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Your data matches 36 different statistics following compositions of up to 3 maps.
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Matching statistic: St000929
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00308: Integer partitions —Bulgarian solitaire⟶ Integer partitions
St000929: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00308: Integer partitions —Bulgarian solitaire⟶ Integer partitions
St000929: 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
[2,1,1]
=> [1,1]
=> [2]
=> 0
[1,1,1,1]
=> [1,1,1]
=> [3]
=> 0
[3,2]
=> [2]
=> [1,1]
=> 1
[3,1,1]
=> [1,1]
=> [2]
=> 0
[2,2,1]
=> [2,1]
=> [2,1]
=> 0
[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
[4,1,1]
=> [1,1]
=> [2]
=> 0
[3,3]
=> [3]
=> [2,1]
=> 0
[3,2,1]
=> [2,1]
=> [2,1]
=> 0
[3,1,1,1]
=> [1,1,1]
=> [3]
=> 0
[2,2,2]
=> [2,2]
=> [2,1,1]
=> 0
[2,2,1,1]
=> [2,1,1]
=> [3,1]
=> 0
[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
[5,1,1]
=> [1,1]
=> [2]
=> 0
[4,3]
=> [3]
=> [2,1]
=> 0
[4,2,1]
=> [2,1]
=> [2,1]
=> 0
[4,1,1,1]
=> [1,1,1]
=> [3]
=> 0
[3,3,1]
=> [3,1]
=> [2,2]
=> 0
[3,2,2]
=> [2,2]
=> [2,1,1]
=> 0
[3,2,1,1]
=> [2,1,1]
=> [3,1]
=> 0
[3,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 0
[2,2,2,1]
=> [2,2,1]
=> [3,1,1]
=> 0
[2,2,1,1,1]
=> [2,1,1,1]
=> [4,1]
=> 0
[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
[6,1,1]
=> [1,1]
=> [2]
=> 0
[5,3]
=> [3]
=> [2,1]
=> 0
[5,2,1]
=> [2,1]
=> [2,1]
=> 0
[5,1,1,1]
=> [1,1,1]
=> [3]
=> 0
[4,4]
=> [4]
=> [3,1]
=> 0
[4,3,1]
=> [3,1]
=> [2,2]
=> 0
[4,2,2]
=> [2,2]
=> [2,1,1]
=> 0
[4,2,1,1]
=> [2,1,1]
=> [3,1]
=> 0
[4,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 0
[3,3,2]
=> [3,2]
=> [2,2,1]
=> 0
[3,3,1,1]
=> [3,1,1]
=> [3,2]
=> 0
[3,2,2,1]
=> [2,2,1]
=> [3,1,1]
=> 0
[3,2,1,1,1]
=> [2,1,1,1]
=> [4,1]
=> 0
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> 0
[2,2,2,2]
=> [2,2,2]
=> [3,1,1,1]
=> 0
[2,2,2,1,1]
=> [2,2,1,1]
=> [4,1,1]
=> 0
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [5,1]
=> 0
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [6]
=> 0
Description
The constant term of the character polynomial of an integer partition.
The definition of the character polynomial can be found in [1]. Indeed, this constant term is $0$ for partitions $\lambda \neq 1^n$ and $1$ for $\lambda = 1^n$.
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: 50% ●values known / values provided: 94%●distinct values known / distinct values provided: 50%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000296: Binary words ⟶ ℤResult quality: 50% ●values known / values provided: 94%●distinct values known / distinct values provided: 50%
Values
[1,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[2,2]
=> [2]
=> []
=> => ? = 1
[2,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 0
[3,2]
=> [2]
=> []
=> => ? = 1
[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
[4,2]
=> [2]
=> []
=> => ? = 1
[4,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[3,3]
=> [3]
=> []
=> => ? = 0
[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
[5,2]
=> [2]
=> []
=> => ? = 1
[5,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[4,3]
=> [3]
=> []
=> => ? = 0
[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
[6,2]
=> [2]
=> []
=> => ? = 1
[6,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[5,3]
=> [3]
=> []
=> => ? = 0
[5,2,1]
=> [2,1]
=> [1]
=> 10 => 0
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 0
[4,4]
=> [4]
=> []
=> => ? = 0
[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
[7,2]
=> [2]
=> []
=> => ? = 1
[7,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[6,3]
=> [3]
=> []
=> => ? = 0
[6,2,1]
=> [2,1]
=> [1]
=> 10 => 0
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 0
[5,4]
=> [4]
=> []
=> => ? = 0
[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
[8,2]
=> [2]
=> []
=> => ? = 1
[7,3]
=> [3]
=> []
=> => ? = 0
[6,4]
=> [4]
=> []
=> => ? = 0
[5,5]
=> [5]
=> []
=> => ? = 0
[9,2]
=> [2]
=> []
=> => ? = 1
[8,3]
=> [3]
=> []
=> => ? = 0
[7,4]
=> [4]
=> []
=> => ? = 0
[6,5]
=> [5]
=> []
=> => ? = 0
[10,2]
=> [2]
=> []
=> => ? = 1
[9,3]
=> [3]
=> []
=> => ? = 0
[8,4]
=> [4]
=> []
=> => ? = 0
[7,5]
=> [5]
=> []
=> => ? = 0
[6,6]
=> [6]
=> []
=> => ? = 0
[11,2]
=> [2]
