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Your data matches 17 different statistics following compositions of up to 3 maps.
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Matching statistic: St000714
St000714: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> 3
[1,1]
=> 1
[3]
=> 4
[2,1]
=> 2
[1,1,1]
=> 0
[4]
=> 5
[3,1]
=> 3
[2,2]
=> 1
[2,1,1]
=> 0
[1,1,1,1]
=> 0
[5]
=> 6
[4,1]
=> 4
[3,2]
=> 2
[3,1,1]
=> 0
[2,2,1]
=> 0
[2,1,1,1]
=> 0
[1,1,1,1,1]
=> 0
[6]
=> 7
[5,1]
=> 5
[4,2]
=> 3
[4,1,1]
=> 0
[3,3]
=> 1
[3,2,1]
=> 0
[3,1,1,1]
=> 0
[2,2,2]
=> 0
[2,2,1,1]
=> 0
[2,1,1,1,1]
=> 0
[1,1,1,1,1,1]
=> 0
[7]
=> 8
[6,1]
=> 6
[5,2]
=> 4
[5,1,1]
=> 0
[4,3]
=> 2
[4,2,1]
=> 0
[4,1,1,1]
=> 0
[3,3,1]
=> 0
[3,2,2]
=> 0
[3,2,1,1]
=> 0
[3,1,1,1,1]
=> 0
[2,2,2,1]
=> 0
[2,2,1,1,1]
=> 0
[2,1,1,1,1,1]
=> 0
[1,1,1,1,1,1,1]
=> 0
[8]
=> 9
[7,1]
=> 7
[6,2]
=> 5
[6,1,1]
=> 0
[5,3]
=> 3
[5,2,1]
=> 0
[5,1,1,1]
=> 0
Description
The number of semistandard Young tableau of given shape, with entries at most 2.
This is also the dimension of the corresponding irreducible representation of GL2.
Matching statistic: St000296
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000296: Binary words ⟶ ℤResult quality: 7% ●values known / values provided: 83%●distinct values known / distinct values provided: 7%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000296: Binary words ⟶ ℤResult quality: 7% ●values known / values provided: 83%●distinct values known / distinct values provided: 7%
Values
[2]
=> []
=> ?
=> ? => ? ∊ {1,3}
[1,1]
=> [1]
=> []
=> => ? ∊ {1,3}
[3]
=> []
=> ?
=> ? => ? ∊ {2,4}
[2,1]
=> [1]
=> []
=> => ? ∊ {2,4}
[1,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[4]
=> []
=> ?
=> ? => ? ∊ {1,3,5}
[3,1]
=> [1]
=> []
=> => ? ∊ {1,3,5}
[2,2]
=> [2]
=> []
=> => ? ∊ {1,3,5}
[2,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 0
[5]
=> []
=> ?
=> ? => ? ∊ {2,4,6}
[4,1]
=> [1]
=> []
=> => ? ∊ {2,4,6}
[3,2]
=> [2]
=> []
=> => ? ∊ {2,4,6}
[3,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[2,2,1]
=> [2,1]
=> [1]
=> 10 => 0
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1110 => 0
[6]
=> []
=> ?
=> ? => ? ∊ {1,3,5,7}
[5,1]
=> [1]
=> []
=> => ? ∊ {1,3,5,7}
[4,2]
=> [2]
=> []
=> => ? ∊ {1,3,5,7}
[4,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[3,3]
=> [3]
=> []
=> => ? ∊ {1,3,5,7}
[3,2,1]
=> [2,1]
=> [1]
=> 10 => 0
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 0
[2,2,2]
=> [2,2]
=> [2]
=> 100 => 0
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 110 => 0
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1110 => 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 0
[7]
=> []
=> ?
=> ? => ? ∊ {2,4,6,8}
[6,1]
=> [1]
=> []
=> => ? ∊ {2,4,6,8}
[5,2]
=> [2]
=> []
=> => ? ∊ {2,4,6,8}
[5,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[4,3]
=> [3]
=> []
=> => ? ∊ {2,4,6,8}
[4,2,1]
=> [2,1]
=> [1]
=> 10 => 0
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 0
[3,3,1]
=> [3,1]
=> [1]
=> 10 => 0
[3,2,2]
=> [2,2]
=> [2]
=> 100 => 0
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 110 => 0
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1110 => 0
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1010 => 0
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1110 => 0
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 0
[8]
=> []
=> ?
=> ? => ? ∊ {1,3,5,7,9}
[7,1]
=> [1]
=> []
=> => ? ∊ {1,3,5,7,9}
[6,2]
=> [2]
=> []
=> => ? ∊ {1,3,5,7,9}
[6,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[5,3]
=> [3]
=> []
=> => ? ∊ {1,3,5,7,9}
[5,2,1]
=> [2,1]
=> [1]
=> 10 => 0
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 0
[4,4]
=> [4]
=> []
=> => ? ∊ {1,3,5,7,9}
[4,3,1]
=> [3,1]
=> [1]
=> 10 => 0
[4,2,2]
=> [2,2]
=> [2]
=> 100 => 0
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> 110 => 0
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1110 => 0
[3,3,2]
=> [3,2]
=> [2]
=> 100 => 0
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> 110 => 0
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1010 => 0
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1110 => 0
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 0
[2,2,2,2]
=> [2,2,2]
=> [2,2]
=> 1100 => 0
[2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> 10110 => 0
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 0
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 0
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 1111110 => 0
[9]
=> []
=> ?
=> ? => ? ∊ {2,4,6,8,10}
[8,1]
=> [1]
=> []
=> => ? ∊ {2,4,6,8,10}
[7,2]
=> [2]
=> []
=> => ? ∊ {2,4,6,8,10}
[7,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[6,3]
=> [3]
=> []
=> => ? ∊ {2,4,6,8,10}
[6,2,1]
=> [2,1]
=> [1]
=> 10 => 0
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 0
[5,4]
=> [4]
=> []
=> => ? ∊ {2,4,6,8,10}
[5,3,1]
=> [3,1]
=> [1]
=> 10 => 0
[5,2,2]
=> [2,2]
=> [2]
=> 100 => 0
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> 110 => 0
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1110 => 0
[4,4,1]
=> [4,1]
=> [1]
=> 10 => 0
[10]
=> []
=> ?
=> ? => ? ∊ {1,3,5,7,9,11}
[9,1]
=> [1]
=> []
=> => ? ∊ {1,3,5,7,9,11}
[8,2]
=> [2]
=> []
=> => ? ∊ {1,3,5,7,9,11}
[7,3]
=> [3]
=> []
=> => ? ∊ {1,3,5,7,9,11}
[6,4]
=> [4]
=> []
=> => ? ∊ {1,3,5,7,9,11}
[5,5]
=> [5]
=> []
=> => ? ∊ {1,3,5,7,9,11}
[11]
=> []
=> ?
=> ? => ? ∊ {2,4,6,8,10,12}
[10,1]
=> [1]
=> []
=> => ? ∊ {2,4,6,8,10,12}
[9,2]
=> [2]
=> []
=> => ? ∊ {2,4,6,8,10,12}
[8,3]
=> [3]
=> []
=> => ? ∊ {2,4,6,8,10,12}
[7,4]
=> [4]
=> []
=> => ? ∊ {2,4,6,8,10,12}
[6,5]
=> [5]
=> []
=> => ? ∊ {2,4,6,8,10,12}
[12]
=> []
=> ?
=> ? => ? ∊ {1,3,5,7,9,11,13}
[11,1]
=> [1]
=> []
=> => ? ∊ {1,3,5,7,9,11,13}
[10,2]
=> [2]
=> []
=> => ? ∊ {1,3,5,7,9,11,13}
[9,3]
=> [3]
=> []
=> => ? ∊ {1,3,5,7,9,11,13}
[8,4]
=> [4]
=> []
=> => ? ∊ {1,3,5,7,9,11,13}
[7,5]
=> [5]
=> []
=> => ? ∊ {1,3,5,7,9,11,13}
[6,6]
=> [6]
=> []
=> => ? ∊ {1,3,5,7,9,11,13}
Description
The length of the symmetric border of a binary word.
