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Your data matches 118 different statistics following compositions of up to 3 maps.
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Matching statistic: St000183
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000183: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000183: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,1]
=> [1,1]
=> [2]
=> []
=> 0
[2,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[2,1,1]
=> [1,1]
=> [2]
=> []
=> 0
[1,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> 0
[3,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[3,1,1]
=> [1,1]
=> [2]
=> []
=> 0
[2,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[2,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> []
=> 0
[4,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[4,1,1]
=> [1,1]
=> [2]
=> []
=> 0
[3,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 1
[3,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[3,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> 0
[2,2,2]
=> [2,2]
=> [2,2]
=> [2]
=> 1
[2,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> []
=> 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> []
=> 0
[5,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[5,1,1]
=> [1,1]
=> [2]
=> []
=> 0
[4,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 1
[4,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[4,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> 0
[3,3,1]
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[3,2,2]
=> [2,2]
=> [2,2]
=> [2]
=> 1
[3,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> []
=> 0
[2,2,2,1]
=> [2,2,1]
=> [3,2]
=> [2]
=> 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [4,1]
=> [1]
=> 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> []
=> 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [6]
=> []
=> 0
[6,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[6,1,1]
=> [1,1]
=> [2]
=> []
=> 0
[5,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 1
[5,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[5,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> 0
[4,4]
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[4,3,1]
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[4,2,2]
=> [2,2]
=> [2,2]
=> [2]
=> 1
[4,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> []
=> 0
[3,3,2]
=> [3,2]
=> [2,2,1]
=> [2,1]
=> 1
[3,3,1,1]
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[3,2,2,1]
=> [2,2,1]
=> [3,2]
=> [2]
=> 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [4,1]
=> [1]
=> 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> []
=> 0
[2,2,2,2]
=> [2,2,2]
=> [3,3]
=> [3]
=> 1
[2,2,2,1,1]
=> [2,2,1,1]
=> [4,2]
=> [2]
=> 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [5,1]
=> [1]
=> 1
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [6]
=> []
=> 0
Description
The side length of the Durfee square of an integer partition.
Given a partition $\lambda = (\lambda_1,\ldots,\lambda_n)$, the Durfee square is the largest partition $(s^s)$ whose diagram fits inside the diagram of $\lambda$. In symbols, $s = \max\{ i \mid \lambda_i \geq i \}$.
This is also known as the Frobenius rank.
Matching statistic: St000292
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,1]
=> [1,1]
=> 110 => 110 => 0
[2,2]
=> [2]
=> 100 => 010 => 1
[2,1,1]
=> [1,1]
=> 110 => 110 => 0
[1,1,1,1]
=> [1,1,1]
=> 1110 => 1110 => 0
[3,2]
=> [2]
=> 100 => 010 => 1
[3,1,1]
=> [1,1]
=> 110 => 110 => 0
[2,2,1]
=> [2,1]
=> 1010 => 0110 => 1
[2,1,1,1]
=> [1,1,1]
=> 1110 => 1110 => 0
[1,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 11110 => 0
[4,2]
=> [2]
=> 100 => 010 => 1
[4,1,1]
=> [1,1]
=> 110 => 110 => 0
[3,3]
=> [3]
=> 1000 => 0010 => 1
[3,2,1]
=> [2,1]
=> 1010 => 0110 => 1
[3,1,1,1]
=> [1,1,1]
=> 1110 => 1110 => 0
[2,2,2]
=> [2,2]
=> 1100 => 1010 => 1
[2,2,1,1]
=> [2,1,1]
=> 10110 => 01110 => 1
[2,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 11110 => 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 111110 => 0
[5,2]
=> [2]
=> 100 => 010 => 1
[5,1,1]
=> [1,1]
=> 110 => 110 => 0
[4,3]
=> [3]
=> 1000 => 0010 => 1
[4,2,1]
=> [2,1]
=> 1010 => 0110 => 1
[4,1,1,1]
=> [1,1,1]
=> 1110 => 1110 => 0
[3,3,1]
=> [3,1]
=> 10010 => 00110 => 1
[3,2,2]
=> [2,2]
=> 1100 => 1010 => 1
[3,2,1,1]
=> [2,1,1]
=> 10110 => 01110 => 1
[3,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 11110 => 0
[2,2,2,1]
=> [2,2,1]
=> 11010 => 10110 => 1
[2,2,1,1,1]
=> [2,1,1,1]
=> 101110 => 011110 => 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 111110 => 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 1111110 => 1111110 => 0
[6,2]
=> [2]
=> 100 => 010 => 1
[6,1,1]
=> [1,1]
=> 110 => 110 => 0
[5,3]
=> [3]
=> 1000 => 0010 => 1
[5,2,1]
=> [2,1]
=> 1010 => 0110 => 1
[5,1,1,1]
=> [1,1,1]
=> 1110 => 1110 => 0
[4,4]
=> [4]
=> 10000 => 00010 => 1
[4,3,1]
=> [3,1]
=> 10010 => 00110 => 1
[4,2,2]
=> [2,2]
=> 1100 => 1010 => 1
[4,2,1,1]
=> [2,1,1]
=> 10110 => 01110 => 1
[4,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 11110 => 0
[3,3,2]
=> [3,2]
=> 10100 => 10010 => 1
[3,3,1,1]
=> [3,1,1]
=> 100110 => 001110 => 1
[3,2,2,1]
=> [2,2,1]
=> 11010 => 10110 => 1
[3,2,1,1,1]
=> [2,1,1,1]
=> 101110 => 011110 => 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 111110 => 0
[2,2,2,2]
=> [2,2,2]
=> 11100 => 11010 => 1
[2,2,2,1,1]
=> [2,2,1,1]
=> 110110 => 101110 => 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> 1011110 => 0111110 => 1
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 1111110 => 1111110 => 0
Description
The number of ascents of a binary word.
Matching statistic: St001031
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001031: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001031: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,1]
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 0
[2,2]
=> [2]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,1,1]
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 0
[1,1,1,1]
=> [1,1,1]
=> [3]
=> [1,0,1,0,1,0]
=> 0
[3,2]
=> [2]
=> [1,1]
=> [1,1,0,0]
=> 1
[3,1,1]
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 0
[2,2,1]
=> [2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,1,1,1]
=> [1,1,1]
=> [3]
=> [1,0,1,0,1,0]
=> 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 0
[4,2]
=> [2]
=> [1,1]
=> [1,1,0,0]
=> 1
[4,1,1]
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 0
[3,3]
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[3,2,1]
=> [2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[3,1,1,1]
=> [1,1,1]
=> [3]
=> [1,0,1,0,1,0]
=> 0
[2,2,2]
=> [2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 1
[2,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[5,2]
=> [2]
=> [1,1]
=> [1,1,0,0]
=> 1
[5,1,1]
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 0
[4,3]
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[4,2,1]
=> [2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[4,1,1,1]
=> [1,1,1]
=> [3]
=> [1,0,1,0,1,0]
=> 0
[3,3,1]
=> [3,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[3,2,2]
=> [2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 1
[3,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 0
[2,2,2,1]
=> [2,2,1]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[6,2]
=> [2]
=> [1,1]
=> [1,1,0,0]
=> 1
[6,1,1]
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 0
[5,3]
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[5,2,1]
=> [2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[5,1,1,1]
=> [1,1,1]
=> [3]
=> [1,0,1,0,1,0]
=> 0
[4,4]
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[4,3,1]
=> [3,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[4,2,2]
=> [2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 1
[4,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 0
[3,3,2]
=> [3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
[3,3,1,1]
=> [3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[3,2,2,1]
=> [2,2,1]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[2,2,2,2]
=> [2,2,2]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> 1
[2,2,2,1,1]
=> [2,2,1,1]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
Description
The height of the bicoloured Motzkin path associated with the Dyck path.
