],[13,14,15,16,17,18],[19,20,21,22,23,24]]
=> ? = 19
[5,5,5,5,5,5]
=> [5,5,5,5,5]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12,13,14,15],[16,17,18,19,20],[21,22,23,24,25]]
=> ? = 21
[7,7,7,7]
=> [7,7,7]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14],[15,16,17,18,19,20,21]]
=> ? = 15
[7,7,7]
=> [7,7]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> ? = 8
[5,5,5,5,5]
=> [5,5,5,5]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12,13,14,15],[16,17,18,19,20]]
=> ? = 16
[8,8,8]
=> [8,8]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12,13,14,15,16]]
=> ? = 9
[6,5,5,5]
=> [5,5,5]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12,13,14,15]]
=> ? = 11
[5,4,4,4,4]
=> [4,4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16]]
=> ? = 13
Description
The first entry in the last row of a standard tableau.
For the last entry in the first row, see [[St000734]].
Matching statistic: St000734
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 74% ●values known / values provided: 82%●distinct values known / distinct values provided: 74%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 74% ●values known / values provided: 82%●distinct values known / distinct values provided: 74%
Values
[1]
=> []
=> []
=> []
=> ? = 0
[2]
=> []
=> []
=> []
=> ? = 0
[1,1]
=> [1]
=> [[1]]
=> [[1]]
=> 1
[3]
=> []
=> []
=> []
=> ? = 0
[2,1]
=> [1]
=> [[1]]
=> [[1]]
=> 1
[1,1,1]
=> [1,1]
=> [[1],[2]]
=> [[1,2]]
=> 2
[4]
=> []
=> []
=> []
=> ? = 0
[3,1]
=> [1]
=> [[1]]
=> [[1]]
=> 1
[2,2]
=> [2]
=> [[1,2]]
=> [[1],[2]]
=> 1
[2,1,1]
=> [1,1]
=> [[1],[2]]
=> [[1,2]]
=> 2
[1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> 3
[5]
=> []
=> []
=> []
=> ? = 0
[4,1]
=> [1]
=> [[1]]
=> [[1]]
=> 1
[3,2]
=> [2]
=> [[1,2]]
=> [[1],[2]]
=> 1
[3,1,1]
=> [1,1]
=> [[1],[2]]
=> [[1,2]]
=> 2
[2,2,1]
=> [2,1]
=> [[1,3],[2]]
=> [[1,2],[3]]
=> 2
[2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> 3
[1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> 4
[6]
=> []
=> []
=> []
=> ? = 0
[5,1]
=> [1]
=> [[1]]
=> [[1]]
=> 1
[4,2]
=> [2]
=> [[1,2]]
=> [[1],[2]]
=> 1
[4,1,1]
=> [1,1]
=> [[1],[2]]
=> [[1,2]]
=> 2
[3,3]
=> [3]
=> [[1,2,3]]
=> [[1],[2],[3]]
=> 1
[3,2,1]
=> [2,1]
=> [[1,3],[2]]
=> [[1,2],[3]]
=> 2
[3,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> 3
[2,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 3
[2,2,1,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 3
[2,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> 4
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [[1,2,3,4,5]]
=> 5
[7]
=> []
=> []
=> []
=> ? = 0
[6,1]
=> [1]
=> [[1]]
=> [[1]]
=> 1
[5,2]
=> [2]
=> [[1,2]]
=> [[1],[2]]
=> 1
[5,1,1]
=> [1,1]
=> [[1],[2]]
=> [[1,2]]
=> 2
[4,3]
=> [3]
=> [[1,2,3]]
=> [[1],[2],[3]]
=> 1
[4,2,1]
=> [2,1]
=> [[1,3],[2]]
=> [[1,2],[3]]
=> 2
[4,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> 3
[3,3,1]
=> [3,1]
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 2
[3,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 3
[3,2,1,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 3
[3,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> 4
