Your data matches 14 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000713
St000713: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[2]
=> 10
[1,1]
=> 5
[3]
=> 20
[2,1]
=> 16
[1,1,1]
=> 0
[4]
=> 35
[3,1]
=> 35
[2,2]
=> 14
[2,1,1]
=> 0
[1,1,1,1]
=> 0
[5]
=> 56
[4,1]
=> 64
[3,2]
=> 40
[3,1,1]
=> 0
[2,2,1]
=> 0
[2,1,1,1]
=> 0
[1,1,1,1,1]
=> 0
[6]
=> 84
[5,1]
=> 105
[4,2]
=> 81
[4,1,1]
=> 0
[3,3]
=> 30
[3,2,1]
=> 0
[3,1,1,1]
=> 0
[2,2,2]
=> 0
[2,2,1,1]
=> 0
[2,1,1,1,1]
=> 0
[1,1,1,1,1,1]
=> 0
[7]
=> 120
[6,1]
=> 160
[5,2]
=> 140
[5,1,1]
=> 0
[4,3]
=> 80
[4,2,1]
=> 0
[4,1,1,1]
=> 0
[3,3,1]
=> 0
[3,2,2]
=> 0
[3,2,1,1]
=> 0
[3,1,1,1,1]
=> 0
[2,2,2,1]
=> 0
[2,2,1,1,1]
=> 0
[2,1,1,1,1,1]
=> 0
[1,1,1,1,1,1,1]
=> 0
[8]
=> 165
[7,1]
=> 231
[6,2]
=> 220
[6,1,1]
=> 0
[5,3]
=> 154
[5,2,1]
=> 0
[5,1,1,1]
=> 0
Description
The dimension of the irreducible representation of Sp(4) labelled by an integer partition. Consider the symplectic group $Sp(2n)$. Then the integer partition $(\mu_1,\dots,\mu_k)$ of length at most $n$ corresponds to the weight vector $(\mu_1-\mu_2,\dots,\mu_{k-2}-\mu_{k-1},\mu_n,0,\dots,0)$. For example, the integer partition $(2)$ labels the symmetric square of the vector representation, whereas the integer partition $(1,1)$ labels the second fundamental representation.
Matching statistic: St000296
Mp00202: Integer partitions first row removalInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
Mp00095: Integer partitions to binary wordBinary words
St000296: Binary words ⟶ ℤResult quality: 2% values known / values provided: 83%distinct values known / distinct values provided: 2%
Values
[2]
=> []
=> ?
=> ? => ? ∊ {5,10}
[1,1]
=> [1]
=> []
=> => ? ∊ {5,10}
[3]
=> []
=> ?
=> ? => ? ∊ {16,20}
[2,1]
=> [1]
=> []
=> => ? ∊ {16,20}
[1,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[4]
=> []
=> ?
=> ? => ? ∊ {14,35,35}
[3,1]
=> [1]
=> []
=> => ? ∊ {14,35,35}
[2,2]
=> [2]
=> []
=> => ? ∊ {14,35,35}
[2,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 0
[5]
=> []
=> ?
=> ? => ? ∊ {40,56,64}
[4,1]
=> [1]
=> []
=> => ? ∊ {40,56,64}
[3,2]
=> [2]
=> []
=> => ? ∊ {40,56,64}
[3,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[2,2,1]
=> [2,1]
=> [1]
=> 10 => 0
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1110 => 0
[6]
=> []
=> ?
=> ? => ? ∊ {30,81,84,105}
[5,1]
=> [1]
=> []
=> => ? ∊ {30,81,84,105}
[4,2]
=> [2]
=> []
=> => ? ∊ {30,81,84,105}
[4,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[3,3]
=> [3]
=> []
=> => ? ∊ {30,81,84,105}
[3,2,1]
=> [2,1]
=> [1]
=> 10 => 0
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 0
[2,2,2]
=> [2,2]
=> [2]
=> 100 => 0
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 110 => 0
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1110 => 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 0
[7]
=> []
=> ?
=> ? => ? ∊ {80,120,140,160}
[6,1]
=> [1]
=> []
=> => ? ∊ {80,120,140,160}
[5,2]
=> [2]
=> []
=> => ? ∊ {80,120,140,160}
[5,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[4,3]
=> [3]
=> []
=> => ? ∊ {80,120,140,160}
[4,2,1]
=> [2,1]
=> [1]
=> 10 => 0
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 0
[3,3,1]
=> [3,1]
=> [1]
=> 10 => 0
[3,2,2]
=> [2,2]
=> [2]
=> 100 => 0
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 110 => 0
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1110 => 0
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1010 => 0
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1110 => 0
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 0
[8]
=> []
=> ?
=> ? => ? ∊ {55,154,165,220,231}
[7,1]
=> [1]
=> []
=> => ? ∊ {55,154,165,220,231}
[6,2]
=> [2]
=> []
=> => ? ∊ {55,154,165,220,231}
[6,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[5,3]
=> [3]
=> []
=> => ? ∊ {55,154,165,220,231}
[5,2,1]
=> [2,1]
=> [1]
=> 10 => 0
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 0
[4,4]
=> [4]
=> []
=> => ? ∊ {55,154,165,220,231}
[4,3,1]
=> [3,1]
=> [1]
=> 10 => 0
[4,2,2]
=> [2,2]
=> [2]
=> 100 => 0
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> 110 => 0
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1110 => 0
[3,3,2]
=> [3,2]
=> [2]
=> 100 => 0
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> 110 => 0
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1010 => 0
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1110 => 0
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 0
[2,2,2,2]
=> [2,2,2]
=> [2,2]
=> 1100 => 0
[2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> 10110 => 0
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 0
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 0
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 1111110 => 0
[9]
=> []
=> ?
=> ? => ? ∊ {140,220,256,320,324}
[8,1]
=> [1]
=> []
=> => ? ∊ {140,220,256,320,324}
[7,2]
=> [2]
=> []
=> => ? ∊ {140,220,256,320,324}
[7,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[6,3]
=> [3]
=> []
=> => ? ∊ {140,220,256,320,324}
[6,2,1]
=> [2,1]
=> [1]
=> 10 => 0
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 0
[5,4]
=> [4]
=> []
=> => ? ∊ {140,220,256,320,324}
[5,3,1]
=> [3,1]
=> [1]
=> 10 => 0
[5,2,2]
=> [2,2]
=> [2]
=> 100 => 0
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> 110 => 0
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1110 => 0
[4,4,1]
=> [4,1]
=> [1]
=> 10 => 0
[10]
=> []
=> ?
=> ? => ? ∊ {91,260,286,390,429,455}
[9,1]
=> [1]
=> []
=> => ? ∊ {91,260,286,390,429,455}
[8,2]
=> [2]
=> []
=> => ? ∊ {91,260,286,390,429,455}
[7,3]
=> [3]
=> []
=> => ? ∊ {91,260,286,390,429,455}
[6,4]
=> [4]
=> []
=> => ? ∊ {91,260,286,390,429,455}
[5,5]
=> [5]
=> []
=> => ? ∊ {91,260,286,390,429,455}
[11]
=> []
=> ?
=> ? => ? ∊ {224,364,420,560,560,616}
[10,1]
=> [1]
=> []
=> => ? ∊ {224,364,420,560,560,616}
[9,2]
=> [2]
=> []
=> => ? ∊ {224,364,420,560,560,616}
[8,3]
=> [3]
=> []
=> => ? ∊ {224,364,420,560,560,616}
[7,4]
=> [4]
=> []
=> => ? ∊ {224,364,420,560,560,616}
[6,5]
=> [5]
=> []
=> => ? ∊ {224,364,420,560,560,616}
[12]
=> []
=> ?