Matching statistic: St001175
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St001175: Integer partitions ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 100%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St001175: Integer partitions ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 100%
Values
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [3,1]
=> 0
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,2]
=> 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [3,1]
=> 0
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [4,1]
=> 0
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [2,2]
=> 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [3,1]
=> 0
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [2,2]
=> 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [4,1]
=> 0
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [5,1]
=> 0
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => [3,2]
=> 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [4,1]
=> 0
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [3,2]
=> 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,2]
=> 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [4,1]
=> 0
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [3,2]
=> 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [3,2]
=> 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,3,4,5,1,6] => [5,1]
=> 0
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [6,1]
=> 0
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,1,2,3,6,4] => [4,2]
=> 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5,6] => [5,1]
=> 0
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [3,2]
=> 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [3,2]
=> 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [4,1]
=> 0
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [3,2]
=> 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [3,2]
=> 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [3,2]
=> 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,3,4,1,5,6] => [5,1]
=> 0
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [3,2]
=> 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,3,4,6,1,5] => [4,2]
=> 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [2,3,4,5,6,1,7] => [6,1]
=> 0
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => [7,1]
=> 0
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [6,1,2,3,4,7,5] => [5,2]
=> 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [5,1,2,3,4,6,7] => [6,1]
=> 0
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [5,1,2,6,3,4] => [4,2]
=> 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [4,1,2,3,6,5] => [4,2]
=> 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [3,1,2,4,5,6] => [5,1]
=> 0
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,6,1,2,3,4] => [4,2]
=> 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [3,2]
=> 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [3,2]
=> 1
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [3,2]
=> 1
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,3,1,4,5,6] => [5,1]
=> 0
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [3,2]
=> 1
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [3,2]
=> 1
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [3,2]
=> 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [2,3,4,1,6,5] => [4,2]
=> 1
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [2,3,4,5,1,6,7] => [6,1]
=> 0
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => [4,2]
=> 1
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,3,5,6,1,4] => [4,2]
=> 1
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [2,3,4,5,7,1,6] => [5,2]
=> 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1,8] => [7,1]
=> 0
[10,1,1]
=> [1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,8,10,11] => ?
=> ? = 0
[8,2,1,1]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,7,9,8] => ?
=> ? = 1
[4,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [2,3,4,5,6,1,7,9,8] => ?
=> ? = 1
[3,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [2,3,4,6,7,1,8,5] => ?
=> ? = 1
[11,2]
=> [1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1
[11,1,1]
=> [1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 0
[10,3]
=> [1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1
[10,2,1]
=> [1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,8,11,10] => ?
=> ? = 1
[10,1,1,1]
=> [1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 0
[9,4]
=> [1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,10,6,7,8] => ?
=> ? = 1
[9,3,1]
=> [1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,10,7,9] => ?
=> ? = 1
[9,2,2]
=> [1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1
[9,2,1,1]
=> [1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6,8,10,9] => ?
=> ? = 1
[8,4,1]
=> [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0,1,0]
=> [7,1,2,3,4,9,5,6,8] => ?
=> ? = 1
[8,2,2,1]
=> [1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,8,9,7] => ?
=> ? = 1
[8,2,1,1,1]
=> [1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [5,1,2,3,4,6,7,9,8] => ?
=> ? = 1
[7,3,2,1]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> [5,1,2,3,4,7,6,8] => ?
=> ? = 1
[3,2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [2,3,5,6,7,1,8,4] => ?
=> ? = 1
[12,2]
=> [1,1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1
[12,1,1]
=> [1,1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 0
[11,3]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1
[11,2,1]
=> [1,1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1
[11,1,1,1]
=> [1,1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 0
[10,4]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1
[10,3,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1
[10,2,2]
=> [1,1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1
[10,2,1,1]
=> [1,1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1
[10,1,1,1,1]
=> [1,1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 0
[9,4,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1
[9,3,2]
=> [1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,9,7,10] => ?
=> ? = 1
[9,3,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1
[9,2,2,1]
=> [1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6,9,10,8] => ?
=> ? = 1
[9,2,1,1,1]
=> [1,1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1
[8,5,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0,1,0]
=> ? => ?
=> ? = 1
[8,4,2]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0,1,0]
=> [7,1,2,3,4,8,5,6,9] => ?
=> ? = 1
[8,4,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0,1,0]
=> ? => ?
=> ? = 1
[8,3,3]
=> [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0,1,0]
=> [7,1,2,3,4,8,9,5,6] => ?
=> ? = 2
[8,3,2,1]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,8,7,9] => ?
=> ? = 1
[8,3,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1
[8,2,2,2]
=> [1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1
[8,2,2,1,1]
=> [1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [5,1,2,3,4,6,8,9,7] => ?
=> ? = 1
[8,2,1,1,1,1]
=> [1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1
[7,4,2,1]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0,1,0]
=> [5,1,2,3,7,4,6,8] => ?
=> ? = 1
[6,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0]
=> ? => ?
=> ? = 1
[5,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
=> ? => ?
=> ? = 1
[4,3,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [2,3,4,6,1,7,5,8] => ?
=> ? = 1
[4,2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [2,3,5,6,1,7,8,4] => ?
=> ? = 1
[4,2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,0,0,1,0,0,0,0,0]
=> ? => ?
=> ? = 1
[3,2,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0]
=> [2,3,4,6,7,8,1,9,5] => ?
=> ? = 1
[13,2]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1
Description
The size of a partition minus the hook length of the base cell.
This is, the number of boxes in the diagram of a partition that are neither in the first row nor in the first column.
Matching statistic: St001176
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 100%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 100%
Values
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [3,1]
=> 1 = 0 + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,2]
=> 2 = 1 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [3,1]
=> 1 = 0 + 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [4,1]
=> 1 = 0 + 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [2,2]
=> 2 = 1 + 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [3,1]
=> 1 = 0 + 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [2,2]
=> 2 = 1 + 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [4,1]
=> 1 = 0 + 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [5,1]
=> 1 = 0 + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => [3,2]
=> 2 = 1 + 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [4,1]
=> 1 = 0 + 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [3,2]
=> 2 = 1 + 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,2]
=> 2 = 1 + 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [4,1]
=> 1 = 0 + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [3,2]
=> 2 = 1 + 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [3,2]
=> 2 = 1 + 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,3,4,5,1,6] => [5,1]
=> 1 = 0 + 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [6,1]
=> 1 = 0 + 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,1,2,3,6,4] => [4,2]
=> 2 = 1 + 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5,6] => [5,1]
=> 1 = 0 + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [3,2]
=> 2 = 1 + 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [3,2]
=> 2 = 1 + 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [4,1]
=> 1 = 0 + 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [3,2]
=> 2 = 1 + 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [3,2]
=> 2 = 1 + 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [3,2]
=> 2 = 1 + 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,3,4,1,5,6] => [5,1]
=> 1 = 0 + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [3,2]
=> 2 = 1 + 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,3,4,6,1,5] => [4,2]
=> 2 = 1 + 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [2,3,4,5,6,1,7] => [6,1]
=> 1 = 0 + 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => [7,1]
=> 1 = 0 + 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [6,1,2,3,4,7,5] => [5,2]
=> 2 = 1 + 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [5,1,2,3,4,6,7] => [6,1]
=> 1 = 0 + 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [5,1,2,6,3,4] => [4,2]
=> 2 = 1 + 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [4,1,2,3,6,5] => [4,2]
=> 2 = 1 + 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [3,1,2,4,5,6] => [5,1]
=> 1 = 0 + 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,6,1,2,3,4] => [4,2]
=> 2 = 1 + 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [3,2]
=> 2 = 1 + 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [3,2]
=> 2 = 1 + 1
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [3,2]
=> 2 = 1 + 1
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,3,1,4,5,6] => [5,1]
=> 1 = 0 + 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [3,2]
=> 2 = 1 + 1
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [3,2]
=> 2 = 1 + 1
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [3,2]
=> 2 = 1 + 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [2,3,4,1,6,5] => [4,2]
=> 2 = 1 + 1
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [2,3,4,5,1,6,7] => [6,1]
=> 1 = 0 + 1
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => [4,2]
=> 2 = 1 + 1
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,3,5,6,1,4] => [4,2]
=> 2 = 1 + 1
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [2,3,4,5,7,1,6] => [5,2]
=> 2 = 1 + 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1,8] => [7,1]
=> 1 = 0 + 1
[10,1,1]
=> [1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,8,10,11] => ?
