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Your data matches 21 different statistics following compositions of up to 3 maps.
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Matching statistic: St000993
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,1]
=> [2]
=> 1
[2,1] => [2]
=> [1,1]
=> 2
[1,2,3] => [1,1,1]
=> [3]
=> 1
[1,3,2] => [2,1]
=> [2,1]
=> 1
[2,1,3] => [2,1]
=> [2,1]
=> 1
[2,3,1] => [3]
=> [1,1,1]
=> 3
[3,1,2] => [3]
=> [1,1,1]
=> 3
[3,2,1] => [2,1]
=> [2,1]
=> 1
[1,2,3,4] => [1,1,1,1]
=> [4]
=> 1
[1,2,4,3] => [2,1,1]
=> [3,1]
=> 1
[1,3,2,4] => [2,1,1]
=> [3,1]
=> 1
[1,3,4,2] => [3,1]
=> [2,1,1]
=> 1
[1,4,2,3] => [3,1]
=> [2,1,1]
=> 1
[1,4,3,2] => [2,1,1]
=> [3,1]
=> 1
[2,1,3,4] => [2,1,1]
=> [3,1]
=> 1
[2,1,4,3] => [2,2]
=> [2,2]
=> 2
[2,3,1,4] => [3,1]
=> [2,1,1]
=> 1
[2,3,4,1] => [4]
=> [1,1,1,1]
=> 4
[2,4,1,3] => [4]
=> [1,1,1,1]
=> 4
[2,4,3,1] => [3,1]
=> [2,1,1]
=> 1
[3,1,2,4] => [3,1]
=> [2,1,1]
=> 1
[3,1,4,2] => [4]
=> [1,1,1,1]
=> 4
[3,2,1,4] => [2,1,1]
=> [3,1]
=> 1
[3,2,4,1] => [3,1]
=> [2,1,1]
=> 1
[3,4,1,2] => [2,2]
=> [2,2]
=> 2
[3,4,2,1] => [4]
=> [1,1,1,1]
=> 4
[4,1,2,3] => [4]
=> [1,1,1,1]
=> 4
[4,1,3,2] => [3,1]
=> [2,1,1]
=> 1
[4,2,1,3] => [3,1]
=> [2,1,1]
=> 1
[4,2,3,1] => [2,1,1]
=> [3,1]
=> 1
[4,3,1,2] => [4]
=> [1,1,1,1]
=> 4
[4,3,2,1] => [2,2]
=> [2,2]
=> 2
[1,2,3,4,5] => [1,1,1,1,1]
=> [5]
=> 1
[1,2,3,5,4] => [2,1,1,1]
=> [4,1]
=> 1
[1,2,4,3,5] => [2,1,1,1]
=> [4,1]
=> 1
[1,2,4,5,3] => [3,1,1]
=> [3,1,1]
=> 1
[1,2,5,3,4] => [3,1,1]
=> [3,1,1]
=> 1
[1,2,5,4,3] => [2,1,1,1]
=> [4,1]
=> 1
[1,3,2,4,5] => [2,1,1,1]
=> [4,1]
=> 1
[1,3,2,5,4] => [2,2,1]
=> [3,2]
=> 1
[1,3,4,2,5] => [3,1,1]
=> [3,1,1]
=> 1
[1,3,4,5,2] => [4,1]
=> [2,1,1,1]
=> 1
[1,3,5,2,4] => [4,1]
=> [2,1,1,1]
=> 1
[1,3,5,4,2] => [3,1,1]
=> [3,1,1]
=> 1
[1,4,2,3,5] => [3,1,1]
=> [3,1,1]
=> 1
[1,4,2,5,3] => [4,1]
=> [2,1,1,1]
=> 1
[1,4,3,2,5] => [2,1,1,1]
=> [4,1]
=> 1
[1,4,3,5,2] => [3,1,1]
=> [3,1,1]
=> 1
[1,4,5,2,3] => [2,2,1]
=> [3,2]
=> 1
[1,4,5,3,2] => [4,1]
=> [2,1,1,1]
=> 1
Description
The multiplicity of the largest part of an integer partition.