The symmetric border of a word is the longest word which is a prefix and its reverse is a suffix.
The statistic value is equal to the length of the word if and only if the word is [[https://en.wikipedia.org/wiki/Palindrome|palindromic]].
Matching statistic: St000629
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000629: Binary words ⟶ ℤResult quality: 7% ●values known / values provided: 83%●distinct values known / distinct values provided: 7%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000629: Binary words ⟶ ℤResult quality: 7% ●values known / values provided: 83%●distinct values known / distinct values provided: 7%
Values
[2]
=> []
=> ?
=> ? => ? ∊ {1,3}
[1,1]
=> [1]
=> []
=> => ? ∊ {1,3}
[3]
=> []
=> ?
=> ? => ? ∊ {2,4}
[2,1]
=> [1]
=> []
=> => ? ∊ {2,4}
[1,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[4]
=> []
=> ?
=> ? => ? ∊ {1,3,5}
[3,1]
=> [1]
=> []
=> => ? ∊ {1,3,5}
[2,2]
=> [2]
=> []
=> => ? ∊ {1,3,5}
[2,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 0
[5]
=> []
=> ?
=> ? => ? ∊ {2,4,6}
[4,1]
=> [1]
=> []
=> => ? ∊ {2,4,6}
[3,2]
=> [2]
=> []
=> => ? ∊ {2,4,6}
[3,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[2,2,1]
=> [2,1]
=> [1]
=> 10 => 0
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1110 => 0
[6]
=> []
=> ?
=> ? => ? ∊ {1,3,5,7}
[5,1]
=> [1]
=> []
=> => ? ∊ {1,3,5,7}
[4,2]
=> [2]
=> []
=> => ? ∊ {1,3,5,7}
[4,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[3,3]
=> [3]
=> []
=> => ? ∊ {1,3,5,7}
[3,2,1]
=> [2,1]
=> [1]
=> 10 => 0
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 0
[2,2,2]
=> [2,2]
=> [2]
=> 100 => 0
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 110 => 0
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1110 => 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 0
[7]
=> []
=> ?
=> ? => ? ∊ {2,4,6,8}
[6,1]
=> [1]
=> []
=> => ? ∊ {2,4,6,8}
[5,2]
=> [2]
=> []
=> => ? ∊ {2,4,6,8}
[5,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[4,3]
=> [3]
=> []
=> => ? ∊ {2,4,6,8}
[4,2,1]
=> [2,1]
=> [1]
=> 10 => 0
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 0
[3,3,1]
=> [3,1]
=> [1]
=> 10 => 0
[3,2,2]
=> [2,2]
=> [2]
=> 100 => 0
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 110 => 0
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1110 => 0
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1010 => 0
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1110 => 0
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 0
[8]
=> []
=> ?
=> ? => ? ∊ {1,3,5,7,9}
[7,1]
=> [1]
=> []
=> => ? ∊ {1,3,5,7,9}
[6,2]
=> [2]
=> []
=> => ? ∊ {1,3,5,7,9}
[6,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[5,3]
=> [3]
=> []
=> => ? ∊ {1,3,5,7,9}
[5,2,1]
=> [2,1]
=> [1]
=> 10 => 0
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 0
[4,4]
=> [4]
=> []
=> => ? ∊ {1,3,5,7,9}
[4,3,1]
=> [3,1]
=> [1]
=> 10 => 0
[4,2,2]
=> [2,2]
=> [2]
=> 100 => 0
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> 110 => 0
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1110 => 0
[3,3,2]
=> [3,2]
=> [2]
=> 100 => 0
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> 110 => 0
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1010 => 0
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1110 => 0
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 0
[2,2,2,2]
=> [2,2,2]
=> [2,2]
=> 1100 => 0
[2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> 10110 => 0
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 0
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 0
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 1111110 => 0
[9]
=> []
=> ?
=> ? => ? ∊ {2,4,6,8,10}
[8,1]
=> [1]
=> []
=> => ? ∊ {2,4,6,8,10}
[7,2]
=> [2]
=> []
=> => ? ∊ {2,4,6,8,10}
[7,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[6,3]
=> [3]
=> []
=> => ? ∊ {2,4,6,8,10}
[6,2,1]
=> [2,1]
=> [1]
=> 10 => 0
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 0
[5,4]
=> [4]
=> []
=> => ? ∊ {2,4,6,8,10}
[5,3,1]
=> [3,1]
=> [1]
=> 10 => 0
[5,2,2]
=> [2,2]
=> [2]
=> 100 => 0
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> 110 => 0
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1110 => 0
[4,4,1]
=> [4,1]
=> [1]
=> 10 => 0
[10]
=> []
=> ?
=> ? => ? ∊ {1,3,5,7,9,11}
[9,1]
=> [1]
=> []
=> => ? ∊ {1,3,5,7,9,11}
[8,2]
=> [2]
=> []
=> => ? ∊ {1,3,5,7,9,11}
[7,3]
=> [3]
=> []
=> => ? ∊ {1,3,5,7,9,11}
[6,4]
=> [4]
=> []
=> => ? ∊ {1,3,5,7,9,11}
[5,5]
=> [5]
=> []
=> => ? ∊ {1,3,5,7,9,11}
[11]
=> []
=> ?
=> ? => ? ∊ {2,4,6,8,10,12}
[10,1]
=> [1]
=> []
=> => ? ∊ {2,4,6,8,10,12}
[9,2]
=> [2]
=> []
=> => ? ∊ {2,4,6,8,10,12}
[8,3]
=> [3]
=> []
=> => ? ∊ {2,4,6,8,10,12}
[7,4]
=> [4]
=> []
=> => ? ∊ {2,4,6,8,10,12}
[6,5]
=> [5]
=> []
=> => ? ∊ {2,4,6,8,10,12}
[12]
=> []
=> ?
=> ? => ? ∊ {1,3,5,7,9,11,13}
[11,1]
=> [1]
=> []
=> => ? ∊ {1,3,5,7,9,11,13}
[10,2]
=> [2]
=> []
=> => ? ∊ {1,3,5,7,9,11,13}
[9,3]
=> [3]
=> []
=> => ? ∊ {1,3,5,7,9,11,13}
[8,4]
=> [4]
=> []
=> => ? ∊ {1,3,5,7,9,11,13}
[7,5]
=> [5]
=> []
=> => ? ∊ {1,3,5,7,9,11,13}
[6,6]
=> [6]
=> []
=> => ? ∊ {1,3,5,7,9,11,13}
Description
The defect of a binary word.
The defect of a finite word w is given by the difference between the maximum possible number and the actual number of palindromic factors contained in w. The maximum possible number of palindromic factors in a word w is |w|+1.
Matching statistic: St000752
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000752: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 83%●distinct values known / distinct values provided: 14%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000752: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 83%●distinct values known / distinct values provided: 14%
Values
[2]
=> []
=> ?
=> ?
=> ? ∊ {1,3}
[1,1]
=> [1]
=> []
=> ?
=> ? ∊ {1,3}
[3]
=> []
=> ?
=> ?
=> ? ∊ {2,4}
[2,1]
=> [1]
=> []
=> ?
=> ? ∊ {2,4}
[1,1,1]
=> [1,1]
=> [1]
=> []
=> 0
[4]
=> []
=> ?
=> ?
=> ? ∊ {1,3,5}
[3,1]
=> [1]
=> []
=> ?
=> ? ∊ {1,3,5}
[2,2]
=> [2]
=> []
=> ?
=> ? ∊ {1,3,5}
[2,1,1]
=> [1,1]
=> [1]
=> []
=> 0
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
[5]
=> []
=> ?
=> ?
=> ? ∊ {2,4,6}
[4,1]
=> [1]
=> []
=> ?