=> ? => ? ∊ {140,405,455,625,715,770,810}
[11,1]
=> [1]
=> []
=> => ? ∊ {140,405,455,625,715,770,810}
[10,2]
=> [2]
=> []
=> => ? ∊ {140,405,455,625,715,770,810}
[9,3]
=> [3]
=> []
=> => ? ∊ {140,405,455,625,715,770,810}
[8,4]
=> [4]
=> []
=> => ? ∊ {140,405,455,625,715,770,810}
[7,5]
=> [5]
=> []
=> => ? ∊ {140,405,455,625,715,770,810}
[6,6]
=> [6]
=> []
=> => ? ∊ {140,405,455,625,715,770,810}
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]].
Mp00202: Integer partitions first row removalInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
Mp00095: Integer partitions to binary wordBinary words
St000629: Binary words ⟶ ℤResult quality: 2% values known / values provided: 83%distinct values known / distinct values provided: 2%
Values
[2]
=> []
=> ?
=> ? => ? ∊ {5,10}
[1,1]
=> [1]
=> []
=> => ? ∊ {5,10}
[3]
=> []
=> ?
=> ? => ? ∊ {16,20}
[2,1]
=> [1]
=> []
=> => ? ∊ {16,20}
[1,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[4]
=> []
=> ?
=> ? => ? ∊ {14,35,35}
[3,1]
=> [1]
=> []
=> => ? ∊ {14,35,35}
[2,2]
=> [2]
=> []
=> => ? ∊ {14,35,35}
[2,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 0
[5]
=> []
=> ?
=> ? => ? ∊ {40,56,64}
[4,1]
=> [1]
=> []
=> => ? ∊ {40,56,64}
[3,2]
=> [2]
=> []
=> => ? ∊ {40,56,64}
[3,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[2,2,1]
=> [2,1]
=> [1]
=> 10 => 0
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1110 => 0
[6]
=> []
=> ?
=> ? => ? ∊ {30,81,84,105}
[5,1]
=> [1]
=> []
=> => ? ∊ {30,81,84,105}
[4,2]
=> [2]
=> []
=> => ? ∊ {30,81,84,105}
[4,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[3,3]
=> [3]
=> []
=> => ? ∊ {30,81,84,105}
[3,2,1]
=> [2,1]
=> [1]
=> 10 => 0
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 0
[2,2,2]
=> [2,2]
=> [2]
=> 100 => 0
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 110 => 0
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1110 => 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 0
[7]
=> []
=> ?
=> ? => ? ∊ {80,120,140,160}
[6,1]
=> [1]
=> []
=> => ? ∊ {80,120,140,160}
[5,2]
=> [2]
=> []
=> => ? ∊ {80,120,140,160}
[5,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[4,3]
=> [3]
=> []
=> => ? ∊ {80,120,140,160}
[4,2,1]
=> [2,1]
=> [1]
=> 10 => 0
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 0
[3,3,1]
=> [3,1]
=> [1]
=> 10 => 0
[3,2,2]
=> [2,2]
=> [2]
=> 100 => 0
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 110 => 0
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1110 => 0
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1010 => 0
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1110 => 0
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 0
[8]
=> []
=> ?
=> ? => ? ∊ {55,154,165,220,231}
[7,1]
=> [1]
=> []
=> => ? ∊ {55,154,165,220,231}
[6,2]
=> [2]
=> []
=> => ? ∊ {55,154,165,220,231}
[6,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[5,3]
=> [3]
=> []
=> => ? ∊ {55,154,165,220,231}
[5,2,1]
=> [2,1]
=> [1]
=> 10 => 0
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 0
[4,4]
=> [4]
=> []
=> => ? ∊ {55,154,165,220,231}
[4,3,1]
=> [3,1]
=> [1]
=> 10 => 0
[4,2,2]
=> [2,2]
=> [2]
=> 100 => 0
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> 110 => 0
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1110 => 0
[3,3,2]
=> [3,2]
=> [2]
=> 100 => 0
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> 110 => 0
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1010 => 0
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1110 => 0
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 0
[2,2,2,2]
=> [2,2,2]
=> [2,2]
=> 1100 => 0
[2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> 10110 => 0
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 0
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 0
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 1111110 => 0
[9]
=> []
=> ?
=> ? => ? ∊ {140,220,256,320,324}
[8,1]
=> [1]
=> []
=> => ? ∊ {140,220,256,320,324}
[7,2]
=> [2]
=> []
=> => ? ∊ {140,220,256,320,324}
[7,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[6,3]
=> [3]
=> []
=> => ? ∊ {140,220,256,320,324}
[6,2,1]
=> [2,1]
=> [1]
=> 10 => 0
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 0
[5,4]
=> [4]
=> []
=> => ? ∊ {140,220,256,320,324}
[5,3,1]
=> [3,1]
=> [1]
=> 10 => 0
[5,2,2]
=> [2,2]
=> [2]
=> 100 => 0
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> 110 => 0
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1110 => 0
[4,4,1]
=> [4,1]
=> [1]
=> 10 => 0
[10]
=> []
=> ?
=> ? => ? ∊ {91,260,286,390,429,455}
[9,1]
=> [1]
=> []
=> => ? ∊ {91,260,286,390,429,455}
[8,2]
=> [2]
=> []
=> => ? ∊ {91,260,286,390,429,455}
[7,3]
=> [3]
=> []
=> => ? ∊ {91,260,286,390,429,455}
[6,4]
=> [4]
=> []
=> => ? ∊ {91,260,286,390,429,455}
[5,5]
=> [5]
=> []
=> => ? ∊ {91,260,286,390,429,455}
[11]
=> []
=> ?
=> ? => ? ∊ {224,364,420,560,560,616}
[10,1]
=> [1]
=> []
=> => ? ∊ {224,364,420,560,560,616}
[9,2]
=> [2]
=> []
=> => ? ∊ {224,364,420,560,560,616}
[8,3]
=> [3]
=> []
=> => ? ∊ {224,364,420,560,560,616}
[7,4]
=> [4]
=> []
=> => ? ∊ {224,364,420,560,560,616}
[6,5]
=> [5]
=> []
=> => ? ∊ {224,364,420,560,560,616}
[12]
=> []
=> ?
=> ? => ? ∊ {140,405,455,625,715,770,810}
[11,1]
=> [1]
=> []
=> => ? ∊ {140,405,455,625,715,770,810}
[10,2]
=> [2]
=> []
=> => ? ∊ {140,405,455,625,715,770,810}
[9,3]
=> [3]
=> []
=> => ? ∊ {140,405,455,625,715,770,810}
[8,4]
=> [4]
=> []
=> => ? ∊ {140,405,455,625,715,770,810}
[7,5]
=> [5]
=> []
=> => ? ∊ {140,405,455,625,715,770,810}
[6,6]
=> [6]
=> []
=> => ? ∊ {140,405,455,625,715,770,810}
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: St000921
Mp00202: Integer partitions first row removalInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
Mp00317: Integer partitions odd partsBinary words
St000921: Binary words ⟶ ℤResult quality: 2% values known / values provided: 82%distinct values known / distinct values provided: 2%
Values
[2]
=> []
=> ?
=> ? => ? ∊ {5,10}
[1,1]
=> [1]
=> []
=> ? => ? ∊ {5,10}
[3]
=> []
=> ?
=> ? => ? ∊ {16,20}
[2,1]
=> [1]
=> []
=> ? => ? ∊ {16,20}
[1,1,1]
=> [1,1]
=> [1]
=> 1 => 0
[4]
=> []
=> ?