Matching statistic: St000326
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 83% ●values known / values provided: 94%●distinct values known / distinct values provided: 83%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 83% ●values known / values provided: 94%●distinct values known / distinct values provided: 83%
Values
[1,2] => [1,1]
=> 110 => 011 => 2 = 1 + 1
[2,1] => [2]
=> 100 => 001 => 3 = 2 + 1
[1,2,3] => [1,1,1]
=> 1110 => 0111 => 2 = 1 + 1
[1,3,2] => [2,1]
=> 1010 => 0101 => 2 = 1 + 1
[2,1,3] => [2,1]
=> 1010 => 0101 => 2 = 1 + 1
[2,3,1] => [3]
=> 1000 => 0001 => 4 = 3 + 1
[3,1,2] => [3]
=> 1000 => 0001 => 4 = 3 + 1
[3,2,1] => [2,1]
=> 1010 => 0101 => 2 = 1 + 1
[1,2,3,4] => [1,1,1,1]
=> 11110 => 01111 => 2 = 1 + 1
[1,2,4,3] => [2,1,1]
=> 10110 => 01101 => 2 = 1 + 1
[1,3,2,4] => [2,1,1]
=> 10110 => 01101 => 2 = 1 + 1
[1,3,4,2] => [3,1]
=> 10010 => 01001 => 2 = 1 + 1
[1,4,2,3] => [3,1]
=> 10010 => 01001 => 2 = 1 + 1
[1,4,3,2] => [2,1,1]
=> 10110 => 01101 => 2 = 1 + 1
[2,1,3,4] => [2,1,1]
=> 10110 => 01101 => 2 = 1 + 1
[2,1,4,3] => [2,2]
=> 1100 => 0011 => 3 = 2 + 1
[2,3,1,4] => [3,1]
=> 10010 => 01001 => 2 = 1 + 1
[2,3,4,1] => [4]
=> 10000 => 00001 => 5 = 4 + 1
[2,4,1,3] => [4]
=> 10000 => 00001 => 5 = 4 + 1
[2,4,3,1] => [3,1]
=> 10010 => 01001 => 2 = 1 + 1
[3,1,2,4] => [3,1]
=> 10010 => 01001 => 2 = 1 + 1
[3,1,4,2] => [4]
=> 10000 => 00001 => 5 = 4 + 1
[3,2,1,4] => [2,1,1]
=> 10110 => 01101 => 2 = 1 + 1
[3,2,4,1] => [3,1]
=> 10010 => 01001 => 2 = 1 + 1
[3,4,1,2] => [2,2]
=> 1100 => 0011 => 3 = 2 + 1
[3,4,2,1] => [4]
=> 10000 => 00001 => 5 = 4 + 1
[4,1,2,3] => [4]
=> 10000 => 00001 => 5 = 4 + 1
[4,1,3,2] => [3,1]
=> 10010 => 01001 => 2 = 1 + 1
[4,2,1,3] => [3,1]
=> 10010 => 01001 => 2 = 1 + 1
[4,2,3,1] => [2,1,1]
=> 10110 => 01101 => 2 = 1 + 1
[4,3,1,2] => [4]
=> 10000 => 00001 => 5 = 4 + 1
[4,3,2,1] => [2,2]
=> 1100 => 0011 => 3 = 2 + 1
[1,2,3,4,5] => [1,1,1,1,1]
=> 111110 => 011111 => 2 = 1 + 1
[1,2,3,5,4] => [2,1,1,1]
=> 101110 => 011101 => 2 = 1 + 1
[1,2,4,3,5] => [2,1,1,1]
=> 101110 => 011101 => 2 = 1 + 1
[1,2,4,5,3] => [3,1,1]
=> 100110 => 011001 => 2 = 1 + 1
[1,2,5,3,4] => [3,1,1]
=> 100110 => 011001 => 2 = 1 + 1
[1,2,5,4,3] => [2,1,1,1]
=> 101110 => 011101 => 2 = 1 + 1
[1,3,2,4,5] => [2,1,1,1]
=> 101110 => 011101 => 2 = 1 + 1
[1,3,2,5,4] => [2,2,1]
=> 11010 => 01011 => 2 = 1 + 1
[1,3,4,2,5] => [3,1,1]
=> 100110 => 011001 => 2 = 1 + 1
[1,3,4,5,2] => [4,1]
=> 100010 => 010001 => 2 = 1 + 1
[1,3,5,2,4] => [4,1]
=> 100010 => 010001 => 2 = 1 + 1
[1,3,5,4,2] => [3,1,1]
