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Your data matches 36 different statistics following compositions of up to 3 maps.
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Matching statistic: St000297
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [1]
=> 10 => 1
[1,1] => [1,1]
=> [2]
=> 100 => 1
[2] => [2]
=> [1,1]
=> 110 => 2
[1,1,1] => [1,1,1]
=> [3]
=> 1000 => 1
[1,2] => [2,1]
=> [2,1]
=> 1010 => 1
[2,1] => [2,1]
=> [2,1]
=> 1010 => 1
[3] => [3]
=> [1,1,1]
=> 1110 => 3
[1,1,1,1] => [1,1,1,1]
=> [4]
=> 10000 => 1
[1,1,2] => [2,1,1]
=> [3,1]
=> 10010 => 1
[1,2,1] => [2,1,1]
=> [3,1]
=> 10010 => 1
[1,3] => [3,1]
=> [2,1,1]
=> 10110 => 1
[2,1,1] => [2,1,1]
=> [3,1]
=> 10010 => 1
[2,2] => [2,2]
=> [2,2]
=> 1100 => 2
[3,1] => [3,1]
=> [2,1,1]
=> 10110 => 1
[4] => [4]
=> [1,1,1,1]
=> 11110 => 4
[1,1,1,1,1] => [1,1,1,1,1]
=> [5]
=> 100000 => 1
[1,1,1,2] => [2,1,1,1]
=> [4,1]
=> 100010 => 1
[1,1,2,1] => [2,1,1,1]
=> [4,1]
=> 100010 => 1
[1,1,3] => [3,1,1]
=> [3,1,1]
=> 100110 => 1
[1,2,1,1] => [2,1,1,1]
=> [4,1]
=> 100010 => 1
[1,2,2] => [2,2,1]
=> [3,2]
=> 10100 => 1
[1,3,1] => [3,1,1]
=> [3,1,1]
=> 100110 => 1
[1,4] => [4,1]
=> [2,1,1,1]
=> 101110 => 1
[2,1,1,1] => [2,1,1,1]
=> [4,1]
=> 100010 => 1
[2,1,2] => [2,2,1]
=> [3,2]
=> 10100 => 1
[2,2,1] => [2,2,1]
=> [3,2]
=> 10100 => 1
[2,3] => [3,2]
=> [2,2,1]
=> 11010 => 2
[3,1,1] => [3,1,1]
=> [3,1,1]
=> 100110 => 1
[3,2] => [3,2]
=> [2,2,1]
=> 11010 => 2
[4,1] => [4,1]
=> [2,1,1,1]
=> 101110 => 1
[5] => [5]
=> [1,1,1,1,1]
=> 111110 => 5
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> [6]
=> 1000000 => 1
[1,1,1,1,2] => [2,1,1,1,1]
=> [5,1]
=> 1000010 => 1
[1,1,1,2,1] => [2,1,1,1,1]
=> [5,1]
=> 1000010 => 1
[1,1,1,3] => [3,1,1,1]
=> [4,1,1]
=> 1000110 => 1
[1,1,2,1,1] => [2,1,1,1,1]
=> [5,1]
=> 1000010 => 1
[1,1,2,2] => [2,2,1,1]
=> [4,2]
=> 100100 => 1
[1,1,3,1] => [3,1,1,1]
=> [4,1,1]
=> 1000110 => 1
[1,1,4] => [4,1,1]
=> [3,1,1,1]
=> 1001110 => 1
[1,2,1,1,1] => [2,1,1,1,1]
=> [5,1]
=> 1000010 => 1
[1,2,1,2] => [2,2,1,1]
=> [4,2]
=> 100100 => 1
[1,2,2,1] => [2,2,1,1]
=> [4,2]
=> 100100 => 1
[1,2,3] => [3,2,1]
=> [3,2,1]
=> 101010 => 1
[1,3,1,1] => [3,1,1,1]
=> [4,1,1]
=> 1000110 => 1
[1,3,2] => [3,2,1]
=> [3,2,1]
=> 101010 => 1
[1,4,1] => [4,1,1]
=> [3,1,1,1]
=> 1001110 => 1
[1,5] => [5,1]
=> [2,1,1,1,1]
=> 1011110 => 1
[2,1,1,1,1] => [2,1,1,1,1]
=> [5,1]
=> 1000010 => 1
[2,1,1,2] => [2,2,1,1]
=> [4,2]
=> 100100 => 1
[2,1,2,1] => [2,2,1,1]
=> [4,2]
=> 100100 => 1
Description
The number of leading ones in a binary word.
Matching statistic: St000326
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00096: Binary words —Foata bijection⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00096: Binary words —Foata bijection⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> 10 => 10 => 1
[1,1] => [1,1]
=> 110 => 110 => 1
[2] => [2]
=> 100 => 010 => 2
[1,1,1] => [1,1,1]
=> 1110 => 1110 => 1
[1,2] => [2,1]
=> 1010 => 1100 => 1
[2,1] => [2,1]
=> 1010 => 1100 => 1
[3] => [3]
=> 1000 => 0010 => 3
[1,1,1,1] => [1,1,1,1]
=> 11110 => 11110 => 1
[1,1,2] => [2,1,1]
=> 10110 => 11010 => 1
[1,2,1] => [2,1,1]
=> 10110 => 11010 => 1
[1,3] => [3,1]
=> 10010 => 10100 => 1
[2,1,1] => [2,1,1]
=> 10110 => 11010 => 1
[2,2] => [2,2]
=> 1100 => 0110 => 2
[3,1] => [3,1]
=> 10010 => 10100 => 1
[4] => [4]
=> 10000 => 00010 => 4
[1,1,1,1,1] => [1,1,1,1,1]
=> 111110 => 111110 => 1
[1,1,1,2] => [2,1,1,1]
=> 101110 => 110110 => 1
[1,1,2,1] => [2,1,1,1]
=> 101110 => 110110 => 1
[1,1,3] => [3,1,1]
=> 100110 => 101010 => 1
[1,2,1,1] => [2,1,1,1]
=> 101110 => 110110 => 1
[1,2,2] => [2,2,1]
=> 11010 => 11100 => 1
[1,3,1] => [3,1,1]
=> 100110 => 101010 => 1
[1,4] => [4,1]
=> 100010 => 100100 => 1
[2,1,1,1] => [2,1,1,1]
=> 101110 => 110110 => 1
[2,1,2] => [2,2,1]
=> 11010 => 11100 => 1
[2,2,1] => [2,2,1]
=> 11010 => 11100 => 1
[2,3] => [3,2]
=> 10100 => 01100 => 2
[3,1,1] => [3,1,1]
=> 100110 => 101010 => 1
[3,2] => [3,2]
=> 10100 => 01100 => 2
[4,1] => [4,1]
=> 100010 => 100100 => 1
[5] => [5]
=> 100000 => 000010 => 5
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> 1111110 => 1111110 => 1
[1,1,1,1,2] => [2,1,1,1,1]
=> 1011110 => 1101110 => 1
[1,1,1,2,1] => [2,1,1,1,1]
=> 1011110 => 1101110 => 1
[1,1,1,3] => [3,1,1,1]
=> 1001110 => 1010110 => 1
[1,1,2,1,1] => [2,1,1,1,1]
=> 1011110 => 1101110 => 1
[1,1,2,2] => [2,2,1,1]
=> 110110 => 111010 => 1
[1,1,3,1] => [3,1,1,1]
=> 1001110 => 1010110 => 1
[1,1,4] => [4,1,1]
=> 1000110 => 1001010 => 1
[1,2,1,1,1] => [2,1,1,1,1]
=> 1011110 => 1101110 => 1
[1,2,1,2] => [2,2,1,1]
=> 110110 => 111010 => 1
[1,2,2,1] => [2,2,1,1]
=> 110110 => 111010 => 1
[1,2,3] => [3,2,1]
=> 101010 => 111000 => 1
[1,3,1,1] => [3,1,1,1]
=> 1001110 => 1010110 => 1
[1,3,2] => [3,2,1]
=> 101010 => 111000 => 1
[1,4,1] => [4,1,1]
=> 1000110 => 1001010 => 1
[1,5] => [5,1]
=> 1000010 => 1000100 => 1
[2,1,1,1,1] => [2,1,1,1,1]
=> 1011110 => 1101110 => 1
[2,1,1,2] => [2,2,1,1]
=> 110110 => 111010 => 1
[2,1,2,1] => [2,2,1,1]
=> 110110 => 111010 => 1
Description
The position of the first one in a binary word after appending a 1 at the end.
Regarding the binary word as a subset of $\{1,\dots,n,n+1\}$ that contains $n+1$, this is the minimal element of the set.
