searching the database
Your data matches 55 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000929
(load all 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
St000929: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[2]
=> 0
[1,1]
=> 1
[3]
=> 0
[2,1]
=> 0
[1,1,1]
=> 1
[4]
=> 0
[3,1]
=> 0
[2,2]
=> 0
[2,1,1]
=> 0
[1,1,1,1]
=> 1
[5]
=> 0
[4,1]
=> 0
[3,2]
=> 0
[3,1,1]
=> 0
[2,2,1]
=> 0
[2,1,1,1]
=> 0
[1,1,1,1,1]
=> 1
[6]
=> 0
[5,1]
=> 0
[4,2]
=> 0
[4,1,1]
=> 0
[3,3]
=> 0
[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]
=> 1
[7]
=> 0
[6,1]
=> 0
[5,2]
=> 0
[5,1,1]
=> 0
[4,3]
=> 0
[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]
=> 1
[8]
=> 0
[7,1]
=> 0
[6,2]
=> 0
[6,1,1]
=> 0
[5,3]
=> 0
[5,2,1]
=> 0
[5,1,1,1]
=> 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
Mp00044: Integer partitions āconjugateā¶ Integer partitions
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
Mp00095: Integer partitions āto binary wordā¶ Binary words
St000296: Binary words ā¶ ā¤Result quality: 50% āvalues known / values provided: 96%ādistinct values known / distinct values provided: 50%
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
Mp00095: Integer partitions āto binary wordā¶ Binary words
St000296: Binary words ā¶ ā¤Result quality: 50% āvalues known / values provided: 96%ādistinct values known / distinct values provided: 50%
Values
[2]
=> [1,1]
=> [1]
=> 10 => 0
[1,1]
=> [2]
=> []
=> => ? = 1
[3]
=> [1,1,1]
=> [1,1]
=> 110 => 0
[2,1]
=> [2,1]
=> [1]
=> 10 => 0
[1,1,1]
=> [3]
=> []
=> => ? = 1
[4]
=> [1,1,1,1]
=> [1,1,1]
=> 1110 => 0
[3,1]
=> [2,1,1]
=> [1,1]
=> 110 => 0
[2,2]
=> [2,2]
=> [2]
=> 100 => 0
[2,1,1]
=> [3,1]
=> [1]
=> 10 => 0
[1,1,1,1]
=> [4]
=> []
=> => ? = 1
[5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 0
[4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1110 => 0
[3,2]
=> [2,2,1]
=> [2,1]
=> 1010 => 0
[3,1,1]
=> [3,1,1]
=> [1,1]
=> 110 => 0
[2,2,1]
=> [3,2]
=> [2]
=> 100 => 0
[2,1,1,1]
=> [4,1]
=> [1]
=> 10 => 0
[1,1,1,1,1]
=> [5]
=> []
=> => ? = 1
[6]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 0
[5,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 0
[4,2]
=> [2,2,1,1]
=> [2,1,1]
=> 10110 => 0
[4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> 1110 => 0
[3,3]
=> [2,2,2]
=> [2,2]
=> 1100 => 0
[3,2,1]
=> [3,2,1]
=> [2,1]
=> 1010 => 0
[3,1,1,1]
=> [4,1,1]
=> [1,1]
=> 110 => 0
[2,2,2]
=> [3,3]
=> [3]
=> 1000 => 0
[2,2,1,1]
=> [4,2]
=> [2]
=> 100 => 0
[2,1,1,1,1]
=> [5,1]
=> [1]
=> 10 => 0
[1,1,1,1,1,1]
=> [6]
=> []
=> => ? = 1
[7]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 1111110 => 0
[6,1]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 0
[5,2]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> 101110 => 0
[5,1,1]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 0
[4,3]
=> [2,2,2,1]
=> [2,2,1]
=> 11010 => 0
[4,2,1]
=> [3,2,1,1]
=> [2,1,1]
=> 10110 => 0
[4,1,1,1]
=> [4,1,1,1]
=> [1,1,1]
=> 1110 => 0
[3,3,1]
=> [3,2,2]
=> [2,2]
=> 1100 => 0
[3,2,2]
=> [3,3,1]
=> [3,1]
=> 10010 => 0
[3,2,1,1]
=> [4,2,1]
=> [2,1]
=> 1010 => 0
[3,1,1,1,1]
=> [5,1,1]
=> [1,1]
=> 110 => 0
[2,2,2,1]
=> [4,3]
=> [3]
=> 1000 => 0
[2,2,1,1,1]
=> [5,2]
=> [2]
=> 100 => 0
[2,1,1,1,1,1]
=> [6,1]
=> [1]
=> 10 => 0
[1,1,1,1,1,1,1]
=> [7]
=> []
=> => ? = 1
[8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> 11111110 => 0
[7,1]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 1111110 => 0
[6,2]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> 1011110 => 0
[6,1,1]
=> [3,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 0
[5,3]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> 110110 => 0
[5,2,1]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> 101110 => 0
[5,1,1,1]
=> [4,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 0
[4,4]
=> [2,2,2,2]
=> [2,2,2]
=> 11100 => 0
[4,3,1]
=> [3,2,2,1]
=> [2,2,1]
=> 11010 => 0
[4,2,2]
=> [3,3,1,1]
=> [3,1,1]
=> 100110 => 0
[4,2,1,1]
=> [4,2,1,1]
=> [2,1,1]
=> 10110 => 0
[4,1,1,1,1]
=> [5,1,1,1]
=> [1,1,1]
=> 1110 => 0
[3,3,2]
=> [3,3,2]
=> [3,2]
=> 10100 => 0
[1,1,1,1,1,1,1,1]
=> [8]
=> []
=> => ? = 1
[1,1,1,1,1,1,1,1,1]
=> [9]
=> []
=> => ? = 1
[1,1,1,1,1,1,1,1,1,1]
=> [10]
=> []
=> => ? = 1
[1,1,1,1,1,1,1,1,1,1,1]
=> [11]
=> []
=> => ? = 1
[12]
=> [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
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [12]
=> []
=> => ? = 1
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)
Mp00044: Integer partitions āconjugateā¶ Integer partitions
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
Mp00095: Integer partitions āto binary wordā¶ Binary words
St000629: Binary words ā¶ ā¤Result quality: 50% āvalues known / values provided: 96%ādistinct values known / distinct values provided: 50%
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
Mp00095: Integer partitions āto binary wordā¶ Binary words
St000629: Binary words ā¶ ā¤Result quality: 50% āvalues known / values provided: 96%ādistinct values known / distinct values provided: 50%
Values
[2]
=> [1,1]
=> [1]
=> 10 => 0
[1,1]
=> [2]
=> []
=> => ? = 1
[3]
=> [1,1,1]
=> [1,1]
=> 110 => 0
[2,1]
=> [2,1]
=> [1]
=> 10 => 0
[1,1,1]
=> [3]
=> []
=> => ? = 1
[4]
=> [1,1,1,1]
=> [1,1,1]
=> 1110 => 0
[3,1]
=> [2,1,1]
=> [1,1]
=> 110 => 0
[2,2]
=> [2,2]
=> [2]
=> 100 => 0
[2,1,1]
=> [3,1]
=> [1]
=> 10 => 0
[1,1,1,1]
=> [4]
=> []
=> => ? = 1
[5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 0
[4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1110 => 0
[3,2]
=> [2,2,1]
=> [2,1]
=> 1010 => 0
[3,1,1]
=> [3,1,1]
=> [1,1]
=> 110 => 0
[2,2,1]
=> [3,2]
=> [2]
=> 100 => 0
[2,1,1,1]
=> [4,1]
=> [1]
=> 10 => 0
[1,1,1,1,1]
=> [5]
=> []
=> => ? = 1
[6]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 0