=> ? ∊ {2,4,6}
[3,2]
=> [2]
=> []
=> ?
=> ? ∊ {2,4,6}
[3,1,1]
=> [1,1]
=> [1]
=> []
=> 0
[2,2,1]
=> [2,1]
=> [1]
=> []
=> 0
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[6]
=> []
=> ?
=> ?
=> ? ∊ {1,3,5,7}
[5,1]
=> [1]
=> []
=> ?
=> ? ∊ {1,3,5,7}
[4,2]
=> [2]
=> []
=> ?
=> ? ∊ {1,3,5,7}
[4,1,1]
=> [1,1]
=> [1]
=> []
=> 0
[3,3]
=> [3]
=> []
=> ?
=> ? ∊ {1,3,5,7}
[3,2,1]
=> [2,1]
=> [1]
=> []
=> 0
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
[2,2,2]
=> [2,2]
=> [2]
=> []
=> 0
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[7]
=> []
=> ?
=> ?
=> ? ∊ {2,4,6,8}
[6,1]
=> [1]
=> []
=> ?
=> ? ∊ {2,4,6,8}
[5,2]
=> [2]
=> []
=> ?
=> ? ∊ {2,4,6,8}
[5,1,1]
=> [1,1]
=> [1]
=> []
=> 0
[4,3]
=> [3]
=> []
=> ?
=> ? ∊ {2,4,6,8}
[4,2,1]
=> [2,1]
=> [1]
=> []
=> 0
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
[3,3,1]
=> [3,1]
=> [1]
=> []
=> 0
[3,2,2]
=> [2,2]
=> [2]
=> []
=> 0
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[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]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[8]
=> []
=> ?
=> ?
=> ? ∊ {1,3,5,7,9}
[7,1]
=> [1]
=> []
=> ?
=> ? ∊ {1,3,5,7,9}
[6,2]
=> [2]
=> []
=> ?
=> ? ∊ {1,3,5,7,9}
[6,1,1]
=> [1,1]
=> [1]
=> []
=> 0
[5,3]
=> [3]
=> []
=> ?
=> ? ∊ {1,3,5,7,9}
[5,2,1]
=> [2,1]
=> [1]
=> []
=> 0
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
[4,4]
=> [4]
=> []
=> ?
=> ? ∊ {1,3,5,7,9}
[4,3,1]
=> [3,1]
=> [1]
=> []
=> 0
[4,2,2]
=> [2,2]
=> [2]
=> []
=> 0
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[3,3,2]
=> [3,2]
=> [2]
=> []
=> 0
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[2,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
[9]
=> []
=> ?
=> ?
=> ? ∊ {2,4,6,8,10}
[8,1]
=> [1]
=> []
=> ?
=> ? ∊ {2,4,6,8,10}
[7,2]
=> [2]
=> []
=> ?
=> ? ∊ {2,4,6,8,10}
[7,1,1]
=> [1,1]
=> [1]
=> []
=> 0
[6,3]
=> [3]
=> []
=> ?
=> ? ∊ {2,4,6,8,10}
[6,2,1]
=> [2,1]
=> [1]
=> []
=> 0
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
[5,4]
=> [4]
=> []
=> ?
=> ? ∊ {2,4,6,8,10}
[5,3,1]
=> [3,1]
=> [1]
=> []
=> 0
[5,2,2]
=> [2,2]
=> [2]
=> []
=> 0
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[4,4,1]
=> [4,1]
=> [1]
=> []
=> 0
[10]
=> []
=> ?
=> ?
=> ? ∊ {1,3,5,7,9,11}
[9,1]
=> [1]
=> []
=> ?
=> ? ∊ {1,3,5,7,9,11}
[8,2]
=> [2]
=> []
=> ?
=> ? ∊ {1,3,5,7,9,11}
[7,3]
=> [3]
=> []
=> ?
=> ? ∊ {1,3,5,7,9,11}
[6,4]
=> [4]
=> []
=> ?
=> ? ∊ {1,3,5,7,9,11}
[5,5]
=> [5]
=> []
=> ?
=> ? ∊ {1,3,5,7,9,11}
[11]
=> []
=> ?
=> ?
=> ? ∊ {2,4,6,8,10,12}
[10,1]
=> [1]
=> []
=> ?
=> ? ∊ {2,4,6,8,10,12}
[9,2]
=> [2]
=> []
=> ?
=> ? ∊ {2,4,6,8,10,12}
[8,3]
=> [3]
=> []
=> ?
=> ? ∊ {2,4,6,8,10,12}
[7,4]
=> [4]
=> []
=> ?
=> ? ∊ {2,4,6,8,10,12}
[6,5]
=> [5]
=> []
=> ?
=> ? ∊ {2,4,6,8,10,12}
[12]
=> []
=> ?
=> ?
=> ? ∊ {0,3,5,7,9,11,13}
[11,1]
=> [1]
=> []
=> ?
=> ? ∊ {0,3,5,7,9,11,13}
[10,2]
=> [2]
=> []
=> ?
=> ? ∊ {0,3,5,7,9,11,13}
[9,3]
=> [3]
=> []
=> ?
=> ? ∊ {0,3,5,7,9,11,13}
[8,4]
=> [4]
=> []
=> ?
=> ? ∊ {0,3,5,7,9,11,13}
[7,5]
=> [5]
=> []
=> ?
=> ? ∊ {0,3,5,7,9,11,13}
[6,6]
=> [6]
=> []
=> ?
=> ? ∊ {0,3,5,7,9,11,13}
Description
The Grundy value for the game 'Couples are forever' on an integer partition.
Two players alternately choose a part of the partition greater than two, and split it into two parts. The player facing a partition with all parts at most two looses.
Matching statistic: St000921
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St000921: Binary words ⟶ ℤResult quality: 7% ●values known / values provided: 82%●distinct values known / distinct values provided: 7%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St000921: Binary words ⟶ ℤResult quality: 7% ●values known / values provided: 82%●distinct values known / distinct values provided: 7%
Values
[2]
=> []
=> ?
=> ? => ? ∊ {1,3}
[1,1]
=> [1]
=> []
=> ? => ? ∊ {1,3}
[3]
=> []
=> ?
=> ? => ? ∊ {2,4}
[2,1]
=> [1]
=> []
=> ? => ? ∊ {2,4}
[1,1,1]
=> [1,1]
=> [1]
=> 1 => 0
[4]
=> []
=> ?
=> ? => ? ∊ {1,3,5}
[3,1]
=> [1]
=> []
=> ? => ? ∊ {1,3,5}
[2,2]
=> [2]
=> []
=> ? => ? ∊ {1,3,5}
[2,1,1]
=> [1,1]
=> [1]
=> 1 => 0
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 11 => 0
[5]
=> []
=> ?
=> ? => ? ∊ {2,4,6}
[4,1]
=> [1]
=> []
=> ? => ? ∊ {2,4,6}
[3,2]
=> [2]
=> []
=> ? => ? ∊ {2,4,6}
[3,1,1]
=> [1,1]
=> [1]
=> 1 => 0
[2,2,1]
=> [2,1]
=> [1]
=> 1 => 0
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 11 => 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 111 => 0
[6]
=> []
=> ?
=> ? => ? ∊ {1,3,5,7}
[5,1]
=> [1]
=> []
=> ? => ? ∊ {1,3,5,7}
[4,2]
=> [2]
=> []
=> ? => ? ∊ {1,3,5,7}
[4,1,1]
=> [1,1]
=> [1]
=> 1 => 0
[3,3]
=> [3]
=> []
=> ? => ? ∊ {1,3,5,7}
[3,2,1]
=> [2,1]
=> [1]
=> 1 => 0
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 11 => 0
[2,2,2]
=> [2,2]
=> [2]
=> 0 => 0
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 11 => 0
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 111 => 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1111 => 0
[7]
=> []
=> ?