Matching statistic: St000993
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [1]
=> ? = 1
[1,1] => [1,1]
=> [2]
=> 1
[2] => [2]
=> [1,1]
=> 2
[1,1,1] => [1,1,1]
=> [3]
=> 1
[1,2] => [2,1]
=> [2,1]
=> 1
[2,1] => [2,1]
=> [2,1]
=> 1
[3] => [3]
=> [1,1,1]
=> 3
[1,1,1,1] => [1,1,1,1]
=> [4]
=> 1
[1,1,2] => [2,1,1]
=> [3,1]
=> 1
[1,2,1] => [2,1,1]
=> [3,1]
=> 1
[1,3] => [3,1]
=> [2,1,1]
=> 1
[2,1,1] => [2,1,1]
=> [3,1]
=> 1
[2,2] => [2,2]
=> [2,2]
=> 2
[3,1] => [3,1]
=> [2,1,1]
=> 1
[4] => [4]
=> [1,1,1,1]
=> 4
[1,1,1,1,1] => [1,1,1,1,1]
=> [5]
=> 1
[1,1,1,2] => [2,1,1,1]
=> [4,1]
=> 1
[1,1,2,1] => [2,1,1,1]
=> [4,1]
=> 1
[1,1,3] => [3,1,1]
=> [3,1,1]
=> 1
[1,2,1,1] => [2,1,1,1]
=> [4,1]
=> 1
[1,2,2] => [2,2,1]
=> [3,2]
=> 1
[1,3,1] => [3,1,1]
=> [3,1,1]
=> 1
[1,4] => [4,1]
=> [2,1,1,1]
=> 1
[2,1,1,1] => [2,1,1,1]
=> [4,1]
=> 1
[2,1,2] => [2,2,1]
=> [3,2]
=> 1
[2,2,1] => [2,2,1]
=> [3,2]
=> 1
[2,3] => [3,2]
=> [2,2,1]
=> 2
[3,1,1] => [3,1,1]
=> [3,1,1]
=> 1
[3,2] => [3,2]
=> [2,2,1]
=> 2
[4,1] => [4,1]
=> [2,1,1,1]
=> 1
[5] => [5]
=> [1,1,1,1,1]
=> 5
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> [6]
=> 1
[1,1,1,1,2] => [2,1,1,1,1]
=> [5,1]
=> 1
[1,1,1,2,1] => [2,1,1,1,1]
=> [5,1]
=> 1
[1,1,1,3] => [3,1,1,1]
=> [4,1,1]
=> 1
[1,1,2,1,1] => [2,1,1,1,1]
=> [5,1]
=> 1
[1,1,2,2] => [2,2,1,1]
=> [4,2]
=> 1
[1,1,3,1] => [3,1,1,1]
=> [4,1,1]
=> 1
[1,1,4] => [4,1,1]
=> [3,1,1,1]
=> 1
[1,2,1,1,1] => [2,1,1,1,1]
=> [5,1]
=> 1
[1,2,1,2] => [2,2,1,1]
=> [4,2]
=> 1
[1,2,2,1] => [2,2,1,1]
=> [4,2]
=> 1
[1,2,3] => [3,2,1]
=> [3,2,1]
=> 1
[1,3,1,1] => [3,1,1,1]
=> [4,1,1]
=> 1
[1,3,2] => [3,2,1]
=> [3,2,1]
=> 1
[1,4,1] => [4,1,1]
=> [3,1,1,1]
=> 1
[1,5] => [5,1]
=> [2,1,1,1,1]
=> 1
[2,1,1,1,1] => [2,1,1,1,1]
=> [5,1]
=> 1
[2,1,1,2] => [2,2,1,1]
=> [4,2]
=> 1
[2,1,2,1] => [2,2,1,1]
=> [4,2]
=> 1
[2,1,3] => [3,2,1]
=> [3,2,1]
=> 1
[1,2,2,2,2,2,2] => [2,2,2,2,2,2,1]
=> [7,6]
=> ? = 1
[2,2,2,2,2,2,1] => [2,2,2,2,2,2,1]
=> [7,6]
=> ? = 1
[1,5,1,5,1] => [5,5,1,1,1]
=> [5,2,2,2,2]
=> ? = 1
[1,4,3,4,1] => [4,4,3,1,1]
=> [5,3,3,2]
=> ? = 1
[1,4,5,2,1] => [5,4,2,1,1]
=> [5,3,2,2,1]
=> ? = 1
[1,2,5,4,1] => [5,4,2,1,1]
=> [5,3,2,2,1]
=> ? = 1
Description
The multiplicity of the largest part of an integer partition.
Matching statistic: St001038
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001038: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 97%●distinct values known / distinct values provided: 60%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001038: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 97%●distinct values known / distinct values provided: 60%
Values
[1] => [1]
=> []
=> []
=> ? = 1
[1,1] => [1,1]
=> [1]
=> [1,0]
=> ? = 1
[2] => [2]
=> []
=> []
=> ? = 2
[1,1,1] => [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,2] => [2,1]
=> [1]
=> [1,0]
=> ? = 1
[2,1] => [2,1]
=> [1]
=> [1,0]
=> ? = 1
[3] => [3]
=> []
=> []
=> ? = 3
[1,1,1,1] => [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,1,2] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,2,1] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,3] => [3,1]
=> [1]
=> [1,0]
=> ? = 1
[2,1,1] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,2] => [2,2]
=> [2]
=> [1,0,1,0]
=> 2
[3,1] => [3,1]
=> [1]
=> [1,0]
=> ? = 1
[4] => [4]
=> []
=> []
=> ? = 4
[1,1,1,1,1] => [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,1,1,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,1,2,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,1,3] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,2,1,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,2,2] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,3,1] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,4] => [4,1]
=> [1]
=> [1,0]
=> ? = 1
[2,1,1,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[2,1,2] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,2,1] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,3] => [3,2]
=> [2]
=> [1,0,1,0]
=> 2
[3,1,1] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[3,2] => [3,2]
=> [2]
=> [1,0,1,0]
=> 2
[4,1] => [4,1]
=> [1]
=> [1,0]
=> ? = 1
[5] => [5]
=> []
=> []
=> ? = 5
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,1,1,1,2] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,1,1,2,1] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,1,1,3] => [3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,1,2,1,1] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,1,2,2] => [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,3,1] => [3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,1,4] => [4,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,2,1,1,1] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,2,1,2] => [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,2,2,1] => [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,2,3] => [3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,3,1,1] => [3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,3,2] => [3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,4,1] => [4,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,5] => [5,1]
=> [1]
=> [1,0]
=> ? = 1
[2,1,1,1,1] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[2,1,1,2] => [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,1,2,1] => [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,1,3] => [3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,2,1,1] => [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,2,2] => [2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
[2,3,1] => [3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,4] => [4,2]
=> [2]
=> [1,0,1,0]
=> 2
[3,1,1,1] => [3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[3,1,2] => [3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[3,2,1] => [3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[3,3] => [3,3]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[4,1,1] => [4,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[4,2] => [4,2]
=> [2]
=> [1,0,1,0]
=> 2
[5,1] => [5,1]
=> [1]
=> [1,0]
=> ? = 1
[6] => [6]
=> []
=> []
=> ? = 6
[1,1,1,1,1,1,1] => [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 1
[1,1,1,1,1,2] => [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,6] => [6,1]
=> [1]
=> [1,0]
=> ? = 1
[6,1] => [6,1]
=> [1]
=> [1,0]
=> ? = 1
[7] => [7]
=> []
=> []
=> ? = 7
[1,7] => [7,1]
=> [1]
=> [1,0]
=> ? = 1
[7,1] => [7,1]
=> [1]
=> [1,0]
=> ? = 1
[8] => [8]
=> []
=> []
=> ? = 8
[1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,8] => [8,1]
=> [1]
=> [1,0]
=> ? = 1
[8,1] => [8,1]
=> [1]
=> [1,0]
=> ? = 1
[9] => [9]
=> []
=> []
=> ? = 9
[1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,1,1,1,1,1,1,2,1] => [2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,1,1,1,1,1,2,1,1] => [2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,1,1,1,1,2,1,1,1] => [2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,1,1,1,2,1,1,1,1] => [2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,1,1,2,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,1,2,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,2,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,9] => [9,1]
=> [1]
=> [1,0]
=> ? = 1
[2,1,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[9,1] => [9,1]
=> [1]
=> [1,0]
=> ? = 1
[10] => [10]
=> []
=> []
=> ? = 10
Description
The minimal height of a column in the parallelogram polyomino associated with the Dyck path.
