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Your data matches 26 different statistics following compositions of up to 3 maps.
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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 (μ1,…,μk) of length at most n corresponds to the weight vector (μ1−μ2,…,μk−2−μk−1,μn,0,…,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 removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000296: Binary words ⟶ ℤResult quality: 2% ●values known / values provided: 83%●distinct values known / distinct values provided: 2%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000296: Binary words ⟶ ℤResult quality: 2% ●values known / values provided: 83%●distinct values known / distinct values provided: 2%
Values
[2]
=> []
=> ?
=> ? => ? = 10
[1,1]
=> [1]
=> []
=> => ? = 5
[3]
=> []
=> ?
=> ? => ? = 20
[2,1]
=> [1]
=> []
=> => ? = 16
[1,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[4]
=> []
=> ?
=> ? => ? = 35
[3,1]
=> [1]
=> []
=> => ? = 35
[2,2]
=> [2]
=> []
=> => ? = 14
[2,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 0
[5]
=> []
=> ?
=> ? => ? = 56
[4,1]
=> [1]
=> []
=> => ? = 64
[3,2]
=> [2]
=> []
=> => ? = 40
[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]
=> []
=> ?
=> ? => ? = 84
[5,1]
=> [1]
=> []
=> => ? = 105
[4,2]
=> [2]
=> []
=> => ? = 81
[4,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[3,3]
=> [3]
=> []
=> => ? = 30
[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]
=> []
=> ?
=> ? => ? = 120
[6,1]
=> [1]
=> []
=> => ? = 160
[5,2]
=> [2]
=> []
=> => ? = 140
[5,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[4,3]
=> [3]
=> []
=> => ? = 80
[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]
=> []
=> ?
=> ? => ? = 165
[7,1]
=> [1]
=> []
=> => ? = 231
[6,2]
=> [2]
=> []
=> => ? = 220
[6,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[5,3]
=> [3]
=> []
=> => ? = 154
[5,2,1]
=> [2,1]
=> [1]
=> 10 => 0
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 0
[4,4]
=> [4]
=> []
=> => ? = 55
[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]
=> []
=> ?
=> ? => ? = 220
[8,1]
=> [1]
=> []
=> => ? = 320
[7,2]
=> [2]
=> []
=> => ? = 324
[7,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[6,3]
=> [3]
=> []
=> => ? = 256
[6,2,1]
=> [2,1]
=> [1]
=> 10 => 0
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 0
[5,4]
=> [4]
=> []
=> => ? = 140
[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]
=> []
=> ?
=> ? => ? = 286
[9,1]
=> [1]
=> []
=> => ? = 429
[8,2]
=> [2]
=> []
=> => ? = 455
[7,3]
=> [3]
=> []
=> => ? = 390
[6,4]
=> [4]
=> []
=> => ? = 260
[5,5]
=> [5]
=> []
=> => ? = 91
[11]
=> []
=> ?
=> ? => ? = 364
[10,1]
=> [1]
=> []
=> => ? = 560
[9,2]
=> [2]
=> []
=> => ? = 616
[8,3]
=> [3]
=> []
=> => ? = 560
[7,4]
=> [4]
=> []
=> => ? = 420
[6,5]
=> [5]
=> []
=> => ? = 224
[12]
=> []
=> ?
=> ? => ? = 455
[11,1]
=> [1]
=> []
=> => ? = 715
[10,2]
=> [2]
=> []
=> => ? = 810
[9,3]
=> [3]
=> []
=> => ? = 770
[8,4]
=> [4]
=> []
=> => ? = 625
[7,5]
=> [5]
=> []
=> => ? = 405
[6,6]
=> [6]
=> []
=> => ? = 140
Description
The length of the symmetric border of a binary word.
The symmetric border of a word is the longest word which is a prefix and its reverse is a suffix.
The statistic value is equal to the length of the word if and only if the word is [[https://en.wikipedia.org/wiki/Palindrome|palindromic]].
Matching statistic: St000629
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000629: Binary words ⟶ ℤResult quality: 2% ●values known / values provided: 83%●distinct values known / distinct values provided: 2%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000629: Binary words ⟶ ℤResult quality: 2% ●values known / values provided: 83%●distinct values known / distinct values provided: 2%
Values
[2]
=> []
=> ?
=> ? => ? = 10
[1,1]
=> [1]
=> []
=> => ? = 5
[3]
=> []
=> ?
=> ? => ? = 20
[2,1]
=> [1]
=> []
=> => ? = 16
[1,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[4]
=> []
=> ?
=> ? => ? = 35
[3,1]
=> [1]
=> []
=> => ? = 35
[2,2]
=> [2]
=> []
=> => ? = 14
[2,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 0
[5]
=> []
=> ?
=> ? => ? = 56
[4,1]
=> [1]
=> []
=> => ? = 64
[3,2]
=> [2]
=> []
=> => ? = 40
[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]
=> []
=> ?
=> ? => ? = 84
[5,1]
=> [1]
=> []
=> => ? = 105
[4,2]
=> [2]
=> []
=> => ? = 81
[4,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[3,3]
=> [3]
=> []
=> => ? = 30
[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]
=> []
=> ?
=> ? => ? = 120
[6,1]
=> [1]
=> []
=> => ? = 160
[5,2]
=> [2]
=> []
=> => ? = 140
[5,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[4,3]
=> [3]
=> []
=> => ? = 80
[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]
=> []
=> ?