=> ? => ? ∊ {2,4,6,8}
[6,1]
=> [1]
=> []
=> ? => ? ∊ {2,4,6,8}
[5,2]
=> [2]
=> []
=> ? => ? ∊ {2,4,6,8}
[5,1,1]
=> [1,1]
=> [1]
=> 1 => 0
[4,3]
=> [3]
=> []
=> ? => ? ∊ {2,4,6,8}
[4,2,1]
=> [2,1]
=> [1]
=> 1 => 0
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 11 => 0
[3,3,1]
=> [3,1]
=> [1]
=> 1 => 0
[3,2,2]
=> [2,2]
=> [2]
=> 0 => 0
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 11 => 0
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 111 => 0
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 01 => 0
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 111 => 0
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1111 => 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 11111 => 0
[8]
=> []
=> ?
=> ? => ? ∊ {1,3,5,7,9}
[7,1]
=> [1]
=> []
=> ? => ? ∊ {1,3,5,7,9}
[6,2]
=> [2]
=> []
=> ? => ? ∊ {1,3,5,7,9}
[6,1,1]
=> [1,1]
=> [1]
=> 1 => 0
[5,3]
=> [3]
=> []
=> ? => ? ∊ {1,3,5,7,9}
[5,2,1]
=> [2,1]
=> [1]
=> 1 => 0
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 11 => 0
[4,4]
=> [4]
=> []
=> ? => ? ∊ {1,3,5,7,9}
[4,3,1]
=> [3,1]
=> [1]
=> 1 => 0
[4,2,2]
=> [2,2]
=> [2]
=> 0 => 0
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> 11 => 0
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 111 => 0
[3,3,2]
=> [3,2]
=> [2]
=> 0 => 0
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> 11 => 0
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 01 => 0
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 111 => 0
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1111 => 0
[2,2,2,2]
=> [2,2,2]
=> [2,2]
=> 00 => 0
[2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> 011 => 0
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 1111 => 0
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 11111 => 0
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 111111 => 0
[9]
=> []
=> ?
=> ? => ? ∊ {2,4,6,8,10}
[8,1]
=> [1]
=> []
=> ? => ? ∊ {2,4,6,8,10}
[7,2]
=> [2]
=> []
=> ? => ? ∊ {2,4,6,8,10}
[7,1,1]
=> [1,1]
=> [1]
=> 1 => 0
[6,3]
=> [3]
=> []
=> ? => ? ∊ {2,4,6,8,10}
[6,2,1]
=> [2,1]
=> [1]
=> 1 => 0
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 11 => 0
[5,4]
=> [4]
=> []
=> ? => ? ∊ {2,4,6,8,10}
[5,3,1]
=> [3,1]
=> [1]
=> 1 => 0
[5,2,2]
=> [2,2]
=> [2]
=> 0 => 0
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> 11 => 0
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 111 => 0
[4,4,1]
=> [4,1]
=> [1]
=> 1 => 0
[10]
=> []
=> ?
=> ? => ? ∊ {1,3,5,7,9,11}
[9,1]
=> [1]
=> []
=> ? => ? ∊ {1,3,5,7,9,11}
[8,2]
=> [2]
=> []
=> ? => ? ∊ {1,3,5,7,9,11}
[7,3]
=> [3]
=> []
=> ? => ? ∊ {1,3,5,7,9,11}
[6,4]
=> [4]
=> []
=> ? => ? ∊ {1,3,5,7,9,11}
[5,5]
=> [5]
=> []
=> ? => ? ∊ {1,3,5,7,9,11}
[11]
=> []
=> ?
=> ? => ? ∊ {2,4,6,8,10,12}
[10,1]
=> [1]
=> []
=> ? => ? ∊ {2,4,6,8,10,12}
[9,2]
=> [2]
=> []
=> ? => ? ∊ {2,4,6,8,10,12}
[8,3]
=> [3]
=> []
=> ? => ? ∊ {2,4,6,8,10,12}
[7,4]
=> [4]
=> []
=> ? => ? ∊ {2,4,6,8,10,12}
[6,5]
=> [5]
=> []
=> ? => ? ∊ {2,4,6,8,10,12}
[12]
=> []
=> ?
=> ? => ? ∊ {0,1,3,5,7,9,11,13}
[11,1]
=> [1]
=> []
=> ? => ? ∊ {0,1,3,5,7,9,11,13}
[10,2]
=> [2]
=> []
=> ? => ? ∊ {0,1,3,5,7,9,11,13}
[9,3]
=> [3]
=> []
=> ? => ? ∊ {0,1,3,5,7,9,11,13}
[8,4]
=> [4]
=> []
=> ? => ? ∊ {0,1,3,5,7,9,11,13}
[7,5]
=> [5]
=> []
=> ? => ? ∊ {0,1,3,5,7,9,11,13}
[6,6]
=> [6]
=> []
=> ? => ? ∊ {0,1,3,5,7,9,11,13}
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> 1111111111 => ? ∊ {0,1,3,5,7,9,11,13}
Description
The number of internal inversions of a binary word.
Let ˉw be the non-decreasing rearrangement of w, that is, ˉw is sorted.
An internal inversion is a pair i<j such that wi>wj and ˉwi=ˉwj. For example, the word 110 has two inversions, but only the second is internal.
Matching statistic: St001371
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St001371: Binary words ⟶ ℤResult quality: 7% ●values known / values provided: 81%●distinct values known / distinct values provided: 7%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St001371: Binary words ⟶ ℤResult quality: 7% ●values known / values provided: 81%●distinct values known / distinct values provided: 7%
Values
[2]
=> []
=> ?
=> ? => ? ∊ {1,3}
[1,1]
=> [1]
=> []
=> => ? ∊ {1,3}
[3]
=> []
=> ?
=> ? => ? ∊ {2,4}
[2,1]
=> [1]
=> []
=> => ? ∊ {2,4}
[1,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[4]
=> []
=> ?
=> ? => ? ∊ {1,3,5}
[3,1]
=> [1]
=> []
=> => ? ∊ {1,3,5}
[2,2]
=> [2]
=> []
=> => ? ∊ {1,3,5}
[2,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 0
[5]
=> []
=> ?
=> ? => ? ∊ {2,4,6}
[4,1]
=> [1]
=> []
=> => ? ∊ {2,4,6}
[3,2]
=> [2]
=> []
=> => ? ∊ {2,4,6}
[3,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[2,2,1]
=> [2,1]
=> [1]
=> 10 => 0
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1110 => 0
[6]
=> []
=> ?
=> ? => ? ∊ {1,3,5,7}
[5,1]
=> [1]
=> []
=> => ? ∊ {1,3,5,7}
[4,2]
=> [2]
=> []
=> => ? ∊ {1,3,5,7}
[4,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[3,3]
=> [3]
=> []
=> => ? ∊ {1,3,5,7}
[3,2,1]
=> [2,1]
=> [1]
=> 10 => 0
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 0
[2,2,2]
=> [2,2]
=> [2]
=> 100 => 0
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 110 => 0
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1110 => 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 0
[7]
=> []
=> ?
=> ? => ? ∊ {2,4,6,8}
[6,1]
=> [1]
=> []
=> => ? ∊ {2,4,6,8}
[5,2]
=> [2]
=> []
=> => ? ∊ {2,4,6,8}
[5,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[4,3]
=> [3]
=> []
=> => ? ∊ {2,4,6,8}
[4,2,1]
=> [2,1]
=> [1]
=> 10 => 0
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 0
[3,3,1]
=> [3,1]
=> [1]
=> 10 => 0
[3,2,2]
=> [2,2]
=> [2]
=> 100 => 0
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 110 => 0
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1110 => 0
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1010 => 0
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1110 => 0
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 0
[8]
=> []
=> ?