Matching statistic: St000733
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [1]
=> [[1]]
=> 1
[1,1] => [1,1]
=> [2]
=> [[1,2]]
=> 1
[2] => [2]
=> [1,1]
=> [[1],[2]]
=> 2
[1,1,1] => [1,1,1]
=> [3]
=> [[1,2,3]]
=> 1
[1,2] => [2,1]
=> [2,1]
=> [[1,3],[2]]
=> 1
[2,1] => [2,1]
=> [2,1]
=> [[1,3],[2]]
=> 1
[3] => [3]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3
[1,1,1,1] => [1,1,1,1]
=> [4]
=> [[1,2,3,4]]
=> 1
[1,1,2] => [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 1
[1,2,1] => [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 1
[1,3] => [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 1
[2,1,1] => [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 1
[2,2] => [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 2
[3,1] => [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 1
[4] => [4]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4
[1,1,1,1,1] => [1,1,1,1,1]
=> [5]
=> [[1,2,3,4,5]]
=> 1
[1,1,1,2] => [2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 1
[1,1,2,1] => [2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 1
[1,1,3] => [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 1
[1,2,1,1] => [2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 1
[1,2,2] => [2,2,1]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 1
[1,3,1] => [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 1
[1,4] => [4,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 1
[2,1,1,1] => [2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 1
[2,1,2] => [2,2,1]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 1
[2,2,1] => [2,2,1]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 1
[2,3] => [3,2]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 2
[3,1,1] => [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 1
[3,2] => [3,2]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 2
[4,1] => [4,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 1
[5] => [5]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 5
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> [6]
=> [[1,2,3,4,5,6]]
=> 1
[1,1,1,1,2] => [2,1,1,1,1]
=> [5,1]
=> [[1,3,4,5,6],[2]]
=> 1
[1,1,1,2,1] => [2,1,1,1,1]
=> [5,1]
=> [[1,3,4,5,6],[2]]
=> 1
[1,1,1,3] => [3,1,1,1]
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> 1
[1,1,2,1,1] => [2,1,1,1,1]
=> [5,1]
=> [[1,3,4,5,6],[2]]
=> 1
[1,1,2,2] => [2,2,1,1]
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> 1
[1,1,3,1] => [3,1,1,1]
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> 1
[1,1,4] => [4,1,1]
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> 1
[1,2,1,1,1] => [2,1,1,1,1]
=> [5,1]
=> [[1,3,4,5,6],[2]]
=> 1
[1,2,1,2] => [2,2,1,1]
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> 1
[1,2,2,1] => [2,2,1,1]
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> 1
[1,2,3] => [3,2,1]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> 1
[1,3,1,1] => [3,1,1,1]
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> 1
[1,3,2] => [3,2,1]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> 1
[1,4,1] => [4,1,1]
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> 1
[1,5] => [5,1]
=> [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> 1
[2,1,1,1,1] => [2,1,1,1,1]
=> [5,1]
=> [[1,3,4,5,6],[2]]
=> 1
[2,1,1,2] => [2,2,1,1]
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> 1
[2,1,2,1] => [2,2,1,1]
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> 1
[1,1,5,5] => [5,5,1,1]
=> [4,2,2,2,2]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ? = 1
[2,2,2,1,2,3] => [3,2,2,2,2,1]
=> [6,5,1]
=> [[1,3,4,5,6,12],[2,8,9,10,11],[7]]
=> ? = 1
[2,2,3,2,1,2] => [3,2,2,2,2,1]
=> [6,5,1]
=> [[1,3,4,5,6,12],[2,8,9,10,11],[7]]
=> ? = 1
[2,1,2,3,2,2] => [3,2,2,2,2,1]
=> [6,5,1]
=> [[1,3,4,5,6,12],[2,8,9,10,11],[7]]
=> ? = 1
[2,1,2,2,2,3] => [3,2,2,2,2,1]
=> [6,5,1]
=> [[1,3,4,5,6,12],[2,8,9,10,11],[7]]
=> ? = 1
[3,3,3,3] => [3,3,3,3]
=> [4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12]]
=> ? = 3
[3,2,1,2,2,2] => [3,2,2,2,2,1]
=> [6,5,1]
=> [[1,3,4,5,6,12],[2,8,9,10,11],[7]]
=> ? = 1
[3,2,2,2,1,2] => [3,2,2,2,2,1]
=> [6,5,1]
=> [[1,3,4,5,6,12],[2,8,9,10,11],[7]]
=> ? = 1
[5,5,1,1] => [5,5,1,1]
=> [4,2,2,2,2]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ? = 1
[5,1,1,5] => [5,5,1,1]
=> [4,2,2,2,2]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ? = 1
[1,2,2,2,2,2,1] => [2,2,2,2,2,1,1]
=> [7,5]
=> [[1,2,3,4,5,11,12],[6,7,8,9,10]]
=> ? = 1
[1,2,2,2,3,2] => [3,2,2,2,2,1]
=> [6,5,1]
=> [[1,3,4,5,6,12],[2,8,9,10,11],[7]]
=> ? = 1
[1,2,2,3,2,2] => [3,2,2,2,2,1]
=> [6,5,1]
=> [[1,3,4,5,6,12],[2,8,9,10,11],[7]]
=> ? = 1
[1,2,3,2,2,2] => [3,2,2,2,2,1]
=> [6,5,1]
=> [[1,3,4,5,6,12],[2,8,9,10,11],[7]]
=> ? = 1
[1,3,2,2,2,2] => [3,2,2,2,2,1]
=> [6,5,1]
=> [[1,3,4,5,6,12],[2,8,9,10,11],[7]]
=> ? = 1
[1,5,5,1] => [5,5,1,1]
=> [4,2,2,2,2]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ? = 1
[1,6,5] => [6,5,1]
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? = 1
[2,3,2,2,2,1] => [3,2,2,2,2,1]
=> [6,5,1]
=> [[1,3,4,5,6,12],[2,8,9,10,11],[7]]
=> ? = 1
[2,3,2,3,2] => [3,3,2,2,2]
=> [5,5,2]
=> [[1,2,5,6,7],[3,4,10,11,12],[8,9]]
=> ? = 2
[2,3,3,2,2] => [3,3,2,2,2]
=> [5,5,2]
=> [[1,2,5,6,7],[3,4,10,11,12],[8,9]]
=> ? = 2
[2,4,2,2,2] => [4,2,2,2,2]
=> [5,5,1,1]
=> [[1,4,5,6,7],[2,9,10,11,12],[3],[8]]
=> ? = 2
[2,2,3,2,2,1] => [3,2,2,2,2,1]
=> [6,5,1]
=> [[1,3,4,5,6,12],[2,8,9,10,11],[7]]
=> ? = 1
[2,2,3,3,2] => [3,3,2,2,2]
=> [5,5,2]
=> [[1,2,5,6,7],[3,4,10,11,12],[8,9]]
=> ? = 2
[2,2,4,2,2] => [4,2,2,2,2]
=> [5,5,1,1]
=> [[1,4,5,6,7],[2,9,10,11,12],[3],[8]]
=> ? = 2
[2,2,2,3,2,1] => [3,2,2,2,2,1]
=> [6,5,1]
=> [[1,3,4,5,6,12],[2,8,9,10,11],[7]]
=> ? = 1
[2,2,2,4,2] => [4,2,2,2,2]
=> [5,5,1,1]
=> [[1,4,5,6,7],[2,9,10,11,12],[3],[8]]
=> ? = 2
[2,2,2,2,3,1] => [3,2,2,2,2,1]
=> [6,5,1]
=> [[1,3,4,5,6,12],[2,8,9,10,11],[7]]
=> ? = 1
[2,2,2,3,3] => [3,3,2,2,2]
=> [5,5,2]
=> [[1,2,5,6,7],[3,4,10,11,12],[8,9]]
=> ? = 2
[2,2,3,2,3] => [3,3,2,2,2]
=> [5,5,2]
=> [[1,2,5,6,7],[3,4,10,11,12],[8,9]]
=> ? = 2
[2,3,2,2,3] => [3,3,2,2,2]
=> [5,5,2]
=> [[1,2,5,6,7],[3,4,10,11,12],[8,9]]
=> ? = 2
[2,5,5] => [5,5,2]
=> [3,3,2,2,2]
=> [[1,2,9],[3,4,12],[5,6],[7,8],[10,11]]
=> ? = 2
[3,6,3] => [6,3,3]
=> [3,3,3,1,1,1]
=> [[1,5,6],[2,8,9],[3,11,12],[4],[7],[10]]
=> ? = 3
[3,3,2,2,2] => [3,3,2,2,2]
=> [5,5,2]
=> [[1,2,5,6,7],[3,4,10,11,12],[8,9]]
=> ? = 2
[3,2,3,2,2] => [3,3,2,2,2]
=> [5,5,2]
=> [[1,2,5,6,7],[3,4,10,11,12],[8,9]]
=> ? = 2
[3,2,2,3,2] => [3,3,2,2,2]
=> [5,5,2]
=> [[1,2,5,6,7],[3,4,10,11,12],[8,9]]
=> ? = 2
[3,2,2,2,3] => [3,3,2,2,2]
=> [5,5,2]
=> [[1,2,5,6,7],[3,4,10,11,12],[8,9]]
=> ? = 2
[4,4,4] => [4,4,4]
=> [3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12]]
=> ? = 4
[5,6,1] => [6,5,1]
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? = 1
[5,5,2] => [5,5,2]
=> [3,3,2,2,2]
=> [[1,2,9],[3,4,12],[5,6],[7,8],[10,11]]
=> ? = 2
[5,2,5] => [5,5,2]
=> [3,3,2,2,2]
=> [[1,2,9],[3,4,12],[5,6],[7,8],[10,11]]
=> ? = 2
[1,2,2,2,2,2] => [2,2,2,2,2,1]
=> [6,5]
=> [[1,2,3,4,5,11],[6,7,8,9,10]]
=> ? = 1
[1,2,2,2,1,3] => [3,2,2,2,1,1]
=> [6,4,1]
=> [[1,3,4,5,10,11],[2,7,8,9],[6]]
=> ? = 1
[1,2,2,1,3,2] => [3,2,2,2,1,1]
=> [6,4,1]
=> [[1,3,4,5,10,11],[2,7,8,9],[6]]
=> ? = 1
[1,2,2,1,2,3] => [3,2,2,2,1,1]
=> [6,4,1]
=> [[1,3,4,5,10,11],[2,7,8,9],[6]]
=> ? = 1
[1,2,1,3,2,2] => [3,2,2,2,1,1]
=> [6,4,1]
=> [[1,3,4,5,10,11],[2,7,8,9],[6]]
=> ? = 1
[1,2,1,3,1,3] => [3,3,2,1,1,1]
=> [6,3,2]
=> [[1,2,5,9,10,11],[3,4,8],[6,7]]
=> ? = 1
[1,2,1,2,3,2] => [3,2,2,2,1,1]
=> [6,4,1]
=> [[1,3,4,5,10,11],[2,7,8,9],[6]]
=> ? = 1
[1,2,1,2,2,3] => [3,2,2,2,1,1]
=> [6,4,1]
=> [[1,3,4,5,10,11],[2,7,8,9],[6]]
=> ? = 1
[1,2,1,1,3,3] => [3,3,2,1,1,1]
=> [6,3,2]
=> [[1,2,5,9,10,11],[3,4,8],[6,7]]
=> ? = 1
[1,1,3,2,2,2] => [3,2,2,2,1,1]
=> [6,4,1]
=> [[1,3,4,5,10,11],[2,7,8,9],[6]]
=> ? = 1
Description
The row containing the largest entry of a standard tableau.