[2,2,2,1]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [[1,2,4],[3,5]]
=> 4
[2,2,1,1,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> 4
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [[1,2,3,4,5]]
=> 5
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [[1,2,3,4,5,6]]
=> 6
[8]
=> []
=> []
=> []
=> ? = 0
[7,1]
=> [1]
=> [[1]]
=> [[1]]
=> 1
[6,2]
=> [2]
=> [[1,2]]
=> [[1],[2]]
=> 1
[6,1,1]
=> [1,1]
=> [[1],[2]]
=> [[1,2]]
=> 2
[5,3]
=> [3]
=> [[1,2,3]]
=> [[1],[2],[3]]
=> 1
[5,2,1]
=> [2,1]
=> [[1,3],[2]]
=> [[1,2],[3]]
=> 2
[5,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> 3
[4,4]
=> [4]
=> [[1,2,3,4]]
=> [[1],[2],[3],[4]]
=> 1
[4,3,1]
=> [3,1]
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 2
[4,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 3
[4,2,1,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 3
[4,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> 4
[3,3,2]
=> [3,2]
=> [[1,2,5],[3,4]]
=> [[1,3],[2,4],[5]]
=> 3
[3,3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> 3
[9]
=> []
=> []
=> []
=> ? = 0
[10]
=> []
=> []
=> []
=> ? = 0
[11]
=> []
=> []
=> []
=> ? = 0
[2,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8,11],[10]]
=> [[1,2,4,6,8,10],[3,5,7,9,11]]
=> ? = 10
[3,3,3,3,2]
=> [3,3,3,2]
=> [[1,2,5],[3,4,8],[6,7,11],[9,10]]
=> [[1,3,6,9],[2,4,7,10],[5,8,11]]
=> ? = 9
[3,3,3,3,1,1]
=> [3,3,3,1,1]
=> [[1,4,5],[2,7,8],[3,10,11],[6],[9]]
=> [[1,2,3,6,9],[4,7,10],[5,8,11]]
=> ? = 9
[3,3,2,2,2,2]
=> [3,2,2,2,2]
=> [[1,2,11],[3,4],[5,6],[7,8],[9,10]]
=> [[1,3,5,7,9],[2,4,6,8,10],[11]]
=> ? = 9
[4,4,4,3]
=> [4,4,3]
=> [[1,2,3,7],[4,5,6,11],[8,9,10]]
=> [[1,4,8],[2,5,9],[3,6,10],[7,11]]
=> ? = 8
[4,4,3,3,1]
=> [4,3,3,1]
=> [[1,3,4,11],[2,6,7],[5,9,10],[8]]
=> [[1,2,5,8],[3,6,9],[4,7,10],[11]]
=> ? = 8
[4,4,3,2,2]
=> [4,3,2,2]
=> [[1,2,7,11],[3,4,10],[5,6],[8,9]]
=> [[1,3,5,8],[2,4,6,9],[7,10],[11]]
=> ? = 8
[4,3,3,3,2]
=> [3,3,3,2]
=> [[1,2,5],[3,4,8],[6,7,11],[9,10]]
=> [[1,3,6,9],[2,4,7,10],[5,8,11]]
=> ? = 9
[3,3,3,3,3]
=> [3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12]]
=> [[1,4,7,10],[2,5,8,11],[3,6,9,12]]
=> ? = 10
[3,3,3,3,2,1]
=> [3,3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8,12],[7,11],[10]]
=> [[1,2,4,7,10],[3,5,8,11],[6,9,12]]
=> ? = 10
[3,3,3,2,2,2]
=> [3,3,2,2,2]
=> [[1,2,9],[3,4,12],[5,6],[7,8],[10,11]]
=> [[1,3,5,7,10],[2,4,6,8,11],[9,12]]
=> ? = 10
[5,5,5,1]
=> [5,5,1]
=> [[1,3,4,5,6],[2,8,9,10,11],[7]]
=> [[1,2,7],[3,8],[4,9],[5,10],[6,11]]
=> ? = 7
[5,5,4,2]
=> [5,4,2]
=> [[1,2,5,6,11],[3,4,9,10],[7,8]]
=> [[1,3,7],[2,4,8],[5,9],[6,10],[11]]
=> ? = 7
[5,5,3,3]
=> [5,3,3]
=> [[1,2,3,10,11],[4,5,6],[7,8,9]]
=> [[1,4,7],[2,5,8],[3,6,9],[10],[11]]
=> ? = 7
[5,4,4,3]
=> [4,4,3]
=> [[1,2,3,7],[4,5,6,11],[8,9,10]]
=> [[1,4,8],[2,5,9],[3,6,10],[7,11]]
=> ? = 8
[4,4,4,4]
=> [4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12]]
=> [[1,5,9],[2,6,10],[3,7,11],[4,8,12]]
=> ? = 9
[4,4,3,3,2]
=> [4,3,3,2]
=> [[1,2,5,12],[3,4,8],[6,7,11],[9,10]]
=> [[1,3,6,9],[2,4,7,10],[5,8,11],[12]]
=> ? = 9
[4,3,3,3,3]
=> [3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12]]
=> [[1,4,7,10],[2,5,8,11],[3,6,9,12]]
=> ? = 10
[3,3,3,3,3,1]