=> ? = 0 + 1
[8,2,1,1]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,7,9,8] => ?
=> ? = 1 + 1
[4,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [2,3,4,5,6,1,7,9,8] => ?
=> ? = 1 + 1
[3,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [2,3,4,6,7,1,8,5] => ?
=> ? = 1 + 1
[11,2]
=> [1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 1
[11,1,1]
=> [1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 0 + 1
[10,3]
=> [1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 1
[10,2,1]
=> [1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,8,11,10] => ?
=> ? = 1 + 1
[10,1,1,1]
=> [1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 0 + 1
[9,4]
=> [1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,10,6,7,8] => ?
=> ? = 1 + 1
[9,3,1]
=> [1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,10,7,9] => ?
=> ? = 1 + 1
[9,2,2]
=> [1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 1
[9,2,1,1]
=> [1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6,8,10,9] => ?
=> ? = 1 + 1
[8,4,1]
=> [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0,1,0]
=> [7,1,2,3,4,9,5,6,8] => ?
=> ? = 1 + 1
[8,2,2,1]
=> [1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,8,9,7] => ?
=> ? = 1 + 1
[8,2,1,1,1]
=> [1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [5,1,2,3,4,6,7,9,8] => ?
=> ? = 1 + 1
[7,3,2,1]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> [5,1,2,3,4,7,6,8] => ?
=> ? = 1 + 1
[3,2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [2,3,5,6,7,1,8,4] => ?
=> ? = 1 + 1
[12,2]
=> [1,1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 1
[12,1,1]
=> [1,1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 0 + 1
[11,3]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 1
[11,2,1]
=> [1,1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 1
[11,1,1,1]
=> [1,1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 0 + 1
[10,4]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 1
[10,3,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 1
[10,2,2]
=> [1,1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 1
[10,2,1,1]
=> [1,1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 1
[10,1,1,1,1]
=> [1,1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 0 + 1
[9,4,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 1
[9,3,2]
=> [1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,9,7,10] => ?
=> ? = 1 + 1
[9,3,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 1
[9,2,2,1]
=> [1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6,9,10,8] => ?
=> ? = 1 + 1
[9,2,1,1,1]
=> [1,1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 1
[8,5,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 1
[8,4,2]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0,1,0]
=> [7,1,2,3,4,8,5,6,9] => ?
=> ? = 1 + 1
[8,4,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 1
[8,3,3]
=> [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0,1,0]
=> [7,1,2,3,4,8,9,5,6] => ?
=> ? = 2 + 1
[8,3,2,1]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,8,7,9] => ?
=> ? = 1 + 1
[8,3,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 1
[8,2,2,2]
=> [1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 1
[8,2,2,1,1]
=> [1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [5,1,2,3,4,6,8,9,7] => ?
=> ? = 1 + 1
[8,2,1,1,1,1]
=> [1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 1
[7,4,2,1]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0,1,0]
=> [5,1,2,3,7,4,6,8] => ?
=> ? = 1 + 1
[6,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0]
=> ? => ?
=> ? = 1 + 1
[5,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
=> ? => ?
=> ? = 1 + 1
[4,3,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [2,3,4,6,1,7,5,8] => ?
=> ? = 1 + 1
[4,2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [2,3,5,6,1,7,8,4] => ?
=> ? = 1 + 1
[4,2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,0,0,1,0,0,0,0,0]
=> ? => ?
=> ? = 1 + 1
[3,2,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0]
=> [2,3,4,6,7,8,1,9,5] => ?
=> ? = 1 + 1
[13,2]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 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: St000814
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000814: Integer partitions ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 100%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000814: Integer partitions ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 100%
Values
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [3,1]
=> 2 = 0 + 2
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,2]
=> 3 = 1 + 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [3,1]
=> 2 = 0 + 2
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [4,1]
=> 2 = 0 + 2
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [2,2]
=> 3 = 1 + 2
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [3,1]
=> 2 = 0 + 2
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [2,2]
=> 3 = 1 + 2
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [4,1]
=> 2 = 0 + 2
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [5,1]
=> 2 = 0 + 2
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => [3,2]
=> 3 = 1 + 2
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [4,1]
=> 2 = 0 + 2
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [3,2]
=> 3 = 1 + 2
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,2]
=> 3 = 1 + 2
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [4,1]
=> 2 = 0 + 2
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [3,2]
=> 3 = 1 + 2
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [3,2]
=> 3 = 1 + 2
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,3,4,5,1,6] => [5,1]
=> 2 = 0 + 2
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [6,1]
=> 2 = 0 + 2
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,1,2,3,6,4] => [4,2]
=> 3 = 1 + 2
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5,6] => [5,1]
=> 2 = 0 + 2
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [3,2]
=> 3 = 1 + 2
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [3,2]
=> 3 = 1 + 2
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [4,1]
=> 2 = 0 + 2
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [3,2]
=> 3 = 1 + 2
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [3,2]
=> 3 = 1 + 2
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [3,2]
=> 3 = 1 + 2
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,3,4,1,5,6] => [5,1]
=> 2 = 0 + 2
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [3,2]
=> 3 = 1 + 2
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,3,4,6,1,5] => [4,2]
=> 3 = 1 + 2
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [2,3,4,5,6,1,7] => [6,1]
=> 2 = 0 + 2
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => [7,1]
=> 2 = 0 + 2
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [6,1,2,3,4,7,5] => [5,2]
=> 3 = 1 + 2
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [5,1,2,3,4,6,7] => [6,1]
=> 2 = 0 + 2
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [5,1,2,6,3,4] => [4,2]
=> 3 = 1 + 2
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [4,1,2,3,6,5] => [4,2]
=> 3 = 1 + 2
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [3,1,2,4,5,6] => [5,1]
=> 2 = 0 + 2
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,6,1,2,3,4] => [4,2]
=> 3 = 1 + 2
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [3,2]
=> 3 = 1 + 2
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [3,2]
=> 3 = 1 + 2
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [3,2]
=> 3 = 1 + 2
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,3,1,4,5,6] => [5,1]
=> 2 = 0 + 2
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [3,2]
=> 3 = 1 + 2
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [3,2]
=> 3 = 1 + 2
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [3,2]
=> 3 = 1 + 2
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [2,3,4,1,6,5] => [4,2]
=> 3 = 1 + 2
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [2,3,4,5,1,6,7] => [6,1]
=> 2 = 0 + 2
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => [4,2]
=> 3 = 1 + 2
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,3,5,6,1,4] => [4,2]
=> 3 = 1 + 2
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [2,3,4,5,7,1,6] => [5,2]
=> 3 = 1 + 2
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1,8] => [7,1]
=> 2 = 0 + 2
[10,1,1]
=> [1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,8,10,11] => ?