=> ? => ? ∊ {14,35,35}
[3,1]
=> [1]
=> []
=> ? => ? ∊ {14,35,35}
[2,2]
=> [2]
=> []
=> ? => ? ∊ {14,35,35}
[2,1,1]
=> [1,1]
=> [1]
=> 1 => 0
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 11 => 0
[5]
=> []
=> ?
=> ? => ? ∊ {40,56,64}
[4,1]
=> [1]
=> []
=> ? => ? ∊ {40,56,64}
[3,2]
=> [2]
=> []
=> ? => ? ∊ {40,56,64}
[3,1,1]
=> [1,1]
=> [1]
=> 1 => 0
[2,2,1]
=> [2,1]
=> [1]
=> 1 => 0
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 11 => 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 111 => 0
[6]
=> []
=> ?
=> ? => ? ∊ {30,81,84,105}
[5,1]
=> [1]
=> []
=> ? => ? ∊ {30,81,84,105}
[4,2]
=> [2]
=> []
=> ? => ? ∊ {30,81,84,105}
[4,1,1]
=> [1,1]
=> [1]
=> 1 => 0
[3,3]
=> [3]
=> []
=> ? => ? ∊ {30,81,84,105}
[3,2,1]
=> [2,1]
=> [1]
=> 1 => 0
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 11 => 0
[2,2,2]
=> [2,2]
=> [2]
=> 0 => 0
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 11 => 0
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 111 => 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1111 => 0
[7]
=> []
=> ?
=> ? => ? ∊ {80,120,140,160}
[6,1]
=> [1]
=> []
=> ? => ? ∊ {80,120,140,160}
[5,2]
=> [2]
=> []
=> ? => ? ∊ {80,120,140,160}
[5,1,1]
=> [1,1]
=> [1]
=> 1 => 0
[4,3]
=> [3]
=> []
=> ? => ? ∊ {80,120,140,160}
[4,2,1]
=> [2,1]
=> [1]
=> 1 => 0
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 11 => 0
[3,3,1]
=> [3,1]
=> [1]
=> 1 => 0
[3,2,2]
=> [2,2]
=> [2]
=> 0 => 0
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 11 => 0
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 111 => 0
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 01 => 0
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 111 => 0
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1111 => 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 11111 => 0
[8]
=> []
=> ?
=> ? => ? ∊ {55,154,165,220,231}
[7,1]
=> [1]
=> []
=> ? => ? ∊ {55,154,165,220,231}
[6,2]
=> [2]
=> []
=> ? => ? ∊ {55,154,165,220,231}
[6,1,1]
=> [1,1]
=> [1]
=> 1 => 0
[5,3]
=> [3]
=> []
=> ? => ? ∊ {55,154,165,220,231}
[5,2,1]
=> [2,1]
=> [1]
=> 1 => 0
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 11 => 0
[4,4]
=> [4]
=> []
=> ? => ? ∊ {55,154,165,220,231}
[4,3,1]
=> [3,1]
=> [1]
=> 1 => 0
[4,2,2]
=> [2,2]
=> [2]
=> 0 => 0
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> 11 => 0
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 111 => 0
[3,3,2]
=> [3,2]
=> [2]
=> 0 => 0
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> 11 => 0
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 01 => 0
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 111 => 0
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1111 => 0
[2,2,2,2]
=> [2,2,2]
=> [2,2]
=> 00 => 0
[2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> 011 => 0
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 1111 => 0
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 11111 => 0
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 111111 => 0
[9]
=> []
=> ?
=> ? => ? ∊ {140,220,256,320,324}
[8,1]
=> [1]
=> []
=> ? => ? ∊ {140,220,256,320,324}
[7,2]
=> [2]
=> []
=> ? => ? ∊ {140,220,256,320,324}
[7,1,1]
=> [1,1]
=> [1]
=> 1 => 0
[6,3]
=> [3]
=> []
=> ? => ? ∊ {140,220,256,320,324}
[6,2,1]
=> [2,1]
=> [1]
=> 1 => 0
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 11 => 0
[5,4]
=> [4]
=> []
=> ? => ? ∊ {140,220,256,320,324}
[5,3,1]
=> [3,1]
=> [1]
=> 1 => 0
[5,2,2]
=> [2,2]
=> [2]
=> 0 => 0
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> 11 => 0
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 111 => 0
[4,4,1]
=> [4,1]
=> [1]
=> 1 => 0
[10]
=> []
=> ?
=> ? => ? ∊ {91,260,286,390,429,455}
[9,1]
=> [1]
=> []
=> ? => ? ∊ {91,260,286,390,429,455}
[8,2]
=> [2]
=> []
=> ? => ? ∊ {91,260,286,390,429,455}
[7,3]
=> [3]
=> []
=> ? => ? ∊ {91,260,286,390,429,455}
[6,4]
=> [4]
=> []
=> ? => ? ∊ {91,260,286,390,429,455}
[5,5]
=> [5]
=> []
=> ? => ? ∊ {91,260,286,390,429,455}
[11]
=> []
=> ?
=> ? => ? ∊ {224,364,420,560,560,616}
[10,1]
=> [1]
=> []
=> ? => ? ∊ {224,364,420,560,560,616}
[9,2]
=> [2]
=> []
=> ? => ? ∊ {224,364,420,560,560,616}
[8,3]
=> [3]
=> []
=> ? => ? ∊ {224,364,420,560,560,616}
[7,4]
=> [4]
=> []
=> ? => ? ∊ {224,364,420,560,560,616}
[6,5]
=> [5]
=> []
=> ? => ? ∊ {224,364,420,560,560,616}
[12]
=> []
=> ?
=> ? => ? ∊ {0,140,405,455,625,715,770,810}
[11,1]
=> [1]
=> []
=> ? => ? ∊ {0,140,405,455,625,715,770,810}
[10,2]
=> [2]
=> []
=> ? => ? ∊ {0,140,405,455,625,715,770,810}
[9,3]
=> [3]
=> []
=> ? => ? ∊ {0,140,405,455,625,715,770,810}
[8,4]
=> [4]
=> []
=> ? => ? ∊ {0,140,405,455,625,715,770,810}
[7,5]
=> [5]
=> []
=> ? => ? ∊ {0,140,405,455,625,715,770,810}
[6,6]
=> [6]
=> []
=> ? => ? ∊ {0,140,405,455,625,715,770,810}
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> 1111111111 => ? ∊ {0,140,405,455,625,715,770,810}
Description
The number of internal inversions of a binary word. Let $\bar w$ be the non-decreasing rearrangement of $w$, that is, $\bar w$ is sorted. An internal inversion is a pair $i < j$ such that $w_i > w_j$ and $\bar w_i = \bar w_j$. For example, the word $110$ has two inversions, but only the second is internal.
Matching statistic: St001371
Mp00202: Integer partitions first row removalInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
Mp00095: Integer partitions to binary wordBinary words
St001371: Binary words ⟶ ℤResult quality: 2% values known / values provided: 81%distinct values known / distinct values provided: 2%
Values
[2]
=> []
=> ?
=> ? => ? ∊ {5,10}
[1,1]
=> [1]
=> []
=> => ? ∊ {5,10}
[3]
=> []
=> ?
=> ? => ? ∊ {16,20}
[2,1]
=> [1]
=> []
=> => ? ∊ {16,20}
[1,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[4]
=> []
=> ?
=> ? => ? ∊ {14,35,35}
[3,1]
=> [1]
=> []
=> => ? ∊ {14,35,35}
[2,2]
=> [2]
=> []
=> => ? ∊ {14,35,35}
[2,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 0
[5]
=> []
=> ?
=> ? => ? ∊ {40,56,64}
[4,1]
=> [1]
=> []
=> => ? ∊ {40,56,64}
[3,2]
=> [2]
=> []
=> => ? ∊ {40,56,64}
[3,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[2,2,1]
=> [2,1]
=> [1]
=> 10 => 0
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1110 => 0
[6]
=> []
=> ?