=> [3,3,3,3,1]
=> [[1,3,4],[2,6,7],[5,9,10],[8,12,13],[11]]
=> [[1,2,5,8,11],[3,6,9,12],[4,7,10,13]]
=> ? = 11
[3,3,3,3,2,2]
=> [3,3,3,2,2]
=> [[1,2,7],[3,4,10],[5,6,13],[8,9],[11,12]]
=> [[1,3,5,8,11],[2,4,6,9,12],[7,10,13]]
=> ? = 11
[3,3,2,2,2,2,2]
=> [3,2,2,2,2,2]
=> [[1,2,13],[3,4],[5,6],[7,8],[9,10],[11,12]]
=> [[1,3,5,7,9,11],[2,4,6,8,10,12],[13]]
=> ? = 11
[6,6,5]
=> [6,5]
=> [[1,2,3,4,5,11],[6,7,8,9,10]]
=> [[1,6],[2,7],[3,8],[4,9],[5,10],[11]]
=> ? = 6
[5,5,4,3]
=> [5,4,3]
=> [[1,2,3,7,12],[4,5,6,11],[8,9,10]]
=> [[1,4,8],[2,5,9],[3,6,10],[7,11],[12]]
=> ? = 8
[5,4,4,4]
=> [4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12]]
=> [[1,5,9],[2,6,10],[3,7,11],[4,8,12]]
=> ? = 9
[4,4,4,4,1]
=> [4,4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6,11,12,13],[10]]
=> [[1,2,6,10],[3,7,11],[4,8,12],[5,9,13]]
=> ? = 10
[4,4,3,3,3]
=> [4,3,3,3]
=> [[1,2,3,13],[4,5,6],[7,8,9],[10,11,12]]
=> [[1,4,7,10],[2,5,8,11],[3,6,9,12],[13]]
=> ? = 10
[3,3,3,3,3,2]
=> [3,3,3,3,2]
=> [[1,2,5],[3,4,8],[6,7,11],[9,10,14],[12,13]]
=> [[1,3,6,9,12],[2,4,7,10,13],[5,8,11,14]]
=> ? = 12
[4,4,4,4,2]
=> [4,4,4,2]
=> [[1,2,5,6],[3,4,9,10],[7,8,13,14],[11,12]]
=> [[1,3,7,11],[2,4,8,12],[5,9,13],[6,10,14]]
=> ? = 11
[3,3,3,3,3,3]
=> [3,3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12],[13,14,15]]
=> [[1,4,7,10,13],[2,5,8,11,14],[3,6,9,12,15]]
=> ? = 13
[3,3,3,3,3,3,3,3]
=> [3,3,3,3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12],[13,14,15],[16,17,18],[19,20,21]]
=> [[1,4,7,10,13,16,19],[2,5,8,11,14,17,20],[3,6,9,12,15,18,21]]
=> ? = 19
[3,3,3,3,3,3,3]
=> [3,3,3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12],[13,14,15],[16,17,18]]
=> [[1,4,7,10,13,16],[2,5,8,11,14,17],[3,6,9,12,15,18]]
=> ? = 16
[4,3,3,3,3,3]
=> [3,3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12],[13,14,15]]
=> [[1,4,7,10,13],[2,5,8,11,14],[3,6,9,12,15]]
=> ? = 13
[4,4,4,4,4]
=> [4,4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16]]
=> [[1,5,9,13],[2,6,10,14],[3,7,11,15],[4,8,12,16]]
=> ? = 13
[4,4,4,4,4,4]
=> [4,4,4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16],[17,18,19,20]]
=> [[1,5,9,13,17],[2,6,10,14,18],[3,7,11,15,19],[4,8,12,16,20]]
=> ? = 17
[5,5,5,5]
=> [5,5,5]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12,13,14,15]]
=> [[1,6,11],[2,7,12],[3,8,13],[4,9,14],[5,10,15]]
=> ? = 11
[6,6,6,6]
=> [6,6,6]
=> [[1,2,3,4,5,6],[7,8,9,10,11,12],[13,14,15,16,17,18]]
=> [[1,7,13],[2,8,14],[3,9,15],[4,10,16],[5,11,17],[6,12,18]]
=> ? = 13
[6,6,6,6,6]
=> [6,6,6,6]
=> [[1,2,3,4,5,6],[7,8,9,10,11,12],[13,14,15,16,17,18],[19,20,21,22,23,24]]
=> [[1,7,13,19],[2,8,14,20],[3,9,15,21],[4,10,16,22],[5,11,17,23],[6,12,18,24]]
=> ? = 19
[5,5,5,5,5,5]
=> [5,5,5,5,5]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12,13,14,15],[16,17,18,19,20],[21,22,23,24,25]]
=> [[1,6,11,16,21],[2,7,12,17,22],[3,8,13,18,23],[4,9,14,19,24],[5,10,15,20,25]]
=> ? = 21
[7,7,7,7]
=> [7,7,7]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14],[15,16,17,18,19,20,21]]
=> [[1,8,15],[2,9,16],[3,10,17],[4,11,18],[5,12,19],[6,13,20],[7,14,21]]
=> ? = 15
Description
The last entry in the first row of a standard tableau.