=> []
=> => ? = 1
[10,3]
=> [3]
=> []
=> => ? = 0
[9,4]
=> [4]
=> []
=> => ? = 0
[8,5]
=> [5]
=> []
=> => ? = 0
[7,6]
=> [6]
=> []
=> => ? = 0
[1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> 111111111110 => ? = 0
[12,2]
=> [2]
=> []
=> => ? = 1
[11,3]
=> [3]
=> []
=> => ? = 0
[10,4]
=> [4]
=> []
=> => ? = 0
[9,5]
=> [5]
=> []
=> => ? = 0
[8,6]
=> [6]
=> []
=> => ? = 0
[7,7]
=> [7]
=> []
=> => ? = 0
[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]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> 111111111110 => ? = 0
[13,2]
=> [2]
=> []
=> => ? = 1
[12,3]
=> [3]
=> []
=> => ? = 0
[11,4]
=> [4]
=> []
=> => ? = 0
[10,5]
=> [5]
=> []
=> => ? = 0
[9,6]
=> [6]
=> []
=> => ? = 0
[8,7]
=> [7]
=> []
=> => ? = 0
[3,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> 111111111110 => ? = 0
[14,2]
=> [2]
=> []
=> => ? = 1
[13,3]
=> [3]
=> []
=> => ? = 0
[12,4]
=> [4]
=> []
=> => ? = 0
[11,5]
=> [5]
=> []
=> => ? = 0
[10,6]
=> [6]
=> []
=> => ? = 0
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: 50% ●values known / values provided: 94%●distinct values known / distinct values provided: 50%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000629: Binary words ⟶ ℤResult quality: 50% ●values known / values provided: 94%●distinct values known / distinct values provided: 50%
Values
[1,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[2,2]
=> [2]
=> []
=> => ? = 1
[2,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 0
[3,2]
=> [2]
=> []
=> => ? = 1
[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
[4,2]
=> [2]
=> []
=> => ? = 1
[4,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[3,3]
=> [3]
=> []
=> => ? = 0
[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
[5,2]
=> [2]
=> []
=> => ? = 1
[5,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[4,3]
=> [3]
=> []
=> => ? = 0
[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
[6,2]
=> [2]
=> []
=> => ? = 1
[6,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[5,3]
=> [3]
=> []
=> => ? = 0
[5,2,1]
=> [2,1]
=> [1]
=> 10 => 0
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 0
[4,4]
=> [4]
=> []
=> => ? = 0
[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
[7,2]
=> [2]
=> []
=> => ? = 1
[7,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[6,3]
=> [3]
=> []
=> => ? = 0
[6,2,1]
=> [2,1]
=> [1]
=> 10 => 0
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 0
[5,4]
=> [4]
=> []
=> => ? = 0
[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
[8,2]
=> [2]
=> []
=> => ? = 1
[7,3]
=> [3]
=> []
=> => ? = 0
[6,4]
=> [4]
=> []
=> => ? = 0
[5,5]
=> [5]
=> []
=> => ? = 0
[9,2]
=> [2]
=> []
=> => ? = 1
[8,3]
=> [3]
=> []
=> => ? = 0
[7,4]
=> [4]
=> []
=> => ? = 0
[6,5]
=> [5]
=> []
=> => ? = 0
[10,2]
=> [2]
=> []
=> => ? = 1
[9,3]
=> [3]
=> []
=> => ? = 0
[8,4]
=> [4]
=> []
=> => ? = 0
[7,5]
=> [5]
=> []
=> => ? = 0
[6,6]
=> [6]
=> []
=> => ? = 0
[11,2]
=> [2]
=> []
=> => ? = 1
[10,3]
=> [3]
=> []
=> => ? = 0
[9,4]
=> [4]
=> []
=> => ? = 0
[8,5]
=> [5]
=> []
=> => ? = 0
[7,6]
=> [6]
=> []
=> => ? = 0
[1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> 111111111110 => ? = 0
[12,2]
=> [2]
=> []
=> => ? = 1
[11,3]
=> [3]
=> []
=> => ? = 0
[10,4]
=> [4]
=> []
=> => ? = 0
[9,5]
=> [5]
=> []
=> => ? = 0
[8,6]
=> [6]
=> []
=> => ? = 0
[7,7]
=> [7]
=> []
=> => ? = 0
[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]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> 111111111110 => ? = 0
[13,2]
=> [2]
=> []
=> => ? = 1
[12,3]
=> [3]
=> []
=> => ? = 0
[11,4]
=> [4]
=> []
=> => ? = 0
[10,5]
=> [5]
=> []
=> => ? = 0
[9,6]
=> [6]
=> []
=> => ? = 0
[8,7]
=> [7]
=> []
=> => ? = 0
[3,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> 111111111110 => ? = 0
[14,2]
=> [2]
=> []
=> => ? = 1
[13,3]
=> [3]
=> []
=> => ? = 0
[12,4]
=> [4]
=> []
=> => ? = 0
[11,5]
=> [5]
=> []
=> => ? = 0
[10,6]
=> [6]
=> []
=> => ? = 0
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: St000326
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 50% ●values known / values provided: 94%●distinct values known / distinct values provided: 50%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 50% ●values known / values provided: 94%●distinct values known / distinct values provided: 50%
Values
[1,1,1]
=> [1,1]
=> [1]