[5,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 0
[4,2]
=> [2,2,1,1]
=> [2,1,1]
=> 10110 => 0
[4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> 1110 => 0
[3,3]
=> [2,2,2]
=> [2,2]
=> 1100 => 0
[3,2,1]
=> [3,2,1]
=> [2,1]
=> 1010 => 0
[3,1,1,1]
=> [4,1,1]
=> [1,1]
=> 110 => 0
[2,2,2]
=> [3,3]
=> [3]
=> 1000 => 0
[2,2,1,1]
=> [4,2]
=> [2]
=> 100 => 0
[2,1,1,1,1]
=> [5,1]
=> [1]
=> 10 => 0
[1,1,1,1,1,1]
=> [6]
=> []
=> => ? = 1
[7]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 1111110 => 0
[6,1]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 0
[5,2]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> 101110 => 0
[5,1,1]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 0
[4,3]
=> [2,2,2,1]
=> [2,2,1]
=> 11010 => 0
[4,2,1]
=> [3,2,1,1]
=> [2,1,1]
=> 10110 => 0
[4,1,1,1]
=> [4,1,1,1]
=> [1,1,1]
=> 1110 => 0
[3,3,1]
=> [3,2,2]
=> [2,2]
=> 1100 => 0
[3,2,2]
=> [3,3,1]
=> [3,1]
=> 10010 => 0
[3,2,1,1]
=> [4,2,1]
=> [2,1]
=> 1010 => 0
[3,1,1,1,1]
=> [5,1,1]
=> [1,1]
=> 110 => 0
[2,2,2,1]
=> [4,3]
=> [3]
=> 1000 => 0
[2,2,1,1,1]
=> [5,2]
=> [2]
=> 100 => 0
[2,1,1,1,1,1]
=> [6,1]
=> [1]
=> 10 => 0
[1,1,1,1,1,1,1]
=> [7]
=> []
=> => ? = 1
[8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> 11111110 => 0
[7,1]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 1111110 => 0
[6,2]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> 1011110 => 0
[6,1,1]
=> [3,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 0
[5,3]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> 110110 => 0
[5,2,1]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> 101110 => 0
[5,1,1,1]
=> [4,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 0
[4,4]
=> [2,2,2,2]
=> [2,2,2]
=> 11100 => 0
[4,3,1]
=> [3,2,2,1]
=> [2,2,1]
=> 11010 => 0
[4,2,2]
=> [3,3,1,1]
=> [3,1,1]
=> 100110 => 0
[4,2,1,1]
=> [4,2,1,1]
=> [2,1,1]
=> 10110 => 0
[4,1,1,1,1]
=> [5,1,1,1]
=> [1,1,1]
=> 1110 => 0
[3,3,2]
=> [3,3,2]
=> [3,2]
=> 10100 => 0
[1,1,1,1,1,1,1,1]
=> [8]
=> []
=> => ? = 1
[1,1,1,1,1,1,1,1,1]
=> [9]
=> []
=> => ? = 1
[1,1,1,1,1,1,1,1,1,1]
=> [10]
=> []
=> => ? = 1
[1,1,1,1,1,1,1,1,1,1,1]
=> [11]
=> []
=> => ? = 1
[12]
=> [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
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [12]
=> []
=> => ? = 1
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
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00044: Integer partitions āconjugateā¶ Integer partitions
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
Mp00095: Integer partitions āto binary wordā¶ Binary words
St000326: Binary words ā¶ ā¤Result quality: 50% āvalues known / values provided: 96%ādistinct values known / distinct values provided: 50%
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
Mp00095: Integer partitions āto binary wordā¶ Binary words
St000326: Binary words ā¶ ā¤Result quality: 50% āvalues known / values provided: 96%ādistinct values known / distinct values provided: 50%
Values
[2]
=> [1,1]
=> [1]
=> 10 => 1 = 0 + 1
[1,1]
=> [2]
=> []
=> => ? = 1 + 1
[3]
=> [1,1,1]
=> [1,1]
=> 110 => 1 = 0 + 1
[2,1]
=> [2,1]
=> [1]
=> 10 => 1 = 0 + 1
[1,1,1]
=> [3]
=> []
=> => ? = 1 + 1
[4]
=> [1,1,1,1]
=> [1,1,1]
=> 1110 => 1 = 0 + 1
[3,1]
=> [2,1,1]
=> [1,1]
=> 110 => 1 = 0 + 1
[2,2]
=> [2,2]
=> [2]
=> 100 => 1 = 0 + 1
[2,1,1]
=> [3,1]
=> [1]
=> 10 => 1 = 0 + 1
[1,1,1,1]
=> [4]
=> []
=> => ? = 1 + 1
[5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 1 = 0 + 1
[4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1110 => 1 = 0 + 1
[3,2]
=> [2,2,1]
=> [2,1]
=> 1010 => 1 = 0 + 1
[3,1,1]
=> [3,1,1]
=> [1,1]
=> 110 => 1 = 0 + 1
[2,2,1]
=> [3,2]
=> [2]
=> 100 => 1 = 0 + 1
[2,1,1,1]
=> [4,1]
=> [1]
=> 10 => 1 = 0 + 1
[1,1,1,1,1]
=> [5]
=> []
=> => ? = 1 + 1
[6]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 1 = 0 + 1
[5,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 1 = 0 + 1
[4,2]
=> [2,2,1,1]
=> [2,1,1]
=> 10110 => 1 = 0 + 1
[4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> 1110 => 1 = 0 + 1
[3,3]
=> [2,2,2]
=> [2,2]
=> 1100 => 1 = 0 + 1
[3,2,1]
=> [3,2,1]
=> [2,1]
=> 1010 => 1 = 0 + 1
[3,1,1,1]
=> [4,1,1]
=> [1,1]
=> 110 => 1 = 0 + 1
[2,2,2]
=> [3,3]
=> [3]
=> 1000 => 1 = 0 + 1
[2,2,1,1]
=> [4,2]
=> [2]
=> 100 => 1 = 0 + 1
[2,1,1,1,1]
=> [5,1]
=> [1]
=> 10 => 1 = 0 + 1
[1,1,1,1,1,1]
=> [6]
=> []
=> => ? = 1 + 1
[7]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 1111110 => 1 = 0 + 1
[6,1]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 1 = 0 + 1
[5,2]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> 101110 => 1 = 0 + 1
[5,1,1]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 1 = 0 + 1
[4,3]
=> [2,2,2,1]
=> [2,2,1]
=> 11010 => 1 = 0 + 1
[4,2,1]
=> [3,2,1,1]
=> [2,1,1]
=> 10110 => 1 = 0 + 1
[4,1,1,1]
=> [4,1,1,1]
=> [1,1,1]
=> 1110 => 1 = 0 + 1
[3,3,1]
=> [3,2,2]
=> [2,2]
=> 1100 => 1 = 0 + 1
[3,2,2]
=> [3,3,1]
=> [3,1]
=> 10010 => 1 = 0 + 1
[3,2,1,1]
=> [4,2,1]
=> [2,1]
=> 1010 => 1 = 0 + 1
[3,1,1,1,1]
=> [5,1,1]
=> [1,1]
=> 110 => 1 = 0 + 1
[2,2,2,1]
=> [4,3]
=> [3]
=> 1000 => 1 = 0 + 1
[2,2,1,1,1]
=> [5,2]
=> [2]
=> 100 => 1 = 0 + 1
[2,1,1,1,1,1]
=> [6,1]
=> [1]
=> 10 => 1 = 0 + 1
[1,1,1,1,1,1,1]
=> [7]
=> []
=> => ? = 1 + 1
[8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> 11111110 => 1 = 0 + 1
[7,1]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 1111110 => 1 = 0 + 1
[6,2]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> 1011110 => 1 = 0 + 1
[6,1,1]
=> [3,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 1 = 0 + 1
[5,3]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> 110110 => 1 = 0 + 1
[5,2,1]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> 101110 => 1 = 0 + 1
[5,1,1,1]