=> ? => ? ∊ {1,3,5,7,9}
[7,1]
=> [1]
=> []
=> => ? ∊ {1,3,5,7,9}
[6,2]
=> [2]
=> []
=> => ? ∊ {1,3,5,7,9}
[6,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[5,3]
=> [3]
=> []
=> => ? ∊ {1,3,5,7,9}
[5,2,1]
=> [2,1]
=> [1]
=> 10 => 0
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 0
[4,4]
=> [4]
=> []
=> => ? ∊ {1,3,5,7,9}
[4,3,1]
=> [3,1]
=> [1]
=> 10 => 0
[4,2,2]
=> [2,2]
=> [2]
=> 100 => 0
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> 110 => 0
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1110 => 0
[3,3,2]
=> [3,2]
=> [2]
=> 100 => 0
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> 110 => 0
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1010 => 0
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1110 => 0
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 0
[2,2,2,2]
=> [2,2,2]
=> [2,2]
=> 1100 => 0
[2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> 10110 => 0
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 0
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 0
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 1111110 => 0
[9]
=> []
=> ?
=> ? => ? ∊ {2,4,6,8,10}
[8,1]
=> [1]
=> []
=> => ? ∊ {2,4,6,8,10}
[7,2]
=> [2]
=> []
=> => ? ∊ {2,4,6,8,10}
[7,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[6,3]
=> [3]
=> []
=> => ? ∊ {2,4,6,8,10}
[6,2,1]
=> [2,1]
=> [1]
=> 10 => 0
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 0
[5,4]
=> [4]
=> []
=> => ? ∊ {2,4,6,8,10}
[5,3,1]
=> [3,1]
=> [1]
=> 10 => 0
[5,2,2]
=> [2,2]
=> [2]
=> 100 => 0
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> 110 => 0
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1110 => 0
[4,4,1]
=> [4,1]
=> [1]
=> 10 => 0
[10]
=> []
=> ?
=> ? => ? ∊ {1,3,5,7,9,11}
[9,1]
=> [1]
=> []
=> => ? ∊ {1,3,5,7,9,11}
[8,2]
=> [2]
=> []
=> => ? ∊ {1,3,5,7,9,11}
[7,3]
=> [3]
=> []
=> => ? ∊ {1,3,5,7,9,11}
[6,4]
=> [4]
=> []
=> => ? ∊ {1,3,5,7,9,11}
[5,5]
=> [5]
=> []
=> => ? ∊ {1,3,5,7,9,11}
[11]
=> []
=> ?
=> ? => ? ∊ {0,2,4,6,8,10,12}
[10,1]
=> [1]
=> []
=> => ? ∊ {0,2,4,6,8,10,12}
[9,2]
=> [2]
=> []
=> => ? ∊ {0,2,4,6,8,10,12}
[8,3]
=> [3]
=> []
=> => ? ∊ {0,2,4,6,8,10,12}
[7,4]
=> [4]
=> []
=> => ? ∊ {0,2,4,6,8,10,12}
[6,5]
=> [5]
=> []
=> => ? ∊ {0,2,4,6,8,10,12}
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> 1111111110 => ? ∊ {0,2,4,6,8,10,12}
[12]
=> []
=> ?
=> ? => ? ∊ {0,0,1,3,5,7,9,11,13}
[11,1]
=> [1]
=> []
=> => ? ∊ {0,0,1,3,5,7,9,11,13}
[10,2]
=> [2]
=> []
=> => ? ∊ {0,0,1,3,5,7,9,11,13}
[9,3]
=> [3]
=> []
=> => ? ∊ {0,0,1,3,5,7,9,11,13}
[8,4]
=> [4]
=> []
=> => ? ∊ {0,0,1,3,5,7,9,11,13}
[7,5]
=> [5]
=> []
=> => ? ∊ {0,0,1,3,5,7,9,11,13}
[6,6]
=> [6]
=> []
=> => ? ∊ {0,0,1,3,5,7,9,11,13}
[2,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> 1111111110 => ? ∊ {0,0,1,3,5,7,9,11,13}
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> 11111111110 => ? ∊ {0,0,1,3,5,7,9,11,13}
Description
The length of the longest Yamanouchi prefix of a binary word.
This is the largest index i such that in each of the prefixes w1, w1w2, w1w2…wi the number of zeros is greater than or equal to the number of ones.
Matching statistic: St001695
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St001695: Standard tableaux ⟶ ℤResult quality: 7% ●values known / values provided: 81%●distinct values known / distinct values provided: 7%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St001695: Standard tableaux ⟶ ℤResult quality: 7% ●values known / values provided: 81%●distinct values known / distinct values provided: 7%
Values
[2]
=> []
=> ?
=> ?
=> ? ∊ {1,3}
[1,1]
=> [1]
=> []
=> []
=> ? ∊ {1,3}
[3]
=> []
=> ?
=> ?
=> ? ∊ {2,4}
[2,1]
=> [1]
=> []
=> []
=> ? ∊ {2,4}
[1,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[4]
=> []
=> ?
=> ?
=> ? ∊ {1,3,5}
[3,1]
=> [1]
=> []
=> []
=> ? ∊ {1,3,5}
[2,2]
=> [2]
=> []
=> []
=> ? ∊ {1,3,5}
[2,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[5]
=> []
=> ?
=> ?
=> ? ∊ {2,4,6}
[4,1]
=> [1]
=> []
=> []
=> ? ∊ {2,4,6}
[3,2]
=> [2]
=> []
=> []
=> ? ∊ {2,4,6}
[3,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[2,2,1]
=> [2,1]
=> [1]
=> [[1]]
=> 0
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[6]
=> []
=> ?
=> ?
=> ? ∊ {1,3,5,7}
[5,1]
=> [1]
=> []
=> []
=> ? ∊ {1,3,5,7}
[4,2]
=> [2]
=> []
=> []
=> ? ∊ {1,3,5,7}
[4,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[3,3]
=> [3]
=> []
=> []
=> ? ∊ {1,3,5,7}
[3,2,1]
=> [2,1]
=> [1]
=> [[1]]
=> 0
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[2,2,2]
=> [2,2]
=> [2]
=> [[1,2]]
=> 0
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[7]
=> []
=> ?
=> ?
=> ? ∊ {2,4,6,8}
[6,1]
=> [1]
=> []
=> []
=> ? ∊ {2,4,6,8}
[5,2]
=> [2]
=> []
=> []
=> ? ∊ {2,4,6,8}
[5,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[4,3]
=> [3]
=> []
=> []
=> ? ∊ {2,4,6,8}
[4,2,1]
=> [2,1]
=> [1]
=> [[1]]
=> 0
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[3,3,1]
=> [3,1]
=> [1]
=> [[1]]
=> 0
[3,2,2]
=> [2,2]
=> [2]
=> [[1,2]]
=> 0
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [[1,2],[3]]
=> 0
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 0
[8]
=> []
=> ?
=> ?
=> ? ∊ {1,3,5,7,9}
[7,1]
=> [1]
=> []
=> []
=> ? ∊ {1,3,5,7,9}
[6,2]
=> [2]
=> []
=> []
=> ? ∊ {1,3,5,7,9}
[6,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[5,3]
=> [3]
=> []
=> []
=> ? ∊ {1,3,5,7,9}
[5,2,1]
=> [2,1]
=> [1]
=> [[1]]
=> 0
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[4,4]
=> [4]
=> []
=> []
=> ? ∊ {1,3,5,7,9}
[4,3,1]
=> [3,1]
=> [1]
=> [[1]]
=> 0
[4,2,2]
=> [2,2]
=> [2]
=> [[1,2]]
=> 0
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[3,3,2]
=> [3,2]
=> [2]
=> [[1,2]]
=> 0
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> [[1,2],[3]]
=> 0
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[2,2,2,2]
=> [2,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 0
[2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 0
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 0
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> 0
[9]
=> []
=> ?
=> ?