=> 10 => 1 = 0 + 1
[2,2]
=> [2]
=> []
=> => ? = 1 + 1
[2,1,1]
=> [1,1]
=> [1]
=> 10 => 1 = 0 + 1
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 1 = 0 + 1
[3,2]
=> [2]
=> []
=> => ? = 1 + 1
[3,1,1]
=> [1,1]
=> [1]
=> 10 => 1 = 0 + 1
[2,2,1]
=> [2,1]
=> [1]
=> 10 => 1 = 0 + 1
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 1 = 0 + 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1110 => 1 = 0 + 1
[4,2]
=> [2]
=> []
=> => ? = 1 + 1
[4,1,1]
=> [1,1]
=> [1]
=> 10 => 1 = 0 + 1
[3,3]
=> [3]
=> []
=> => ? = 0 + 1
[3,2,1]
=> [2,1]
=> [1]
=> 10 => 1 = 0 + 1
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 1 = 0 + 1
[2,2,2]
=> [2,2]
=> [2]
=> 100 => 1 = 0 + 1
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 110 => 1 = 0 + 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1110 => 1 = 0 + 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 1 = 0 + 1
[5,2]
=> [2]
=> []
=> => ? = 1 + 1
[5,1,1]
=> [1,1]
=> [1]
=> 10 => 1 = 0 + 1
[4,3]
=> [3]
=> []
=> => ? = 0 + 1
[4,2,1]
=> [2,1]
=> [1]
=> 10 => 1 = 0 + 1
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 1 = 0 + 1
[3,3,1]
=> [3,1]
=> [1]
=> 10 => 1 = 0 + 1
[3,2,2]
=> [2,2]
=> [2]
=> 100 => 1 = 0 + 1
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 110 => 1 = 0 + 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1110 => 1 = 0 + 1
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1010 => 1 = 0 + 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1110 => 1 = 0 + 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 1 = 0 + 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 1 = 0 + 1
[6,2]
=> [2]
=> []
=> => ? = 1 + 1
[6,1,1]
=> [1,1]
=> [1]
=> 10 => 1 = 0 + 1
[5,3]
=> [3]
=> []
=> => ? = 0 + 1
[5,2,1]
=> [2,1]
=> [1]
=> 10 => 1 = 0 + 1
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 1 = 0 + 1
[4,4]
=> [4]
=> []
=> => ? = 0 + 1
[4,3,1]
=> [3,1]
=> [1]
=> 10 => 1 = 0 + 1
[4,2,2]
=> [2,2]
=> [2]
=> 100 => 1 = 0 + 1
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> 110 => 1 = 0 + 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1110 => 1 = 0 + 1
[3,3,2]
=> [3,2]
=> [2]
=> 100 => 1 = 0 + 1
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> 110 => 1 = 0 + 1
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1010 => 1 = 0 + 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1110 => 1 = 0 + 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 1 = 0 + 1
[2,2,2,2]
=> [2,2,2]
=> [2,2]
=> 1100 => 1 = 0 + 1
[2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> 10110 => 1 = 0 + 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 1 = 0 + 1
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 1 = 0 + 1
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 1111110 => 1 = 0 + 1
[7,2]
=> [2]
=> []
=> => ? = 1 + 1
[7,1,1]
=> [1,1]
=> [1]
=> 10 => 1 = 0 + 1
[6,3]
=> [3]
=> []
=> => ? = 0 + 1
[6,2,1]
=> [2,1]
=> [1]
=> 10 => 1 = 0 + 1
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 1 = 0 + 1
[5,4]
=> [4]
=> []
=> => ? = 0 + 1
[5,3,1]
=> [3,1]
=> [1]
=> 10 => 1 = 0 + 1
[5,2,2]
=> [2,2]
=> [2]
=> 100 => 1 = 0 + 1
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> 110 => 1 = 0 + 1
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1110 => 1 = 0 + 1
[4,4,1]
=> [4,1]
=> [1]
=> 10 => 1 = 0 + 1
[8,2]
=> [2]
=> []
=> => ? = 1 + 1
[7,3]
=> [3]
=> []
=> => ? = 0 + 1
[6,4]
=> [4]
=> []
=> => ? = 0 + 1
[5,5]
=> [5]
=> []
=> => ? = 0 + 1
[9,2]
=> [2]
=> []
=> => ? = 1 + 1
[8,3]
=> [3]
=> []
=> => ? = 0 + 1
[7,4]
=> [4]
=> []
=> => ? = 0 + 1
[6,5]
=> [5]
=> []
=> => ? = 0 + 1
[10,2]
=> [2]
=> []
=> => ? = 1 + 1
[9,3]
=> [3]
=> []
=> => ? = 0 + 1
[8,4]
=> [4]
=> []
=> => ? = 0 + 1
[7,5]
=> [5]
=> []
=> => ? = 0 + 1
[6,6]
=> [6]
=> []
=> => ? = 0 + 1
[11,2]
=> [2]
=> []
=> => ? = 1 + 1
[10,3]
=> [3]
=> []
=> => ? = 0 + 1
[9,4]
=> [4]
=> []
=> => ? = 0 + 1
[8,5]
=> [5]
=> []
=> => ? = 0 + 1
[7,6]
=> [6]
=> []
=> => ? = 0 + 1
[1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> 111111111110 => ? = 0 + 1
[12,2]
=> [2]
=> []
=> => ? = 1 + 1
[11,3]
=> [3]
=> []
=> => ? = 0 + 1
[10,4]
=> [4]
=> []
=> => ? = 0 + 1
[9,5]
=> [5]
=> []
=> => ? = 0 + 1
[8,6]
=> [6]
=> []
=> => ? = 0 + 1
[7,7]
=> [7]
=> []
=> => ? = 0 + 1
[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]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> 111111111110 => ? = 0 + 1