Matching statistic: St000382
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 100%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [[1]]
=> [1] => 1
[1,1] => [1,1]
=> [[1],[2]]
=> [1,1] => 1
[2] => [2]
=> [[1,2]]
=> [2] => 2
[1,1,1] => [1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => 1
[1,2] => [2,1]
=> [[1,3],[2]]
=> [1,2] => 1
[2,1] => [2,1]
=> [[1,3],[2]]
=> [1,2] => 1
[3] => [3]
=> [[1,2,3]]
=> [3] => 3
[1,1,1,1] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => 1
[1,1,2] => [2,1,1]
=> [[1,4],[2],[3]]
=> [1,1,2] => 1
[1,2,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> [1,1,2] => 1
[1,3] => [3,1]
=> [[1,3,4],[2]]
=> [1,3] => 1
[2,1,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> [1,1,2] => 1
[2,2] => [2,2]
=> [[1,2],[3,4]]
=> [2,2] => 2
[3,1] => [3,1]
=> [[1,3,4],[2]]
=> [1,3] => 1
[4] => [4]
=> [[1,2,3,4]]
=> [4] => 4
[1,1,1,1,1] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => 1
[1,1,1,2] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [1,1,1,2] => 1
[1,1,2,1] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [1,1,1,2] => 1
[1,1,3] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [1,1,3] => 1
[1,2,1,1] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [1,1,1,2] => 1
[1,2,2] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [1,2,2] => 1
[1,3,1] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [1,1,3] => 1
[1,4] => [4,1]
=> [[1,3,4,5],[2]]
=> [1,4] => 1
[2,1,1,1] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [1,1,1,2] => 1
[2,1,2] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [1,2,2] => 1
[2,2,1] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [1,2,2] => 1
[2,3] => [3,2]
=> [[1,2,5],[3,4]]
=> [2,3] => 2
[3,1,1] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [1,1,3] => 1
[3,2] => [3,2]
=> [[1,2,5],[3,4]]
=> [2,3] => 2
[4,1] => [4,1]
=> [[1,3,4,5],[2]]
=> [1,4] => 1
[5] => [5]
=> [[1,2,3,4,5]]
=> [5] => 5
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1] => 1
[1,1,1,1,2] => [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [1,1,1,1,2] => 1
[1,1,1,2,1] => [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [1,1,1,1,2] => 1
[1,1,1,3] => [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [1,1,1,3] => 1
[1,1,2,1,1] => [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [1,1,1,1,2] => 1
[1,1,2,2] => [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [1,1,2,2] => 1
[1,1,3,1] => [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [1,1,1,3] => 1
[1,1,4] => [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [1,1,4] => 1
[1,2,1,1,1] => [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [1,1,1,1,2] => 1
[1,2,1,2] => [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [1,1,2,2] => 1
[1,2,2,1] => [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [1,1,2,2] => 1
[1,2,3] => [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [1,2,3] => 1
[1,3,1,1] => [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [1,1,1,3] => 1
[1,3,2] => [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [1,2,3] => 1
[1,4,1] => [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [1,1,4] => 1
[1,5] => [5,1]
=> [[1,3,4,5,6],[2]]
=> [1,5] => 1
[2,1,1,1,1] => [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [1,1,1,1,2] => 1
[2,1,1,2] => [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [1,1,2,2] => 1
[2,1,2,1] => [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [1,1,2,2] => 1
[1,5,4,1] => [5,4,1,1]
=> [[1,4,5,6,11],[2,8,9,10],[3],[7]]
=> [1,1,4,5] => ? = 1
[1,4,5,1] => [5,4,1,1]
=> [[1,4,5,6,11],[2,8,9,10],[3],[7]]
=> [1,1,4,5] => ? = 1
[2,2,2,1,2,3] => [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? = 1
[2,2,3,2,1,2] => [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? = 1
[2,1,2,3,2,2] => [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? = 1
[2,1,2,2,2,3] => [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? = 1
[2,1,4,5] => [5,4,2,1]
=> [[1,3,6,7,12],[2,5,10,11],[4,9],[8]]
=> [1,2,4,5] => ? = 1
[3,3,3,3] => [3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12]]
=> ? => ? = 3
[3,2,1,2,2,2] => [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? = 1
[3,2,2,2,1,2] => [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? = 1
[3,1,3,5] => [5,3,3,1]
=> [[1,3,4,11,12],[2,6,7],[5,9,10],[8]]
=> [1,3,3,5] => ? = 1
[4,1,2,5] => [5,4,2,1]
=> [[1,3,6,7,12],[2,5,10,11],[4,9],[8]]
=> [1,2,4,5] => ? = 1
[5,4,1,2] => [5,4,2,1]
=> [[1,3,6,7,12],[2,5,10,11],[4,9],[8]]
=> [1,2,4,5] => ? = 1
[5,3,1,3] => [5,3,3,1]
=> [[1,3,4,11,12],[2,6,7],[5,9,10],[8]]
=> [1,3,3,5] => ? = 1
[5,2,1,4] => [5,4,2,1]
=> [[1,3,6,7,12],[2,5,10,11],[4,9],[8]]
=> [1,2,4,5] => ? = 1
[1,2,2,2,2,2,1] => [2,2,2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5,10],[7,12],[9],[11]]
=> ? => ? = 1
[1,2,2,2,3,2] => [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? = 1
[1,2,2,3,2,2] => [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? = 1
[1,2,2,4,3] => [4,3,2,2,1]
=> [[1,3,8,12],[2,5,11],[4,7],[6,10],[9]]
=> [1,2,2,3,4] => ? = 1
[1,2,3,4,2] => [4,3,2,2,1]
=> [[1,3,8,12],[2,5,11],[4,7],[6,10],[9]]
=> [1,2,2,3,4] => ? = 1
[1,2,3,2,2,2] => [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? = 1
[1,2,4,4,1] => [4,4,2,1,1]
=> [[1,4,7,8],[2,6,11,12],[3,10],[5],[9]]
=> [1,1,2,4,4] => ? = 1
[1,2,4,3,2] => [4,3,2,2,1]
=> [[1,3,8,12],[2,5,11],[4,7],[6,10],[9]]
=> [1,2,2,3,4] => ? = 1
[1,2,4,2,3] => [4,3,2,2,1]
=> [[1,3,8,12],[2,5,11],[4,7],[6,10],[9]]
=> [1,2,2,3,4] => ? = 1
[1,2,5,4] => [5,4,2,1]
=> [[1,3,6,7,12],[2,5,10,11],[4,9],[8]]
=> [1,2,4,5] => ? = 1
[1,3,4,3,1] => [4,3,3,1,1]
=> [[1,4,5,12],[2,7,8],[3,10,11],[6],[9]]
=> [1,1,3,3,4] => ? = 1
[1,3,4,2,2] => [4,3,2,2,1]
=> [[1,3,8,12],[2,5,11],[4,7],[6,10],[9]]
=> [1,2,2,3,4] => ? = 1
[1,3,5,3] => [5,3,3,1]
=> [[1,3,4,11,12],[2,6,7],[5,9,10],[8]]
=> [1,3,3,5] => ? = 1
[1,3,2,4,2] => [4,3,2,2,1]
=> [[1,3,8,12],[2,5,11],[4,7],[6,10],[9]]
=> [1,2,2,3,4] => ? = 1
[1,3,2,2,2,2] => [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? = 1
[1,3,3,4,1] => [4,3,3,1,1]
=> [[1,4,5,12],[2,7,8],[3,10,11],[6],[9]]