=> ? => ? ∊ {30,81,84,105}
[5,1]
=> [1]
=> []
=> => ? ∊ {30,81,84,105}
[4,2]
=> [2]
=> []
=> => ? ∊ {30,81,84,105}
[4,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[3,3]
=> [3]
=> []
=> => ? ∊ {30,81,84,105}
[3,2,1]
=> [2,1]
=> [1]
=> 10 => 0
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 0
[2,2,2]
=> [2,2]
=> [2]
=> 100 => 0
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 110 => 0
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1110 => 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 0
[7]
=> []
=> ?
=> ? => ? ∊ {80,120,140,160}
[6,1]
=> [1]
=> []
=> => ? ∊ {80,120,140,160}
[5,2]
=> [2]
=> []
=> => ? ∊ {80,120,140,160}
[5,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[4,3]
=> [3]
=> []
=> => ? ∊ {80,120,140,160}
[4,2,1]
=> [2,1]
=> [1]
=> 10 => 0
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 0
[3,3,1]
=> [3,1]
=> [1]
=> 10 => 0
[3,2,2]
=> [2,2]
=> [2]
=> 100 => 0
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 110 => 0
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1110 => 0
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1010 => 0
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1110 => 0
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 0
[8]
=> []
=> ?
=> ? => ? ∊ {55,154,165,220,231}
[7,1]
=> [1]
=> []
=> => ? ∊ {55,154,165,220,231}
[6,2]
=> [2]
=> []
=> => ? ∊ {55,154,165,220,231}
[6,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[5,3]
=> [3]
=> []
=> => ? ∊ {55,154,165,220,231}
[5,2,1]
=> [2,1]
=> [1]
=> 10 => 0
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 0
[4,4]
=> [4]
=> []
=> => ? ∊ {55,154,165,220,231}
[4,3,1]
=> [3,1]
=> [1]
=> 10 => 0
[4,2,2]
=> [2,2]
=> [2]
=> 100 => 0
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> 110 => 0
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1110 => 0
[3,3,2]
=> [3,2]
=> [2]
=> 100 => 0
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> 110 => 0
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1010 => 0
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1110 => 0
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 0
[2,2,2,2]
=> [2,2,2]
=> [2,2]
=> 1100 => 0
[2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> 10110 => 0
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 0
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 0
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 1111110 => 0
[9]
=> []
=> ?
=> ? => ? ∊ {140,220,256,320,324}
[8,1]
=> [1]
=> []
=> => ? ∊ {140,220,256,320,324}
[7,2]
=> [2]
=> []
=> => ? ∊ {140,220,256,320,324}
[7,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[6,3]
=> [3]
=> []
=> => ? ∊ {140,220,256,320,324}
[6,2,1]
=> [2,1]
=> [1]
=> 10 => 0
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 0
[5,4]
=> [4]
=> []
=> => ? ∊ {140,220,256,320,324}
[5,3,1]
=> [3,1]
=> [1]
=> 10 => 0
[5,2,2]
=> [2,2]
=> [2]
=> 100 => 0
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> 110 => 0
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1110 => 0
[4,4,1]
=> [4,1]
=> [1]
=> 10 => 0
[10]
=> []
=> ?
=> ? => ? ∊ {91,260,286,390,429,455}
[9,1]
=> [1]
=> []
=> => ? ∊ {91,260,286,390,429,455}
[8,2]
=> [2]
=> []
=> => ? ∊ {91,260,286,390,429,455}
[7,3]
=> [3]
=> []
=> => ? ∊ {91,260,286,390,429,455}
[6,4]
=> [4]
=> []
=> => ? ∊ {91,260,286,390,429,455}
[5,5]
=> [5]
=> []
=> => ? ∊ {91,260,286,390,429,455}
[11]
=> []
=> ?
=> ? => ? ∊ {0,224,364,420,560,560,616}
[10,1]
=> [1]
=> []
=> => ? ∊ {0,224,364,420,560,560,616}
[9,2]
=> [2]
=> []
=> => ? ∊ {0,224,364,420,560,560,616}
[8,3]
=> [3]
=> []
=> => ? ∊ {0,224,364,420,560,560,616}
[7,4]
=> [4]
=> []
=> => ? ∊ {0,224,364,420,560,560,616}
[6,5]
=> [5]
=> []
=> => ? ∊ {0,224,364,420,560,560,616}
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> 1111111110 => ? ∊ {0,224,364,420,560,560,616}
[12]
=> []
=> ?
=> ? => ? ∊ {0,0,140,405,455,625,715,770,810}
[11,1]
=> [1]
=> []
=> => ? ∊ {0,0,140,405,455,625,715,770,810}
[10,2]
=> [2]
=> []
=> => ? ∊ {0,0,140,405,455,625,715,770,810}
[9,3]
=> [3]
=> []
=> => ? ∊ {0,0,140,405,455,625,715,770,810}
[8,4]
=> [4]
=> []
=> => ? ∊ {0,0,140,405,455,625,715,770,810}
[7,5]
=> [5]
=> []
=> => ? ∊ {0,0,140,405,455,625,715,770,810}
[6,6]
=> [6]
=> []
=> => ? ∊ {0,0,140,405,455,625,715,770,810}
[2,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> 1111111110 => ? ∊ {0,0,140,405,455,625,715,770,810}
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> 11111111110 => ? ∊ {0,0,140,405,455,625,715,770,810}
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: St001695
Mp00202: Integer partitions first row removalInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
Mp00042: Integer partitions initial tableauStandard tableaux
St001695: Standard tableaux ⟶ ℤResult quality: 2% values known / values provided: 81%distinct values known / distinct values provided: 2%
Values
[2]
=> []
=> ?
=> ?
=> ? ∊ {5,10}
[1,1]
=> [1]
=> []
=> []
=> ? ∊ {5,10}
[3]
=> []
=> ?
=> ?
=> ? ∊ {16,20}
[2,1]
=> [1]
=> []
=> []
=> ? ∊ {16,20}
[1,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[4]
=> []
=> ?
=> ?
=> ? ∊ {14,35,35}
[3,1]
=> [1]
=> []
=> []
=> ? ∊ {14,35,35}
[2,2]
=> [2]
=> []
=> []
=> ? ∊ {14,35,35}
[2,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[5]
=> []
=> ?
=> ?
=> ? ∊ {40,56,64}
[4,1]
=> [1]
=> []
=> []
=> ? ∊ {40,56,64}
[3,2]
=> [2]
=> []
=> []
=> ? ∊ {40,56,64}
[3,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[2,2,1]
=> [2,1]
=> [1]
=> [[1]]
=> 0
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[6]
=> []
=> ?
=> ?
=> ? ∊ {30,81,84,105}
[5,1]
=> [1]
=> []
=> []
=> ? ∊ {30,81,84,105}
[4,2]
=> [2]
=> []
=> []
=> ? ∊ {30,81,84,105}
[4,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[3,3]
=> [3]
=> []
=> []
=> ? ∊ {30,81,84,105}
[3,2,1]
=> [2,1]
=> [1]
=> [[1]]
=> 0
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[2,2,2]
=> [2,2]
=> [2]
=> [[1,2]]
=> 0
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[7]
=> []
=> ?
=> ?