[13,2]
=> [2]
=> []
=> => ? = 1 + 1
[12,3]
=> [3]
=> []
=> => ? = 0 + 1
[11,4]
=> [4]
=> []
=> => ? = 0 + 1
[10,5]
=> [5]
=> []
=> => ? = 0 + 1
[9,6]
=> [6]
=> []
=> => ? = 0 + 1
[8,7]
=> [7]
=> []
=> => ? = 0 + 1
[3,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> 111111111110 => ? = 0 + 1
[14,2]
=> [2]
=> []
=> => ? = 1 + 1
[13,3]
=> [3]
=> []
=> => ? = 0 + 1
[12,4]
=> [4]
=> []
=> => ? = 0 + 1
[11,5]
=> [5]
=> []
=> => ? = 0 + 1
[10,6]
=> [6]
=> []
=> => ? = 0 + 1
Description
The position of the first one in a binary word after appending a 1 at the end.
Regarding the binary word as a subset of $\{1,\dots,n,n+1\}$ that contains $n+1$, this is the minimal element of the set.
Matching statistic: 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: 50% ●values known / values provided: 90%●distinct values known / distinct values provided: 50%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St001371: Binary words ⟶ ℤResult quality: 50% ●values known / values provided: 90%●distinct values known / distinct values provided: 50%
Values
[1,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[2,2]
=> [2]
=> []
=> => ? = 1
[2,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 0
[3,2]
=> [2]
=> []
=> => ? = 1
[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
[4,2]
=> [2]
=> []
=> => ? = 1
[4,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[3,3]
=> [3]
=> []
=> => ? = 0
[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
[5,2]
=> [2]
=> []
=> => ? = 1
[5,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[4,3]
=> [3]
=> []
=> => ? = 0
[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
[6,2]
=> [2]
=> []
=> => ? = 1
[6,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[5,3]
=> [3]
=> []
=> => ? = 0
[5,2,1]
=> [2,1]
=> [1]
=> 10 => 0
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 0
[4,4]
=> [4]
=> []
=> => ? = 0
[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
[7,2]
=> [2]
=> []
=> => ? = 1
[7,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[6,3]
=> [3]
=> []
=> => ? = 0
[6,2,1]
=> [2,1]
=> [1]
=> 10 => 0
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 0
[5,4]
=> [4]
=> []
=> => ? = 0
[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
[8,2]
=> [2]
=> []
=> => ? = 1
[7,3]
=> [3]
=> []
=> => ? = 0
[6,4]
=> [4]
=> []
=> => ? = 0
[5,5]
=> [5]
=> []
=> => ? = 0
[9,2]
=> [2]
=> []
=> => ? = 1
[8,3]
=> [3]
=> []
=> => ? = 0
[7,4]
=> [4]
=> []
=> => ? = 0
[6,5]
=> [5]
=> []
=> => ? = 0
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> 1111111110 => ? = 0
[10,2]
=> [2]
=> []
=> => ? = 1
[9,3]
=> [3]
=> []
=> => ? = 0
[8,4]
=> [4]
=> []
=> => ? = 0
[7,5]
=> [5]
=> []
=> => ? = 0
[6,6]
=> [6]
=> []
=> => ? = 0
[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
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> 11111111110 => ? = 0
[11,2]
=> [2]
=> []
=> => ? = 1
[10,3]
=> [3]
=> []
=> => ? = 0
[9,4]
=> [4]
=> []
=> => ? = 0
[8,5]
=> [5]
=> []
=> => ? = 0
[7,6]
=> [6]
=> []
=> => ? = 0
[3,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> 1111111110 => ? = 0
[2,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> 1011111110 => ? = 0
[2,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> 1111111110 => ? = 0
[2,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> 11111111110 => ? = 0
[1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> 111111111110 => ? = 0
[12,2]
=> [2]
=> []
=> => ? = 1
[11,3]
=> [3]
=> []
=> => ? = 0
[10,4]
=> [4]
=> []
=> => ? = 0
[9,5]
=> [5]
=> []
=> => ? = 0
[8,6]
=> [6]
=> []
=> => ? = 0
[7,7]
=> [7]
=> []
=> => ? = 0
[4,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> 1111111110 => ? = 0
[3,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> 1011111110 => ? = 0
[3,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> 1111111110 => ? = 0
[3,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> 11111111110 => ? = 0
[2,2,2,2,1,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1]
=> 1101111110 => ? = 0
[2,2,2,1,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1]
=> 10111111110 => ? = 0
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 $w_1$, $w_1w_2$, $w_1w_2\dots w_i$ the number of zeros is greater than or equal to the number of ones.