=> ? => ? = 165
[7,1]
=> [1]
=> []
=> => ? = 231
[6,2]
=> [2]
=> []
=> => ? = 220
[6,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[5,3]
=> [3]
=> []
=> => ? = 154
[5,2,1]
=> [2,1]
=> [1]
=> 10 => 0
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 0
[4,4]
=> [4]
=> []
=> => ? = 55
[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]
=> []
=> ?
=> ? => ? = 220
[8,1]
=> [1]
=> []
=> => ? = 320
[7,2]
=> [2]
=> []
=> => ? = 324
[7,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[6,3]
=> [3]
=> []
=> => ? = 256
[6,2,1]
=> [2,1]
=> [1]
=> 10 => 0
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 0
[5,4]
=> [4]
=> []
=> => ? = 140
[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]
=> []
=> ?
=> ? => ? = 286
[9,1]
=> [1]
=> []
=> => ? = 429
[8,2]
=> [2]
=> []
=> => ? = 455
[7,3]
=> [3]
=> []
=> => ? = 390
[6,4]
=> [4]
=> []
=> => ? = 260
[5,5]
=> [5]
=> []
=> => ? = 91
[11]
=> []
=> ?
=> ? => ? = 364
[10,1]
=> [1]
=> []
=> => ? = 560
[9,2]
=> [2]
=> []
=> => ? = 616
[8,3]
=> [3]
=> []
=> => ? = 560
[7,4]
=> [4]
=> []
=> => ? = 420
[6,5]
=> [5]
=> []
=> => ? = 224
[12]
=> []
=> ?
=> ? => ? = 455
[11,1]
=> [1]
=> []
=> => ? = 715
[10,2]
=> [2]
=> []
=> => ? = 810
[9,3]
=> [3]
=> []
=> => ? = 770
[8,4]
=> [4]
=> []
=> => ? = 625
[7,5]
=> [5]
=> []
=> => ? = 405
[6,6]
=> [6]
=> []
=> => ? = 140
Description
The defect of a binary word.
The defect of a finite word w is given by the difference between the maximum possible number and the actual number of palindromic factors contained in w. The maximum possible number of palindromic factors in a word w is |w|+1.
Matching statistic: St000326
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 2% ●values known / values provided: 83%●distinct values known / distinct values provided: 2%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 2% ●values known / values provided: 83%●distinct values known / distinct values provided: 2%
Values
[2]
=> []
=> ?
=> ? => ? = 10 + 1
[1,1]
=> [1]
=> []
=> => ? = 5 + 1
[3]
=> []
=> ?
=> ? => ? = 20 + 1
[2,1]
=> [1]
=> []
=> => ? = 16 + 1
[1,1,1]
=> [1,1]
=> [1]
=> 10 => 1 = 0 + 1
[4]
=> []
=> ?
=> ? => ? = 35 + 1
[3,1]
=> [1]
=> []
=> => ? = 35 + 1
[2,2]
=> [2]
=> []
=> => ? = 14 + 1
[2,1,1]
=> [1,1]
=> [1]
=> 10 => 1 = 0 + 1
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 1 = 0 + 1
[5]
=> []
=> ?
=> ? => ? = 56 + 1
[4,1]
=> [1]
=> []
=> => ? = 64 + 1
[3,2]
=> [2]
=> []
=> => ? = 40 + 1
[3,1,1]
=> [1,1]
=> [1]
=> 10 => 1 = 0 + 1
[2,2,1]
=> [2,1]
=> [1]
=> 10 => 1 = 0 + 1
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 1 = 0 + 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1110 => 1 = 0 + 1
[6]
=> []
=> ?
=> ? => ? = 84 + 1
[5,1]
=> [1]
=> []
=> => ? = 105 + 1
[4,2]
=> [2]
=> []
=> => ? = 81 + 1
[4,1,1]
=> [1,1]
=> [1]
=> 10 => 1 = 0 + 1
[3,3]
=> [3]
=> []
=> => ? = 30 + 1
[3,2,1]
=> [2,1]
=> [1]
=> 10 => 1 = 0 + 1
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 1 = 0 + 1
[2,2,2]
=> [2,2]
=> [2]
=> 100 => 1 = 0 + 1
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 110 => 1 = 0 + 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1110 => 1 = 0 + 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 1 = 0 + 1
[7]
=> []
=> ?
=> ? => ? = 120 + 1
[6,1]
=> [1]
=> []
=> => ? = 160 + 1
[5,2]
=> [2]
=> []
=> => ? = 140 + 1
[5,1,1]
=> [1,1]
=> [1]
=> 10 => 1 = 0 + 1
[4,3]
=> [3]
=> []
=> => ? = 80 + 1
[4,2,1]
=> [2,1]
=> [1]
=> 10 => 1 = 0 + 1
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 1 = 0 + 1
[3,3,1]
=> [3,1]
=> [1]
=> 10 => 1 = 0 + 1
[3,2,2]
=> [2,2]
=> [2]
=> 100 => 1 = 0 + 1
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 110 => 1 = 0 + 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1110 => 1 = 0 + 1
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1010 => 1 = 0 + 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1110 => 1 = 0 + 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 1 = 0 + 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 1 = 0 + 1
[8]
=> []
=> ?