Matching statistic: St000054
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 53% ●values known / values provided: 80%●distinct values known / distinct values provided: 53%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 53% ●values known / values provided: 80%●distinct values known / distinct values provided: 53%
Values
[1]
=> []
=> []
=> [] => ? = 0
[2]
=> []
=> []
=> [] => ? = 0
[1,1]
=> [1]
=> [[1]]
=> [1] => 1
[3]
=> []
=> []
=> [] => ? = 0
[2,1]
=> [1]
=> [[1]]
=> [1] => 1
[1,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 2
[4]
=> []
=> []
=> [] => ? = 0
[3,1]
=> [1]
=> [[1]]
=> [1] => 1
[2,2]
=> [2]
=> [[1,2]]
=> [1,2] => 1
[2,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 2
[1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 3
[5]
=> []
=> []
=> [] => ? = 0
[4,1]
=> [1]
=> [[1]]
=> [1] => 1
[3,2]
=> [2]
=> [[1,2]]
=> [1,2] => 1
[3,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 2
[2,2,1]
=> [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 2
[2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 3
[1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 4
[6]
=> []
=> []
=> [] => ? = 0
[5,1]
=> [1]
=> [[1]]
=> [1] => 1
[4,2]
=> [2]
=> [[1,2]]
=> [1,2] => 1
[4,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 2
[3,3]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 1
[3,2,1]
=> [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 2
[3,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 3
[2,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 3
[2,2,1,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 3
[2,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 4
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 5
[7]
=> []
=> []
=> [] => ? = 0
[6,1]
=> [1]
=> [[1]]
=> [1] => 1
[5,2]
=> [2]
=> [[1,2]]
=> [1,2] => 1
[5,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 2
[4,3]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 1
[4,2,1]
=> [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 2
[4,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 3
[3,3,1]
=> [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 2
[3,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 3
[3,2,1,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 3
[3,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 4
[2,2,2,1]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 4
[2,2,1,1,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 4
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 5
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => 6
[8]
=> []
=> []
=> [] => ? = 0
[7,1]
=> [1]
=> [[1]]
=> [1] => 1
[6,2]
=> [2]
=> [[1,2]]
=> [1,2] => 1
[6,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 2
[5,3]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 1
[5,2,1]
=> [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 2
[5,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 3
[4,4]
=> [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1
[4,3,1]
=> [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 2
[4,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 3
[4,2,1,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 3
[4,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 4
[3,3,2]
=> [3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => 3
[3,3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 3
[9]
=> []
=> []
=> [] => ? = 0
[10]
=> []
=> []
=> [] => ? = 0
[11]
=> []
=> []
=> [] => ? = 0
[2,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? = 10
[3,3,3,3,2]
=> [3,3,3,2]
=> [[1,2,5],[3,4,8],[6,7,11],[9,10]]
=> [9,10,6,7,11,3,4,8,1,2,5] => ? = 9
[3,3,3,3,1,1]
=> [3,3,3,1,1]
=> [[1,4,5],[2,7,8],[3,10,11],[6],[9]]
=> [9,6,3,10,11,2,7,8,1,4,5] => ? = 9
[3,3,2,2,2,2]