Matching statistic: St001107
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001107: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 89%●distinct values known / distinct values provided: 50%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001107: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 89%●distinct values known / distinct values provided: 50%
Values
[1,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[2,2]
=> [2]
=> []
=> []
=> ? = 1
[2,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[3,2]
=> [2]
=> []
=> []
=> ? = 1
[3,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[2,2,1]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0
[4,2]
=> [2]
=> []
=> []
=> ? = 1
[4,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[3,3]
=> [3]
=> []
=> []
=> ? = 0
[3,2,1]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[2,2,2]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[5,2]
=> [2]
=> []
=> []
=> ? = 1
[5,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[4,3]
=> [3]
=> []
=> []
=> ? = 0
[4,2,1]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[3,3,1]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[3,2,2]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 0
[6,2]
=> [2]
=> []
=> []
=> ? = 1
[6,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[5,3]
=> [3]
=> []
=> []
=> ? = 0
[5,2,1]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[4,4]
=> [4]
=> []
=> []
=> ? = 0
[4,3,1]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[4,2,2]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0
[3,3,2]
=> [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[2,2,2,2]
=> [2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 0
[2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
[7,2]
=> [2]
=> []
=> []
=> ? = 1
[7,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[6,3]
=> [3]
=> []
=> []
=> ? = 0
[6,2,1]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[5,4]
=> [4]
=> []
=> []
=> ? = 0
[5,3,1]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[5,2,2]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0
[4,4,1]
=> [4,1]
=> [1]
=> [1,0,1,0]
=> 0
[8,2]
=> [2]
=> []
=> []
=> ? = 1
[7,3]
=> [3]
=> []
=> []
=> ? = 0
[6,4]
=> [4]
=> []
=> []
=> ? = 0
[5,5]
=> [5]
=> []
=> []
=> ? = 0
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[9,2]
=> [2]
=> []
=> []
=> ? = 1
[8,3]
=> [3]
=> []
=> []
=> ? = 0
[7,4]
=> [4]
=> []
=> []
=> ? = 0
[6,5]
=> [5]
=> []
=> []
=> ? = 0
[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]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 0
[10,2]
=> [2]
=> []
=> []
=> ? = 1
[9,3]
=> [3]
=> []
=> []
=> ? = 0
[8,4]
=> [4]
=> []
=> []
=> ? = 0
[7,5]
=> [5]
=> []
=> []
=> ? = 0
[6,6]
=> [6]
=> []
=> []
=> ? = 0
[3,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[2,2,1,1,1,1,1,1,1,1]
=> [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]
=> ? = 0
[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]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 0
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 0
[11,2]
=> [2]
=> []
=> []
=> ? = 1
[10,3]
=> [3]
=> []
=> []
=> ? = 0
[9,4]
=> [4]
=> []
=> []
=> ? = 0
[8,5]
=> [5]
=> []
=> []
=> ? = 0
[7,6]
=> [6]
=> []
=> []
=> ? = 0
[4,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[3,2,1,1,1,1,1,1,1,1]
=> [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]
=> ? = 0
[3,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 0
[2,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 0
[2,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 0
[2,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 0
[1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 0
[12,2]
=> [2]
=> []
=> []
=> ? = 1
[11,3]
=> [3]
=> []
=> []
=> ? = 0
[10,4]
=> [4]
=> []
=> []
=> ? = 0
[9,5]
=> [5]
=> []
=> []
=> ? = 0
[8,6]
=> [6]
=> []
=> []
=> ? = 0
[7,7]
=> [7]
=> []
=> []
=> ? = 0
Description
The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path.
In other words, this is the lowest height of a valley of a Dyck path, or its semilength in case of the unique path without valleys.