=> [1,1,3,3,4] => ? = 1
[1,4,4,2,1] => [4,4,2,1,1]
=> [[1,4,7,8],[2,6,11,12],[3,10],[5],[9]]
=> [1,1,2,4,4] => ? = 1
[1,4,5,2] => [5,4,2,1]
=> [[1,3,6,7,12],[2,5,10,11],[4,9],[8]]
=> [1,2,4,5] => ? = 1
[1,4,3,3,1] => [4,3,3,1,1]
=> [[1,4,5,12],[2,7,8],[3,10,11],[6],[9]]
=> [1,1,3,3,4] => ? = 1
[1,4,3,2,2] => [4,3,2,2,1]
=> [[1,3,8,12],[2,5,11],[4,7],[6,10],[9]]
=> [1,2,2,3,4] => ? = 1
[1,4,2,4,1] => [4,4,2,1,1]
=> [[1,4,7,8],[2,6,11,12],[3,10],[5],[9]]
=> [1,1,2,4,4] => ? = 1
[1,4,2,3,2] => [4,3,2,2,1]
=> [[1,3,8,12],[2,5,11],[4,7],[6,10],[9]]
=> [1,2,2,3,4] => ? = 1
[1,4,2,2,3] => [4,3,2,2,1]
=> [[1,3,8,12],[2,5,11],[4,7],[6,10],[9]]
=> [1,2,2,3,4] => ? = 1
[1,5,4,2] => [5,4,2,1]
=> [[1,3,6,7,12],[2,5,10,11],[4,9],[8]]
=> [1,2,4,5] => ? = 1
[1,5,3,3] => [5,3,3,1]
=> [[1,3,4,11,12],[2,6,7],[5,9,10],[8]]
=> [1,3,3,5] => ? = 1
[1,5,2,4] => [5,4,2,1]
=> [[1,3,6,7,12],[2,5,10,11],[4,9],[8]]
=> [1,2,4,5] => ? = 1
[1,6,5] => [6,5,1]
=> [[1,3,4,5,6,12],[2,8,9,10,11],[7]]
=> ? => ? = 1
[2,3,2,2,2,1] => [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? = 1
[2,3,2,3,2] => [3,3,2,2,2]
=> [[1,2,9],[3,4,12],[5,6],[7,8],[10,11]]
=> ? => ? = 2
[2,3,3,2,2] => [3,3,2,2,2]
=> [[1,2,9],[3,4,12],[5,6],[7,8],[10,11]]
=> ? => ? = 2
[2,4,3,2,1] => [4,3,2,2,1]
=> [[1,3,8,12],[2,5,11],[4,7],[6,10],[9]]
=> [1,2,2,3,4] => ? = 1
[2,4,2,3,1] => [4,3,2,2,1]
=> [[1,3,8,12],[2,5,11],[4,7],[6,10],[9]]
=> [1,2,2,3,4] => ? = 1
[2,4,2,2,2] => [4,2,2,2,2]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ? => ? = 2
[2,5,4,1] => [5,4,2,1]
=> [[1,3,6,7,12],[2,5,10,11],[4,9],[8]]
=> [1,2,4,5] => ? = 1
[2,5,3,2] => [5,3,2,2]
=> [[1,2,7,11,12],[3,4,10],[5,6],[8,9]]
=> [2,2,3,5] => ? = 2
Description
The first part of an integer composition.
Matching statistic: St000383
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 81% ●values known / values provided: 81%●distinct values known / distinct values provided: 100%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 81% ●values known / values provided: 81%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> 10 => [1,1] => 1
[1,1] => [1,1]
=> 110 => [2,1] => 1
[2] => [2]
=> 100 => [1,2] => 2
[1,1,1] => [1,1,1]
=> 1110 => [3,1] => 1
[1,2] => [2,1]
=> 1010 => [1,1,1,1] => 1
[2,1] => [2,1]
=> 1010 => [1,1,1,1] => 1
[3] => [3]
=> 1000 => [1,3] => 3
[1,1,1,1] => [1,1,1,1]
=> 11110 => [4,1] => 1
[1,1,2] => [2,1,1]
=> 10110 => [1,1,2,1] => 1
[1,2,1] => [2,1,1]
=> 10110 => [1,1,2,1] => 1
[1,3] => [3,1]
=> 10010 => [1,2,1,1] => 1
[2,1,1] => [2,1,1]
=> 10110 => [1,1,2,1] => 1
[2,2] => [2,2]
=> 1100 => [2,2] => 2
[3,1] => [3,1]
=> 10010 => [1,2,1,1] => 1
[4] => [4]
=> 10000 => [1,4] => 4
[1,1,1,1,1] => [1,1,1,1,1]
=> 111110 => [5,1] => 1
[1,1,1,2] => [2,1,1,1]
=> 101110 => [1,1,3,1] => 1
[1,1,2,1] => [2,1,1,1]
=> 101110 => [1,1,3,1] => 1
[1,1,3] => [3,1,1]
=> 100110 => [1,2,2,1] => 1
[1,2,1,1] => [2,1,1,1]
=> 101110 => [1,1,3,1] => 1
[1,2,2] => [2,2,1]
=> 11010 => [2,1,1,1] => 1
[1,3,1] => [3,1,1]
=> 100110 => [1,2,2,1] => 1
[1,4] => [4,1]
=> 100010 => [1,3,1,1] => 1
[2,1,1,1] => [2,1,1,1]
=> 101110 => [1,1,3,1] => 1
[2,1,2] => [2,2,1]
=> 11010 => [2,1,1,1] => 1
[2,2,1] => [2,2,1]
=> 11010 => [2,1,1,1] => 1
[2,3] => [3,2]
=> 10100 => [1,1,1,2] => 2
[3,1,1] => [3,1,1]
=> 100110 => [1,2,2,1] => 1
[3,2] => [3,2]
=> 10100 => [1,1,1,2] => 2
[4,1] => [4,1]
=> 100010 => [1,3,1,1] => 1
[5] => [5]
=> 100000 => [1,5] => 5
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> 1111110 => [6,1] => 1
[1,1,1,1,2] => [2,1,1,1,1]
=> 1011110 => [1,1,4,1] => 1
[1,1,1,2,1] => [2,1,1,1,1]
=> 1011110 => [1,1,4,1] => 1
[1,1,1,3] => [3,1,1,1]
=> 1001110 => [1,2,3,1] => 1
[1,1,2,1,1] => [2,1,1,1,1]
=> 1011110 => [1,1,4,1] => 1
[1,1,2,2] => [2,2,1,1]
=> 110110 => [2,1,2,1] => 1
[1,1,3,1] => [3,1,1,1]
=> 1001110 => [1,2,3,1] => 1
[1,1,4] => [4,1,1]
=> 1000110 => [1,3,2,1] => 1
[1,2,1,1,1] => [2,1,1,1,1]
=> 1011110 => [1,1,4,1] => 1
[1,2,1,2] => [2,2,1,1]
=> 110110 => [2,1,2,1] => 1
[1,2,2,1] => [2,2,1,1]
=> 110110 => [2,1,2,1] => 1
[1,2,3] => [3,2,1]
=> 101010 => [1,1,1,1,1,1] => 1
[1,3,1,1] => [3,1,1,1]
=> 1001110 => [1,2,3,1] => 1
[1,3,2] => [3,2,1]
=> 101010 => [1,1,1,1,1,1] => 1
[1,4,1] => [4,1,1]
=> 1000110 => [1,3,2,1] => 1
[1,5] => [5,1]
=> 1000010 => [1,4,1,1] => 1
[2,1,1,1,1] => [2,1,1,1,1]
=> 1011110 => [1,1,4,1] => 1
[2,1,1,2] => [2,2,1,1]
=> 110110 => [2,1,2,1] => 1
[2,1,2,1] => [2,2,1,1]
=> 110110 => [2,1,2,1] => 1
[1,5,4,1] => [5,4,1,1]
=> 101000110 => [1,1,1,3,2,1] => ? = 1
[1,4,5,1] => [5,4,1,1]
=> 101000110 => [1,1,1,3,2,1] => ? = 1
[1,1,2,2,3,3] => [3,3,2,2,1,1]
=> 110110110 => [2,1,2,1,2,1] => ? = 1
[1,1,3,3,2,2] => [3,3,2,2,1,1]
=> 110110110 => [2,1,2,1,2,1] => ? = 1
[1,1,3,2,2,3] => [3,3,2,2,1,1]
=> 110110110 => [2,1,2,1,2,1] => ? = 1
[1,1,5,5] => [5,5,1,1]
=> 110000110 => [2,4,2,1] => ? = 1
[2,2,1,1,3,3] => [3,3,2,2,1,1]
=> 110110110 => [2,1,2,1,2,1] => ? = 1
[2,2,2,2,2,2] => [2,2,2,2,2,2]
=> 11111100 => [6,2] => ? = 2
[2,2,2,1,2,3] => [3,2,2,2,2,1]
=> 101111010 => [1,1,4,1,1,1] => ? = 1
[2,2,3,3,1,1] => [3,3,2,2,1,1]
=> 110110110 => [2,1,2,1,2,1] => ? = 1
[2,2,3,2,1,2] => [3,2,2,2,2,1]
=> 101111010 => [1,1,4,1,1,1] => ? = 1
[2,2,3,1,1,3] => [3,3,2,2,1,1]
=> 110110110 => [2,1,2,1,2,1] => ? = 1
[2,1,1,2,3,3] => [3,3,2,2,1,1]
=> 110110110 => [2,1,2,1,2,1] => ? = 1
[2,1,2,3,2,2] => [3,2,2,2,2,1]
=> 101111010 => [1,1,4,1,1,1] => ? = 1
[2,1,2,2,2,3] => [3,2,2,2,2,1]
=> 101111010 => [1,1,4,1,1,1] => ? = 1
[2,1,3,3,1,2] => [3,3,2,2,1,1]
=> 110110110 => [2,1,2,1,2,1] => ? = 1
[2,1,3,2,1,3] => [3,3,2,2,1,1]
=> 110110110 => [2,1,2,1,2,1] => ? = 1
[2,1,4,5] => [5,4,2,1]
=> 101001010 => [1,1,1,2,1,1,1,1] => ? = 1
[3,3,1,1,2,2] => [3,3,2,2,1,1]
=> 110110110 => [2,1,2,1,2,1] => ? = 1
[3,3,2,2,1,1] => [3,3,2,2,1,1]
=> 110110110 => [2,1,2,1,2,1] => ? = 1
[3,3,2,1,1,2] => [3,3,2,2,1,1]
=> 110110110 => [2,1,2,1,2,1] => ? = 1
[3,2,1,2,2,2] => [3,2,2,2,2,1]
=> 101111010 => [1,1,4,1,1,1] => ? = 1
[3,2,1,1,2,3] => [3,3,2,2,1,1]
=> 110110110 => [2,1,2,1,2,1] => ? = 1
[3,2,2,3,1,1] => [3,3,2,2,1,1]
=> 110110110 => [2,1,2,1,2,1] => ? = 1