=> [4,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 1 = 0 + 1
[4,4]
=> [2,2,2,2]
=> [2,2,2]
=> 11100 => 1 = 0 + 1
[4,3,1]
=> [3,2,2,1]
=> [2,2,1]
=> 11010 => 1 = 0 + 1
[4,2,2]
=> [3,3,1,1]
=> [3,1,1]
=> 100110 => 1 = 0 + 1
[4,2,1,1]
=> [4,2,1,1]
=> [2,1,1]
=> 10110 => 1 = 0 + 1
[4,1,1,1,1]
=> [5,1,1,1]
=> [1,1,1]
=> 1110 => 1 = 0 + 1
[3,3,2]
=> [3,3,2]
=> [3,2]
=> 10100 => 1 = 0 + 1
[1,1,1,1,1,1,1,1]
=> [8]
=> []
=> => ? = 1 + 1
[1,1,1,1,1,1,1,1,1]
=> [9]
=> []
=> => ? = 1 + 1
[1,1,1,1,1,1,1,1,1,1]
=> [10]
=> []
=> => ? = 1 + 1
[1,1,1,1,1,1,1,1,1,1,1]
=> [11]
=> []
=> => ? = 1 + 1
[12]
=> [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
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [12]
=> []
=> => ? = 1 + 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
Mp00044: Integer partitions āconjugateā¶ Integer partitions
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
Mp00095: Integer partitions āto binary wordā¶ Binary words
St001371: Binary words ā¶ ā¤Result quality: 50% āvalues known / values provided: 92%ādistinct values known / distinct values provided: 50%
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
Mp00095: Integer partitions āto binary wordā¶ Binary words
St001371: Binary words ā¶ ā¤Result quality: 50% āvalues known / values provided: 92%ādistinct values known / distinct values provided: 50%
Values
[2]
=> [1,1]
=> [1]
=> 10 => 0
[1,1]
=> [2]
=> []
=> => ? = 1
[3]
=> [1,1,1]
=> [1,1]
=> 110 => 0
[2,1]
=> [2,1]
=> [1]
=> 10 => 0
[1,1,1]
=> [3]
=> []
=> => ? = 1
[4]
=> [1,1,1,1]
=> [1,1,1]
=> 1110 => 0
[3,1]
=> [2,1,1]
=> [1,1]
=> 110 => 0
[2,2]
=> [2,2]
=> [2]
=> 100 => 0
[2,1,1]
=> [3,1]
=> [1]
=> 10 => 0
[1,1,1,1]
=> [4]
=> []
=> => ? = 1
[5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 0
[4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1110 => 0
[3,2]
=> [2,2,1]
=> [2,1]
=> 1010 => 0
[3,1,1]
=> [3,1,1]
=> [1,1]
=> 110 => 0
[2,2,1]
=> [3,2]
=> [2]
=> 100 => 0
[2,1,1,1]
=> [4,1]
=> [1]
=> 10 => 0
[1,1,1,1,1]
=> [5]
=> []
=> => ? = 1
[6]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 0
[5,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 0
[4,2]
=> [2,2,1,1]
=> [2,1,1]
=> 10110 => 0
[4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> 1110 => 0
[3,3]
=> [2,2,2]
=> [2,2]
=> 1100 => 0
[3,2,1]
=> [3,2,1]
=> [2,1]
=> 1010 => 0
[3,1,1,1]
=> [4,1,1]
=> [1,1]
=> 110 => 0
[2,2,2]
=> [3,3]
=> [3]
=> 1000 => 0
[2,2,1,1]
=> [4,2]
=> [2]
=> 100 => 0
[2,1,1,1,1]
=> [5,1]
=> [1]
=> 10 => 0
[1,1,1,1,1,1]
=> [6]
=> []
=> => ? = 1
[7]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 1111110 => 0
[6,1]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 0
[5,2]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> 101110 => 0
[5,1,1]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 0
[4,3]
=> [2,2,2,1]
=> [2,2,1]
=> 11010 => 0
[4,2,1]
=> [3,2,1,1]
=> [2,1,1]
=> 10110 => 0
[4,1,1,1]
=> [4,1,1,1]
=> [1,1,1]
=> 1110 => 0
[3,3,1]
=> [3,2,2]
=> [2,2]
=> 1100 => 0
[3,2,2]
=> [3,3,1]
=> [3,1]
=> 10010 => 0
[3,2,1,1]
=> [4,2,1]
=> [2,1]
=> 1010 => 0
[3,1,1,1,1]
=> [5,1,1]
=> [1,1]
=> 110 => 0
[2,2,2,1]
=> [4,3]
=> [3]
=> 1000 => 0
[2,2,1,1,1]
=> [5,2]
=> [2]
=> 100 => 0
[2,1,1,1,1,1]
=> [6,1]
=> [1]
=> 10 => 0
[1,1,1,1,1,1,1]
=> [7]
=> []
=> => ? = 1
[8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> 11111110 => 0
[7,1]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 1111110 => 0
[6,2]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> 1011110 => 0
[6,1,1]
=> [3,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 0
[5,3]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> 110110 => 0
[5,2,1]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> 101110 => 0
[5,1,1,1]
=> [4,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 0
[4,4]
=> [2,2,2,2]
=> [2,2,2]
=> 11100 => 0
[4,3,1]
=> [3,2,2,1]
=> [2,2,1]
=> 11010 => 0
[4,2,2]
=> [3,3,1,1]
=> [3,1,1]
=> 100110 => 0
[4,2,1,1]
=> [4,2,1,1]
=> [2,1,1]
=> 10110 => 0
[4,1,1,1,1]
=> [5,1,1,1]
=> [1,1,1]
=> 1110 => 0
[3,3,2]
=> [3,3,2]
=> [3,2]
=> 10100 => 0
[1,1,1,1,1,1,1,1]
=> [8]
=> []
=> => ? = 1
[1,1,1,1,1,1,1,1,1]
=> [9]
=> []
=> => ? = 1
[10]
=> [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]
=> [10]
=> []
=> => ? = 1
[11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> 11111111110 => ? = 0
[10,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> 1111111110 => ? = 0
[9,2]
=> [2,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> 1011111110 => ? = 0
[1,1,1,1,1,1,1,1,1,1,1]
=> [11]
=> []
=> => ? = 1
[12]
=> [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
[11,1]
=> [2,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> 11111111110 => ? = 0
[10,2]
=> [2,2,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1]
=> 10111111110 => ? = 0
[10,1,1]
=> [3,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> 1111111110 => ? = 0
[9,3]
=> [2,2,2,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1]
=> 1101111110 => ? = 0
[9,2,1]
=> [3,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> 1011111110 => ? = 0
[8,2,2]
=> [3,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> 1001111110 => ? = 0
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [12]
=> []
=> => ? = 1
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
Mp00044: Integer partitions āconjugateā¶ Integer partitions
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
Mp00043: Integer partitions āto Dyck pathā¶ Dyck paths
St001107: Dyck paths ā¶ ā¤Result quality: 50% āvalues known / values provided: 91%ādistinct values known / distinct values provided: 50%
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
Mp00043: Integer partitions āto Dyck pathā¶ Dyck paths
St001107: Dyck paths ā¶ ā¤Result quality: 50% āvalues known / values provided: 91%ādistinct values known / distinct values provided: 50%