=> ? ∊ {80,120,140,160}
[6,1]
=> [1]
=> []
=> []
=> ? ∊ {80,120,140,160}
[5,2]
=> [2]
=> []
=> []
=> ? ∊ {80,120,140,160}
[5,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[4,3]
=> [3]
=> []
=> []
=> ? ∊ {80,120,140,160}
[4,2,1]
=> [2,1]
=> [1]
=> [[1]]
=> 0
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[3,3,1]
=> [3,1]
=> [1]
=> [[1]]
=> 0
[3,2,2]
=> [2,2]
=> [2]
=> [[1,2]]
=> 0
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [[1,2],[3]]
=> 0
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 0
[8]
=> []
=> ?
=> ?
=> ? ∊ {55,154,165,220,231}
[7,1]
=> [1]
=> []
=> []
=> ? ∊ {55,154,165,220,231}
[6,2]
=> [2]
=> []
=> []
=> ? ∊ {55,154,165,220,231}
[6,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[5,3]
=> [3]
=> []
=> []
=> ? ∊ {55,154,165,220,231}
[5,2,1]
=> [2,1]
=> [1]
=> [[1]]
=> 0
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[4,4]
=> [4]
=> []
=> []
=> ? ∊ {55,154,165,220,231}
[4,3,1]
=> [3,1]
=> [1]
=> [[1]]
=> 0
[4,2,2]
=> [2,2]
=> [2]
=> [[1,2]]
=> 0
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[3,3,2]
=> [3,2]
=> [2]
=> [[1,2]]
=> 0
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> [[1,2],[3]]
=> 0
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[2,2,2,2]
=> [2,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 0
[2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 0
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 0
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> 0
[9]
=> []
=> ?
=> ?
=> ? ∊ {140,220,256,320,324}
[8,1]
=> [1]
=> []
=> []
=> ? ∊ {140,220,256,320,324}
[7,2]
=> [2]
=> []
=> []
=> ? ∊ {140,220,256,320,324}
[7,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[6,3]
=> [3]
=> []
=> []
=> ? ∊ {140,220,256,320,324}
[6,2,1]
=> [2,1]
=> [1]
=> [[1]]
=> 0
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[5,4]
=> [4]
=> []
=> []
=> ? ∊ {140,220,256,320,324}
[5,3,1]
=> [3,1]
=> [1]
=> [[1]]
=> 0
[5,2,2]
=> [2,2]
=> [2]
=> [[1,2]]
=> 0
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[4,4,1]
=> [4,1]
=> [1]
=> [[1]]
=> 0
[10]
=> []
=> ?
=> ?
=> ? ∊ {91,260,286,390,429,455}
[9,1]
=> [1]
=> []
=> []
=> ? ∊ {91,260,286,390,429,455}
[8,2]
=> [2]
=> []
=> []
=> ? ∊ {91,260,286,390,429,455}
[7,3]
=> [3]
=> []
=> []
=> ? ∊ {91,260,286,390,429,455}
[6,4]
=> [4]
=> []
=> []
=> ? ∊ {91,260,286,390,429,455}
[5,5]
=> [5]
=> []
=> []
=> ? ∊ {91,260,286,390,429,455}
[11]
=> []
=> ?
=> ?
=> ? ∊ {0,224,364,420,560,560,616}
[10,1]
=> [1]
=> []
=> []
=> ? ∊ {0,224,364,420,560,560,616}
[9,2]
=> [2]
=> []
=> []
=> ? ∊ {0,224,364,420,560,560,616}
[8,3]
=> [3]
=> []
=> []
=> ? ∊ {0,224,364,420,560,560,616}
[7,4]
=> [4]
=> []
=> []
=> ? ∊ {0,224,364,420,560,560,616}
[6,5]
=> [5]
=> []
=> []
=> ? ∊ {0,224,364,420,560,560,616}
[1,1,1,1,1,1,1,1,1,1,1]
=> [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]]
=> ? ∊ {0,224,364,420,560,560,616}
[12]
=> []
=> ?
=> ?
=> ? ∊ {0,0,140,405,455,625,715,770,810}
[11,1]
=> [1]
=> []
=> []
=> ? ∊ {0,0,140,405,455,625,715,770,810}
[10,2]
=> [2]
=> []
=> []
=> ? ∊ {0,0,140,405,455,625,715,770,810}
[9,3]
=> [3]
=> []
=> []
=> ? ∊ {0,0,140,405,455,625,715,770,810}
[8,4]
=> [4]
=> []
=> []
=> ? ∊ {0,0,140,405,455,625,715,770,810}
[7,5]
=> [5]
=> []
=> []
=> ? ∊ {0,0,140,405,455,625,715,770,810}
[6,6]
=> [6]
=> []
=> []
=> ? ∊ {0,0,140,405,455,625,715,770,810}
[2,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? ∊ {0,0,140,405,455,625,715,770,810}
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? ∊ {0,0,140,405,455,625,715,770,810}
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 removalInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
Mp00042: Integer partitions initial tableauStandard tableaux
St001698: Standard tableaux ⟶ ℤResult quality: 2% values known / values provided: 81%distinct values known / distinct values provided: 2%
Values
[2]
=> []
=> ?
=> ?
=> ? ∊ {5,10}
[1,1]
=> [1]
=> []
=> []
=> ? ∊ {5,10}
[3]
=> []
=> ?
=> ?
=> ? ∊ {16,20}
[2,1]
=> [1]
=> []
=> []
=> ? ∊ {16,20}
[1,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[4]
=> []
=> ?
=> ?
=> ? ∊ {14,35,35}
[3,1]
=> [1]
=> []
=> []
=> ? ∊ {14,35,35}
[2,2]
=> [2]
=> []
=> []
=> ? ∊ {14,35,35}
[2,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[5]
=> []
=> ?
=> ?
=> ? ∊ {40,56,64}
[4,1]
=> [1]
=> []
=> []
=> ? ∊ {40,56,64}
[3,2]
=> [2]
=> []
=> []
=> ? ∊ {40,56,64}
[3,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[2,2,1]
=> [2,1]
=> [1]
=> [[1]]
=> 0
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[6]
=> []
=> ?
=> ?
=> ? ∊ {30,81,84,105}
[5,1]
=> [1]
=> []
=> []
=> ? ∊ {30,81,84,105}
[4,2]
=> [2]
=> []
=> []
=> ? ∊ {30,81,84,105}
[4,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[3,3]
=> [3]
=> []
=> []
=> ? ∊ {30,81,84,105}
[3,2,1]
=> [2,1]
=> [1]
=> [[1]]
=> 0
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[2,2,2]
=> [2,2]
=> [2]
=> [[1,2]]
=> 0
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[7]
=> []
=> ?
=> ?
=> ? ∊ {80,120,140,160}
[6,1]
=> [1]
=> []
=> []
=> ? ∊ {80,120,140,160}
[5,2]
=> [2]
=> []
=> []
=> ? ∊ {80,120,140,160}
[5,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[4,3]
=> [3]
=> []
=> []
=> ? ∊ {80,120,140,160}
[4,2,1]
=> [2,1]
=> [1]
=> [[1]]
=> 0
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[3,3,1]
=> [3,1]
=> [1]
=> [[1]]
=> 0
[3,2,2]
=> [2,2]
=> [2]
=> [[1,2]]
=> 0
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [[1,2],[3]]
=> 0
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 0
[8]
=> []
=> ?
=> ?
=> ? ∊ {55,154,165,220,231}
[7,1]
=> [1]
=> []
=> []
=> ? ∊ {55,154,165,220,231}
[6,2]
=> [2]
=> []
=> []
=> ? ∊ {55,154,165,220,231}
[6,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[5,3]
=> [3]
=> []
=> []
=> ? ∊ {55,154,165,220,231}
[5,2,1]
=> [2,1]
=> [1]
=> [[1]]
=> 0
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[4,4]
=> [4]
=> []
=> []
=> ? ∊ {55,154,165,220,231}
[4,3,1]
=> [3,1]
=> [1]
=> [[1]]
=> 0
[4,2,2]
=> [2,2]
=> [2]
=> [[1,2]]
=> 0
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[3,3,2]
=> [3,2]
=> [2]
=> [[1,2]]
=> 0
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> [[1,2],[3]]
=> 0
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[2,2,2,2]
=> [2,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 0
[2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 0
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 0
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> 0
[9]
=> []
=> ?