=> ? => ? = 165 + 1
[7,1]
=> [1]
=> []
=> => ? = 231 + 1
[6,2]
=> [2]
=> []
=> => ? = 220 + 1
[6,1,1]
=> [1,1]
=> [1]
=> 10 => 1 = 0 + 1
[5,3]
=> [3]
=> []
=> => ? = 154 + 1
[5,2,1]
=> [2,1]
=> [1]
=> 10 => 1 = 0 + 1
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 1 = 0 + 1
[4,4]
=> [4]
=> []
=> => ? = 55 + 1
[4,3,1]
=> [3,1]
=> [1]
=> 10 => 1 = 0 + 1
[4,2,2]
=> [2,2]
=> [2]
=> 100 => 1 = 0 + 1
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> 110 => 1 = 0 + 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1110 => 1 = 0 + 1
[3,3,2]
=> [3,2]
=> [2]
=> 100 => 1 = 0 + 1
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> 110 => 1 = 0 + 1
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1010 => 1 = 0 + 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1110 => 1 = 0 + 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 1 = 0 + 1
[2,2,2,2]
=> [2,2,2]
=> [2,2]
=> 1100 => 1 = 0 + 1
[2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> 10110 => 1 = 0 + 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 1 = 0 + 1
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 1 = 0 + 1
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 1111110 => 1 = 0 + 1
[9]
=> []
=> ?
=> ? => ? = 220 + 1
[8,1]
=> [1]
=> []
=> => ? = 320 + 1
[7,2]
=> [2]
=> []
=> => ? = 324 + 1
[7,1,1]
=> [1,1]
=> [1]
=> 10 => 1 = 0 + 1
[6,3]
=> [3]
=> []
=> => ? = 256 + 1
[6,2,1]
=> [2,1]
=> [1]
=> 10 => 1 = 0 + 1
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 1 = 0 + 1
[5,4]
=> [4]
=> []
=> => ? = 140 + 1
[5,3,1]
=> [3,1]
=> [1]
=> 10 => 1 = 0 + 1
[5,2,2]
=> [2,2]
=> [2]
=> 100 => 1 = 0 + 1
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> 110 => 1 = 0 + 1
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1110 => 1 = 0 + 1
[4,4,1]
=> [4,1]
=> [1]
=> 10 => 1 = 0 + 1
[10]
=> []
=> ?
=> ? => ? = 286 + 1
[9,1]
=> [1]
=> []
=> => ? = 429 + 1
[8,2]
=> [2]
=> []
=> => ? = 455 + 1
[7,3]
=> [3]
=> []
=> => ? = 390 + 1
[6,4]
=> [4]
=> []
=> => ? = 260 + 1
[5,5]
=> [5]
=> []
=> => ? = 91 + 1
[11]
=> []
=> ?
=> ? => ? = 364 + 1
[10,1]
=> [1]
=> []
=> => ? = 560 + 1
[9,2]
=> [2]
=> []
=> => ? = 616 + 1
[8,3]
=> [3]
=> []
=> => ? = 560 + 1
[7,4]
=> [4]
=> []
=> => ? = 420 + 1
[6,5]
=> [5]
=> []
=> => ? = 224 + 1
[12]
=> []
=> ?
=> ? => ? = 455 + 1
[11,1]
=> [1]
=> []
=> => ? = 715 + 1
[10,2]
=> [2]
=> []
=> => ? = 810 + 1
[9,3]
=> [3]
=> []
=> => ? = 770 + 1
[8,4]
=> [4]
=> []
=> => ? = 625 + 1
[7,5]
=> [5]
=> []
=> => ? = 405 + 1
[6,6]
=> [6]
=> []
=> => ? = 140 + 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,…,n,n+1} that contains n+1, this is the minimal element of the set.
Matching statistic: St000913
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000913: Integer partitions ⟶ ℤResult quality: 2% ●values known / values provided: 83%●distinct values known / distinct values provided: 2%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000913: Integer partitions ⟶ ℤResult quality: 2% ●values known / values provided: 83%●distinct values known / distinct values provided: 2%
Values
[2]
=> []
=> ?
=> ?
=> ? = 10 + 1
[1,1]
=> [1]
=> []
=> ?
=> ? = 5 + 1
[3]
=> []
=> ?
=> ?
=> ? = 20 + 1
[2,1]
=> [1]
=> []
=> ?
=> ? = 16 + 1
[1,1,1]
=> [1,1]
=> [1]
=> []
=> 1 = 0 + 1
[4]
=> []
=> ?
=> ?
=> ? = 35 + 1
[3,1]
=> [1]
=> []
=> ?
=> ? = 35 + 1
[2,2]
=> [2]
=> []
=> ?
=> ? = 14 + 1
[2,1,1]
=> [1,1]
=> [1]
=> []
=> 1 = 0 + 1
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[5]
=> []
=> ?
=> ?
=> ? = 56 + 1
[4,1]
=> [1]
=> []
=> ?
=> ? = 64 + 1
[3,2]
=> [2]
=> []
=> ?
=> ? = 40 + 1
[3,1,1]
=> [1,1]
=> [1]
=> []
=> 1 = 0 + 1
[2,2,1]
=> [2,1]
=> [1]
=> []
=> 1 = 0 + 1
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[6]
=> []
=> ?