Values
[2]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[1,1]
=> [2]
=> []
=> []
=> ? = 1
[3]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[2,1]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[1,1,1]
=> [3]
=> []
=> []
=> ? = 1
[4]
=> [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0
[3,1]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[2,2]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[2,1,1]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[1,1,1,1]
=> [4]
=> []
=> []
=> ? = 1
[5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[4,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0
[3,2]
=> [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0
[3,1,1]
=> [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[2,2,1]
=> [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[2,1,1,1]
=> [4,1]
=> [1]
=> [1,0,1,0]
=> 0
[1,1,1,1,1]
=> [5]
=> []
=> []
=> ? = 1
[6]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 0
[5,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[4,2]
=> [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0
[3,3]
=> [2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 0
[3,2,1]
=> [3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0
[3,1,1,1]
=> [4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[2,2,2]
=> [3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 0
[2,2,1,1]
=> [4,2]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[2,1,1,1,1]
=> [5,1]
=> [1]
=> [1,0,1,0]
=> 0
[1,1,1,1,1,1]
=> [6]
=> []
=> []
=> ? = 1
[7]
=> [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
[6,1]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 0
[5,2]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 0
[5,1,1]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[4,3]
=> [2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[4,2,1]
=> [3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[4,1,1,1]
=> [4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0
[3,3,1]
=> [3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 0
[3,2,2]
=> [3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 0
[3,2,1,1]
=> [4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0
[3,1,1,1,1]
=> [5,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[2,2,2,1]
=> [4,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 0
[2,2,1,1,1]
=> [5,2]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[2,1,1,1,1,1]
=> [6,1]
=> [1]
=> [1,0,1,0]
=> 0
[1,1,1,1,1,1,1]
=> [7]
=> []
=> []
=> ? = 1
[8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 0
[7,1]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
[6,2]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 0
[6,1,1]
=> [3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 0
[5,3]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 0
[5,2,1]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 0
[5,1,1,1]
=> [4,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[4,4]
=> [2,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0
[4,3,1]
=> [3,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[4,2,2]
=> [3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 0
[4,2,1,1]
=> [4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[4,1,1,1,1]
=> [5,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0
[3,3,2]
=> [3,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 0
[1,1,1,1,1,1,1,1]
=> [8]
=> []
=> []
=> ? = 1
[9]
=> [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]
=> [9]
=> []
=> []
=> ? = 1
[10]
=> [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
[9,1]
=> [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
[1,1,1,1,1,1,1,1,1,1]
=> [10]
=> []
=> []
=> ? = 1
[11]
=> [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
[10,1]
=> [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
[9,2]
=> [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
[9,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,1,1,1,1,1,1,1,1,1,1]
=> [11]
=> []
=> []
=> ? = 1
[12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 0
[11,1]
=> [2,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 0
[10,2]
=> [2,2,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> ? = 0
[10,1,1]
=> [3,1,1,1,1,1,1,1,1,1]
=> [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
[9,3]
=> [2,2,2,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> ? = 0
[9,2,1]
=> [3,2,1,1,1,1,1,1,1]
=> [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
[9,1,1,1]
=> [4,1,1,1,1,1,1,1,1]
=> [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]
=> [12]
=> []
=> []
=> ? = 1
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
Mp00044: Integer partitions āconjugateā¶ Integer partitions
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
St000687: Dyck paths ā¶ ā¤Result quality: 50% āvalues known / values provided: 87%ādistinct values known / distinct values provided: 50%
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
St000687: Dyck paths ā¶ ā¤Result quality: 50% āvalues known / values provided: 87%ādistinct values known / distinct values provided: 50%
Values
[2]
=> [1,1]
=> [1]
=> [1,0]
=> 0
[1,1]
=> [2]
=> []
=> []
=> ? = 1
[3]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[2,1]
=> [2,1]
=> [1]
=> [1,0]
=> 0
[1,1,1]
=> [3]
=> []
=> []
=> ? = 1
[4]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[3,1]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[2,2]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 0
[2,1,1]
=> [3,1]
=> [1]
=> [1,0]
=> 0
[1,1,1,1]
=> [4]
=> []
=> []
=> ? = 1
[5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[4,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[3,2]
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