=> ? ∊ {2,4,6,8,10}
[8,1]
=> [1]
=> []
=> []
=> ? ∊ {2,4,6,8,10}
[7,2]
=> [2]
=> []
=> []
=> ? ∊ {2,4,6,8,10}
[7,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[6,3]
=> [3]
=> []
=> []
=> ? ∊ {2,4,6,8,10}
[6,2,1]
=> [2,1]
=> [1]
=> [[1]]
=> 0
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[5,4]
=> [4]
=> []
=> []
=> ? ∊ {2,4,6,8,10}
[5,3,1]
=> [3,1]
=> [1]
=> [[1]]
=> 0
[5,2,2]
=> [2,2]
=> [2]
=> [[1,2]]
=> 0
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[4,4,1]
=> [4,1]
=> [1]
=> [[1]]
=> 0
[10]
=> []
=> ?
=> ?
=> ? ∊ {1,3,5,7,9,11}
[9,1]
=> [1]
=> []
=> []
=> ? ∊ {1,3,5,7,9,11}
[8,2]
=> [2]
=> []
=> []
=> ? ∊ {1,3,5,7,9,11}
[7,3]
=> [3]
=> []
=> []
=> ? ∊ {1,3,5,7,9,11}
[6,4]
=> [4]
=> []
=> []
=> ? ∊ {1,3,5,7,9,11}
[5,5]
=> [5]
=> []
=> []
=> ? ∊ {1,3,5,7,9,11}
[11]
=> []
=> ?
=> ?
=> ? ∊ {0,2,4,6,8,10,12}
[10,1]
=> [1]
=> []
=> []
=> ? ∊ {0,2,4,6,8,10,12}
[9,2]
=> [2]
=> []
=> []
=> ? ∊ {0,2,4,6,8,10,12}
[8,3]
=> [3]
=> []
=> []
=> ? ∊ {0,2,4,6,8,10,12}
[7,4]
=> [4]
=> []
=> []
=> ? ∊ {0,2,4,6,8,10,12}
[6,5]
=> [5]
=> []
=> []
=> ? ∊ {0,2,4,6,8,10,12}
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? ∊ {0,2,4,6,8,10,12}
[12]
=> []
=> ?
=> ?
=> ? ∊ {0,0,1,3,5,7,9,11,13}
[11,1]
=> [1]
=> []
=> []
=> ? ∊ {0,0,1,3,5,7,9,11,13}
[10,2]
=> [2]
=> []
=> []
=> ? ∊ {0,0,1,3,5,7,9,11,13}
[9,3]
=> [3]
=> []
=> []
=> ? ∊ {0,0,1,3,5,7,9,11,13}
[8,4]
=> [4]
=> []
=> []
=> ? ∊ {0,0,1,3,5,7,9,11,13}
[7,5]
=> [5]
=> []
=> []
=> ? ∊ {0,0,1,3,5,7,9,11,13}
[6,6]
=> [6]
=> []
=> []
=> ? ∊ {0,0,1,3,5,7,9,11,13}
[2,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? ∊ {0,0,1,3,5,7,9,11,13}
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? ∊ {0,0,1,3,5,7,9,11,13}
Description
The natural comajor index of a standard Young tableau.
A natural descent of a standard tableau T is an entry i such that i+1 appears in a higher row than i in English notation.
The natural comajor index of a tableau of size n with natural descent set D is then ∑d∈Dn−d.
Matching statistic: St001698
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St001698: Standard tableaux ⟶ ℤResult quality: 7% ●values known / values provided: 81%●distinct values known / distinct values provided: 7%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St001698: Standard tableaux ⟶ ℤResult quality: 7% ●values known / values provided: 81%●distinct values known / distinct values provided: 7%
Values
[2]
=> []
=> ?
=> ?
=> ? ∊ {1,3}
[1,1]
=> [1]
=> []
=> []
=> ? ∊ {1,3}
[3]
=> []
=> ?
=> ?
=> ? ∊ {2,4}
[2,1]
=> [1]
=> []
=> []
=> ? ∊ {2,4}
[1,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[4]
=> []
=> ?
=> ?
=> ? ∊ {1,3,5}
[3,1]
=> [1]
=> []
=> []
=> ? ∊ {1,3,5}
[2,2]
=> [2]
=> []
=> []
=> ? ∊ {1,3,5}
[2,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[5]
=> []
=> ?
=> ?
=> ? ∊ {2,4,6}
[4,1]
=> [1]
=> []
=> []
=> ? ∊ {2,4,6}
[3,2]
=> [2]
=> []
=> []
=> ? ∊ {2,4,6}
[3,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[2,2,1]
=> [2,1]
=> [1]
=> [[1]]
=> 0
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[6]
=> []
=> ?
=> ?
=> ? ∊ {1,3,5,7}
[5,1]
=> [1]
=> []
=> []
=> ? ∊ {1,3,5,7}
[4,2]
=> [2]
=> []
=> []
=> ? ∊ {1,3,5,7}
[4,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[3,3]
=> [3]
=> []
=> []
=> ? ∊ {1,3,5,7}
[3,2,1]
=> [2,1]
=> [1]
=> [[1]]
=> 0
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[2,2,2]
=> [2,2]
=> [2]
=> [[1,2]]
=> 0
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[7]
=> []
=> ?
=> ?
=> ? ∊ {2,4,6,8}
[6,1]
=> [1]
=> []
=> []
=> ? ∊ {2,4,6,8}
[5,2]
=> [2]
=> []
=> []
=> ? ∊ {2,4,6,8}
[5,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[4,3]
=> [3]
=> []
=> []
=> ? ∊ {2,4,6,8}
[4,2,1]
=> [2,1]
=> [1]
=> [[1]]
=> 0
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[3,3,1]
=> [3,1]
=> [1]
=> [[1]]
=> 0
[3,2,2]
=> [2,2]
=> [2]
=> [[1,2]]
=> 0
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [[1,2],[3]]
=> 0
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 0
[8]
=> []
=> ?
=> ?
=> ? ∊ {1,3,5,7,9}
[7,1]
=> [1]
=> []
=> []
=> ? ∊ {1,3,5,7,9}
[6,2]
=> [2]
=> []
=> []
=> ? ∊ {1,3,5,7,9}
[6,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[5,3]
=> [3]
=> []
=> []
=> ? ∊ {1,3,5,7,9}
[5,2,1]
=> [2,1]
=> [1]
=> [[1]]
=> 0
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[4,4]
=> [4]
=> []
=> []
=> ? ∊ {1,3,5,7,9}
[4,3,1]
=> [3,1]
=> [1]
=> [[1]]
=> 0
[4,2,2]
=> [2,2]
=> [2]
=> [[1,2]]
=> 0
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[3,3,2]
=> [3,2]
=> [2]
=> [[1,2]]
=> 0
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> [[1,2],[3]]
=> 0
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[2,2,2,2]
=> [2,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 0
[2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 0
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 0
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> 0
[9]
=> []
=> ?
=> ?
=> ? ∊ {2,4,6,8,10}
[8,1]
=> [1]
=> []
=> []
=> ? ∊ {2,4,6,8,10}
[7,2]
=> [2]
=> []
=> []
=> ? ∊ {2,4,6,8,10}
[7,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[6,3]
=> [3]
=> []
=> []
=> ? ∊ {2,4,6,8,10}
[6,2,1]
=> [2,1]
=> [1]
=> [[1]]
=> 0
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[5,4]
=> [4]
=> []
=> []
=> ? ∊ {2,4,6,8,10}
[5,3,1]
=> [3,1]
=> [1]
=> [[1]]
=> 0
[5,2,2]
=> [2,2]
=> [2]
=> [[1,2]]
=> 0
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[4,4,1]
=> [4,1]
=> [1]
=> [[1]]
=> 0
[10]
=> []
=> ?
=> ?
=> ? ∊ {1,3,5,7,9,11}
[9,1]
=> [1]
=> []
=> []
=> ? ∊ {1,3,5,7,9,11}
[8,2]
=> [2]
=> []
=> []
=> ? ∊ {1,3,5,7,9,11}
[7,3]
=> [3]
=> []
=> []
=> ? ∊ {1,3,5,7,9,11}
[6,4]
=> [4]
=> []
=> []
=> ? ∊ {1,3,5,7,9,11}
[5,5]
=> [5]
=> []
=> []
=> ? ∊ {1,3,5,7,9,11}
[11]
=> []
=> ?