Matching statistic: St000687
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000687: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 85%●distinct values known / distinct values provided: 50%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000687: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 85%●distinct values known / distinct values provided: 50%
Values
[1,1,1]
=> [1,1]
=> [1]
=> [1,0]
=> 0
[2,2]
=> [2]
=> []
=> []
=> ? = 1
[2,1,1]
=> [1,1]
=> [1]
=> [1,0]
=> 0
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[3,2]
=> [2]
=> []
=> []
=> ? = 1
[3,1,1]
=> [1,1]
=> [1]
=> [1,0]
=> 0
[2,2,1]
=> [2,1]
=> [1]
=> [1,0]
=> 0
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[4,2]
=> [2]
=> []
=> []
=> ? = 1
[4,1,1]
=> [1,1]
=> [1]
=> [1,0]
=> 0
[3,3]
=> [3]
=> []
=> []
=> ? = 0
[3,2,1]
=> [2,1]
=> [1]
=> [1,0]
=> 0
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[2,2,2]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 0
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[5,2]
=> [2]
=> []
=> []
=> ? = 1
[5,1,1]
=> [1,1]
=> [1]
=> [1,0]
=> 0
[4,3]
=> [3]
=> []
=> []
=> ? = 0
[4,2,1]
=> [2,1]
=> [1]
=> [1,0]
=> 0
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[3,3,1]
=> [3,1]
=> [1]
=> [1,0]
=> 0
[3,2,2]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 0
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[6,2]
=> [2]
=> []
=> []
=> ? = 1
[6,1,1]
=> [1,1]
=> [1]
=> [1,0]
=> 0
[5,3]
=> [3]
=> []
=> []
=> ? = 0
[5,2,1]
=> [2,1]
=> [1]
=> [1,0]
=> 0
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[4,4]
=> [4]
=> []
=> []
=> ? = 0
[4,3,1]
=> [3,1]
=> [1]
=> [1,0]
=> 0
[4,2,2]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 0
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[3,3,2]
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 0
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[2,2,2,2]
=> [2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0
[2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 0
[7,2]
=> [2]
=> []
=> []
=> ? = 1
[7,1,1]
=> [1,1]
=> [1]
=> [1,0]
=> 0
[6,3]
=> [3]
=> []
=> []
=> ? = 0
[6,2,1]
=> [2,1]
=> [1]
=> [1,0]
=> 0
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[5,4]
=> [4]
=> []
=> []
=> ? = 0
[5,3,1]
=> [3,1]
=> [1]
=> [1,0]
=> 0
[5,2,2]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 0
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[4,4,1]
=> [4,1]
=> [1]
=> [1,0]
=> 0
[8,2]
=> [2]
=> []
=> []
=> ? = 1
[7,3]
=> [3]
=> []
=> []
=> ? = 0
[6,4]
=> [4]
=> []
=> []
=> ? = 0
[5,5]
=> [5]
=> []
=> []
=> ? = 0
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[9,2]
=> [2]
=> []
=> []
=> ? = 1
[8,3]
=> [3]
=> []
=> []
=> ? = 0
[7,4]
=> [4]
=> []
=> []
=> ? = 0
[6,5]
=> [5]
=> []
=> []
=> ? = 0
[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]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[10,2]
=> [2]
=> []
=> []
=> ? = 1
[9,3]
=> [3]
=> []
=> []
=> ? = 0
[8,4]
=> [4]
=> []
=> []
=> ? = 0
[7,5]
=> [5]
=> []
=> []
=> ? = 0
[6,6]
=> [6]
=> []
=> []
=> ? = 0
[3,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[2,2,2,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[2,2,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[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]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[11,2]
=> [2]
=> []
=> []
=> ? = 1
[10,3]
=> [3]
=> []
=> []
=> ? = 0
[9,4]
=> [4]
=> []
=> []
=> ? = 0
[8,5]
=> [5]
=> []
=> []
=> ? = 0
[7,6]
=> [6]
=> []
=> []
=> ? = 0
[4,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[3,2,2,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[3,2,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[3,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[2,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[2,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[2,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[2,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[12,2]
=> [2]
=> []
=> []
=> ? = 1
[11,3]
=> [3]
=> []
=> []
=> ? = 0
[10,4]
=> [4]
=> []
=> []
=> ? = 0
Description
The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path.
In this expression, $I$ is the direct sum of all injective non-projective indecomposable modules and $P$ is the direct sum of all projective non-injective indecomposable modules.
This statistic was discussed in [Theorem 5.7, 1].
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: 50% ●values known / values provided: 85%●distinct values known / distinct values provided: 50%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St001695: Standard tableaux ⟶ ℤResult quality: 50% ●values known / values provided: 85%●distinct values known / distinct values provided: 50%
Values
[1,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[2,2]
=> [2]
=> []
=> []
=> ? = 1
[2,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[3,2]
=> [2]
=> []
=> []
=> ? = 1
[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
[4,2]
=> [2]
=> []
=> []
=> ? = 1
[4,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[3,3]
=> [3]
=> []
=> []
=> ? = 0
[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
[5,2]
=> [2]
=> []
=> []
=> ? = 1
[5,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[4,3]
=> [3]
=> []
=> []
=> ? = 0
[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
[6,2]
=> [2]
=> []
=> []
=> ? = 1
[6,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[5,3]
=> [3]
=> []
=> []
=> ? = 0
[5,2,1]
=> [2,1]
=> [1]
=> [[1]]
=> 0
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[4,4]