[3,2,2,2,1,2] => [3,2,2,2,2,1]
=> 101111010 => [1,1,4,1,1,1] => ? = 1
[3,2,2,1,1,3] => [3,3,2,2,1,1]
=> 110110110 => [2,1,2,1,2,1] => ? = 1
[3,1,1,3,2,2] => [3,3,2,2,1,1]
=> 110110110 => [2,1,2,1,2,1] => ? = 1
[3,1,1,2,2,3] => [3,3,2,2,1,1]
=> 110110110 => [2,1,2,1,2,1] => ? = 1
[3,1,2,3,1,2] => [3,3,2,2,1,1]
=> 110110110 => [2,1,2,1,2,1] => ? = 1
[3,1,2,2,1,3] => [3,3,2,2,1,1]
=> 110110110 => [2,1,2,1,2,1] => ? = 1
[3,1,3,5] => [5,3,3,1]
=> 100110010 => [1,2,2,2,1,1] => ? = 1
[4,1,2,5] => [5,4,2,1]
=> 101001010 => [1,1,1,2,1,1,1,1] => ? = 1
[5,5,1,1] => [5,5,1,1]
=> 110000110 => [2,4,2,1] => ? = 1
[5,4,1,2] => [5,4,2,1]
=> 101001010 => [1,1,1,2,1,1,1,1] => ? = 1
[5,3,1,3] => [5,3,3,1]
=> 100110010 => [1,2,2,2,1,1] => ? = 1
[5,2,1,4] => [5,4,2,1]
=> 101001010 => [1,1,1,2,1,1,1,1] => ? = 1
[5,1,1,5] => [5,5,1,1]
=> 110000110 => [2,4,2,1] => ? = 1
[6,6] => [6,6]
=> 11000000 => [2,6] => ? = 6
[1,2,2,2,2,2,1] => [2,2,2,2,2,1,1]
=> 111110110 => [5,1,2,1] => ? = 1
[1,2,2,2,3,2] => [3,2,2,2,2,1]
=> 101111010 => [1,1,4,1,1,1] => ? = 1
[1,2,2,3,3,1] => [3,3,2,2,1,1]
=> 110110110 => [2,1,2,1,2,1] => ? = 1
[1,2,2,3,2,2] => [3,2,2,2,2,1]
=> 101111010 => [1,1,4,1,1,1] => ? = 1
[1,2,2,4,3] => [4,3,2,2,1]
=> 101011010 => [1,1,1,1,2,1,1,1] => ? = 1
[1,2,3,3,2,1] => [3,3,2,2,1,1]
=> 110110110 => [2,1,2,1,2,1] => ? = 1
[1,2,3,4,2] => [4,3,2,2,1]
=> 101011010 => [1,1,1,1,2,1,1,1] => ? = 1
[1,2,3,2,3,1] => [3,3,2,2,1,1]
=> 110110110 => [2,1,2,1,2,1] => ? = 1
[1,2,3,2,2,2] => [3,2,2,2,2,1]
=> 101111010 => [1,1,4,1,1,1] => ? = 1
[1,2,3,3,3] => [3,3,3,2,1]
=> 11101010 => [3,1,1,1,1,1] => ? = 1
[1,2,4,4,1] => [4,4,2,1,1]
=> 110010110 => [2,2,1,1,2,1] => ? = 1
[1,2,4,3,2] => [4,3,2,2,1]
=> 101011010 => [1,1,1,1,2,1,1,1] => ? = 1
Description
The last part of an integer composition.
Matching statistic: St000745
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 100%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [[1]]
=> [[1]]
=> 1
[1,1] => [1,1]
=> [[1],[2]]
=> [[1,2]]
=> 1
[2] => [2]
=> [[1,2]]
=> [[1],[2]]
=> 2
[1,1,1] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> 1
[1,2] => [2,1]
=> [[1,3],[2]]
=> [[1,2],[3]]
=> 1
[2,1] => [2,1]
=> [[1,3],[2]]
=> [[1,2],[3]]
=> 1
[3] => [3]
=> [[1,2,3]]
=> [[1],[2],[3]]
=> 3
[1,1,1,1] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> 1
[1,1,2] => [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 1
[1,2,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 1
[1,3] => [3,1]
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 1
[2,1,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 1
[2,2] => [2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 2
[3,1] => [3,1]
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 1
[4] => [4]
=> [[1,2,3,4]]
=> [[1],[2],[3],[4]]
=> 4
[1,1,1,1,1] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [[1,2,3,4,5]]
=> 1
[1,1,1,2] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> 1
[1,1,2,1] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> 1
[1,1,3] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> 1
[1,2,1,1] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> 1
[1,2,2] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [[1,2,4],[3,5]]
=> 1
[1,3,1] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> 1
[1,4] => [4,1]
=> [[1,3,4,5],[2]]
=> [[1,2],[3],[4],[5]]
=> 1
[2,1,1,1] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> 1
[2,1,2] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [[1,2,4],[3,5]]
=> 1
[2,2,1] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [[1,2,4],[3,5]]
=> 1
[2,3] => [3,2]
=> [[1,2,5],[3,4]]
=> [[1,3],[2,4],[5]]
=> 2
[3,1,1] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> 1
[3,2] => [3,2]
=> [[1,2,5],[3,4]]
=> [[1,3],[2,4],[5]]
=> 2
[4,1] => [4,1]
=> [[1,3,4,5],[2]]
=> [[1,2],[3],[4],[5]]
=> 1
[5] => [5]
=> [[1,2,3,4,5]]
=> [[1],[2],[3],[4],[5]]
=> 5
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [[1,2,3,4,5,6]]
=> 1
[1,1,1,1,2] => [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [[1,2,3,4,5],[6]]
=> 1
[1,1,1,2,1] => [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [[1,2,3,4,5],[6]]
=> 1
[1,1,1,3] => [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [[1,2,3,4],[5],[6]]
=> 1
[1,1,2,1,1] => [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [[1,2,3,4,5],[6]]
=> 1
[1,1,2,2] => [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [[1,2,3,5],[4,6]]
=> 1
[1,1,3,1] => [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [[1,2,3,4],[5],[6]]
=> 1
[1,1,4] => [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [[1,2,3],[4],[5],[6]]
=> 1
[1,2,1,1,1] => [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [[1,2,3,4,5],[6]]
=> 1
[1,2,1,2] => [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [[1,2,3,5],[4,6]]
=> 1
[1,2,2,1] => [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [[1,2,3,5],[4,6]]
=> 1
[1,2,3] => [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [[1,2,4],[3,5],[6]]
=> 1
[1,3,1,1] => [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [[1,2,3,4],[5],[6]]
=> 1
[1,3,2] => [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [[1,2,4],[3,5],[6]]
=> 1
[1,4,1] => [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [[1,2,3],[4],[5],[6]]
=> 1
[1,5] => [5,1]
=> [[1,3,4,5,6],[2]]
=> [[1,2],[3],[4],[5],[6]]
=> 1
[2,1,1,1,1] => [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [[1,2,3,4,5],[6]]
=> 1
[2,1,1,2] => [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [[1,2,3,5],[4,6]]
=> 1
[2,1,2,1] => [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [[1,2,3,5],[4,6]]
=> 1
[1,5,4,1] => [5,4,1,1]
=> [[1,4,5,6,11],[2,8,9,10],[3],[7]]
=> [[1,2,3,7],[4,8],[5,9],[6,10],[11]]
=> ? = 1
[1,4,5,1] => [5,4,1,1]
=> [[1,4,5,6,11],[2,8,9,10],[3],[7]]
=> [[1,2,3,7],[4,8],[5,9],[6,10],[11]]
=> ? = 1
[1,1,2,2,3,3] => [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
[1,1,3,3,2,2] => [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
[1,1,3,2,2,3] => [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
[1,1,5,5] => [5,5,1,1]
=> [[1,4,5,6,7],[2,9,10,11,12],[3],[8]]
=> ?