=> ?
=> ? ∊ {0,2,4,6,8,10,12}
[10,1]
=> [1]
=> []
=> []
=> ? ∊ {0,2,4,6,8,10,12}
[9,2]
=> [2]
=> []
=> []
=> ? ∊ {0,2,4,6,8,10,12}
[8,3]
=> [3]
=> []
=> []
=> ? ∊ {0,2,4,6,8,10,12}
[7,4]
=> [4]
=> []
=> []
=> ? ∊ {0,2,4,6,8,10,12}
[6,5]
=> [5]
=> []
=> []
=> ? ∊ {0,2,4,6,8,10,12}
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? ∊ {0,2,4,6,8,10,12}
[12]
=> []
=> ?
=> ?
=> ? ∊ {0,0,1,3,5,7,9,11,13}
[11,1]
=> [1]
=> []
=> []
=> ? ∊ {0,0,1,3,5,7,9,11,13}
[10,2]
=> [2]
=> []
=> []
=> ? ∊ {0,0,1,3,5,7,9,11,13}
[9,3]
=> [3]
=> []
=> []
=> ? ∊ {0,0,1,3,5,7,9,11,13}
[8,4]
=> [4]
=> []
=> []
=> ? ∊ {0,0,1,3,5,7,9,11,13}
[7,5]
=> [5]
=> []
=> []
=> ? ∊ {0,0,1,3,5,7,9,11,13}
[6,6]
=> [6]
=> []
=> []
=> ? ∊ {0,0,1,3,5,7,9,11,13}
[2,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? ∊ {0,0,1,3,5,7,9,11,13}
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? ∊ {0,0,1,3,5,7,9,11,13}
Description
The comajor index of a standard tableau minus the weighted size of its shape.
Matching statistic: St001699
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St001699: Standard tableaux ⟶ ℤResult quality: 7% ●values known / values provided: 81%●distinct values known / distinct values provided: 7%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St001699: Standard tableaux ⟶ ℤResult quality: 7% ●values known / values provided: 81%●distinct values known / distinct values provided: 7%
Values
[2]
=> []
=> ?
=> ?
=> ? ∊ {1,3}
[1,1]
=> [1]
=> []
=> []
=> ? ∊ {1,3}
[3]
=> []
=> ?
=> ?
=> ? ∊ {2,4}
[2,1]
=> [1]
=> []
=> []
=> ? ∊ {2,4}
[1,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[4]
=> []
=> ?
=> ?
=> ? ∊ {1,3,5}
[3,1]
=> [1]
=> []
=> []
=> ? ∊ {1,3,5}
[2,2]
=> [2]
=> []
=> []
=> ? ∊ {1,3,5}
[2,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[5]
=> []
=> ?
=> ?
=> ? ∊ {2,4,6}
[4,1]
=> [1]
=> []
=> []
=> ? ∊ {2,4,6}
[3,2]
=> [2]
=> []
=> []
=> ? ∊ {2,4,6}
[3,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[2,2,1]
=> [2,1]
=> [1]
=> [[1]]
=> 0
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[6]
=> []
=> ?
=> ?
=> ? ∊ {1,3,5,7}
[5,1]
=> [1]
=> []
=> []
=> ? ∊ {1,3,5,7}
[4,2]
=> [2]
=> []
=> []
=> ? ∊ {1,3,5,7}
[4,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[3,3]
=> [3]
=> []
=> []
=> ? ∊ {1,3,5,7}
[3,2,1]
=> [2,1]
=> [1]
=> [[1]]
=> 0
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[2,2,2]
=> [2,2]
=> [2]
=> [[1,2]]
=> 0
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[7]
=> []
=> ?
=> ?
=> ? ∊ {2,4,6,8}
[6,1]
=> [1]
=> []
=> []
=> ? ∊ {2,4,6,8}
[5,2]
=> [2]
=> []
=> []
=> ? ∊ {2,4,6,8}
[5,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[4,3]
=> [3]
=> []
=> []
=> ? ∊ {2,4,6,8}
[4,2,1]
=> [2,1]
=> [1]
=> [[1]]
=> 0
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[3,3,1]
=> [3,1]
=> [1]
=> [[1]]
=> 0
[3,2,2]
=> [2,2]
=> [2]
=> [[1,2]]
=> 0
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [[1,3],[2]]
=> 0
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 0
[8]
=> []
=> ?
=> ?
=> ? ∊ {1,3,5,7,9}
[7,1]
=> [1]
=> []
=> []
=> ? ∊ {1,3,5,7,9}
[6,2]
=> [2]
=> []
=> []
=> ? ∊ {1,3,5,7,9}
[6,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[5,3]
=> [3]
=> []
=> []
=> ? ∊ {1,3,5,7,9}
[5,2,1]
=> [2,1]
=> [1]
=> [[1]]
=> 0
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[4,4]
=> [4]
=> []
=> []
=> ? ∊ {1,3,5,7,9}
[4,3,1]
=> [3,1]
=> [1]
=> [[1]]
=> 0
[4,2,2]
=> [2,2]
=> [2]
=> [[1,2]]
=> 0
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[3,3,2]
=> [3,2]
=> [2]
=> [[1,2]]
=> 0
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> [[1,3],[2]]
=> 0
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[2,2,2,2]
=> [2,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 0
[2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 0
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 0
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> 0
[9]
=> []
=> ?
=> ?
=> ? ∊ {2,4,6,8,10}
[8,1]
=> [1]
=> []
=> []
=> ? ∊ {2,4,6,8,10}
[7,2]
=> [2]
=> []
=> []
=> ? ∊ {2,4,6,8,10}
[7,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[6,3]
=> [3]
=> []
=> []
=> ? ∊ {2,4,6,8,10}
[6,2,1]
=> [2,1]
=> [1]
=> [[1]]
=> 0
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[5,4]
=> [4]
=> []
=> []
=> ? ∊ {2,4,6,8,10}
[5,3,1]
=> [3,1]
=> [1]
=> [[1]]
=> 0
[5,2,2]
=> [2,2]
=> [2]
=> [[1,2]]
=> 0
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[4,4,1]
=> [4,1]
=> [1]
=> [[1]]
=> 0
[10]
=> []
=> ?
=> ?
=> ? ∊ {1,3,5,7,9,11}
[9,1]
=> [1]
=> []
=> []
=> ? ∊ {1,3,5,7,9,11}
[8,2]
=> [2]
=> []
=> []
=> ? ∊ {1,3,5,7,9,11}
[7,3]
=> [3]
=> []
=> []
=> ? ∊ {1,3,5,7,9,11}
[6,4]
=> [4]
=> []
=> []
=> ? ∊ {1,3,5,7,9,11}
[5,5]
=> [5]
=> []
=> []
=> ? ∊ {1,3,5,7,9,11}
[11]
=> []
=> ?
=> ?
=> ? ∊ {0,2,4,6,8,10,12}
[10,1]
=> [1]
=> []
=> []
=> ? ∊ {0,2,4,6,8,10,12}
[9,2]
=> [2]
=> []
=> []
=> ? ∊ {0,2,4,6,8,10,12}
[8,3]
=> [3]
=> []
=> []
=> ? ∊ {0,2,4,6,8,10,12}
[7,4]
=> [4]
=> []
=> []
=> ? ∊ {0,2,4,6,8,10,12}
[6,5]
=> [5]
=> []
=> []
=> ? ∊ {0,2,4,6,8,10,12}
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? ∊ {0,2,4,6,8,10,12}
[12]
=> []
=> ?
=> ?