=> ?
=> ? = 84 + 1
[5,1]
=> [1]
=> []
=> ?
=> ? = 105 + 1
[4,2]
=> [2]
=> []
=> ?
=> ? = 81 + 1
[4,1,1]
=> [1,1]
=> [1]
=> []
=> 1 = 0 + 1
[3,3]
=> [3]
=> []
=> ?
=> ? = 30 + 1
[3,2,1]
=> [2,1]
=> [1]
=> []
=> 1 = 0 + 1
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[2,2,2]
=> [2,2]
=> [2]
=> []
=> 1 = 0 + 1
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[7]
=> []
=> ?
=> ?
=> ? = 120 + 1
[6,1]
=> [1]
=> []
=> ?
=> ? = 160 + 1
[5,2]
=> [2]
=> []
=> ?
=> ? = 140 + 1
[5,1,1]
=> [1,1]
=> [1]
=> []
=> 1 = 0 + 1
[4,3]
=> [3]
=> []
=> ?
=> ? = 80 + 1
[4,2,1]
=> [2,1]
=> [1]
=> []
=> 1 = 0 + 1
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[3,3,1]
=> [3,1]
=> [1]
=> []
=> 1 = 0 + 1
[3,2,2]
=> [2,2]
=> [2]
=> []
=> 1 = 0 + 1
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1 = 0 + 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1 = 0 + 1
[8]
=> []
=> ?
=> ?
=> ? = 165 + 1
[7,1]
=> [1]
=> []
=> ?
=> ? = 231 + 1
[6,2]
=> [2]
=> []
=> ?
=> ? = 220 + 1
[6,1,1]
=> [1,1]
=> [1]
=> []
=> 1 = 0 + 1
[5,3]
=> [3]
=> []
=> ?
=> ? = 154 + 1
[5,2,1]
=> [2,1]
=> [1]
=> []
=> 1 = 0 + 1
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[4,4]
=> [4]
=> []
=> ?
=> ? = 55 + 1
[4,3,1]
=> [3,1]
=> [1]
=> []
=> 1 = 0 + 1
[4,2,2]
=> [2,2]
=> [2]
=> []
=> 1 = 0 + 1
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[3,3,2]
=> [3,2]
=> [2]
=> []
=> 1 = 0 + 1
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1 = 0 + 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[2,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1 = 0 + 1
[2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1 = 0 + 1
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1 = 0 + 1
[9]
=> []
=> ?
=> ?
=> ? = 220 + 1
[8,1]
=> [1]
=> []
=> ?
=> ? = 320 + 1
[7,2]
=> [2]
=> []
=> ?
=> ? = 324 + 1
[7,1,1]
=> [1,1]
=> [1]
=> []
=> 1 = 0 + 1
[6,3]
=> [3]
=> []
=> ?
=> ? = 256 + 1
[6,2,1]
=> [2,1]
=> [1]
=> []
=> 1 = 0 + 1
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[5,4]
=> [4]
=> []
=> ?
=> ? = 140 + 1
[5,3,1]
=> [3,1]
=> [1]
=> []
=> 1 = 0 + 1
[5,2,2]
=> [2,2]
=> [2]
=> []
=> 1 = 0 + 1
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[4,4,1]
=> [4,1]
=> [1]
=> []
=> 1 = 0 + 1
[10]
=> []
=> ?
=> ?
=> ? = 286 + 1
[9,1]
=> [1]
=> []
=> ?
=> ? = 429 + 1
[8,2]
=> [2]
=> []
=> ?
=> ? = 455 + 1
[7,3]
=> [3]
=> []
=> ?
=> ? = 390 + 1
[6,4]
=> [4]
=> []
=> ?
=> ? = 260 + 1
[5,5]
=> [5]
=> []
=> ?
=> ? = 91 + 1
[11]
=> []
=> ?
=> ?
=> ? = 364 + 1
[10,1]
=> [1]
=> []
=> ?
=> ? = 560 + 1
[9,2]
=> [2]
=> []
=> ?
=> ? = 616 + 1
[8,3]
=> [3]
=> []
=> ?
=> ? = 560 + 1
[7,4]
=> [4]
=> []
=> ?
=> ? = 420 + 1
[6,5]
=> [5]
=> []
=> ?
=> ? = 224 + 1
[12]
=> []
=> ?
=> ?
=> ? = 455 + 1
[11,1]
=> [1]
=> []
=> ?
=> ? = 715 + 1
[10,2]
=> [2]
=> []
=> ?
=> ? = 810 + 1
[9,3]
=> [3]
=> []
=> ?
=> ? = 770 + 1
[8,4]
=> [4]
=> []
=> ?
=> ? = 625 + 1
[7,5]
=> [5]
=> []
=> ?
=> ? = 405 + 1
[6,6]
=> [6]
=> []
=> ?
=> ? = 140 + 1
Description
The number of ways to refine the partition into singletons.
For example there is only one way to refine [2,2]: [2,2]>[2,1,1]>[1,1,1,1]. However, there are two ways to refine [3,2]: [3,2]>[2,2,1]>[2,1,1,1]>[1,1,1,1,1 and [3,2]>[3,1,1]>[2,1,1,1]>[1,1,1,1,1].