=> ? = 1
[2,2,1,1,3,3] => [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
[2,2,2,1,2,3] => [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> [[1,2,4,6,8,10],[3,5,7,9,11],[12]]
=> ? = 1
[2,2,3,3,1,1] => [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
[2,2,3,2,1,2] => [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> [[1,2,4,6,8,10],[3,5,7,9,11],[12]]
=> ? = 1
[2,2,3,1,1,3] => [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
[2,2,4,4] => [4,4,2,2]
=> [[1,2,7,8],[3,4,11,12],[5,6],[9,10]]
=> [[1,3,5,9],[2,4,6,10],[7,11],[8,12]]
=> ? = 2
[2,1,1,2,3,3] => [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
[2,1,2,3,2,2] => [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> [[1,2,4,6,8,10],[3,5,7,9,11],[12]]
=> ? = 1
[2,1,2,2,2,3] => [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> [[1,2,4,6,8,10],[3,5,7,9,11],[12]]
=> ? = 1
[2,1,3,3,1,2] => [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
[2,1,3,2,1,3] => [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
[2,1,4,5] => [5,4,2,1]
=> [[1,3,6,7,12],[2,5,10,11],[4,9],[8]]
=> [[1,2,4,8],[3,5,9],[6,10],[7,11],[12]]
=> ? = 1
[3,3,1,1,2,2] => [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
[3,3,2,2,1,1] => [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
[3,3,2,1,1,2] => [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
[3,3,3,3] => [3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12]]
=> [[1,4,7,10],[2,5,8,11],[3,6,9,12]]
=> ? = 3
[3,2,1,2,2,2] => [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> [[1,2,4,6,8,10],[3,5,7,9,11],[12]]
=> ? = 1
[3,2,1,1,2,3] => [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
[3,2,2,3,1,1] => [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
[3,2,2,2,1,2] => [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> [[1,2,4,6,8,10],[3,5,7,9,11],[12]]
=> ? = 1
[3,2,2,1,1,3] => [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
[3,2,3,4] => [4,3,3,2]
=> [[1,2,5,12],[3,4,8],[6,7,11],[9,10]]
=> [[1,3,6,9],[2,4,7,10],[5,8,11],[12]]
=> ? = 2
[3,1,1,3,2,2] => [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
[3,1,1,2,2,3] => [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
[3,1,2,3,1,2] => [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
[3,1,2,2,1,3] => [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
[3,1,3,5] => [5,3,3,1]
=> [[1,3,4,11,12],[2,6,7],[5,9,10],[8]]
=> [[1,2,5,8],[3,6,9],[4,7,10],[11],[12]]
=> ? = 1
[4,4,2,2] => [4,4,2,2]
=> [[1,2,7,8],[3,4,11,12],[5,6],[9,10]]
=> [[1,3,5,9],[2,4,6,10],[7,11],[8,12]]
=> ? = 2
[4,3,2,3] => [4,3,3,2]
=> [[1,2,5,12],[3,4,8],[6,7,11],[9,10]]
=> [[1,3,6,9],[2,4,7,10],[5,8,11],[12]]
=> ? = 2
[4,2,2,4] => [4,4,2,2]
=> [[1,2,7,8],[3,4,11,12],[5,6],[9,10]]
=> [[1,3,5,9],[2,4,6,10],[7,11],[8,12]]
=> ? = 2
[4,1,2,5] => [5,4,2,1]
=> [[1,3,6,7,12],[2,5,10,11],[4,9],[8]]
=> [[1,2,4,8],[3,5,9],[6,10],[7,11],[12]]
=> ? = 1
[5,5,1,1] => [5,5,1,1]
=> [[1,4,5,6,7],[2,9,10,11,12],[3],[8]]
=> ?
=> ? = 1
[5,4,1,2] => [5,4,2,1]
=> [[1,3,6,7,12],[2,5,10,11],[4,9],[8]]
=> [[1,2,4,8],[3,5,9],[6,10],[7,11],[12]]
=> ? = 1
[5,3,1,3] => [5,3,3,1]
=> [[1,3,4,11,12],[2,6,7],[5,9,10],[8]]
=> [[1,2,5,8],[3,6,9],[4,7,10],[11],[12]]
=> ? = 1
[5,2,1,4] => [5,4,2,1]
=> [[1,3,6,7,12],[2,5,10,11],[4,9],[8]]
=> [[1,2,4,8],[3,5,9],[6,10],[7,11],[12]]
=> ? = 1
[5,1,1,5] => [5,5,1,1]
=> [[1,4,5,6,7],[2,9,10,11,12],[3],[8]]
=> ?
=> ? = 1
[1,2,2,2,2,2,1] => [2,2,2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5,10],[7,12],[9],[11]]
=> ?
=> ? = 1
[1,2,2,2,3,2] => [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> [[1,2,4,6,8,10],[3,5,7,9,11],[12]]
=> ? = 1
[1,2,2,3,3,1] => [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
[1,2,2,3,2,2] => [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> [[1,2,4,6,8,10],[3,5,7,9,11],[12]]
=> ? = 1
[1,2,2,4,3] => [4,3,2,2,1]
=> [[1,3,8,12],[2,5,11],[4,7],[6,10],[9]]
=> [[1,2,4,6,9],[3,5,7,10],[8,11],[12]]
=> ? = 1
[1,2,3,3,2,1] => [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
[1,2,3,4,2] => [4,3,2,2,1]
=> [[1,3,8,12],[2,5,11],[4,7],[6,10],[9]]
=> [[1,2,4,6,9],[3,5,7,10],[8,11],[12]]
=> ? = 1
[1,2,3,2,3,1] => [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
Description
The index of the last row whose first entry is the row number in a standard Young tableau.