=> ? = 0 + 2
[8,2,1,1]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,7,9,8] => ?
=> ? = 1 + 2
[4,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [2,3,4,5,6,1,7,9,8] => ?
=> ? = 1 + 2
[3,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [2,3,4,6,7,1,8,5] => ?
=> ? = 1 + 2
[11,2]
=> [1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 2
[11,1,1]
=> [1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 0 + 2
[10,3]
=> [1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 2
[10,2,1]
=> [1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,8,11,10] => ?
=> ? = 1 + 2
[10,1,1,1]
=> [1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 0 + 2
[9,4]
=> [1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,10,6,7,8] => ?
=> ? = 1 + 2
[9,3,1]
=> [1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,10,7,9] => ?
=> ? = 1 + 2
[9,2,2]
=> [1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 2
[9,2,1,1]
=> [1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6,8,10,9] => ?
=> ? = 1 + 2
[8,4,1]
=> [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0,1,0]
=> [7,1,2,3,4,9,5,6,8] => ?
=> ? = 1 + 2
[8,2,2,1]
=> [1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,8,9,7] => ?
=> ? = 1 + 2
[8,2,1,1,1]
=> [1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [5,1,2,3,4,6,7,9,8] => ?
=> ? = 1 + 2
[7,3,2,1]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> [5,1,2,3,4,7,6,8] => ?
=> ? = 1 + 2
[3,2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [2,3,5,6,7,1,8,4] => ?
=> ? = 1 + 2
[12,2]
=> [1,1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 2
[12,1,1]
=> [1,1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 0 + 2
[11,3]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 2
[11,2,1]
=> [1,1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 2
[11,1,1,1]
=> [1,1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 0 + 2
[10,4]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 2
[10,3,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 2
[10,2,2]
=> [1,1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 2
[10,2,1,1]
=> [1,1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 2
[10,1,1,1,1]
=> [1,1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 0 + 2
[9,4,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 2
[9,3,2]
=> [1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,9,7,10] => ?
=> ? = 1 + 2
[9,3,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 2
[9,2,2,1]
=> [1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6,9,10,8] => ?
=> ? = 1 + 2
[9,2,1,1,1]
=> [1,1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 2
[8,5,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 2
[8,4,2]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0,1,0]
=> [7,1,2,3,4,8,5,6,9] => ?
=> ? = 1 + 2
[8,4,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 2
[8,3,3]
=> [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0,1,0]
=> [7,1,2,3,4,8,9,5,6] => ?
=> ? = 2 + 2
[8,3,2,1]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,8,7,9] => ?
=> ? = 1 + 2
[8,3,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 2
[8,2,2,2]
=> [1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 2
[8,2,2,1,1]
=> [1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [5,1,2,3,4,6,8,9,7] => ?
=> ? = 1 + 2
[8,2,1,1,1,1]
=> [1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 2
[7,4,2,1]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0,1,0]
=> [5,1,2,3,7,4,6,8] => ?
=> ? = 1 + 2
[6,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0]
=> ? => ?
=> ? = 1 + 2
[5,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
=> ? => ?
=> ? = 1 + 2
[4,3,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [2,3,4,6,1,7,5,8] => ?
=> ? = 1 + 2
[4,2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [2,3,5,6,1,7,8,4] => ?
=> ? = 1 + 2
[4,2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,0,0,1,0,0,0,0,0]
=> ? => ?
=> ? = 1 + 2
[3,2,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0]
=> [2,3,4,6,7,8,1,9,5] => ?
=> ? = 1 + 2
[13,2]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 2
Description
The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions.
For example, $e_{22} = s_{1111} + s_{211} + s_{22}$, so the statistic on the partition $22$ is $3$.
Matching statistic: St000185
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000185: Integer partitions ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 100%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000185: Integer partitions ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 100%
Values
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [3,1]
=> 1 = 0 + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,2]
=> 2 = 1 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [3,1]
=> 1 = 0 + 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [4,1]
=> 1 = 0 + 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [2,2]
=> 2 = 1 + 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [3,1]
=> 1 = 0 + 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [2,2]
=> 2 = 1 + 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [4,1]
=> 1 = 0 + 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [5,1]
=> 1 = 0 + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => [3,2]
=> 2 = 1 + 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [4,1]
=> 1 = 0 + 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [3,2]
=> 2 = 1 + 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,2]
=> 2 = 1 + 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [4,1]
=> 1 = 0 + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [3,2]
=> 2 = 1 + 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [3,2]
=> 2 = 1 + 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,3,4,5,1,6] => [5,1]
=> 1 = 0 + 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [6,1]
=> 1 = 0 + 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,1,2,3,6,4] => [4,2]
=> 2 = 1 + 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5,6] => [5,1]
=> 1 = 0 + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [3,2]
=> 2 = 1 + 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [3,2]
=> 2 = 1 + 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [4,1]
=> 1 = 0 + 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [3,2]
=> 2 = 1 + 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [3,2]
=> 2 = 1 + 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [3,2]
=> 2 = 1 + 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,3,4,1,5,6] => [5,1]
=> 1 = 0 + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [3,2]
=> 2 = 1 + 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,3,4,6,1,5] => [4,2]
=> 2 = 1 + 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [2,3,4,5,6,1,7] => [6,1]
=> 1 = 0 + 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => [7,1]
=> 1 = 0 + 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [6,1,2,3,4,7,5] => [5,2]
=> 2 = 1 + 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [5,1,2,3,4,6,7] => [6,1]
=> 1 = 0 + 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [5,1,2,6,3,4] => [4,2]
=> 2 = 1 + 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [4,1,2,3,6,5] => [4,2]
=> 2 = 1 + 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [3,1,2,4,5,6] => [5,1]
=> 1 = 0 + 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,6,1,2,3,4] => [4,2]
=> 2 = 1 + 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [3,2]
=> 2 = 1 + 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [3,2]
=> 2 = 1 + 1
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [3,2]
=> 2 = 1 + 1
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,3,1,4,5,6] => [5,1]
=> 1 = 0 + 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [3,2]
=> 2 = 1 + 1
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [3,2]
=> 2 = 1 + 1
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [3,2]
=> 2 = 1 + 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [2,3,4,1,6,5] => [4,2]
=> 2 = 1 + 1
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [2,3,4,5,1,6,7] => [6,1]
=> 1 = 0 + 1
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => [4,2]
=> 2 = 1 + 1
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,3,5,6,1,4] => [4,2]
=> 2 = 1 + 1
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [2,3,4,5,7,1,6] => [5,2]
=> 2 = 1 + 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1,8] => [7,1]
=> 1 = 0 + 1
[10,2]
=> [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,0]
=> [10,1,2,3,4,5,6,7,8,11,9] => [9,2]
=> ? = 1 + 1
[10,1,1]
=> [1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,8,10,11] => ?
=> ? = 0 + 1
[8,2,1,1]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,7,9,8] => ?
=> ? = 1 + 1
[4,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [2,3,4,5,6,1,7,9,8] => ?
=> ? = 1 + 1
[3,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [2,3,4,6,7,1,8,5] => ?