=> [3,2,2,2,2]
=> [[1,2,11],[3,4],[5,6],[7,8],[9,10]]
=> ? => ? = 9
[2,2,2,2,2,2,2]
=> [2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]
=> [11,12,9,10,7,8,5,6,3,4,1,2] => ? = 11
[4,4,4,3]
=> [4,4,3]
=> [[1,2,3,7],[4,5,6,11],[8,9,10]]
=> [8,9,10,4,5,6,11,1,2,3,7] => ? = 8
[4,4,3,3,1]
=> [4,3,3,1]
=> [[1,3,4,11],[2,6,7],[5,9,10],[8]]
=> [8,5,9,10,2,6,7,1,3,4,11] => ? = 8
[4,4,3,2,2]
=> [4,3,2,2]
=> [[1,2,7,11],[3,4,10],[5,6],[8,9]]
=> [8,9,5,6,3,4,10,1,2,7,11] => ? = 8
[4,3,3,3,2]
=> [3,3,3,2]
=> [[1,2,5],[3,4,8],[6,7,11],[9,10]]
=> [9,10,6,7,11,3,4,8,1,2,5] => ? = 9
[3,3,3,3,3]
=> [3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12]]
=> ? => ? = 10
[3,3,3,3,2,1]
=> [3,3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8,12],[7,11],[10]]
=> [10,7,11,4,8,12,2,5,9,1,3,6] => ? = 10
[3,3,3,2,2,2]
=> [3,3,2,2,2]
=> [[1,2,9],[3,4,12],[5,6],[7,8],[10,11]]
=> ? => ? = 10
[3,2,2,2,2,2,2]
=> [2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]
=> [11,12,9,10,7,8,5,6,3,4,1,2] => ? = 11
[5,5,5,1]
=> [5,5,1]
=> [[1,3,4,5,6],[2,8,9,10,11],[7]]
=> ? => ? = 7
[5,5,4,2]
=> [5,4,2]
=> [[1,2,5,6,11],[3,4,9,10],[7,8]]
=> [7,8,3,4,9,10,1,2,5,6,11] => ? = 7
[5,5,3,3]
=> [5,3,3]
=> [[1,2,3,10,11],[4,5,6],[7,8,9]]
=> [7,8,9,4,5,6,1,2,3,10,11] => ? = 7
[5,4,4,3]
=> [4,4,3]
=> [[1,2,3,7],[4,5,6,11],[8,9,10]]
=> [8,9,10,4,5,6,11,1,2,3,7] => ? = 8
[4,4,4,4]
=> [4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12]]
=> ? => ? = 9
[4,4,3,3,2]
=> [4,3,3,2]
=> [[1,2,5,12],[3,4,8],[6,7,11],[9,10]]
=> [9,10,6,7,11,3,4,8,1,2,5,12] => ? = 9
[4,3,3,3,3]
=> [3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12]]
=> ? => ? = 10
[3,3,3,3,3,1]
=> [3,3,3,3,1]
=> [[1,3,4],[2,6,7],[5,9,10],[8,12,13],[11]]
=> ? => ? = 11
[3,3,3,3,2,2]
=> [3,3,3,2,2]
=> [[1,2,7],[3,4,10],[5,6,13],[8,9],[11,12]]
=> ? => ? = 11
[3,3,2,2,2,2,2]
=> [3,2,2,2,2,2]
=> [[1,2,13],[3,4],[5,6],[7,8],[9,10],[11,12]]
=> ? => ? = 11
[2,2,2,2,2,2,2,2]
=> [2,2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14]]
=> [13,14,11,12,9,10,7,8,5,6,3,4,1,2] => ? = 13
[6,6,5]
=> [6,5]
=> [[1,2,3,4,5,11],[6,7,8,9,10]]
=> ? => ? = 6
[5,5,4,3]
=> [5,4,3]
=> [[1,2,3,7,12],[4,5,6,11],[8,9,10]]
=> [8,9,10,4,5,6,11,1,2,3,7,12] => ? = 8
[5,4,4,4]
=> [4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12]]
=> ? => ? = 9
[4,4,4,4,1]
=> [4,4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6,11,12,13],[10]]
=> ? => ? = 10
[4,4,3,3,3]
=> [4,3,3,3]
=> [[1,2,3,13],[4,5,6],[7,8,9],[10,11,12]]
=> ? => ? = 10
[3,3,3,3,3,2]
=> [3,3,3,3,2]
=> [[1,2,5],[3,4,8],[6,7,11],[9,10,14],[12,13]]
=> ? => ? = 12
[3,2,2,2,2,2,2,2]
=> [2,2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14]]
=> [13,14,11,12,9,10,7,8,5,6,3,4,1,2] => ? = 13
[4,4,4,4,2]
=> [4,4,4,2]
=> [[1,2,5,6],[3,4,9,10],[7,8,13,14],[11,12]]
=> ? => ? = 11
[2,2,2,2,2,2,2,2,2]
=> [2,2,2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14],[15,16]]
=> [15,16,13,14,11,12,9,10,7,8,5,6,3,4,1,2] => ? = 15
[3,3,3,3,3,3]
=> [3,3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12],[13,14,15]]
=> ? => ? = 13
[3,3,3,3,3,3,3,3]
=> [3,3,3,3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12],[13,14,15],[16,17,18],[19,20,21]]
=> ? => ? = 19
[2,2,2,2,2,2,2,2,2,2]
=> [2,2,2,2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14],[15,16],[17,18]]
=> [17,18,15,16,13,14,11,12,9,10,7,8,5,6,3,4,1,2] => ? = 17
[3,3,3,3,3,3,3]
=> [3,3,3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12],[13,14,15],[16,17,18]]
=> ? => ? = 16
[4,3,3,3,3,3]
=> [3,3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12],[13,14,15]]
=> ? => ? = 13
[4,4,4,4,4]
=> [4,4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16]]
=> ? => ? = 13
Description
The first entry of the permutation.