[3,1,1]
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[2,2,1]
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 0
[2,1,1,1]
=> [4,1]
=> [1]
=> [1,0]
=> 0
[1,1,1,1,1]
=> [5]
=> []
=> []
=> ? = 1
[6]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[5,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[4,2]
=> [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[3,3]
=> [2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0
[3,2,1]
=> [3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
[3,1,1,1]
=> [4,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[2,2,2]
=> [3,3]
=> [3]
=> [1,0,1,0,1,0]
=> 0
[2,2,1,1]
=> [4,2]
=> [2]
=> [1,0,1,0]
=> 0
[2,1,1,1,1]
=> [5,1]
=> [1]
=> [1,0]
=> 0
[1,1,1,1,1,1]
=> [6]
=> []
=> []
=> ? = 1
[7]
=> [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
[6,1]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[5,2]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0
[5,1,1]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[4,3]
=> [2,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 0
[4,2,1]
=> [3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[4,1,1,1]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[3,3,1]
=> [3,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0
[3,2,2]
=> [3,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[3,2,1,1]
=> [4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
[3,1,1,1,1]
=> [5,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[2,2,2,1]
=> [4,3]
=> [3]
=> [1,0,1,0,1,0]
=> 0
[2,2,1,1,1]
=> [5,2]
=> [2]
=> [1,0,1,0]
=> 0
[2,1,1,1,1,1]
=> [6,1]
=> [1]
=> [1,0]
=> 0
[1,1,1,1,1,1,1]
=> [7]
=> []
=> []
=> ? = 1
[8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 0
[7,1]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 0
[6,2]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 0
[6,1,1]
=> [3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[5,3]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
[5,2,1]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0
[5,1,1,1]
=> [4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[4,4]
=> [2,2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 0
[4,3,1]
=> [3,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 0
[4,2,2]
=> [3,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[4,2,1,1]
=> [4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[4,1,1,1,1]
=> [5,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[3,3,2]
=> [3,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 0
[1,1,1,1,1,1,1,1]
=> [8]
=> []
=> []
=> ? = 1
[9]
=> [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]
=> [9]
=> []
=> []
=> ? = 1
[10]
=> [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
[9,1]
=> [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
[8,2]
=> [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
[1,1,1,1,1,1,1,1,1,1]
=> [10]
=> []
=> []
=> ? = 1
[11]
=> [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
[10,1]
=> [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
[9,2]
=> [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
[9,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> [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
[8,3]
=> [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
[8,2,1]
=> [3,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[7,2,2]
=> [3,3,1,1,1,1,1]
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[1,1,1,1,1,1,1,1,1,1,1]
=> [11]
=> []
=> []
=> ? = 1
[12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[11,1]
=> [2,1,1,1,1,1,1,1,1,1,1]
=> [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
[10,2]
=> [2,2,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[10,1,1]
=> [3,1,1,1,1,1,1,1,1,1]
=> [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
[9,3]
=> [2,2,2,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[9,2,1]
=> [3,2,1,1,1,1,1,1,1]
=> [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
[9,1,1,1]
=> [4,1,1,1,1,1,1,1,1]
=> [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
[8,4]
=> [2,2,2,2,1,1,1,1]
=> [2,2,2,1,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[8,3,1]
=> [3,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[8,2,2]
=> [3,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[8,2,1,1]
=> [4,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[7,3,2]
=> [3,3,2,1,1,1,1]
=> [3,2,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[7,2,2,1]
=> [4,3,1,1,1,1,1]
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[6,2,2,2]
=> [4,4,1,1,1,1]
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 0
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [12]
=> []
=> []
=> ? = 1
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
Mp00044: Integer partitions āconjugateā¶ Integer partitions
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
Mp00042: Integer partitions āinitial tableauā¶ Standard tableaux
St001695: Standard tableaux ā¶ ā¤Result quality: 50% āvalues known / values provided: 86%ādistinct values known / distinct values provided: 50%
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
Mp00042: Integer partitions āinitial tableauā¶ Standard tableaux
St001695: Standard tableaux ā¶ ā¤Result quality: 50% āvalues known / values provided: 86%ādistinct values known / distinct values provided: 50%
Values
[2]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[1,1]
=> [2]
=> []
=> []
=> ? = 1
[3]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[2,1]
=> [2,1]
=> [1]
=> [[1]]
=> 0
[1,1,1]
=> [3]
=> []
=> []
=> ? = 1
[4]
=> [1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[3,1]
=> [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[2,2]
=> [2,2]
=> [2]
=> [[1,2]]
=> 0
[2,1,1]
=> [3,1]
=> [1]
=> [[1]]
=> 0