=> ? = 1 + 1
[11,2]
=> [1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 1
[11,1,1]
=> [1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 0 + 1
[10,3]
=> [1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 1
[10,2,1]
=> [1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,8,11,10] => ?
=> ? = 1 + 1
[10,1,1,1]
=> [1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 0 + 1
[9,4]
=> [1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,10,6,7,8] => ?
=> ? = 1 + 1
[9,3,1]
=> [1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,10,7,9] => ?
=> ? = 1 + 1
[9,2,2]
=> [1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 1
[9,2,1,1]
=> [1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6,8,10,9] => ?
=> ? = 1 + 1
[8,4,1]
=> [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0,1,0]
=> [7,1,2,3,4,9,5,6,8] => ?
=> ? = 1 + 1
[8,2,2,1]
=> [1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,8,9,7] => ?
=> ? = 1 + 1
[8,2,1,1,1]
=> [1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [5,1,2,3,4,6,7,9,8] => ?
=> ? = 1 + 1
[7,3,2,1]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> [5,1,2,3,4,7,6,8] => ?
=> ? = 1 + 1
[3,2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [2,3,5,6,7,1,8,4] => ?
=> ? = 1 + 1
[12,2]
=> [1,1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 1
[12,1,1]
=> [1,1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 0 + 1
[11,3]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 1
[11,2,1]
=> [1,1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 1
[11,1,1,1]
=> [1,1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 0 + 1
[10,4]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 1
[10,3,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 1
[10,2,2]
=> [1,1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 1
[10,2,1,1]
=> [1,1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 1
[10,1,1,1,1]
=> [1,1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 0 + 1
[9,4,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 1
[9,3,2]
=> [1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,9,7,10] => ?
=> ? = 1 + 1
[9,3,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 1
[9,2,2,1]
=> [1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6,9,10,8] => ?
=> ? = 1 + 1
[9,2,1,1,1]
=> [1,1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 1
[8,5,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 1
[8,4,2]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0,1,0]
=> [7,1,2,3,4,8,5,6,9] => ?
=> ? = 1 + 1
[8,4,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 1
[8,3,3]
=> [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0,1,0]
=> [7,1,2,3,4,8,9,5,6] => ?
=> ? = 2 + 1
[8,3,2,1]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,8,7,9] => ?
=> ? = 1 + 1
[8,3,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 1
[8,2,2,2]
=> [1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 1
[8,2,2,1,1]
=> [1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [5,1,2,3,4,6,8,9,7] => ?
=> ? = 1 + 1
[8,2,1,1,1,1]
=> [1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 1
[7,4,2,1]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0,1,0]
=> [5,1,2,3,7,4,6,8] => ?
=> ? = 1 + 1
[6,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0]
=> ? => ?
=> ? = 1 + 1
[5,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
=> ? => ?
=> ? = 1 + 1
[4,3,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [2,3,4,6,1,7,5,8] => ?
=> ? = 1 + 1
[4,2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [2,3,5,6,1,7,8,4] => ?
=> ? = 1 + 1
[4,2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,0,0,1,0,0,0,0,0]
=> ? => ?
=> ? = 1 + 1
[3,2,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0]
=> [2,3,4,6,7,8,1,9,5] => ?
=> ? = 1 + 1
Description
The weighted size of a partition.
Let $\lambda = (\lambda_0\geq\lambda_1 \geq \dots\geq\lambda_m)$ be an integer partition. Then the weighted size of $\lambda$ is
$$\sum_{i=0}^m i \cdot \lambda_i.$$
This is also the sum of the leg lengths of the cells in $\lambda$, or
$$
\sum_i \binom{\lambda^{\prime}_i}{2}
$$
where $\lambda^{\prime}$ is the conjugate partition of $\lambda$.
This is the minimal number of inversions a permutation with the given shape can have, see [1, cor.2.2].
This is also the smallest possible sum of the entries of a semistandard tableau (allowing 0 as a part) of shape $\lambda=(\lambda_0,\lambda_1,\ldots,\lambda_m)$, obtained uniquely by placing $i-1$ in all the cells of the $i$th row of $\lambda$, see [2, eq.7.103].
Matching statistic: St001214
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St001214: Integer partitions ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 100%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St001214: Integer partitions ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 100%
Values
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [3,1]
=> 1 = 0 + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,2]
=> 2 = 1 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [3,1]
=> 1 = 0 + 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [4,1]
=> 1 = 0 + 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [2,2]
=> 2 = 1 + 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [3,1]
=> 1 = 0 + 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [2,2]
=> 2 = 1 + 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [4,1]
=> 1 = 0 + 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [5,1]
=> 1 = 0 + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => [3,2]
=> 2 = 1 + 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [4,1]
=> 1 = 0 + 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [3,2]
=> 2 = 1 + 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,2]
=> 2 = 1 + 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [4,1]
=> 1 = 0 + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [3,2]
=> 2 = 1 + 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [3,2]
=> 2 = 1 + 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,3,4,5,1,6] => [5,1]
=> 1 = 0 + 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [6,1]
=> 1 = 0 + 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,1,2,3,6,4] => [4,2]
=> 2 = 1 + 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5,6] => [5,1]
=> 1 = 0 + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [3,2]
=> 2 = 1 + 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [3,2]
=> 2 = 1 + 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [4,1]
=> 1 = 0 + 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [3,2]
=> 2 = 1 + 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [3,2]
=> 2 = 1 + 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [3,2]
=> 2 = 1 + 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,3,4,1,5,6] => [5,1]
=> 1 = 0 + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [3,2]
=> 2 = 1 + 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,3,4,6,1,5] => [4,2]
=> 2 = 1 + 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [2,3,4,5,6,1,7] => [6,1]
=> 1 = 0 + 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => [7,1]
=> 1 = 0 + 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [6,1,2,3,4,7,5] => [5,2]
=> 2 = 1 + 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [5,1,2,3,4,6,7] => [6,1]
=> 1 = 0 + 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [5,1,2,6,3,4] => [4,2]
=> 2 = 1 + 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [4,1,2,3,6,5] => [4,2]
=> 2 = 1 + 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [3,1,2,4,5,6] => [5,1]
=> 1 = 0 + 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,6,1,2,3,4] => [4,2]
=> 2 = 1 + 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [3,2]
=> 2 = 1 + 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [3,2]
=> 2 = 1 + 1
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [3,2]
=> 2 = 1 + 1
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,3,1,4,5,6] => [5,1]
=> 1 = 0 + 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [3,2]
=> 2 = 1 + 1
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [3,2]
=> 2 = 1 + 1
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [3,2]
=> 2 = 1 + 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [2,3,4,1,6,5] => [4,2]
=> 2 = 1 + 1
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [2,3,4,5,1,6,7] => [6,1]
=> 1 = 0 + 1
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => [4,2]
=> 2 = 1 + 1
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,3,5,6,1,4] => [4,2]
=> 2 = 1 + 1
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [2,3,4,5,7,1,6] => [5,2]
=> 2 = 1 + 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1,8] => [7,1]
=> 1 = 0 + 1
[10,2]
=> [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,0]
=> [10,1,2,3,4,5,6,7,8,11,9] => [9,2]
=> ? = 1 + 1
[10,1,1]
=> [1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,8,10,11] => ?
=> ? = 0 + 1
[8,2,1,1]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,7,9,8] => ?
=> ? = 1 + 1
[4,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [2,3,4,5,6,1,7,9,8] => ?