Matching statistic: St000990
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000990: Permutations ⟶ ℤResult quality: 50% ●values known / values provided: 61%●distinct values known / distinct values provided: 50%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000990: Permutations ⟶ ℤResult quality: 50% ●values known / values provided: 61%●distinct values known / distinct values provided: 50%
Values
[1] => [1]
=> [1,0,1,0]
=> [1,2] => 1
[1,1] => [1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[2] => [2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 2
[1,1,1] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
[1,2] => [2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 1
[2,1] => [2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 1
[3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[1,1,1,1] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 1
[1,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1
[1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1
[1,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 1
[2,1,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1
[2,2] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
[3,1] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 1
[4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 4
[1,1,1,1,1] => [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => 1
[1,1,1,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => 1
[1,1,2,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => 1
[1,1,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1
[1,2,1,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => 1
[1,2,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
[1,3,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1
[1,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,2,1,5] => 1
[2,1,1,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => 1
[2,1,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
[2,2,1] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
[2,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 2
[3,1,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1
[3,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 2
[4,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,2,1,5] => 1
[5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,4,3,2,1,6] => 5
[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]
=> [1,7,6,5,4,3,2] => ? = 1
[1,1,1,1,2] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,5,6,4,3,2] => 1
[1,1,1,2,1] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,5,6,4,3,2] => 1
[1,1,1,3] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 1
[1,1,2,1,1] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,5,6,4,3,2] => 1
[1,1,2,2] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 1
[1,1,3,1] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 1
[1,1,4] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => 1
[1,2,1,1,1] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,5,6,4,3,2] => 1
[1,2,1,2] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 1
[1,2,2,1] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 1
[1,2,3] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 1
[1,3,1,1] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 1
[1,3,2] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 1
[1,4,1] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => 1
[1,5] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [4,5,3,2,1,6] => 1
[2,1,1,1,1] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,5,6,4,3,2] => 1
[2,1,1,2] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 1
[2,1,2,1] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 1
[2,1,3] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 1
[6] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => ? = 6
[1,1,1,1,1,1,1] => [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,7,6,5,4,3,2] => ? = 1
[1,1,1,1,1,2] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,6,7,5,4,3,2] => ? = 1
[1,1,1,1,2,1] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,6,7,5,4,3,2] => ? = 1
[1,1,1,2,1,1] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,6,7,5,4,3,2] => ? = 1
[1,1,2,1,1,1] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,6,7,5,4,3,2] => ? = 1
[1,2,1,1,1,1] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,6,7,5,4,3,2] => ? = 1
[1,6] => [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [5,6,4,3,2,1,7] => ? = 1
[2,1,1,1,1,1] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,6,7,5,4,3,2] => ? = 1
[6,1] => [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [5,6,4,3,2,1,7] => ? = 1
[7] => [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7,6,5,4,3,2,1,8] => ? = 7
[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]
=> [1,9,8,7,6,5,4,3,2] => ? = 1
[1,1,1,1,1,1,2] => [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,7,8,6,5,4,3,2] => ? = 1
[1,1,1,1,1,2,1] => [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,7,8,6,5,4,3,2] => ? = 1
[1,1,1,1,1,3] => [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,6,5,7,4,3,2] => ? = 1
[1,1,1,1,2,1,1] => [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,7,8,6,5,4,3,2] => ? = 1
[1,1,1,1,2,2] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,5,7,6,4,3,2] => ? = 1
[1,1,1,1,3,1] => [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,6,5,7,4,3,2] => ? = 1
[1,1,1,2,1,1,1] => [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,7,8,6,5,4,3,2] => ? = 1
[1,1,1,2,1,2] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,5,7,6,4,3,2] => ? = 1
[1,1,1,2,2,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,5,7,6,4,3,2] => ? = 1
[1,1,1,3,1,1] => [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,6,5,7,4,3,2] => ? = 1
[1,1,2,1,1,1,1] => [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,7,8,6,5,4,3,2] => ? = 1
[1,1,2,1,1,2] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,5,7,6,4,3,2] => ? = 1
[1,1,2,1,2,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,5,7,6,4,3,2] => ? = 1
[1,1,2,2,1,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,5,7,6,4,3,2] => ? = 1
[1,1,3,1,1,1] => [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,6,5,7,4,3,2] => ? = 1
[1,1,6] => [6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [4,6,5,3,2,1,7] => ? = 1
[1,2,1,1,1,1,1] => [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,7,8,6,5,4,3,2] => ? = 1
[1,2,1,1,1,2] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,5,7,6,4,3,2] => ? = 1
[1,2,1,1,2,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,5,7,6,4,3,2] => ? = 1
[1,2,1,2,1,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,5,7,6,4,3,2] => ? = 1
[1,2,2,1,1,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,5,7,6,4,3,2] => ? = 1
[1,3,1,1,1,1] => [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,6,5,7,4,3,2] => ? = 1
[1,6,1] => [6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [4,6,5,3,2,1,7] => ? = 1
[1,7] => [7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [6,7,5,4,3,2,1,8] => ? = 1
[2,1,1,1,1,1,1] => [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,7,8,6,5,4,3,2] => ? = 1
[2,1,1,1,1,2] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,5,7,6,4,3,2] => ? = 1
[2,1,1,1,2,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,5,7,6,4,3,2] => ? = 1
[2,1,1,2,1,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,5,7,6,4,3,2] => ? = 1
[2,1,2,1,1,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,5,7,6,4,3,2] => ? = 1
[2,2,1,1,1,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,5,7,6,4,3,2] => ? = 1
[2,6] => [6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [5,4,6,3,2,1,7] => ? = 2
[3,1,1,1,1,1] => [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,6,5,7,4,3,2] => ? = 1
[6,1,1] => [6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [4,6,5,3,2,1,7] => ? = 1
[6,2] => [6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [5,4,6,3,2,1,7] => ? = 2
[7,1] => [7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [6,7,5,4,3,2,1,8] => ? = 1
[8] => [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8,7,6,5,4,3,2,1,9] => ? = 8
[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]
=> [1,10,9,8,7,6,5,4,3,2] => ? = 1
Description
The first ascent of a permutation.
For a permutation $\pi$, this is the smallest index such that $\pi(i) < \pi(i+1)$.
For the first descent, see [[St000654]].
Matching statistic: St000657
(load all 91 compositions to match this statistic)
(load all 91 compositions to match this statistic)
St000657: Integer compositions ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 90%
Values
[1] => 1
[1,1] => 1
[2] => 2
[1,1,1] => 1
[1,2] => 1
[2,1] => 1
[3] => 3
[1,1,1,1] => 1
[1,1,2] => 1
[1,2,1] => 1
[1,3] => 1
[2,1,1] => 1
[2,2] => 2
[3,1] => 1
[4] => 4
[1,1,1,1,1] => 1
[1,1,1,2] => 1
[1,1,2,1] => 1
[1,1,3] => 1
[1,2,1,1] => 1
[1,2,2] => 1
[1,3,1] => 1
[1,4] => 1
[2,1,1,1] => 1
[2,1,2] => 1
[2,2,1] => 1
[2,3] => 2
[3,1,1] => 1
[3,2] => 2
[4,1] => 1
[5] => 5
[1,1,1,1,1,1] => 1
[1,1,1,1,2] => 1
[1,1,1,2,1] => 1
[1,1,1,3] => 1
[1,1,2,1,1] => 1
[1,1,2,2] => 1
[1,1,3,1] => 1
[1,1,4] => 1
[1,2,1,1,1] => 1
[1,2,1,2] => 1
[1,2,2,1] => 1
[1,2,3] => 1
[1,3,1,1] => 1
[1,3,2] => 1
[1,4,1] => 1
[1,5] => 1
[2,1,1,1,1] => 1
[2,1,1,2] => 1
[2,1,2,1] => 1
[1,1,1,1,1,1,1,1,1,1] => ? = 1
[1,1,1,1,1,1,1,1,2] => ? = 1
[1,1,1,1,1,1,1,2,1] => ? = 1
[1,1,1,1,1,1,1,3] => ? = 1
[1,1,1,1,1,1,2,1,1] => ? = 1
[1,1,1,1,1,1,2,2] => ? = 1
[1,1,1,1,1,1,3,1] => ? = 1
[1,1,1,1,1,1,4] => ? = 1
[1,1,1,1,1,2,1,1,1] => ? = 1
[1,1,1,1,1,2,1,2] => ? = 1
[1,1,1,1,1,2,2,1] => ? = 1
[1,1,1,1,1,2,3] => ? = 1
[1,1,1,1,1,3,1,1] => ? = 1
[1,1,1,1,1,3,2] => ? = 1
[1,1,1,1,1,4,1] => ? = 1
[1,1,1,1,1,5] => ? = 1
[1,1,1,1,2,1,1,1,1] => ? = 1
[1,1,1,1,2,1,1,2] => ? = 1
[1,1,1,1,2,1,2,1] => ? = 1
[1,1,1,1,2,1,3] => ? = 1
[1,1,1,1,2,2,1,1] => ? = 1
[1,1,1,1,2,2,2] => ? = 1
[1,1,1,1,2,3,1] => ? = 1
[1,1,1,1,2,4] => ? = 1
[1,1,1,1,3,1,1,1] => ? = 1
[1,1,1,1,3,1,2] => ? = 1
[1,1,1,1,3,2,1] => ? = 1
[1,1,1,1,3,3] => ? = 1
[1,1,1,1,4,1,1] => ? = 1
[1,1,1,1,4,2] => ? = 1
[1,1,1,1,5,1] => ? = 1
[1,1,1,1,6] => ? = 1
[1,1,1,2,1,1,1,1,1] => ? = 1
[1,1,1,2,1,1,1,2] => ? = 1
[1,1,1,2,1,1,2,1] => ? = 1
[1,1,1,2,1,1,3] => ? = 1
[1,1,1,2,1,2,1,1] => ? = 1
[1,1,1,2,1,2,2] => ? = 1
[1,1,1,2,1,3,1] => ? = 1
[1,1,1,2,1,4] => ? = 1
[1,1,1,2,2,1,1,1] => ? = 1
[1,1,1,2,2,1,2] => ? = 1
[1,1,1,2,2,2,1] => ? = 1
[1,1,1,2,2,3] => ? = 1
[1,1,1,2,3,1,1] => ? = 1
[1,1,1,2,3,2] => ? = 1
[1,1,1,2,4,1] => ? = 1
[1,1,1,2,5] => ? = 1
[1,1,1,3,1,1,1,1] => ? = 1
[1,1,1,3,1,1,2] => ? = 1
Description
The smallest part of an integer composition.
The following 26 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000655The length of the minimal rise of a Dyck path. St000617The number of global maxima of a Dyck path. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000700The protection number of an ordered tree. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001481The minimal height of a peak of a Dyck path. St000654The first descent of a permutation. St001075The minimal size of a block of a set partition. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St001829The common independence number of a graph. St001119The length of a shortest maximal path in a graph. St001316The domatic number of a graph. St000908The length of the shortest maximal antichain in a poset. St001322The size of a minimal independent dominating set in a graph. 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. St000210Minimum over maximum difference of elements in cycles. St000487The length of the shortest cycle of a permutation. St000906The length of the shortest maximal chain in a poset. St000090The variation of a composition. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St000314The number of left-to-right-maxima of a permutation. St000310The minimal degree of a vertex of a graph. St000699The toughness times the least common multiple of 1,. St000260The radius of a connected graph. St000456The monochromatic index of a connected graph. St000455The second largest eigenvalue of a graph if it is integral.
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