This can be described as 1 plus the number of occurrences of the vincular pattern ([2,1], {(0,0),(0,1),(0,2)}), i.e., the first column is shaded, see [1].
This statistic is related to the number of deficiencies [[St000703]] as follows: consider the arc diagram of a permutation $\pi$ of $n$, together with its rotations, obtained by conjugating with the long cycle $(1,\dots,n)$. Drawing the labels $1$ to $n$ in this order on a circle, and the arcs $(i, \pi(i))$ as straight lines, the rotation of $\pi$ is obtained by replacing each number $i$ by $(i\bmod n) +1$. Then, $\pi(1)-1$ is the number of rotations of $\pi$ where the arc $(1, \pi(1))$ is a deficiency. In particular, if $O(\pi)$ is the orbit of rotations of $\pi$, then the number of deficiencies of $\pi$ equals
$$
\frac{1}{|O(\pi)|}\sum_{\sigma\in O(\pi)} (\sigma(1)-1).
$$
Matching statistic: St000141
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 53% ●values known / values provided: 80%●distinct values known / distinct values provided: 53%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 53% ●values known / values provided: 80%●distinct values known / distinct values provided: 53%
Values
[1]
=> []
=> []
=> [] => ? = 0 - 1
[2]
=> []
=> []
=> [] => ? = 0 - 1
[1,1]
=> [1]
=> [[1]]
=> [1] => 0 = 1 - 1
[3]
=> []
=> []
=> [] => ? = 0 - 1
[2,1]
=> [1]
=> [[1]]
=> [1] => 0 = 1 - 1
[1,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 1 = 2 - 1
[4]
=> []
=> []
=> [] => ? = 0 - 1
[3,1]
=> [1]
=> [[1]]
=> [1] => 0 = 1 - 1
[2,2]
=> [2]
=> [[1,2]]
=> [1,2] => 0 = 1 - 1
[2,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 1 = 2 - 1
[1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 2 = 3 - 1
[5]
=> []
=> []
=> [] => ? = 0 - 1
[4,1]
=> [1]
=> [[1]]
=> [1] => 0 = 1 - 1
[3,2]
=> [2]
=> [[1,2]]
=> [1,2] => 0 = 1 - 1
[3,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 1 = 2 - 1
[2,2,1]
=> [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 1 = 2 - 1
[2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 2 = 3 - 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 3 = 4 - 1
[6]
=> []
=> []
=> [] => ? = 0 - 1
[5,1]
=> [1]
=> [[1]]
=> [1] => 0 = 1 - 1
[4,2]
=> [2]
=> [[1,2]]
=> [1,2] => 0 = 1 - 1
[4,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 1 = 2 - 1
[3,3]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 0 = 1 - 1
[3,2,1]
=> [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 1 = 2 - 1
[3,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 2 = 3 - 1
[2,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2 = 3 - 1
[2,2,1,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 2 = 3 - 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 3 = 4 - 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 4 = 5 - 1
[7]
=> []
=> []
=> [] => ? = 0 - 1
[6,1]
=> [1]
=> [[1]]
=> [1] => 0 = 1 - 1
[5,2]
=> [2]
=> [[1,2]]
=> [1,2] => 0 = 1 - 1
[5,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 1 = 2 - 1
[4,3]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 0 = 1 - 1
[4,2,1]
=> [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 1 = 2 - 1
[4,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 2 = 3 - 1
[3,3,1]
=> [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 1 = 2 - 1
[3,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2 = 3 - 1