=> ? = 1 + 1
[3,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [2,3,4,6,7,1,8,5] => ?
=> ? = 1 + 1
[11,2]
=> [1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 1
[11,1,1]
=> [1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 0 + 1
[10,3]
=> [1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 1
[10,2,1]
=> [1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,8,11,10] => ?
=> ? = 1 + 1
[10,1,1,1]
=> [1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 0 + 1
[9,4]
=> [1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,10,6,7,8] => ?
=> ? = 1 + 1
[9,3,1]
=> [1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,10,7,9] => ?
=> ? = 1 + 1
[9,2,2]
=> [1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 1
[9,2,1,1]
=> [1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6,8,10,9] => ?
=> ? = 1 + 1
[8,4,1]
=> [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0,1,0]
=> [7,1,2,3,4,9,5,6,8] => ?
=> ? = 1 + 1
[8,2,2,1]
=> [1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,8,9,7] => ?
=> ? = 1 + 1
[8,2,1,1,1]
=> [1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [5,1,2,3,4,6,7,9,8] => ?
=> ? = 1 + 1
[7,3,2,1]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> [5,1,2,3,4,7,6,8] => ?
=> ? = 1 + 1
[3,2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [2,3,5,6,7,1,8,4] => ?
=> ? = 1 + 1
[12,2]
=> [1,1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 1
[12,1,1]
=> [1,1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 0 + 1
[11,3]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 1
[11,2,1]
=> [1,1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 1
[11,1,1,1]
=> [1,1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 0 + 1
[10,4]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 1
[10,3,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 1
[10,2,2]
=> [1,1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 1
[10,2,1,1]
=> [1,1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 1
[10,1,1,1,1]
=> [1,1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 0 + 1
[9,4,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 1
[9,3,2]
=> [1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,9,7,10] => ?
=> ? = 1 + 1
[9,3,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 1
[9,2,2,1]
=> [1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6,9,10,8] => ?
=> ? = 1 + 1
[9,2,1,1,1]
=> [1,1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 1
[8,5,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 1
[8,4,2]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0,1,0]
=> [7,1,2,3,4,8,5,6,9] => ?
=> ? = 1 + 1
[8,4,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 1
[8,3,3]
=> [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0,1,0]
=> [7,1,2,3,4,8,9,5,6] => ?
=> ? = 2 + 1
[8,3,2,1]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,8,7,9] => ?
=> ? = 1 + 1
[8,3,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 1
[8,2,2,2]
=> [1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 1
[8,2,2,1,1]
=> [1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [5,1,2,3,4,6,8,9,7] => ?
=> ? = 1 + 1
[8,2,1,1,1,1]
=> [1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 1
[7,4,2,1]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0,1,0]
=> [5,1,2,3,7,4,6,8] => ?
=> ? = 1 + 1
[6,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0]
=> ? => ?
=> ? = 1 + 1
[5,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
=> ? => ?
=> ? = 1 + 1
[4,3,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [2,3,4,6,1,7,5,8] => ?
=> ? = 1 + 1
[4,2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [2,3,5,6,1,7,8,4] => ?
=> ? = 1 + 1
[4,2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,0,0,1,0,0,0,0,0]
=> ? => ?
=> ? = 1 + 1
[3,2,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0]
=> [2,3,4,6,7,8,1,9,5] => ?
=> ? = 1 + 1
Description
The aft of an integer partition.
The aft is the size of the partition minus the length of the first row or column, whichever is larger.
See also [[St000784]].
Matching statistic: St000336
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000336: Standard tableaux ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 100%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000336: Standard tableaux ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 100%
Values
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [[1,3,4],[2]]
=> 1 = 0 + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [[1,2],[3,4]]
=> 2 = 1 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [[1,3,4],[2]]
=> 1 = 0 + 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [[1,3,4,5],[2]]
=> 1 = 0 + 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [[1,2],[3,4]]
=> 2 = 1 + 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [[1,3,4],[2]]
=> 1 = 0 + 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [[1,3],[2,4]]
=> 2 = 1 + 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [[1,3,4,5],[2]]
=> 1 = 0 + 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [[1,3,4,5,6],[2]]
=> 1 = 0 + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => [[1,2,3],[4,5]]
=> 2 = 1 + 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [[1,2,4,5],[3]]
=> 1 = 0 + 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [[1,2,3],[4,5]]
=> 2 = 1 + 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [[1,3],[2,4]]
=> 2 = 1 + 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [[1,3,4,5],[2]]
=> 1 = 0 + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [[1,2,5],[3,4]]
=> 2 = 1 + 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [[1,3,4],[2,5]]
=> 2 = 1 + 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,3,4,5,1,6] => [[1,3,4,5,6],[2]]
=> 1 = 0 + 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [[1,3,4,5,6,7],[2]]
=> 1 = 0 + 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,1,2,3,6,4] => [[1,2,3,4],[5,6]]
=> 2 = 1 + 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5,6] => [[1,2,3,5,6],[4]]
=> 1 = 0 + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [[1,2,3],[4,5]]
=> 2 = 1 + 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [[1,2,4],[3,5]]
=> 2 = 1 + 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [[1,3,4,5],[2]]
=> 1 = 0 + 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [[1,2,4],[3,5]]
=> 2 = 1 + 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [[1,2,5],[3,4]]
=> 2 = 1 + 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [[1,3,4],[2,5]]
=> 2 = 1 + 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,3,4,1,5,6] => [[1,3,4,5,6],[2]]
=> 1 = 0 + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [[1,3,5],[2,4]]
=> 2 = 1 + 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,3,4,6,1,5] => [[1,3,4,5],[2,6]]
=> 2 = 1 + 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [2,3,4,5,6,1,7] => [[1,3,4,5,6,7],[2]]
=> 1 = 0 + 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => [[1,3,4,5,6,7,8],[2]]
=> 1 = 0 + 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [6,1,2,3,4,7,5] => [[1,2,3,4,5],[6,7]]
=> 2 = 1 + 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [5,1,2,3,4,6,7] => [[1,2,3,4,6,7],[5]]
=> 1 = 0 + 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [5,1,2,6,3,4] => [[1,2,3,4],[5,6]]
=> 2 = 1 + 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [4,1,2,3,6,5] => [[1,2,3,5],[4,6]]
=> 2 = 1 + 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [3,1,2,4,5,6] => [[1,2,4,5,6],[3]]
=> 1 = 0 + 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,6,1,2,3,4] => [[1,2,3,4],[5,6]]
=> 2 = 1 + 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [[1,2,4],[3,5]]
=> 2 = 1 + 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [[1,2,5],[3,4]]
=> 2 = 1 + 1
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [[1,3,4],[2,5]]
=> 2 = 1 + 1
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,3,1,4,5,6] => [[1,3,4,5,6],[2]]
=> 1 = 0 + 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [[1,2,5],[3,4]]
=> 2 = 1 + 1
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [[1,3,4],[2,5]]
=> 2 = 1 + 1
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [[1,3,5],[2,4]]
=> 2 = 1 + 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [2,3,4,1,6,5] => [[1,3,4,5],[2,6]]
=> 2 = 1 + 1
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [2,3,4,5,1,6,7] => [[1,3,4,5,6,7],[2]]
=> 1 = 0 + 1
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => [[1,2,5,6],[3,4]]
=> 2 = 1 + 1
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,3,5,6,1,4] => [[1,3,4,6],[2,5]]
=> 2 = 1 + 1
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [2,3,4,5,7,1,6] => [[1,3,4,5,6],[2,7]]
=> 2 = 1 + 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1,8] => [[1,3,4,5,6,7,8],[2]]
=> 1 = 0 + 1
[8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6,8,9] => [[1,2,3,4,5,6,8,9],[7]]
=> ? = 0 + 1
[2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,9,1,8] => [[1,3,4,5,6,7,8],[2,9]]
=> ? = 1 + 1
[8,1,1,1]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,7,8,9] => [[1,2,3,4,5,7,8,9],[6]]
=> ? = 0 + 1
[3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1,9,8] => [[1,3,4,5,6,7,8],[2,9]]
=> ? = 1 + 1
[2,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,8,9,1,7] => [[1,3,4,5,6,7,9],[2,8]]
=> ? = 1 + 1
[10,2]
=> [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,0]
=> [10,1,2,3,4,5,6,7,8,11,9] => [[1,2,3,4,5,6,7,8,9],[10,11]]
=> ? = 1 + 1
[10,1,1]
=> [1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,8,10,11] => ?