In other words, this is the number of saturated chains in the refinement order from the bottom element to the given partition.
The sequence of values on the partitions with only one part is [[A002846]].
Matching statistic: St000921
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St000921: Binary words ⟶ ℤResult quality: 2% ●values known / values provided: 82%●distinct values known / distinct values provided: 2%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St000921: Binary words ⟶ ℤResult quality: 2% ●values known / values provided: 82%●distinct values known / distinct values provided: 2%
Values
[2]
=> []
=> ?
=> ? => ? = 10
[1,1]
=> [1]
=> []
=> ? => ? = 5
[3]
=> []
=> ?
=> ? => ? = 20
[2,1]
=> [1]
=> []
=> ? => ? = 16
[1,1,1]
=> [1,1]
=> [1]
=> 1 => 0
[4]
=> []
=> ?
=> ? => ? = 35
[3,1]
=> [1]
=> []
=> ? => ? = 35
[2,2]
=> [2]
=> []
=> ? => ? = 14
[2,1,1]
=> [1,1]
=> [1]
=> 1 => 0
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 11 => 0
[5]
=> []
=> ?
=> ? => ? = 56
[4,1]
=> [1]
=> []
=> ? => ? = 64
[3,2]
=> [2]
=> []
=> ? => ? = 40
[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]
=> []
=> ?
=> ? => ? = 84
[5,1]
=> [1]
=> []
=> ? => ? = 105
[4,2]
=> [2]
=> []
=> ? => ? = 81
[4,1,1]
=> [1,1]
=> [1]
=> 1 => 0
[3,3]
=> [3]
=> []
=> ? => ? = 30
[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]
=> []
=> ?
=> ? => ? = 120
[6,1]
=> [1]
=> []
=> ? => ? = 160
[5,2]
=> [2]
=> []
=> ? => ? = 140
[5,1,1]
=> [1,1]
=> [1]
=> 1 => 0
[4,3]
=> [3]
=> []
=> ? => ? = 80
[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]
=> []
=> ?
=> ? => ? = 165
[7,1]
=> [1]
=> []
=> ? => ? = 231
[6,2]
=> [2]
=> []
=> ? => ? = 220
[6,1,1]
=> [1,1]
=> [1]
=> 1 => 0
[5,3]
=> [3]
=> []
=> ? => ? = 154
[5,2,1]
=> [2,1]
=> [1]
=> 1 => 0
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 11 => 0
[4,4]
=> [4]
=> []
=> ? => ? = 55
[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]
=> []
=> ?
=> ? => ? = 220
[8,1]
=> [1]
=> []
=> ? => ? = 320
[7,2]
=> [2]
=> []
=> ? => ? = 324
[7,1,1]
=> [1,1]
=> [1]
=> 1 => 0
[6,3]
=> [3]
=> []
=> ? => ? = 256
[6,2,1]
=> [2,1]
=> [1]
=> 1 => 0
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 11 => 0
[5,4]
=> [4]
=> []
=> ? => ? = 140
[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]
=> []
=> ?
=> ? => ? = 286
[9,1]
=> [1]
=> []
=> ? => ? = 429
[8,2]
=> [2]
=> []
=> ? => ? = 455
[7,3]
=> [3]
=> []
=> ? => ? = 390
[6,4]
=> [4]
=> []
=> ? => ? = 260
[5,5]
=> [5]
=> []
=> ? => ? = 91
[11]
=> []
=> ?
=> ? => ? = 364
[10,1]
=> [1]
=> []
=> ? => ? = 560
[9,2]
=> [2]
=> []
=> ? => ? = 616
[8,3]
=> [3]
=> []
=> ? => ? = 560
[7,4]
=> [4]
=> []
=> ? => ? = 420
[6,5]
=> [5]
=> []
=> ? => ? = 224
[12]
=> []
=> ?
=> ? => ? = 455
[11,1]
=> [1]
=> []
=> ? => ? = 715
[10,2]
=> [2]
=> []
=> ? => ? = 810
[9,3]
=> [3]
=> []
=> ? => ? = 770
[8,4]
=> [4]
=> []
=> ? => ? = 625
[7,5]
=> [5]
=> []
=> ? => ? = 405
[6,6]
=> [6]
=> []
=> ? => ? = 140
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> 1111111111 => ? = 0
Description
The number of internal inversions of a binary word.
Let ˉw be the non-decreasing rearrangement of w, that is, ˉw is sorted.
An internal inversion is a pair i<j such that wi>wj and ˉwi=ˉwj. For example, the word 110 has two inversions, but only the second is internal.
Matching statistic: St001371
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St001371: Binary words ⟶ ℤResult quality: 2% ●values known / values provided: 81%●distinct values known / distinct values provided: 2%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St001371: Binary words ⟶ ℤResult quality: 2% ●values known / values provided: 81%●distinct values known / distinct values provided: 2%
Values
[2]
=> []
=> ?
=> ? => ? = 10
[1,1]
=> [1]
=> []
=> => ? = 5
[3]
=> []
=> ?
=> ? => ? = 20
[2,1]
=> [1]
=> []
=> => ? = 16
[1,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[4]
=> []
=> ?
=> ? => ? = 35
[3,1]
=> [1]
=> []
=> => ? = 35
[2,2]
=> [2]
=> []
=> => ? = 14
[2,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 0
[5]
=> []
=> ?