[3,2,1,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 2 = 3 - 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 3 = 4 - 1
[2,2,2,1]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 3 = 4 - 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 3 = 4 - 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 4 = 5 - 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => 5 = 6 - 1
[8]
=> []
=> []
=> [] => ? = 0 - 1
[7,1]
=> [1]
=> [[1]]
=> [1] => 0 = 1 - 1
[6,2]
=> [2]
=> [[1,2]]
=> [1,2] => 0 = 1 - 1
[6,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 1 = 2 - 1
[5,3]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 0 = 1 - 1
[5,2,1]
=> [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 1 = 2 - 1
[5,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 2 = 3 - 1
[4,4]
=> [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0 = 1 - 1
[4,3,1]
=> [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 1 = 2 - 1
[4,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2 = 3 - 1
[4,2,1,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 2 = 3 - 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 3 = 4 - 1
[3,3,2]
=> [3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => 2 = 3 - 1
[3,3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 2 = 3 - 1
[9]
=> []
=> []
=> [] => ? = 0 - 1
[10]
=> []
=> []
=> [] => ? = 0 - 1
[11]
=> []
=> []
=> [] => ? = 0 - 1
[2,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? = 10 - 1
[3,3,3,3,2]
=> [3,3,3,2]
=> [[1,2,5],[3,4,8],[6,7,11],[9,10]]
=> [9,10,6,7,11,3,4,8,1,2,5] => ? = 9 - 1
[3,3,3,3,1,1]
=> [3,3,3,1,1]
=> [[1,4,5],[2,7,8],[3,10,11],[6],[9]]
=> [9,6,3,10,11,2,7,8,1,4,5] => ? = 9 - 1
[3,3,2,2,2,2]
=> [3,2,2,2,2]
=> [[1,2,11],[3,4],[5,6],[7,8],[9,10]]
=> ? => ? = 9 - 1
[2,2,2,2,2,2,2]
=> [2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]
=> [11,12,9,10,7,8,5,6,3,4,1,2] => ? = 11 - 1
[4,4,4,3]
=> [4,4,3]
=> [[1,2,3,7],[4,5,6,11],[8,9,10]]
=> [8,9,10,4,5,6,11,1,2,3,7] => ? = 8 - 1
[4,4,3,3,1]
=> [4,3,3,1]
=> [[1,3,4,11],[2,6,7],[5,9,10],[8]]
=> [8,5,9,10,2,6,7,1,3,4,11] => ? = 8 - 1
[4,4,3,2,2]
=> [4,3,2,2]
=> [[1,2,7,11],[3,4,10],[5,6],[8,9]]
=> [8,9,5,6,3,4,10,1,2,7,11] => ? = 8 - 1
[4,3,3,3,2]
=> [3,3,3,2]
=> [[1,2,5],[3,4,8],[6,7,11],[9,10]]
=> [9,10,6,7,11,3,4,8,1,2,5] => ? = 9 - 1
[3,3,3,3,3]
=> [3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12]]
=> ? => ? = 10 - 1
[3,3,3,3,2,1]
=> [3,3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8,12],[7,11],[10]]
=> [10,7,11,4,8,12,2,5,9,1,3,6] => ? = 10 - 1
[3,3,3,2,2,2]
=> [3,3,2,2,2]
=> [[1,2,9],[3,4,12],[5,6],[7,8],[10,11]]
=> ? => ? = 10 - 1
[3,2,2,2,2,2,2]
=> [2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]
=> [11,12,9,10,7,8,5,6,3,4,1,2] => ? = 11 - 1
[5,5,5,1]
=> [5,5,1]
=> [[1,3,4,5,6],[2,8,9,10,11],[7]]
=> ? => ? = 7 - 1
[5,5,4,2]
=> [5,4,2]
=> [[1,2,5,6,11],[3,4,9,10],[7,8]]
=> [7,8,3,4,9,10,1,2,5,6,11] => ? = 7 - 1
[5,5,3,3]
=> [5,3,3]
=> [[1,2,3,10,11],[4,5,6],[7,8,9]]
=> [7,8,9,4,5,6,1,2,3,10,11] => ? = 7 - 1
[5,4,4,3]
=> [4,4,3]
=> [[1,2,3,7],[4,5,6,11],[8,9,10]]
=> [8,9,10,4,5,6,11,1,2,3,7] => ? = 8 - 1
[4,4,4,4]
=> [4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12]]
=> ? => ? = 9 - 1
[4,4,3,3,2]
=> [4,3,3,2]
=> [[1,2,5,12],[3,4,8],[6,7,11],[9,10]]
=> [9,10,6,7,11,3,4,8,1,2,5,12] => ? = 9 - 1
[4,3,3,3,3]
=> [3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12]]
=> ? => ? = 10 - 1