=> ?
=> ? ∊ {140,220,256,320,324}
[8,1]
=> [1]
=> []
=> []
=> ? ∊ {140,220,256,320,324}
[7,2]
=> [2]
=> []
=> []
=> ? ∊ {140,220,256,320,324}
[7,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[6,3]
=> [3]
=> []
=> []
=> ? ∊ {140,220,256,320,324}
[6,2,1]
=> [2,1]
=> [1]
=> [[1]]
=> 0
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[5,4]
=> [4]
=> []
=> []
=> ? ∊ {140,220,256,320,324}
[5,3,1]
=> [3,1]
=> [1]
=> [[1]]
=> 0
[5,2,2]
=> [2,2]
=> [2]
=> [[1,2]]
=> 0
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[4,4,1]
=> [4,1]
=> [1]
=> [[1]]
=> 0
[10]
=> []
=> ?
=> ?
=> ? ∊ {91,260,286,390,429,455}
[9,1]
=> [1]
=> []
=> []
=> ? ∊ {91,260,286,390,429,455}
[8,2]
=> [2]
=> []
=> []
=> ? ∊ {91,260,286,390,429,455}
[7,3]
=> [3]
=> []
=> []
=> ? ∊ {91,260,286,390,429,455}
[6,4]
=> [4]
=> []
=> []
=> ? ∊ {91,260,286,390,429,455}
[5,5]
=> [5]
=> []
=> []
=> ? ∊ {91,260,286,390,429,455}
[11]
=> []
=> ?
=> ?
=> ? ∊ {0,224,364,420,560,560,616}
[10,1]
=> [1]
=> []
=> []
=> ? ∊ {0,224,364,420,560,560,616}
[9,2]
=> [2]
=> []
=> []
=> ? ∊ {0,224,364,420,560,560,616}
[8,3]
=> [3]
=> []
=> []
=> ? ∊ {0,224,364,420,560,560,616}
[7,4]
=> [4]
=> []
=> []
=> ? ∊ {0,224,364,420,560,560,616}
[6,5]
=> [5]
=> []
=> []
=> ? ∊ {0,224,364,420,560,560,616}
[1,1,1,1,1,1,1,1,1,1,1]
=> [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]]
=> ? ∊ {0,224,364,420,560,560,616}
[12]
=> []
=> ?
=> ?
=> ? ∊ {0,0,140,405,455,625,715,770,810}
[11,1]
=> [1]
=> []
=> []
=> ? ∊ {0,0,140,405,455,625,715,770,810}
[10,2]
=> [2]
=> []
=> []
=> ? ∊ {0,0,140,405,455,625,715,770,810}
[9,3]
=> [3]
=> []
=> []
=> ? ∊ {0,0,140,405,455,625,715,770,810}
[8,4]
=> [4]
=> []
=> []
=> ? ∊ {0,0,140,405,455,625,715,770,810}
[7,5]
=> [5]
=> []
=> []
=> ? ∊ {0,0,140,405,455,625,715,770,810}
[6,6]
=> [6]
=> []
=> []
=> ? ∊ {0,0,140,405,455,625,715,770,810}
[2,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? ∊ {0,0,140,405,455,625,715,770,810}
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? ∊ {0,0,140,405,455,625,715,770,810}
Description
The comajor index of a standard tableau minus the weighted size of its shape.
Matching statistic: St001699
Mp00202: Integer partitions first row removalInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
Mp00045: Integer partitions reading tableauStandard tableaux
St001699: Standard tableaux ⟶ ℤResult quality: 2% values known / values provided: 81%distinct values known / distinct values provided: 2%
Values
[2]
=> []
=> ?
=> ?
=> ? ∊ {5,10}
[1,1]
=> [1]
=> []
=> []
=> ? ∊ {5,10}
[3]
=> []
=> ?
=> ?
=> ? ∊ {16,20}
[2,1]
=> [1]
=> []
=> []
=> ? ∊ {16,20}
[1,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[4]
=> []
=> ?
=> ?
=> ? ∊ {14,35,35}
[3,1]
=> [1]
=> []
=> []
=> ? ∊ {14,35,35}
[2,2]
=> [2]
=> []
=> []
=> ? ∊ {14,35,35}
[2,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[5]
=> []
=> ?
=> ?
=> ? ∊ {40,56,64}
[4,1]
=> [1]
=> []
=> []
=> ? ∊ {40,56,64}
[3,2]
=> [2]
=> []
=> []
=> ? ∊ {40,56,64}
[3,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[2,2,1]
=> [2,1]
=> [1]
=> [[1]]
=> 0
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[6]
=> []
=> ?
=> ?
=> ? ∊ {30,81,84,105}
[5,1]
=> [1]
=> []
=> []
=> ? ∊ {30,81,84,105}
[4,2]
=> [2]
=> []
=> []
=> ? ∊ {30,81,84,105}
[4,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[3,3]
=> [3]
=> []
=> []
=> ? ∊ {30,81,84,105}
[3,2,1]
=> [2,1]
=> [1]
=> [[1]]
=> 0
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[2,2,2]
=> [2,2]
=> [2]
=> [[1,2]]
=> 0
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[7]
=> []
=> ?
=> ?
=> ? ∊ {80,120,140,160}
[6,1]
=> [1]
=> []
=> []
=> ? ∊ {80,120,140,160}
[5,2]
=> [2]
=> []
=> []
=> ? ∊ {80,120,140,160}
[5,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[4,3]
=> [3]
=> []
=> []
=> ? ∊ {80,120,140,160}
[4,2,1]
=> [2,1]
=> [1]
=> [[1]]
=> 0
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[3,3,1]
=> [3,1]
=> [1]
=> [[1]]
=> 0
[3,2,2]
=> [2,2]
=> [2]
=> [[1,2]]
=> 0
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [[1,3],[2]]
=> 0
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 0
[8]
=> []
=> ?
=> ?
=> ? ∊ {55,154,165,220,231}
[7,1]
=> [1]
=> []
=> []
=> ? ∊ {55,154,165,220,231}
[6,2]
=> [2]
=> []
=> []
=> ? ∊ {55,154,165,220,231}
[6,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[5,3]
=> [3]
=> []
=> []
=> ? ∊ {55,154,165,220,231}
[5,2,1]
=> [2,1]
=> [1]
=> [[1]]
=> 0
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[4,4]
=> [4]
=> []
=> []
=> ? ∊ {55,154,165,220,231}
[4,3,1]
=> [3,1]
=> [1]
=> [[1]]
=> 0
[4,2,2]
=> [2,2]
=> [2]
=> [[1,2]]
=> 0
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[3,3,2]
=> [3,2]
=> [2]
=> [[1,2]]
=> 0
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> [[1,3],[2]]
=> 0
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[2,2,2,2]
=> [2,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 0
[2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 0
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 0
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> 0
[9]
=> []
=> ?
=> ?