[1,1,1,1]
=> [4]
=> []
=> []
=> ? = 1
[5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[4,1]
=> [2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[3,2]
=> [2,2,1]
=> [2,1]
=> [[1,2],[3]]
=> 0
[3,1,1]
=> [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[2,2,1]
=> [3,2]
=> [2]
=> [[1,2]]
=> 0
[2,1,1,1]
=> [4,1]
=> [1]
=> [[1]]
=> 0
[1,1,1,1,1]
=> [5]
=> []
=> []
=> ? = 1
[6]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 0
[5,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[4,2]
=> [2,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 0
[4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[3,3]
=> [2,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 0
[3,2,1]
=> [3,2,1]
=> [2,1]
=> [[1,2],[3]]
=> 0
[3,1,1,1]
=> [4,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[2,2,2]
=> [3,3]
=> [3]
=> [[1,2,3]]
=> 0
[2,2,1,1]
=> [4,2]
=> [2]
=> [[1,2]]
=> 0
[2,1,1,1,1]
=> [5,1]
=> [1]
=> [[1]]
=> 0
[1,1,1,1,1,1]
=> [6]
=> []
=> []
=> ? = 1
[7]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> 0
[6,1]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 0
[5,2]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 0
[5,1,1]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[4,3]
=> [2,2,2,1]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 0
[4,2,1]
=> [3,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 0
[4,1,1,1]
=> [4,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[3,3,1]
=> [3,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 0
[3,2,2]
=> [3,3,1]
=> [3,1]
=> [[1,2,3],[4]]
=> 0
[3,2,1,1]
=> [4,2,1]
=> [2,1]
=> [[1,2],[3]]
=> 0
[3,1,1,1,1]
=> [5,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[2,2,2,1]
=> [4,3]
=> [3]
=> [[1,2,3]]
=> 0
[2,2,1,1,1]
=> [5,2]
=> [2]
=> [[1,2]]
=> 0
[2,1,1,1,1,1]
=> [6,1]
=> [1]
=> [[1]]
=> 0
[1,1,1,1,1,1,1]
=> [7]
=> []
=> []
=> ? = 1
[8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> 0
[7,1]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> 0
[6,2]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> 0
[6,1,1]
=> [3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 0
[5,3]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> 0
[5,2,1]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 0
[5,1,1,1]
=> [4,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[4,4]
=> [2,2,2,2]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> 0
[4,3,1]
=> [3,2,2,1]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 0
[4,2,2]
=> [3,3,1,1]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 0
[4,2,1,1]
=> [4,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 0
[4,1,1,1,1]
=> [5,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[3,3,2]
=> [3,3,2]
=> [3,2]
=> [[1,2,3],[4,5]]
=> 0
[1,1,1,1,1,1,1,1]
=> [8]
=> []
=> []
=> ? = 1
[1,1,1,1,1,1,1,1,1]
=> [9]
=> []
=> []
=> ? = 1
[10]
=> [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]
=> [10]
=> []
=> []
=> ? = 1
[11]
=> [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
[10,1]
=> [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
[9,2]
=> [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
[8,3]
=> [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
[7,4]
=> [2,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9]]
=> ? = 0
[6,5]
=> [2,2,2,2,2,1]
=> [2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> ? = 0
[1,1,1,1,1,1,1,1,1,1,1]
=> [11]
=> []
=> []
=> ? = 1
[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],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 0
[11,1]
=> [2,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? = 0
[10,2]
=> [2,2,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? = 0
[10,1,1]
=> [3,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 0
[9,3]
=> [2,2,2,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10]]
=> ? = 0
[9,2,1]
=> [3,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 0
[8,4]
=> [2,2,2,2,1,1,1,1]
=> [2,2,2,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9],[10]]
=> ? = 0
[8,3,1]
=> [3,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9]]
=> ? = 0
[8,2,2]
=> [3,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8],[9]]
=> ? = 0
[7,5]
=> [2,2,2,2,2,1,1]
=> [2,2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10]]
=> ? = 0
[7,4,1]
=> [3,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9]]
=> ? = 0
[7,3,2]
=> [3,3,2,1,1,1,1]
=> [3,2,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9]]
=> ? = 0
[6,6]
=> [2,2,2,2,2,2]
=> [2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> ? = 0
[6,5,1]
=> [3,2,2,2,2,1]
=> [2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> ? = 0
[6,4,2]
=> [3,3,2,2,1,1]
=> [3,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9]]
=> ? = 0
[6,3,3]
=> [3,3,3,1,1,1]
=> [3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> ? = 0
[5,5,2]
=> [3,3,2,2,2]
=> [3,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9]]
=> ? = 0
[5,4,3]
=> [3,3,3,2,1]
=> [3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> ? = 0
[4,4,4]
=> [3,3,3,3]
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> ? = 0
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [12]
=> []
=> []
=> ? = 1
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
Mp00044: Integer partitions āconjugateā¶ Integer partitions
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
Mp00042: Integer partitions āinitial tableauā¶ Standard tableaux
St001698: Standard tableaux ā¶ ā¤Result quality: 50% āvalues known / values provided: 86%ādistinct values known / distinct values provided: 50%
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
Mp00042: Integer partitions āinitial tableauā¶ Standard tableaux
St001698: Standard tableaux ā¶ ā¤Result quality: 50% āvalues known / values provided: 86%ādistinct values known / distinct values provided: 50%