=> ? ∊ {0,0,1,3,5,7,9,11,13}
[11,1]
=> [1]
=> []
=> []
=> ? ∊ {0,0,1,3,5,7,9,11,13}
[10,2]
=> [2]
=> []
=> []
=> ? ∊ {0,0,1,3,5,7,9,11,13}
[9,3]
=> [3]
=> []
=> []
=> ? ∊ {0,0,1,3,5,7,9,11,13}
[8,4]
=> [4]
=> []
=> []
=> ? ∊ {0,0,1,3,5,7,9,11,13}
[7,5]
=> [5]
=> []
=> []
=> ? ∊ {0,0,1,3,5,7,9,11,13}
[6,6]
=> [6]
=> []
=> []
=> ? ∊ {0,0,1,3,5,7,9,11,13}
[2,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? ∊ {0,0,1,3,5,7,9,11,13}
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? ∊ {0,0,1,3,5,7,9,11,13}
Description
The major index of a standard tableau minus the weighted size of its shape.
Matching statistic: St001712
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St001712: Standard tableaux ⟶ ℤResult quality: 7% ●values known / values provided: 81%●distinct values known / distinct values provided: 7%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St001712: Standard tableaux ⟶ ℤResult quality: 7% ●values known / values provided: 81%●distinct values known / distinct values provided: 7%
Values
[2]
=> []
=> ?
=> ?
=> ? ∊ {1,3}
[1,1]
=> [1]
=> []
=> []
=> ? ∊ {1,3}
[3]
=> []
=> ?
=> ?
=> ? ∊ {2,4}
[2,1]
=> [1]
=> []
=> []
=> ? ∊ {2,4}
[1,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[4]
=> []
=> ?
=> ?
=> ? ∊ {1,3,5}
[3,1]
=> [1]
=> []
=> []
=> ? ∊ {1,3,5}
[2,2]
=> [2]
=> []
=> []
=> ? ∊ {1,3,5}
[2,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[5]
=> []
=> ?
=> ?
=> ? ∊ {2,4,6}
[4,1]
=> [1]
=> []
=> []
=> ? ∊ {2,4,6}
[3,2]
=> [2]
=> []
=> []
=> ? ∊ {2,4,6}
[3,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[2,2,1]
=> [2,1]
=> [1]
=> [[1]]
=> 0
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[6]
=> []
=> ?
=> ?
=> ? ∊ {1,3,5,7}
[5,1]
=> [1]
=> []
=> []
=> ? ∊ {1,3,5,7}
[4,2]
=> [2]
=> []
=> []
=> ? ∊ {1,3,5,7}
[4,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[3,3]
=> [3]
=> []
=> []
=> ? ∊ {1,3,5,7}
[3,2,1]
=> [2,1]
=> [1]
=> [[1]]
=> 0
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[2,2,2]
=> [2,2]
=> [2]
=> [[1,2]]
=> 0
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[7]
=> []
=> ?
=> ?
=> ? ∊ {2,4,6,8}
[6,1]
=> [1]
=> []
=> []
=> ? ∊ {2,4,6,8}
[5,2]
=> [2]
=> []
=> []
=> ? ∊ {2,4,6,8}
[5,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[4,3]
=> [3]
=> []
=> []
=> ? ∊ {2,4,6,8}
[4,2,1]
=> [2,1]
=> [1]
=> [[1]]
=> 0
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[3,3,1]
=> [3,1]
=> [1]
=> [[1]]
=> 0
[3,2,2]
=> [2,2]
=> [2]
=> [[1,2]]
=> 0
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [[1,2],[3]]
=> 0
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 0
[8]
=> []
=> ?
=> ?
=> ? ∊ {1,3,5,7,9}
[7,1]
=> [1]
=> []
=> []
=> ? ∊ {1,3,5,7,9}
[6,2]
=> [2]
=> []
=> []
=> ? ∊ {1,3,5,7,9}
[6,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[5,3]
=> [3]
=> []
=> []
=> ? ∊ {1,3,5,7,9}
[5,2,1]
=> [2,1]
=> [1]
=> [[1]]
=> 0
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[4,4]
=> [4]
=> []
=> []
=> ? ∊ {1,3,5,7,9}
[4,3,1]
=> [3,1]
=> [1]
=> [[1]]
=> 0
[4,2,2]
=> [2,2]
=> [2]
=> [[1,2]]
=> 0
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[3,3,2]
=> [3,2]
=> [2]
=> [[1,2]]
=> 0
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> [[1,2],[3]]
=> 0
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[2,2,2,2]
=> [2,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 0
[2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 0
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 0
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> 0
[9]
=> []
=> ?
=> ?
=> ? ∊ {2,4,6,8,10}
[8,1]
=> [1]
=> []
=> []
=> ? ∊ {2,4,6,8,10}
[7,2]
=> [2]
=> []
=> []
=> ? ∊ {2,4,6,8,10}
[7,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[6,3]
=> [3]
=> []
=> []
=> ? ∊ {2,4,6,8,10}
[6,2,1]
=> [2,1]
=> [1]
=> [[1]]
=> 0
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[5,4]
=> [4]
=> []
=> []
=> ? ∊ {2,4,6,8,10}
[5,3,1]
=> [3,1]
=> [1]
=> [[1]]
=> 0
[5,2,2]
=> [2,2]
=> [2]
=> [[1,2]]
=> 0
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[4,4,1]
=> [4,1]
=> [1]
=> [[1]]
=> 0
[10]
=> []
=> ?
=> ?
=> ? ∊ {1,3,5,7,9,11}
[9,1]
=> [1]
=> []
=> []
=> ? ∊ {1,3,5,7,9,11}
[8,2]
=> [2]
=> []
=> []
=> ? ∊ {1,3,5,7,9,11}
[7,3]
=> [3]
=> []
=> []
=> ? ∊ {1,3,5,7,9,11}
[6,4]
=> [4]
=> []
=> []
=> ? ∊ {1,3,5,7,9,11}
[5,5]
=> [5]
=> []
=> []
=> ? ∊ {1,3,5,7,9,11}
[11]
=> []
=> ?
=> ?
=> ? ∊ {0,2,4,6,8,10,12}
[10,1]
=> [1]
=> []
=> []
=> ? ∊ {0,2,4,6,8,10,12}
[9,2]
=> [2]
=> []
=> []
=> ? ∊ {0,2,4,6,8,10,12}
[8,3]
=> [3]
=> []
=> []
=> ? ∊ {0,2,4,6,8,10,12}
[7,4]
=> [4]
=> []
=> []
=> ? ∊ {0,2,4,6,8,10,12}
[6,5]
=> [5]
=> []
=> []
=> ? ∊ {0,2,4,6,8,10,12}
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? ∊ {0,2,4,6,8,10,12}
[12]
=> []
=> ?
=> ?
=> ? ∊ {0,0,1,3,5,7,9,11,13}
[11,1]
=> [1]
=> []
=> []
=> ? ∊ {0,0,1,3,5,7,9,11,13}
[10,2]
=> [2]
=> []
=> []
=> ? ∊ {0,0,1,3,5,7,9,11,13}
[9,3]
=> [3]
=> []
=> []
=> ? ∊ {0,0,1,3,5,7,9,11,13}
[8,4]
=> [4]
=> []
=> []
=> ? ∊ {0,0,1,3,5,7,9,11,13}
[7,5]
=> [5]
=> []
=> []
=> ? ∊ {0,0,1,3,5,7,9,11,13}
[6,6]
=> [6]
=> []
=> []
=> ? ∊ {0,0,1,3,5,7,9,11,13}
[2,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? ∊ {0,0,1,3,5,7,9,11,13}
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? ∊ {0,0,1,3,5,7,9,11,13}
Description
The number of natural descents of a standard Young tableau.
A natural descent of a standard tableau T is an entry i such that i+1 appears in a higher row than i in English notation.
The following 7 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St000687The dimension of Hom(I,P) for the LNakayama algebra of a Dyck path. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001498The normalised height of a Nakayama algebra with magnitude 1.
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