=> ? ∊ {140,220,256,320,324}
[8,1]
=> [1]
=> []
=> []
=> ? ∊ {140,220,256,320,324}
[7,2]
=> [2]
=> []
=> []
=> ? ∊ {140,220,256,320,324}
[7,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[6,3]
=> [3]
=> []
=> []
=> ? ∊ {140,220,256,320,324}
[6,2,1]
=> [2,1]
=> [1]
=> [[1]]
=> 0
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[5,4]
=> [4]
=> []
=> []
=> ? ∊ {140,220,256,320,324}
[5,3,1]
=> [3,1]
=> [1]
=> [[1]]
=> 0
[5,2,2]
=> [2,2]
=> [2]
=> [[1,2]]
=> 0
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[4,4,1]
=> [4,1]
=> [1]
=> [[1]]
=> 0
[10]
=> []
=> ?
=> ?
=> ? ∊ {91,260,286,390,429,455}
[9,1]
=> [1]
=> []
=> []
=> ? ∊ {91,260,286,390,429,455}
[8,2]
=> [2]
=> []
=> []
=> ? ∊ {91,260,286,390,429,455}
[7,3]
=> [3]
=> []
=> []
=> ? ∊ {91,260,286,390,429,455}
[6,4]
=> [4]
=> []
=> []
=> ? ∊ {91,260,286,390,429,455}
[5,5]
=> [5]
=> []
=> []
=> ? ∊ {91,260,286,390,429,455}
[11]
=> []
=> ?
=> ?
=> ? ∊ {0,224,364,420,560,560,616}
[10,1]
=> [1]
=> []
=> []
=> ? ∊ {0,224,364,420,560,560,616}
[9,2]
=> [2]
=> []
=> []
=> ? ∊ {0,224,364,420,560,560,616}
[8,3]
=> [3]
=> []
=> []
=> ? ∊ {0,224,364,420,560,560,616}
[7,4]
=> [4]
=> []
=> []
=> ? ∊ {0,224,364,420,560,560,616}
[6,5]
=> [5]
=> []
=> []
=> ? ∊ {0,224,364,420,560,560,616}
[1,1,1,1,1,1,1,1,1,1,1]
=> [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]]
=> ? ∊ {0,224,364,420,560,560,616}
[12]
=> []
=> ?
=> ?
=> ? ∊ {0,0,140,405,455,625,715,770,810}
[11,1]
=> [1]
=> []
=> []
=> ? ∊ {0,0,140,405,455,625,715,770,810}
[10,2]
=> [2]
=> []
=> []
=> ? ∊ {0,0,140,405,455,625,715,770,810}
[9,3]
=> [3]
=> []
=> []
=> ? ∊ {0,0,140,405,455,625,715,770,810}
[8,4]
=> [4]
=> []
=> []
=> ? ∊ {0,0,140,405,455,625,715,770,810}
[7,5]
=> [5]
=> []
=> []
=> ? ∊ {0,0,140,405,455,625,715,770,810}
[6,6]
=> [6]
=> []
=> []
=> ? ∊ {0,0,140,405,455,625,715,770,810}
[2,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? ∊ {0,0,140,405,455,625,715,770,810}
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? ∊ {0,0,140,405,455,625,715,770,810}
Description
The major index of a standard tableau minus the weighted size of its shape.
Matching statistic: St001712
Mp00202: Integer partitions first row removalInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
Mp00042: Integer partitions initial tableauStandard tableaux
St001712: Standard tableaux ⟶ ℤResult quality: 2% values known / values provided: 81%distinct values known / distinct values provided: 2%
Values
[2]
=> []
=> ?
=> ?
=> ? ∊ {5,10}
[1,1]
=> [1]
=> []
=> []
=> ? ∊ {5,10}
[3]
=> []
=> ?
=> ?
=> ? ∊ {16,20}
[2,1]
=> [1]
=> []
=> []
=> ? ∊ {16,20}
[1,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[4]
=> []
=> ?
=> ?
=> ? ∊ {14,35,35}
[3,1]
=> [1]
=> []
=> []
=> ? ∊ {14,35,35}
[2,2]
=> [2]
=> []
=> []
=> ? ∊ {14,35,35}
[2,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[5]
=> []
=> ?
=> ?
=> ? ∊ {40,56,64}
[4,1]
=> [1]
=> []
=> []
=> ? ∊ {40,56,64}
[3,2]
=> [2]
=> []
=> []
=> ? ∊ {40,56,64}
[3,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[2,2,1]
=> [2,1]
=> [1]
=> [[1]]
=> 0
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[6]
=> []
=> ?
=> ?
=> ? ∊ {30,81,84,105}
[5,1]
=> [1]
=> []
=> []
=> ? ∊ {30,81,84,105}
[4,2]
=> [2]
=> []
=> []
=> ? ∊ {30,81,84,105}
[4,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[3,3]
=> [3]
=> []
=> []
=> ? ∊ {30,81,84,105}
[3,2,1]
=> [2,1]
=> [1]
=> [[1]]
=> 0
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[2,2,2]
=> [2,2]
=> [2]
=> [[1,2]]
=> 0
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[7]
=> []
=> ?
=> ?
=> ? ∊ {80,120,140,160}
[6,1]
=> [1]
=> []
=> []
=> ? ∊ {80,120,140,160}
[5,2]
=> [2]
=> []
=> []
=> ? ∊ {80,120,140,160}
[5,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[4,3]
=> [3]
=> []
=> []
=> ? ∊ {80,120,140,160}
[4,2,1]
=> [2,1]
=> [1]
=> [[1]]
=> 0
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[3,3,1]
=> [3,1]
=> [1]
=> [[1]]
=> 0
[3,2,2]
=> [2,2]
=> [2]
=> [[1,2]]
=> 0
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [[1,2],[3]]
=> 0
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 0
[8]
=> []
=> ?
=> ?
=> ? ∊ {55,154,165,220,231}
[7,1]
=> [1]
=> []
=> []
=> ? ∊ {55,154,165,220,231}
[6,2]
=> [2]
=> []
=> []
=> ? ∊ {55,154,165,220,231}
[6,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[5,3]
=> [3]
=> []
=> []
=> ? ∊ {55,154,165,220,231}
[5,2,1]
=> [2,1]
=> [1]
=> [[1]]
=> 0
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[4,4]
=> [4]
=> []
=> []
=> ? ∊ {55,154,165,220,231}
[4,3,1]
=> [3,1]
=> [1]
=> [[1]]
=> 0
[4,2,2]
=> [2,2]
=> [2]
=> [[1,2]]
=> 0
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[3,3,2]
=> [3,2]
=> [2]
=> [[1,2]]
=> 0
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> [[1,2],[3]]
=> 0
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[2,2,2,2]
=> [2,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 0
[2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 0
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 0
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> 0
[9]
=> []
=> ?
=> ?
=> ? ∊ {140,220,256,320,324}
[8,1]
=> [1]
=> []
=> []
=> ? ∊ {140,220,256,320,324}
[7,2]
=> [2]
=> []
=> []
=> ? ∊ {140,220,256,320,324}
[7,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[6,3]
=> [3]
=> []
=> []
=> ? ∊ {140,220,256,320,324}
[6,2,1]
=> [2,1]
=> [1]
=> [[1]]
=> 0
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[5,4]
=> [4]
=> []
=> []
=> ? ∊ {140,220,256,320,324}
[5,3,1]
=> [3,1]
=> [1]
=> [[1]]
=> 0
[5,2,2]
=> [2,2]
=> [2]
=> [[1,2]]
=> 0
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[4,4,1]
=> [4,1]
=> [1]
=> [[1]]
=> 0
[10]
=> []
=> ?
=> ?
=> ? ∊ {91,260,286,390,429,455}
[9,1]
=> [1]
=> []
=> []
=> ? ∊ {91,260,286,390,429,455}
[8,2]
=> [2]
=> []
=> []
=> ? ∊ {91,260,286,390,429,455}
[7,3]
=> [3]
=> []
=> []
=> ? ∊ {91,260,286,390,429,455}
[6,4]
=> [4]
=> []
=> []
=> ? ∊ {91,260,286,390,429,455}
[5,5]
=> [5]
=> []
=> []
=> ? ∊ {91,260,286,390,429,455}
[11]
=> []
=> ?