=> ? => ? = 56
[4,1]
=> [1]
=> []
=> => ? = 64
[3,2]
=> [2]
=> []
=> => ? = 40
[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]
=> []
=> ?
=> ? => ? = 84
[5,1]
=> [1]
=> []
=> => ? = 105
[4,2]
=> [2]
=> []
=> => ? = 81
[4,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[3,3]
=> [3]
=> []
=> => ? = 30
[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]
=> []
=> ?
=> ? => ? = 120
[6,1]
=> [1]
=> []
=> => ? = 160
[5,2]
=> [2]
=> []
=> => ? = 140
[5,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[4,3]
=> [3]
=> []
=> => ? = 80
[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]
=> []
=> ?
=> ? => ? = 165
[7,1]
=> [1]
=> []
=> => ? = 231
[6,2]
=> [2]
=> []
=> => ? = 220
[6,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[5,3]
=> [3]
=> []
=> => ? = 154
[5,2,1]
=> [2,1]
=> [1]
=> 10 => 0
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 0
[4,4]
=> [4]
=> []
=> => ? = 55
[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]
=> []
=> ?
=> ? => ? = 220
[8,1]
=> [1]
=> []
=> => ? = 320
[7,2]
=> [2]
=> []
=> => ? = 324
[7,1,1]
=> [1,1]
=> [1]
=> 10 => 0
[6,3]
=> [3]
=> []
=> => ? = 256
[6,2,1]
=> [2,1]
=> [1]
=> 10 => 0
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 110 => 0
[5,4]
=> [4]
=> []
=> => ? = 140
[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]
=> []
=> ?
=> ? => ? = 286
[9,1]
=> [1]
=> []
=> => ? = 429
[8,2]
=> [2]
=> []
=> => ? = 455
[7,3]
=> [3]
=> []
=> => ? = 390
[6,4]
=> [4]
=> []
=> => ? = 260
[5,5]
=> [5]
=> []
=> => ? = 91
[11]
=> []
=> ?
=> ? => ? = 364
[10,1]
=> [1]
=> []
=> => ? = 560
[9,2]
=> [2]
=> []
=> => ? = 616
[8,3]
=> [3]
=> []
=> => ? = 560
[7,4]
=> [4]
=> []
=> => ? = 420
[6,5]
=> [5]
=> []
=> => ? = 224
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> 1111111110 => ? = 0
[12]
=> []
=> ?
=> ? => ? = 455
[11,1]
=> [1]
=> []
=> => ? = 715
[10,2]
=> [2]
=> []
=> => ? = 810
[9,3]
=> [3]
=> []
=> => ? = 770
[8,4]
=> [4]
=> []
=> => ? = 625
[7,5]
=> [5]
=> []
=> => ? = 405
[6,6]
=> [6]
=> []
=> => ? = 140
[2,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> 1111111110 => ? = 0
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> 11111111110 => ? = 0
Description
The length of the longest Yamanouchi prefix of a binary word.
This is the largest index i such that in each of the prefixes w1, w1w2, w1w2…wi the number of zeros is greater than or equal to the number of ones.
Matching statistic: St001695
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St001695: Standard tableaux ⟶ ℤResult quality: 2% ●values known / values provided: 81%●distinct values known / distinct values provided: 2%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St001695: Standard tableaux ⟶ ℤResult quality: 2% ●values known / values provided: 81%●distinct values known / distinct values provided: 2%
Values
[2]
=> []
=> ?
=> ?
=> ? = 10
[1,1]
=> [1]
=> []
=> []
=> ? = 5
[3]
=> []
=> ?
=> ?
=> ? = 20
[2,1]
=> [1]
=> []
=> []
=> ? = 16
[1,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[4]
=> []
=> ?
=> ?
=> ? = 35
[3,1]
=> [1]
=> []
=> []
=> ? = 35
[2,2]
=> [2]
=> []
=> []
=> ? = 14
[2,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[5]
=> []
=> ?
=> ?
=> ? = 56
[4,1]
=> [1]
=> []
=> []
=> ? = 64
[3,2]
=> [2]
=> []
=> []
=> ? = 40
[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]
=> []
=> ?
=> ?
=> ? = 84
[5,1]
=> [1]
=> []
=> []
=> ? = 105
[4,2]
=> [2]
=> []
=> []
=> ? = 81
[4,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[3,3]
=> [3]
=> []
=> []
=> ? = 30
[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]
=> []
=> ?
=> ?
=> ? = 120
[6,1]
=> [1]
=> []
=> []
=> ? = 160
[5,2]
=> [2]
=> []
=> []
=> ? = 140
[5,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[4,3]
=> [3]
=> []
=> []
=> ? = 80
[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]
=> []
=> ?
=> ?
=> ? = 165
[7,1]
=> [1]
=> []
=> []
=> ? = 231
[6,2]
=> [2]
=> []
=> []
=> ? = 220
[6,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[5,3]
=> [3]
=> []
=> []
=> ? = 154
[5,2,1]
=> [2,1]
=> [1]
=> [[1]]
=> 0
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[4,4]
=> [4]
=> []
=> []
=> ? = 55
[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]
=> []
=> ?
=> ?