=> [4]
=> []
=> []
=> ? = 0
[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
[7,2]
=> [2]
=> []
=> []
=> ? = 1
[7,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[6,3]
=> [3]
=> []
=> []
=> ? = 0
[6,2,1]
=> [2,1]
=> [1]
=> [[1]]
=> 0
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[5,4]
=> [4]
=> []
=> []
=> ? = 0
[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
[8,2]
=> [2]
=> []
=> []
=> ? = 1
[7,3]
=> [3]
=> []
=> []
=> ? = 0
[6,4]
=> [4]
=> []
=> []
=> ? = 0
[5,5]
=> [5]
=> []
=> []
=> ? = 0
[9,2]
=> [2]
=> []
=> []
=> ? = 1
[8,3]
=> [3]
=> []
=> []
=> ? = 0
[7,4]
=> [4]
=> []
=> []
=> ? = 0
[6,5]
=> [5]
=> []
=> []
=> ? = 0
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 0
[10,2]
=> [2]
=> []
=> []
=> ? = 1
[9,3]
=> [3]
=> []
=> []
=> ? = 0
[8,4]
=> [4]
=> []
=> []
=> ? = 0
[7,5]
=> [5]
=> []
=> []
=> ? = 0
[6,6]
=> [6]
=> []
=> []
=> ? = 0
[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
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [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
[11,2]
=> [2]
=> []
=> []
=> ? = 1
[10,3]
=> [3]
=> []
=> []
=> ? = 0
[9,4]
=> [4]
=> []
=> []
=> ? = 0
[8,5]
=> [5]
=> []
=> []
=> ? = 0
[7,6]
=> [6]
=> []
=> []
=> ? = 0
[3,1,1,1,1,1,1,1,1,1,1]
=> [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
[2,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> ? = 0
[2,2,2,2,2,1,1,1]
=> [2,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9]]
=> ? = 0
[2,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9]]
=> ? = 0
[2,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 0
[2,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 0
[2,1,1,1,1,1,1,1,1,1,1,1]
=> [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
[1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [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],[11]]
=> ? = 0
[12,2]
=> [2]
=> []
=> []
=> ? = 1
[11,3]
=> [3]
=> []
=> []
=> ? = 0
[10,4]
=> [4]
=> []
=> []
=> ? = 0
[9,5]
=> [5]
=> []
=> []
=> ? = 0
[8,6]
=> [6]
=> []
=> []
=> ? = 0
[7,7]
=> [7]
=> []
=> []
=> ? = 0
[4,1,1,1,1,1,1,1,1,1,1]
=> [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
[3,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> ? = 0
[3,2,2,2,2,1,1,1]
=> [2,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9]]
=> ? = 0
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 $\sum_{d\in D} n-d$.
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: 50% ●values known / values provided: 85%●distinct values known / distinct values provided: 50%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St001698: Standard tableaux ⟶ ℤResult quality: 50% ●values known / values provided: 85%●distinct values known / distinct values provided: 50%
Values
[1,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[2,2]
=> [2]
=> []
=> []
=> ? = 1
[2,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[3,2]
=> [2]
=> []
=> []
=> ? = 1
[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
[4,2]
=> [2]
=> []
=> []
=> ? = 1
[4,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[3,3]
=> [3]
=> []
=> []
=> ? = 0
[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
[5,2]
=> [2]
=> []
=> []
=> ? = 1
[5,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[4,3]
=> [3]
=> []
=> []
=> ? = 0
[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
[6,2]
=> [2]
=> []
=> []
=> ? = 1
[6,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[5,3]
=> [3]
=> []
=> []
=> ? = 0
[5,2,1]
=> [2,1]
=> [1]
=> [[1]]
=> 0
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[4,4]
=> [4]
=> []
=> []
=> ? = 0
[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
[7,2]
=> [2]
=> []
=> []
=> ? = 1
[7,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[6,3]
=> [3]
=> []
=> []
=> ? = 0
[6,2,1]
=> [2,1]
=> [1]
=> [[1]]
=> 0
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[5,4]
=> [4]
=> []
=> []
=> ? = 0
[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
[8,2]
=> [2]
=> []
=> []
=> ? = 1
[7,3]
=> [3]
=> []
=> []
=> ? = 0
[6,4]
=> [4]
=> []
=> []
=> ? = 0
[5,5]
=> [5]
=> []
=> []
=> ? = 0
[9,2]
=> [2]
=> []
=> []
=> ? = 1
[8,3]
=> [3]
=> []
=> []
=> ? = 0
[7,4]
=> [4]
=> []
=> []
=> ? = 0
[6,5]
=> [5]
=> []
=> []
=> ? = 0
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 0
[10,2]
=> [2]
=> []
=> []
=> ? = 1
[9,3]
=> [3]
=> []
=> []
=> ? = 0
[8,4]
=> [4]
=> []
=> []
=> ? = 0
[7,5]
=> [5]
=> []
=> []
=> ? = 0
[6,6]
=> [6]
=> []
=> []
=> ? = 0
[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
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [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
[11,2]
=> [2]
=> []
=> []
=> ? = 1
[10,3]
=> [3]
=> []
=> []
=> ? = 0
[9,4]
=> [4]
=> []
=> []
=> ? = 0
[8,5]
=> [5]
=> []
=> []
=> ? = 0
[7,6]
=> [6]
=> []
=> []
=> ? = 0
[3,1,1,1,1,1,1,1,1,1,1]
=> [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
[2,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> ? = 0
[2,2,2,2,2,1,1,1]
=> [2,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9]]
=> ? = 0
[2,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9]]
=> ? = 0
[2,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 0
[2,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 0
[2,1,1,1,1,1,1,1,1,1,1,1]
=> [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
[1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [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],[11]]
=> ? = 0
[12,2]
=> [2]
=> []
=> []
=> ? = 1
[11,3]
=> [3]
=> []
=> []
=> ? = 0
[10,4]
=> [4]
=> []
=> []