Values
[2]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[1,1]
=> [2]
=> []
=> []
=> ? = 1
[3]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[2,1]
=> [2,1]
=> [1]
=> [[1]]
=> 0
[1,1,1]
=> [3]
=> []
=> []
=> ? = 1
[4]
=> [1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[3,1]
=> [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[2,2]
=> [2,2]
=> [2]
=> [[1,2]]
=> 0
[2,1,1]
=> [3,1]
=> [1]
=> [[1]]
=> 0
[1,1,1,1]
=> [4]
=> []
=> []
=> ? = 1
[5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[4,1]
=> [2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[3,2]
=> [2,2,1]
=> [2,1]
=> [[1,2],[3]]
=> 0
[3,1,1]
=> [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[2,2,1]
=> [3,2]
=> [2]
=> [[1,2]]
=> 0
[2,1,1,1]
=> [4,1]
=> [1]
=> [[1]]
=> 0
[1,1,1,1,1]
=> [5]
=> []
=> []
=> ? = 1
[6]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 0
[5,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[4,2]
=> [2,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 0
[4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[3,3]
=> [2,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 0
[3,2,1]
=> [3,2,1]
=> [2,1]
=> [[1,2],[3]]
=> 0
[3,1,1,1]
=> [4,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[2,2,2]
=> [3,3]
=> [3]
=> [[1,2,3]]
=> 0
[2,2,1,1]
=> [4,2]
=> [2]
=> [[1,2]]
=> 0
[2,1,1,1,1]
=> [5,1]
=> [1]
=> [[1]]
=> 0
[1,1,1,1,1,1]
=> [6]
=> []
=> []
=> ? = 1
[7]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> 0
[6,1]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 0
[5,2]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 0
[5,1,1]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[4,3]
=> [2,2,2,1]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 0
[4,2,1]
=> [3,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 0
[4,1,1,1]
=> [4,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[3,3,1]
=> [3,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 0
[3,2,2]
=> [3,3,1]
=> [3,1]
=> [[1,2,3],[4]]
=> 0
[3,2,1,1]
=> [4,2,1]
=> [2,1]
=> [[1,2],[3]]
=> 0
[3,1,1,1,1]
=> [5,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[2,2,2,1]
=> [4,3]
=> [3]
=> [[1,2,3]]
=> 0
[2,2,1,1,1]
=> [5,2]
=> [2]
=> [[1,2]]
=> 0
[2,1,1,1,1,1]
=> [6,1]
=> [1]
=> [[1]]
=> 0
[1,1,1,1,1,1,1]
=> [7]
=> []
=> []
=> ? = 1
[8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> 0
[7,1]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> 0
[6,2]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> 0
[6,1,1]
=> [3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 0
[5,3]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> 0
[5,2,1]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 0
[5,1,1,1]
=> [4,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[4,4]
=> [2,2,2,2]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> 0
[4,3,1]
=> [3,2,2,1]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 0
[4,2,2]
=> [3,3,1,1]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 0
[4,2,1,1]
=> [4,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 0
[4,1,1,1,1]
=> [5,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[3,3,2]
=> [3,3,2]
=> [3,2]
=> [[1,2,3],[4,5]]
=> 0
[1,1,1,1,1,1,1,1]
=> [8]
=> []
=> []
=> ? = 1
[1,1,1,1,1,1,1,1,1]
=> [9]
=> []
=> []
=> ? = 1
[10]
=> [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]
=> [10]
=> []
=> []
=> ? = 1
[11]
=> [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
[10,1]
=> [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
[9,2]
=> [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
[8,3]
=> [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
[7,4]
=> [2,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9]]
=> ? = 0
[6,5]
=> [2,2,2,2,2,1]
=> [2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> ? = 0
[1,1,1,1,1,1,1,1,1,1,1]
=> [11]
=> []
=> []
=> ? = 1
[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],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 0
[11,1]
=> [2,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? = 0
[10,2]
=> [2,2,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? = 0
[10,1,1]
=> [3,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 0
[9,3]
=> [2,2,2,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10]]
=> ? = 0
[9,2,1]
=> [3,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 0
[8,4]
=> [2,2,2,2,1,1,1,1]
=> [2,2,2,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9],[10]]
=> ? = 0
[8,3,1]
=> [3,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9]]
=> ? = 0
[8,2,2]
=> [3,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8],[9]]
=> ? = 0
[7,5]
=> [2,2,2,2,2,1,1]
=> [2,2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10]]
=> ? = 0
[7,4,1]
=> [3,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9]]
=> ? = 0
[7,3,2]
=> [3,3,2,1,1,1,1]
=> [3,2,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9]]
=> ? = 0
[6,6]
=> [2,2,2,2,2,2]
=> [2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> ? = 0
[6,5,1]
=> [3,2,2,2,2,1]
=> [2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> ? = 0
[6,4,2]
=> [3,3,2,2,1,1]
=> [3,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9]]
=> ? = 0
[6,3,3]
=> [3,3,3,1,1,1]
=> [3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> ? = 0
[5,5,2]
=> [3,3,2,2,2]
=> [3,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9]]
=> ? = 0
[5,4,3]
=> [3,3,3,2,1]
=> [3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> ? = 0
[4,4,4]
=> [3,3,3,3]
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> ? = 0
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [12]
=> []
=> []
=> ? = 1
Description
The comajor index of a standard tableau minus the weighted size of its shape.