=> ? = 220
[8,1]
=> [1]
=> []
=> []
=> ? = 320
[7,2]
=> [2]
=> []
=> []
=> ? = 324
[7,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[6,3]
=> [3]
=> []
=> []
=> ? = 256
[6,2,1]
=> [2,1]
=> [1]
=> [[1]]
=> 0
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[5,4]
=> [4]
=> []
=> []
=> ? = 140
[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]
=> []
=> ?
=> ?
=> ? = 286
[9,1]
=> [1]
=> []
=> []
=> ? = 429
[8,2]
=> [2]
=> []
=> []
=> ? = 455
[7,3]
=> [3]
=> []
=> []
=> ? = 390
[6,4]
=> [4]
=> []
=> []
=> ? = 260
[5,5]
=> [5]
=> []
=> []
=> ? = 91
[11]
=> []
=> ?
=> ?
=> ? = 364
[10,1]
=> [1]
=> []
=> []
=> ? = 560
[9,2]
=> [2]
=> []
=> []
=> ? = 616
[8,3]
=> [3]
=> []
=> []
=> ? = 560
[7,4]
=> [4]
=> []
=> []
=> ? = 420
[6,5]
=> [5]
=> []
=> []
=> ? = 224
[1,1,1,1,1,1,1,1,1,1,1]
=> [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
[12]
=> []
=> ?
=> ?
=> ? = 455
[11,1]
=> [1]
=> []
=> []
=> ? = 715
[10,2]
=> [2]
=> []
=> []
=> ? = 810
[9,3]
=> [3]
=> []
=> []
=> ? = 770
[8,4]
=> [4]
=> []
=> []
=> ? = 625
[7,5]
=> [5]
=> []
=> []
=> ? = 405
[6,6]
=> [6]
=> []
=> []
=> ? = 140
[2,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 0
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? = 0
Description
The natural comajor index of a standard Young tableau.
A natural descent of a standard tableau T is an entry i such that i+1 appears in a higher row than i in English notation.
The natural comajor index of a tableau of size n with natural descent set D is then ∑d∈Dn−d.
Matching statistic: St001698
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St001698: Standard tableaux ⟶ ℤResult quality: 2% ●values known / values provided: 81%●distinct values known / distinct values provided: 2%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St001698: Standard tableaux ⟶ ℤResult quality: 2% ●values known / values provided: 81%●distinct values known / distinct values provided: 2%
Values
[2]
=> []
=> ?
=> ?
=> ? = 10
[1,1]
=> [1]
=> []
=> []
=> ? = 5
[3]
=> []
=> ?
=> ?
=> ? = 20
[2,1]
=> [1]
=> []
=> []
=> ? = 16
[1,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[4]
=> []
=> ?
=> ?
=> ? = 35
[3,1]
=> [1]
=> []
=> []
=> ? = 35
[2,2]
=> [2]
=> []
=> []
=> ? = 14
[2,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[5]
=> []
=> ?
=> ?
=> ? = 56
[4,1]
=> [1]
=> []
=> []
=> ? = 64
[3,2]
=> [2]
=> []
=> []
=> ? = 40
[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]
=> []
=> ?
=> ?
=> ? = 84
[5,1]
=> [1]
=> []
=> []
=> ? = 105
[4,2]
=> [2]
=> []
=> []
=> ? = 81
[4,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[3,3]
=> [3]
=> []
=> []
=> ? = 30
[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]
=> []
=> ?
=> ?
=> ? = 120
[6,1]
=> [1]
=> []
=> []
=> ? = 160
[5,2]
=> [2]
=> []
=> []
=> ? = 140
[5,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[4,3]
=> [3]
=> []
=> []
=> ? = 80
[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]
=> []
=> ?
=> ?
=> ? = 165
[7,1]
=> [1]
=> []
=> []
=> ? = 231
[6,2]
=> [2]
=> []
=> []
=> ? = 220
[6,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[5,3]
=> [3]
=> []
=> []
=> ? = 154
[5,2,1]
=> [2,1]
=> [1]
=> [[1]]
=> 0
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[4,4]
=> [4]
=> []
=> []
=> ? = 55
[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]
=> []
=> ?
=> ?
=> ? = 220
[8,1]
=> [1]
=> []
=> []
=> ? = 320
[7,2]
=> [2]
=> []
=> []
=> ? = 324
[7,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[6,3]
=> [3]
=> []
=> []
=> ? = 256
[6,2,1]
=> [2,1]
=> [1]
=> [[1]]
=> 0
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[5,4]
=> [4]
=> []
=> []
=> ? = 140
[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]
=> []
=> ?
=> ?
=> ? = 286
[9,1]
=> [1]
=> []
=> []
=> ? = 429
[8,2]
=> [2]
=> []
=> []
=> ? = 455
[7,3]
=> [3]
=> []
=> []
=> ? = 390
[6,4]
=> [4]
=> []
=> []
=> ? = 260
[5,5]
=> [5]
=> []
=> []
=> ? = 91
[11]
=> []
=> ?
=> ?
=> ? = 364
[10,1]
=> [1]
=> []
=> []
=> ? = 560
[9,2]
=> [2]
=> []
=> []
=> ? = 616
[8,3]
=> [3]
=> []
=> []
=> ? = 560
[7,4]
=> [4]
=> []
=> []
=> ? = 420
[6,5]
=> [5]
=> []
=> []
=> ? = 224
[1,1,1,1,1,1,1,1,1,1,1]
=> [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
[12]
=> []
=> ?