=> ? = 0 + 1
[8,2,2]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,8,9,6] => [[1,2,3,4,5,6,9],[7,8]]
=> ? = 1 + 1
[8,2,1,1]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,7,9,8] => ?
=> ? = 1 + 1
[8,1,1,1,1]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [5,1,2,3,4,6,7,8,9] => [[1,2,3,4,6,7,8,9],[5]]
=> ? = 0 + 1
[4,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [2,3,4,5,6,1,7,9,8] => ?
=> ? = 1 + 1
[3,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [2,3,4,6,7,1,8,5] => ?
=> ? = 1 + 1
[3,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> [2,3,4,5,6,8,1,9,7] => [[1,3,4,5,6,7,9],[2,8]]
=> ? = 1 + 1
[2,2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,7,8,9,1,6] => [[1,3,4,5,6,8,9],[2,7]]
=> ? = 1 + 1
[11,2]
=> [1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 1
[11,1,1]
=> [1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 0 + 1
[10,3]
=> [1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 1
[10,2,1]
=> [1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,8,11,10] => ?
=> ? = 1 + 1
[10,1,1,1]
=> [1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 0 + 1
[9,4]
=> [1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,10,6,7,8] => ?
=> ? = 1 + 1
[9,3,1]
=> [1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,10,7,9] => ?
=> ? = 1 + 1
[9,2,2]
=> [1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 1
[9,2,1,1]
=> [1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6,8,10,9] => ?
=> ? = 1 + 1
[8,4,1]
=> [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0,1,0]
=> [7,1,2,3,4,9,5,6,8] => ?
=> ? = 1 + 1
[8,3,2]
=> [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,8,6,9] => [[1,2,3,4,5,6,9],[7,8]]
=> ? = 1 + 1
[8,3,1,1]
=> [1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,9,7,8] => [[1,2,3,4,5,7,8],[6,9]]
=> ? = 1 + 1
[8,2,2,1]
=> [1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,8,9,7] => ?
=> ? = 1 + 1
[8,2,1,1,1]
=> [1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [5,1,2,3,4,6,7,9,8] => ?
=> ? = 1 + 1
[8,1,1,1,1,1]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [4,1,2,3,5,6,7,8,9] => [[1,2,3,5,6,7,8,9],[4]]
=> ? = 0 + 1
[7,3,2,1]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> [5,1,2,3,4,7,6,8] => ?
=> ? = 1 + 1
[5,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
=> [2,3,4,5,1,6,7,9,8] => [[1,3,4,5,6,7,8],[2,9]]
=> ? = 1 + 1
[4,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
=> [2,3,4,5,6,1,8,9,7] => [[1,3,4,5,6,7,9],[2,8]]
=> ? = 1 + 1
[3,2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [2,3,5,6,7,1,8,4] => ?
=> ? = 1 + 1
[3,2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0]
=> [2,3,4,5,7,8,1,9,6] => [[1,3,4,5,6,8,9],[2,7]]
=> ? = 1 + 1
[2,2,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,6,7,8,9,1,5] => [[1,3,4,5,7,8,9],[2,6]]
=> ? = 1 + 1
[12,2]
=> [1,1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 1
[12,1,1]
=> [1,1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 0 + 1
[11,3]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 1
[11,2,1]
=> [1,1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 1
[11,1,1,1]
=> [1,1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 0 + 1
[10,4]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 1
[10,3,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 1
[10,2,2]
=> [1,1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 1
[10,2,1,1]
=> [1,1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 1
[10,1,1,1,1]
=> [1,1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 0 + 1
[9,4,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 1
[9,3,2]
=> [1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,9,7,10] => ?
=> ? = 1 + 1
[9,3,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 1
[9,2,2,1]
=> [1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6,9,10,8] => ?
=> ? = 1 + 1
[9,2,1,1,1]
=> [1,1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> ? => ?
=> ? = 1 + 1
Description
The leg major index of a standard tableau.
The leg length of a cell is the number of cells strictly below in the same column. This statistic is the sum of all leg lengths. Therefore, this is actually a statistic on the underlying integer partition.
It happens to coincide with the (leg) major index of a tabloid restricted to standard Young tableaux, defined as follows: the descent set of a tabloid is the set of cells, not in the top row, whose entry is strictly larger than the entry directly above it. The leg major index is the sum of the leg lengths of the descents plus the number of descents.