Matching statistic: St001699
Mp00044: Integer partitions āconjugateā¶ Integer partitions
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
Mp00045: Integer partitions āreading tableauā¶ Standard tableaux
St001699: Standard tableaux ā¶ ā¤Result quality: 50% āvalues known / values provided: 86%ādistinct values known / distinct values provided: 50%
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
Mp00045: Integer partitions āreading tableauā¶ Standard tableaux
St001699: Standard tableaux ā¶ ā¤Result quality: 50% āvalues known / values provided: 86%ādistinct values known / distinct values provided: 50%
Values
[2]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[1,1]
=> [2]
=> []
=> []
=> ? = 1
[3]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[2,1]
=> [2,1]
=> [1]
=> [[1]]
=> 0
[1,1,1]
=> [3]
=> []
=> []
=> ? = 1
[4]
=> [1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[3,1]
=> [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[2,2]
=> [2,2]
=> [2]
=> [[1,2]]
=> 0
[2,1,1]
=> [3,1]
=> [1]
=> [[1]]
=> 0
[1,1,1,1]
=> [4]
=> []
=> []
=> ? = 1
[5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[4,1]
=> [2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[3,2]
=> [2,2,1]
=> [2,1]
=> [[1,3],[2]]
=> 0
[3,1,1]
=> [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[2,2,1]
=> [3,2]
=> [2]
=> [[1,2]]
=> 0
[2,1,1,1]
=> [4,1]
=> [1]
=> [[1]]
=> 0
[1,1,1,1,1]
=> [5]
=> []
=> []
=> ? = 1
[6]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 0
[5,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[4,2]
=> [2,2,1,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 0
[4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[3,3]
=> [2,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 0
[3,2,1]
=> [3,2,1]
=> [2,1]
=> [[1,3],[2]]
=> 0
[3,1,1,1]
=> [4,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[2,2,2]
=> [3,3]
=> [3]
=> [[1,2,3]]
=> 0
[2,2,1,1]
=> [4,2]
=> [2]
=> [[1,2]]
=> 0
[2,1,1,1,1]
=> [5,1]
=> [1]
=> [[1]]
=> 0
[1,1,1,1,1,1]
=> [6]
=> []
=> []
=> ? = 1
[7]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> 0
[6,1]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 0
[5,2]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 0
[5,1,1]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[4,3]
=> [2,2,2,1]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 0
[4,2,1]
=> [3,2,1,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 0
[4,1,1,1]
=> [4,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[3,3,1]
=> [3,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 0
[3,2,2]
=> [3,3,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 0
[3,2,1,1]
=> [4,2,1]
=> [2,1]
=> [[1,3],[2]]
=> 0
[3,1,1,1,1]
=> [5,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[2,2,2,1]
=> [4,3]
=> [3]
=> [[1,2,3]]
=> 0
[2,2,1,1,1]
=> [5,2]
=> [2]
=> [[1,2]]
=> 0
[2,1,1,1,1,1]
=> [6,1]
=> [1]
=> [[1]]
=> 0
[1,1,1,1,1,1,1]
=> [7]
=> []
=> []
=> ? = 1
[8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> 0
[7,1]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> 0
[6,2]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> 0
[6,1,1]
=> [3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 0
[5,3]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> 0
[5,2,1]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 0
[5,1,1,1]
=> [4,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[4,4]
=> [2,2,2,2]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> 0
[4,3,1]
=> [3,2,2,1]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 0
[4,2,2]
=> [3,3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 0
[4,2,1,1]
=> [4,2,1,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 0
[4,1,1,1,1]
=> [5,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[3,3,2]
=> [3,3,2]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 0
[1,1,1,1,1,1,1,1]
=> [8]
=> []
=> []
=> ? = 1
[1,1,1,1,1,1,1,1,1]
=> [9]
=> []
=> []
=> ? = 1
[10]
=> [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]
=> [10]
=> []
=> []
=> ? = 1
[11]
=> [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
[10,1]
=> [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
[9,2]
=> [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
[8,3]
=> [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
[7,4]
=> [2,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [[1,5],[2,7],[3,9],[4],[6],[8]]
=> ? = 0
[6,5]
=> [2,2,2,2,2,1]
=> [2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8]]
=> ? = 0
[1,1,1,1,1,1,1,1,1,1,1]
=> [11]
=> []
=> []
=> ? = 1
[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],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 0
[11,1]
=> [2,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? = 0
[10,2]
=> [2,2,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1]
=> [[1,10],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 0
[10,1,1]
=> [3,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 0
[9,3]
=> [2,2,2,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1]
=> [[1,8],[2,10],[3],[4],[5],[6],[7],[9]]
=> ? = 0
[9,2,1]
=> [3,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [[1,9],[2],[3],[4],[5],[6],[7],[8]]
=> ? = 0
[8,4]
=> [2,2,2,2,1,1,1,1]
=> [2,2,2,1,1,1,1]
=> [[1,6],[2,8],[3,10],[4],[5],[7],[9]]
=> ? = 0
[8,3,1]
=> [3,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [[1,7],[2,9],[3],[4],[5],[6],[8]]
=> ? = 0
[8,2,2]
=> [3,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> [[1,8,9],[2],[3],[4],[5],[6],[7]]
=> ? = 0
[7,5]
=> [2,2,2,2,2,1,1]
=> [2,2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5,10],[7],[9]]
=> ? = 0
[7,4,1]
=> [3,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [[1,5],[2,7],[3,9],[4],[6],[8]]
=> ? = 0
[7,3,2]
=> [3,3,2,1,1,1,1]
=> [3,2,1,1,1,1]
=> [[1,6,9],[2,8],[3],[4],[5],[7]]
=> ? = 0
[6,6]
=> [2,2,2,2,2,2]
=> [2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> ? = 0
[6,5,1]
=> [3,2,2,2,2,1]
=> [2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8]]
=> ? = 0
[6,4,2]
=> [3,3,2,2,1,1]
=> [3,2,2,1,1]
=> [[1,4,9],[2,6],[3,8],[5],[7]]
=> ? = 0
[6,3,3]
=> [3,3,3,1,1,1]
=> [3,3,1,1,1]
=> [[1,5,6],[2,8,9],[3],[4],[7]]
=> ? = 0
[5,5,2]
=> [3,3,2,2,2]
=> [3,2,2,2]
=> [[1,2,9],[3,4],[5,6],[7,8]]
=> ? = 0
[5,4,3]
=> [3,3,3,2,1]
=> [3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8],[7]]
=> ? = 0
[4,4,4]
=> [3,3,3,3]
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> ? = 0
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [12]
=> []
=> []
=> ? = 1
Description
The major index of a standard tableau minus the weighted size of its shape.
The following 45 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. St001139The number of occurrences of hills of size 2 in a Dyck path. St001256Number of simple reflexive modules that are 2-stable reflexive. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000993The multiplicity of the largest part of an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St000658The number of rises of length 2 of a Dyck path. St000264The girth of a graph, which is not a tree. St000297The number of leading ones in a binary word. St000546The number of global descents of a permutation. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St000260The radius of a connected graph. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St000759The smallest missing part in an integer partition. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001498The normalised height of a Nakayama algebra with magnitude 1. 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 $eA$. St000214The number of adjacencies of a permutation. St001141The number of occurrences of hills of size 3 in a Dyck path. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St000990The first ascent of a permutation. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{nā1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St000843The decomposition number of a perfect matching. St001049The smallest label in the subtree not containing 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St000153The number of adjacent cycles of a permutation. St001568The smallest positive integer that does not appear twice in the partition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001271The competition number of a graph. St000234The number of global ascents of a permutation. St000699The toughness times the least common multiple of 1,. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001831The multiplicity of the non-nesting perfect matching in the chord expansion of a perfect matching. St000056The decomposition (or block) number of a permutation. St000455The second largest eigenvalue of a graph if it is integral.
Sorry, this statistic was not found in the database
or
add this statistic to the database ā it's very simple and we need your support!