[3,3,3,3,3,1]
=> [3,3,3,3,1]
=> [[1,3,4],[2,6,7],[5,9,10],[8,12,13],[11]]
=> ? => ? = 11 - 1
[3,3,3,3,2,2]
=> [3,3,3,2,2]
=> [[1,2,7],[3,4,10],[5,6,13],[8,9],[11,12]]
=> ? => ? = 11 - 1
[3,3,2,2,2,2,2]
=> [3,2,2,2,2,2]
=> [[1,2,13],[3,4],[5,6],[7,8],[9,10],[11,12]]
=> ? => ? = 11 - 1
[2,2,2,2,2,2,2,2]
=> [2,2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14]]
=> [13,14,11,12,9,10,7,8,5,6,3,4,1,2] => ? = 13 - 1
[6,6,5]
=> [6,5]
=> [[1,2,3,4,5,11],[6,7,8,9,10]]
=> ? => ? = 6 - 1
[5,5,4,3]
=> [5,4,3]
=> [[1,2,3,7,12],[4,5,6,11],[8,9,10]]
=> [8,9,10,4,5,6,11,1,2,3,7,12] => ? = 8 - 1
[5,4,4,4]
=> [4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12]]
=> ? => ? = 9 - 1
[4,4,4,4,1]
=> [4,4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6,11,12,13],[10]]
=> ? => ? = 10 - 1
[4,4,3,3,3]
=> [4,3,3,3]
=> [[1,2,3,13],[4,5,6],[7,8,9],[10,11,12]]
=> ? => ? = 10 - 1
[3,3,3,3,3,2]
=> [3,3,3,3,2]
=> [[1,2,5],[3,4,8],[6,7,11],[9,10,14],[12,13]]
=> ? => ? = 12 - 1
[3,2,2,2,2,2,2,2]
=> [2,2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14]]
=> [13,14,11,12,9,10,7,8,5,6,3,4,1,2] => ? = 13 - 1
[4,4,4,4,2]
=> [4,4,4,2]
=> [[1,2,5,6],[3,4,9,10],[7,8,13,14],[11,12]]
=> ? => ? = 11 - 1
[2,2,2,2,2,2,2,2,2]
=> [2,2,2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14],[15,16]]
=> [15,16,13,14,11,12,9,10,7,8,5,6,3,4,1,2] => ? = 15 - 1
[3,3,3,3,3,3]
=> [3,3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12],[13,14,15]]
=> ? => ? = 13 - 1
[3,3,3,3,3,3,3,3]
=> [3,3,3,3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12],[13,14,15],[16,17,18],[19,20,21]]
=> ? => ? = 19 - 1
[2,2,2,2,2,2,2,2,2,2]
=> [2,2,2,2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14],[15,16],[17,18]]
=> [17,18,15,16,13,14,11,12,9,10,7,8,5,6,3,4,1,2] => ? = 17 - 1
[3,3,3,3,3,3,3]
=> [3,3,3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12],[13,14,15],[16,17,18]]
=> ? => ? = 16 - 1
[4,3,3,3,3,3]
=> [3,3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12],[13,14,15]]
=> ? => ? = 13 - 1
[4,4,4,4,4]
=> [4,4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16]]
=> ? => ? = 13 - 1
Description
The maximum drop size of a permutation.
The maximum drop size of a permutation $\pi$ of $[n]=\{1,2,\ldots, n\}$ is defined to be the maximum value of $i-\pi(i)$.
The following 19 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001726The number of visible inversions of a permutation. St000029The depth of a permutation. St000224The sorting index of a permutation. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St000839The largest opener of a set partition. St000877The depth of the binary word interpreted as a path. St000209Maximum difference of elements in cycles. St000316The number of non-left-to-right-maxima of a permutation. St000653The last descent of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000956The maximal displacement of a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000868The aid statistic in the sense of Shareshian-Wachs. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St000028The number of stack-sorts needed to sort a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000454The largest eigenvalue of a graph if it is integral. St001330The hat guessing number of a graph. St001812The biclique partition number of a graph.
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