=> ?
=> ? ∊ {0,224,364,420,560,560,616}
[10,1]
=> [1]
=> []
=> []
=> ? ∊ {0,224,364,420,560,560,616}
[9,2]
=> [2]
=> []
=> []
=> ? ∊ {0,224,364,420,560,560,616}
[8,3]
=> [3]
=> []
=> []
=> ? ∊ {0,224,364,420,560,560,616}
[7,4]
=> [4]
=> []
=> []
=> ? ∊ {0,224,364,420,560,560,616}
[6,5]
=> [5]
=> []
=> []
=> ? ∊ {0,224,364,420,560,560,616}
[1,1,1,1,1,1,1,1,1,1,1]
=> [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]]
=> ? ∊ {0,224,364,420,560,560,616}
[12]
=> []
=> ?
=> ?
=> ? ∊ {0,0,140,405,455,625,715,770,810}
[11,1]
=> [1]
=> []
=> []
=> ? ∊ {0,0,140,405,455,625,715,770,810}
[10,2]
=> [2]
=> []
=> []
=> ? ∊ {0,0,140,405,455,625,715,770,810}
[9,3]
=> [3]
=> []
=> []
=> ? ∊ {0,0,140,405,455,625,715,770,810}
[8,4]
=> [4]
=> []
=> []
=> ? ∊ {0,0,140,405,455,625,715,770,810}
[7,5]
=> [5]
=> []
=> []
=> ? ∊ {0,0,140,405,455,625,715,770,810}
[6,6]
=> [6]
=> []
=> []
=> ? ∊ {0,0,140,405,455,625,715,770,810}
[2,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? ∊ {0,0,140,405,455,625,715,770,810}
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? ∊ {0,0,140,405,455,625,715,770,810}
Description
The number of natural descents of a standard Young tableau. A natural descent of a standard tableau $T$ is an entry $i$ such that $i+1$ appears in a higher row than $i$ in English notation.
Matching statistic: St001107
Mp00202: Integer partitions first row removalInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
Mp00043: Integer partitions to Dyck pathDyck paths
St001107: Dyck paths ⟶ ℤResult quality: 2% values known / values provided: 80%distinct values known / distinct values provided: 2%
Values
[2]
=> []
=> ?
=> ?
=> ? ∊ {5,10}
[1,1]
=> [1]
=> []
=> []
=> ? ∊ {5,10}
[3]
=> []
=> ?
=> ?
=> ? ∊ {16,20}
[2,1]
=> [1]
=> []
=> []
=> ? ∊ {16,20}
[1,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[4]
=> []
=> ?
=> ?
=> ? ∊ {14,35,35}
[3,1]
=> [1]
=> []
=> []
=> ? ∊ {14,35,35}
[2,2]
=> [2]
=> []
=> []
=> ? ∊ {14,35,35}
[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
[5]
=> []
=> ?
=> ?
=> ? ∊ {40,56,64}
[4,1]
=> [1]
=> []
=> []
=> ? ∊ {40,56,64}
[3,2]
=> [2]
=> []
=> []
=> ? ∊ {40,56,64}
[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
[6]
=> []
=> ?
=> ?
=> ? ∊ {30,81,84,105}
[5,1]
=> [1]
=> []
=> []
=> ? ∊ {30,81,84,105}
[4,2]
=> [2]
=> []
=> []
=> ? ∊ {30,81,84,105}
[4,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[3,3]
=> [3]
=> []
=> []
=> ? ∊ {30,81,84,105}
[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
[7]
=> []
=> ?
=> ?
=> ? ∊ {80,120,140,160}
[6,1]
=> [1]
=> []
=> []
=> ? ∊ {80,120,140,160}
[5,2]
=> [2]
=> []
=> []
=> ? ∊ {80,120,140,160}
[5,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[4,3]
=> [3]
=> []
=> []
=> ? ∊ {80,120,140,160}
[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
[8]
=> []
=> ?
=> ?
=> ? ∊ {55,154,165,220,231}
[7,1]
=> [1]
=> []
=> []
=> ? ∊ {55,154,165,220,231}
[6,2]
=> [2]
=> []
=> []
=> ? ∊ {55,154,165,220,231}
[6,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[5,3]
=> [3]
=> []
=> []
=> ? ∊ {55,154,165,220,231}
[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]
=> []
=> []
=> ? ∊ {55,154,165,220,231}
[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
[9]
=> []
=> ?
=> ?
=> ? ∊ {140,220,256,320,324}
[8,1]
=> [1]
=> []
=> []
=> ? ∊ {140,220,256,320,324}
[7,2]
=> [2]
=> []
=> []
=> ? ∊ {140,220,256,320,324}
[7,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[6,3]
=> [3]
=> []
=> []
=> ? ∊ {140,220,256,320,324}
[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]
=> []
=> []
=> ? ∊ {140,220,256,320,324}
[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
[10]
=> []
=> ?
=> ?
=> ? ∊ {0,91,260,286,390,429,455}
[9,1]
=> [1]
=> []
=> []
=> ? ∊ {0,91,260,286,390,429,455}
[8,2]
=> [2]
=> []
=> []
=> ? ∊ {0,91,260,286,390,429,455}
[7,3]
=> [3]
=> []
=> []
=> ? ∊ {0,91,260,286,390,429,455}
[6,4]
=> [4]
=> []
=> []
=> ? ∊ {0,91,260,286,390,429,455}
[5,5]
=> [5]
=> []
=> []
=> ? ∊ {0,91,260,286,390,429,455}
[1,1,1,1,1,1,1,1,1,1]
=> [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]
=> ? ∊ {0,91,260,286,390,429,455}
[11]
=> []
=> ?
=> ?
=> ? ∊ {0,0,224,364,420,560,560,616}
[10,1]
=> [1]
=> []
=> []
=> ? ∊ {0,0,224,364,420,560,560,616}
[9,2]
=> [2]
=> []
=> []
=> ? ∊ {0,0,224,364,420,560,560,616}
[8,3]
=> [3]
=> []
=> []
=> ? ∊ {0,0,224,364,420,560,560,616}
[7,4]
=> [4]
=> []
=> []
=> ? ∊ {0,0,224,364,420,560,560,616}
[6,5]
=> [5]
=> []
=> []
=> ? ∊ {0,0,224,364,420,560,560,616}
[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]
=> ? ∊ {0,0,224,364,420,560,560,616}
[1,1,1,1,1,1,1,1,1,1,1]
=> [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]
=> ? ∊ {0,0,224,364,420,560,560,616}
[12]
=> []
=> ?
=> ?
=> ? ∊ {0,0,0,0,140,405,455,625,715,770,810}
[11,1]
=> [1]
=> []
=> []
=> ? ∊ {0,0,0,0,140,405,455,625,715,770,810}
[10,2]
=> [2]
=> []
=> []
=> ? ∊ {0,0,0,0,140,405,455,625,715,770,810}
[9,3]
=> [3]
=> []
=> []
=> ? ∊ {0,0,0,0,140,405,455,625,715,770,810}
[8,4]
=> [4]
=> []
=> []
=> ? ∊ {0,0,0,0,140,405,455,625,715,770,810}
[7,5]
=> [5]
=> []
=> []
=> ? ∊ {0,0,0,0,140,405,455,625,715,770,810}
[6,6]
=> [6]
=> []
=> []
=> ? ∊ {0,0,0,0,140,405,455,625,715,770,810}
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.
The following 4 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001498The normalised height of a Nakayama algebra with magnitude 1.