=> ?
=> ? = 455
[11,1]
=> [1]
=> []
=> []
=> ? = 715
[10,2]
=> [2]
=> []
=> []
=> ? = 810
[9,3]
=> [3]
=> []
=> []
=> ? = 770
[8,4]
=> [4]
=> []
=> []
=> ? = 625
[7,5]
=> [5]
=> []
=> []
=> ? = 405
[6,6]
=> [6]
=> []
=> []
=> ? = 140
[2,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 0
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? = 0
Description
The comajor index of a standard tableau minus the weighted size of its shape.
Matching statistic: St001699
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St001699: Standard tableaux ⟶ ℤResult quality: 2% ●values known / values provided: 81%●distinct values known / distinct values provided: 2%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St001699: Standard tableaux ⟶ ℤResult quality: 2% ●values known / values provided: 81%●distinct values known / distinct values provided: 2%
Values
[2]
=> []
=> ?
=> ?
=> ? = 10
[1,1]
=> [1]
=> []
=> []
=> ? = 5
[3]
=> []
=> ?
=> ?
=> ? = 20
[2,1]
=> [1]
=> []
=> []
=> ? = 16
[1,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[4]
=> []
=> ?
=> ?
=> ? = 35
[3,1]
=> [1]
=> []
=> []
=> ? = 35
[2,2]
=> [2]
=> []
=> []
=> ? = 14
[2,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[5]
=> []
=> ?
=> ?
=> ? = 56
[4,1]
=> [1]
=> []
=> []
=> ? = 64
[3,2]
=> [2]
=> []
=> []
=> ? = 40
[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]
=> []
=> ?
=> ?
=> ? = 84
[5,1]
=> [1]
=> []
=> []
=> ? = 105
[4,2]
=> [2]
=> []
=> []
=> ? = 81
[4,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[3,3]
=> [3]
=> []
=> []
=> ? = 30
[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]
=> []
=> ?
=> ?
=> ? = 120
[6,1]
=> [1]
=> []
=> []
=> ? = 160
[5,2]
=> [2]
=> []
=> []
=> ? = 140
[5,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[4,3]
=> [3]
=> []
=> []
=> ? = 80
[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]
=> []
=> ?
=> ?
=> ? = 165
[7,1]
=> [1]
=> []
=> []
=> ? = 231
[6,2]
=> [2]
=> []
=> []
=> ? = 220
[6,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[5,3]
=> [3]
=> []
=> []
=> ? = 154
[5,2,1]
=> [2,1]
=> [1]
=> [[1]]
=> 0
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[4,4]
=> [4]
=> []
=> []
=> ? = 55
[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]
=> []
=> ?
=> ?
=> ? = 220
[8,1]
=> [1]
=> []
=> []
=> ? = 320
[7,2]
=> [2]
=> []
=> []
=> ? = 324
[7,1,1]
=> [1,1]
=> [1]
=> [[1]]
=> 0
[6,3]
=> [3]
=> []
=> []
=> ? = 256
[6,2,1]
=> [2,1]
=> [1]
=> [[1]]
=> 0
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 0
[5,4]
=> [4]
=> []
=> []
=> ? = 140
[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]
=> []
=> ?
=> ?
=> ? = 286
[9,1]
=> [1]
=> []
=> []
=> ? = 429
[8,2]
=> [2]
=> []
=> []
=> ? = 455
[7,3]
=> [3]
=> []
=> []
=> ? = 390
[6,4]
=> [4]
=> []
=> []
=> ? = 260
[5,5]
=> [5]
=> []
=> []
=> ? = 91
[11]
=> []
=> ?
=> ?
=> ? = 364
[10,1]
=> [1]
=> []
=> []
=> ? = 560
[9,2]
=> [2]
=> []
=> []
=> ? = 616
[8,3]
=> [3]
=> []
=> []
=> ? = 560
[7,4]
=> [4]
=> []
=> []
=> ? = 420
[6,5]
=> [5]
=> []
=> []
=> ? = 224
[1,1,1,1,1,1,1,1,1,1,1]
=> [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
[12]
=> []
=> ?
=> ?
=> ? = 455
[11,1]
=> [1]
=> []
=> []
=> ? = 715
[10,2]
=> [2]
=> []
=> []
=> ? = 810
[9,3]
=> [3]
=> []
=> []
=> ? = 770
[8,4]
=> [4]
=> []
=> []
=> ? = 625
[7,5]
=> [5]
=> []
=> []
=> ? = 405
[6,6]
=> [6]
=> []
=> []
=> ? = 140
[2,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 0
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? = 0
Description
The major index of a standard tableau minus the weighted size of its shape.
The following 16 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. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St000687The dimension of Hom(I,P) for the LNakayama algebra of a Dyck path. St001722The number of minimal chains with small intervals between a binary word and the top element. St001256Number of simple reflexive modules that are 2-stable reflexive. 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. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St000781The number of proper colouring schemes of a Ferrers diagram. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St000264The girth of a graph, which is not a tree. St001498The normalised height of a Nakayama algebra with magnitude 1. St001199The dominant dimension of eAe for the corresponding Nakayama algebra A with minimal faithful projective-injective module eA.
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