Matching statistic: St001044
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St001044: Perfect matchings ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 75%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St001044: Perfect matchings ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 75%
Values
[1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 0 + 1
[2,2]
=> [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 2 = 1 + 1
[2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 0 + 1
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> 1 = 0 + 1
[3,2]
=> [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 2 = 1 + 1
[3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 0 + 1
[2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 2 = 1 + 1
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> 1 = 0 + 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 1 = 0 + 1
[4,2]
=> [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 2 = 1 + 1
[4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 0 + 1
[3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> 2 = 1 + 1
[3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 2 = 1 + 1
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> 1 = 0 + 1
[2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> 2 = 1 + 1
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> 2 = 1 + 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 1 = 0 + 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> ? = 0 + 1
[5,2]
=> [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 2 = 1 + 1
[5,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 0 + 1
[4,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> 2 = 1 + 1
[4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 2 = 1 + 1
[4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> 1 = 0 + 1
[3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> 2 = 1 + 1
[3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> 2 = 1 + 1
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> 2 = 1 + 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 1 = 0 + 1
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> 2 = 1 + 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 2 = 1 + 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> ? = 0 + 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 0 + 1
[6,2]
=> [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 2 = 1 + 1
[6,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 0 + 1
[5,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> 2 = 1 + 1
[5,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 2 = 1 + 1
[5,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> 1 = 0 + 1
[4,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> 2 = 1 + 1
[4,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> 2 = 1 + 1
[4,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> 2 = 1 + 1
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> 2 = 1 + 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 1 = 0 + 1
[3,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> 2 = 1 + 1
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> 2 = 1 + 1
[3,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> 2 = 1 + 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 2 = 1 + 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> ? = 0 + 1
[2,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> 2 = 1 + 1
[2,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> 2 = 1 + 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,7),(8,9)]
=> ? = 1 + 1
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 0 + 1
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [(1,2),(3,16),(4,15),(5,14),(6,13),(7,12),(8,11),(9,10)]
=> ? = 0 + 1
[7,2]
=> [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 2 = 1 + 1
[7,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 0 + 1
[6,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> 2 = 1 + 1
[6,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 2 = 1 + 1
[6,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> 1 = 0 + 1
[5,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> 2 = 1 + 1
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> ? = 0 + 1
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,7),(8,9)]
=> ? = 1 + 1
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 0 + 1
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,6),(7,10),(8,9)]
=> ? = 1 + 1
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,8),(9,10)]
=> ? = 1 + 1
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [(1,2),(3,16),(4,15),(5,14),(6,13),(7,12),(8,11),(9,10)]
=> ? = 0 + 1
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [(1,2),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,11)]
=> ? = 0 + 1
[5,5]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6),(11,12)]
=> ? = 1 + 1
[5,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> ? = 0 + 1
[4,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,7),(8,9)]
=> ? = 1 + 1
[4,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 0 + 1
[3,3,1,1,1,1]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [(1,2),(3,12),(4,11),(5,8),(6,7),(9,10)]
=> ? = 1 + 1
[3,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,6),(7,10),(8,9)]
=> ? = 1 + 1
[3,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,8),(9,10)]
=> ? = 1 + 1
[3,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [(1,2),(3,16),(4,15),(5,14),(6,13),(7,12),(8,11),(9,10)]
=> ? = 0 + 1
[2,2,2,2,2]
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [(1,4),(2,3),(5,12),(6,11),(7,10),(8,9)]
=> ? = 1 + 1
[2,2,2,2,1,1]
=> [2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [(1,2),(3,12),(4,5),(6,11),(7,10),(8,9)]
=> ? = 1 + 1
[2,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,7),(8,11),(9,10)]
=> ? = 1 + 1
[2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [(1,2),(3,16),(4,15),(5,14),(6,13),(7,12),(8,9),(10,11)]
=> ? = 1 + 1
[2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [(1,2),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,11)]
=> ? = 0 + 1
[6,5]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6),(11,12)]
=> ? = 1 + 1
[6,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> ? = 0 + 1
[5,5,1]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7),(11,12)]
=> ? = 1 + 1
[5,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,7),(8,9)]
=> ? = 1 + 1
[5,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 0 + 1
[4,3,1,1,1,1]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [(1,2),(3,12),(4,11),(5,8),(6,7),(9,10)]
=> ? = 1 + 1
[4,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,6),(7,10),(8,9)]
=> ? = 1 + 1
[4,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,8),(9,10)]
=> ? = 1 + 1
[4,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [(1,2),(3,16),(4,15),(5,14),(6,13),(7,12),(8,11),(9,10)]
=> ? = 0 + 1
[3,3,2,1,1,1]
=> [3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [(1,2),(3,12),(4,11),(5,6),(7,8),(9,10)]
=> ? = 1 + 1
[3,3,1,1,1,1,1]
=> [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,9),(7,8),(10,11)]
=> ? = 1 + 1
[3,2,2,2,2]
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [(1,4),(2,3),(5,12),(6,11),(7,10),(8,9)]
=> ? = 1 + 1
[3,2,2,2,1,1]
=> [2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [(1,2),(3,12),(4,5),(6,11),(7,10),(8,9)]
=> ? = 1 + 1
[3,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,7),(8,11),(9,10)]
=> ? = 1 + 1
[3,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [(1,2),(3,16),(4,15),(5,14),(6,13),(7,12),(8,9),(10,11)]
=> ? = 1 + 1
[3,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [(1,2),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,11)]
=> ? = 0 + 1
[2,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,6),(7,12),(8,11),(9,10)]
=> ? = 1 + 1
[2,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [(1,2),(3,16),(4,15),(5,14),(6,13),(7,8),(9,12),(10,11)]
=> ? = 1 + 1
[7,5]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6),(11,12)]
=> ? = 1 + 1
[7,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> ? = 0 + 1
[6,6]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,14)]
=> ? = 1 + 1
[6,5,1]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7),(11,12)]
=> ? = 1 + 1
[6,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,7),(8,9)]
=> ? = 1 + 1
Description
The number of pairs whose larger element is at most one more than half the size of the perfect matching.
Under the bijection between perfect matchings of $\{1,\dots,2n\}$ and rooted unordered binary trees with $n+1$ labelled leaves described in Example 5.2.6 of [1], this is the number of nodes having two leaves as children.
The number of perfect matchings of $\{1,\dots,2n\}$ with $j$ such pairs, computed in [2], is
$$
\frac{(2j-3)!! (n+1)!}{2^j j!}\binom{n-1}{2j-2}.
$$
The following 108 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000251The number of nonsingleton blocks of a set partition. St000362The size of a minimal vertex cover of a graph. St000387The matching number of a graph. St000291The number of descents of a binary word. St000390The number of runs of ones in a binary word. St000352The Elizalde-Pak rank of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000386The number of factors DDU in a Dyck path. St000829The Ulam distance of a permutation to the identity permutation. St000035The number of left outer peaks of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000884The number of isolated descents of a permutation. St001083The number of boxed occurrences of 132 in a permutation. St000201The number of leaf nodes in a binary tree. St000522The number of 1-protected nodes of a rooted tree. St000071The number of maximal chains in a poset. St000068The number of minimal elements in a poset. St000527The width of the poset. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001874Lusztig's a-function for the symmetric group. St000731The number of double exceedences of a permutation. St000862The number of parts of the shifted shape of a permutation. St000007The number of saliances of the permutation. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000632The jump number of the poset. St001840The number of descents of a set partition. St000647The number of big descents of a permutation. St000354The number of recoils of a permutation. St000646The number of big ascents of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001729The number of visible descents of a permutation. St001928The number of non-overlapping descents in a permutation. St000470The number of runs in a permutation. St000619The number of cyclic descents of a permutation. St001394The genus of a permutation. St000758The length of the longest staircase fitting into an integer composition. St000779The tier of a permutation. St000711The number of big exceedences of a permutation. St001728The number of invisible descents of a permutation. St000710The number of big deficiencies of a permutation. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St000021The number of descents of a permutation. St000023The number of inner peaks of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000523The number of 2-protected nodes of a rooted tree. St000099The number of valleys of a permutation, including the boundary. St000325The width of the tree associated to a permutation. St000451The length of the longest pattern of the form k 1 2. St000028The number of stack-sorts needed to sort a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001335The cardinality of a minimal cycle-isolating set of a graph. St001220The width of a permutation. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001597The Frobenius rank of a skew partition. St000091The descent variation of a composition. St000534The number of 2-rises of a permutation. St000535The rank-width of a graph. St000552The number of cut vertices of a graph. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001331The size of the minimal feedback vertex set. St001333The cardinality of a minimal edge-isolating set of a graph. St001393The induced matching number of a graph. St001638The book thickness of a graph. St001743The discrepancy of a graph. St001839The number of excedances of a set partition. St000272The treewidth of a graph. St000396The register function (or Horton-Strahler number) of a binary tree. St000482The (zero)-forcing number of a graph. St000536The pathwidth of a graph. St000544The cop number of a graph. St000778The metric dimension of a graph. St001111The weak 2-dynamic chromatic number of a graph. St001261The Castelnuovo-Mumford regularity of a graph. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001716The 1-improper chromatic number of a graph. St001792The arboricity of a graph. St001962The proper pathwidth of a graph. St001093The detour number of a graph. St001118The acyclic chromatic index of a graph. St001580The acyclic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St000486The number of cycles of length at least 3 of a permutation. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St000703The number of deficiencies of a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St001644The dimension of a graph. St000298The order dimension or Dushnik-Miller dimension of a poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000307The number of rowmotion orbits of a poset. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St001960The number of descents of a permutation minus one if its first entry is not one. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001665The number of pure excedances of a permutation. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001330The hat guessing number of a graph. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
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