=> ? = 0
[9,5]
=> [5]
=> []
=> []
=> ? = 0
[8,6]
=> [6]
=> []
=> []
=> ? = 0
[7,7]
=> [7]
=> []
=> []
=> ? = 0
[4,1,1,1,1,1,1,1,1,1,1]
=> [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
[3,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> ? = 0
[3,2,2,2,2,1,1,1]
=> [2,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9]]
=> ? = 0
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: 50% ●values known / values provided: 85%●distinct values known / distinct values provided: 50%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St001699: Standard tableaux ⟶ ℤResult quality: 50% ●values known / values provided: 85%●distinct values known / distinct values provided: 50%
Values
[1,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[2,2]
=> [2]
=> []
=> []
=> ? = 1
[2,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[3,2]
=> [2]
=> []
=> []
=> ? = 1
[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
[4,2]
=> [2]
=> []
=> []
=> ? = 1
[4,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[3,3]
=> [3]
=> []
=> []
=> ? = 0
[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
[5,2]
=> [2]
=> []
=> []
=> ? = 1
[5,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[4,3]
=> [3]
=> []
=> []
=> ? = 0
[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
[6,2]
=> [2]
=> []
=> []
=> ? = 1
[6,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[5,3]
=> [3]
=> []
=> []
=> ? = 0
[5,2,1]
=> [2,1]
=> [1]
=> [[1]]
=> 0
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[4,4]
=> [4]
=> []
=> []
=> ? = 0
[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
[7,2]
=> [2]
=> []
=> []
=> ? = 1
[7,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[6,3]
=> [3]
=> []
=> []
=> ? = 0
[6,2,1]
=> [2,1]
=> [1]
=> [[1]]
=> 0
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[5,4]
=> [4]
=> []
=> []
=> ? = 0
[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
[8,2]
=> [2]
=> []
=> []
=> ? = 1
[7,3]
=> [3]
=> []
=> []
=> ? = 0
[6,4]
=> [4]
=> []
=> []
=> ? = 0
[5,5]
=> [5]
=> []
=> []
=> ? = 0
[9,2]
=> [2]
=> []
=> []
=> ? = 1
[8,3]
=> [3]
=> []
=> []
=> ? = 0
[7,4]
=> [4]
=> []
=> []
=> ? = 0
[6,5]
=> [5]
=> []
=> []
=> ? = 0
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 0
[10,2]
=> [2]
=> []
=> []
=> ? = 1
[9,3]
=> [3]
=> []
=> []
=> ? = 0
[8,4]
=> [4]
=> []
=> []
=> ? = 0
[7,5]
=> [5]
=> []
=> []
=> ? = 0
[6,6]
=> [6]
=> []
=> []
=> ? = 0
[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
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [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
[11,2]
=> [2]
=> []
=> []
=> ? = 1
[10,3]
=> [3]
=> []
=> []
=> ? = 0
[9,4]
=> [4]
=> []
=> []
=> ? = 0
[8,5]
=> [5]
=> []
=> []
=> ? = 0
[7,6]
=> [6]
=> []
=> []
=> ? = 0
[3,1,1,1,1,1,1,1,1,1,1]
=> [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
[2,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8]]
=> ? = 0
[2,2,2,2,2,1,1,1]
=> [2,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [[1,5],[2,7],[3,9],[4],[6],[8]]
=> ? = 0
[2,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [[1,7],[2,9],[3],[4],[5],[6],[8]]
=> ? = 0
[2,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [[1,9],[2],[3],[4],[5],[6],[7],[8]]
=> ? = 0
[2,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 0
[2,1,1,1,1,1,1,1,1,1,1,1]
=> [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
[1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [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],[11]]
=> ? = 0
[12,2]
=> [2]
=> []
=> []
=> ? = 1
[11,3]
=> [3]
=> []
=> []
=> ? = 0
[10,4]
=> [4]
=> []
=> []
=> ? = 0
[9,5]
=> [5]
=> []
=> []
=> ? = 0
[8,6]
=> [6]
=> []
=> []
=> ? = 0
[7,7]
=> [7]
=> []
=> []
=> ? = 0
[4,1,1,1,1,1,1,1,1,1,1]
=> [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
[3,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8]]
=> ? = 0
[3,2,2,2,2,1,1,1]
=> [2,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [[1,5],[2,7],[3,9],[4],[6],[8]]
=> ? = 0
Description
The major index of a standard tableau minus the weighted size of its shape.
The following 26 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001712The number of natural descents of a standard Young tableau. St001722The number of minimal chains with small intervals between a binary word and the top element. St001256Number of simple reflexive modules that are 2-stable reflexive. St000260The radius of a connected graph. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St000264The girth of a graph, which is not a tree. 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 $eA$. St001141The number of occurrences of hills of size 3 in a Dyck path. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001139The number of occurrences of hills of size 2 in a Dyck path. St001111The weak 2-dynamic chromatic number of a graph. St000259The diameter of a connected graph. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001498The normalised height of a Nakayama algebra with magnitude 1. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000124The cardinality of the preimage of the Simion-Schmidt map. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St000782The indicator function of whether a given perfect matching is an L & P matching. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000883The number of longest increasing subsequences of a permutation. St001330The hat guessing number of a graph. St000454The largest eigenvalue of a graph if it is integral.
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