=> 100110 => 011001 => 2 = 1 + 1
[1,4,2,3,5] => [3,1,1]
=> 100110 => 011001 => 2 = 1 + 1
[1,4,2,5,3] => [4,1]
=> 100010 => 010001 => 2 = 1 + 1
[1,4,3,2,5] => [2,1,1,1]
=> 101110 => 011101 => 2 = 1 + 1
[1,4,3,5,2] => [3,1,1]
=> 100110 => 011001 => 2 = 1 + 1
[1,4,5,2,3] => [2,2,1]
=> 11010 => 01011 => 2 = 1 + 1
[1,4,5,3,2] => [4,1]
=> 100010 => 010001 => 2 = 1 + 1
[1,2,3,4,5,6,7,8,9,11,10] => [2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => 011111111101 => ? = 1 + 1
[2,1,3,4,5,6,7,8,9,10,11] => [2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => 011111111101 => ? = 1 + 1
[11,1,2,3,4,5,6,7,8,9,10] => [11]
=> 100000000000 => 000000000001 => ? = 11 + 1
[10,1,2,3,4,5,6,7,8,11,9] => [11]
=> 100000000000 => 000000000001 => ? = 11 + 1
[11,1,2,3,4,5,6,7,10,8,9] => [11]
=> 100000000000 => 000000000001 => ? = 11 + 1
[11,1,2,3,4,5,6,10,7,8,9] => [9,2]
=> 10000000100 => 00100000001 => ? = 2 + 1
[11,1,2,3,10,4,5,6,7,8,9] => [11]
=> 100000000000 => 000000000001 => ? = 11 + 1
[10,1,2,11,3,4,5,6,7,8,9] => [11]
=> 100000000000 => 000000000001 => ? = 11 + 1
[11,1,10,2,3,4,5,6,7,8,9] => [11]
=> 100000000000 => 000000000001 => ? = 11 + 1
[2,3,4,5,6,7,8,9,10,11,1] => [11]
=> 100000000000 => 000000000001 => ? = 11 + 1
[2,4,6,8,10,12,1,3,5,7,9,11] => [12]
=> 1000000000000 => 0000000000001 => ? = 12 + 1
[2,4,6,8,11,12,1,3,5,7,9,10] => [9,3]
=> 10000001000 => 00010000001 => ? = 3 + 1
[2,4,6,9,11,12,1,3,5,7,8,10] => [12]
=> 1000000000000 => 0000000000001 => ? = 12 + 1
[2,4,7,9,10,12,1,3,5,6,8,11] => [12]
=> 1000000000000 => 0000000000001 => ? = 12 + 1
[2,4,7,9,11,12,1,3,5,6,8,10] => [9,3]
=> 10000001000 => 00010000001 => ? = 3 + 1
[2,4,7,10,11,12,1,3,5,6,8,9] => [12]
=> 1000000000000 => 0000000000001 => ? = 12 + 1
[2,4,8,9,10,12,1,3,5,6,7,11] => [10,2]
=> 100000000100 => 001000000001 => ? = 2 + 1
[2,4,8,10,11,12,1,3,5,6,7,9] => [10,2]
=> 100000000100 => 001000000001 => ? = 2 + 1
[2,4,9,10,11,12,1,3,5,6,7,8] => [12]
=> 1000000000000 => 0000000000001 => ? = 12 + 1
[2,5,6,8,11,12,1,3,4,7,9,10] => [12]
=> 1000000000000 => 0000000000001 => ? = 12 + 1
[2,5,6,9,11,12,1,3,4,7,8,10] => [10,2]
=> 100000000100 => 001000000001 => ? = 2 + 1
[2,5,6,10,11,12,1,3,4,7,8,9] => [12]
=> 1000000000000 => 0000000000001 => ? = 12 + 1
[2,5,7,8,10,12,1,3,4,6,9,11] => [12]
=> 1000000000000 => 0000000000001 => ? = 12 + 1
[2,5,7,8,11,12,1,3,4,6,9,10] => [9,3]
=> 10000001000 => 00010000001 => ? = 3 + 1
[2,5,7,9,10,12,1,3,4,6,8,11] => [10,2]
=> 100000000100 => 001000000001 => ? = 2 + 1
[2,5,8,9,10,12,1,3,4,6,7,11] => [8,2,2]
=> 10000001100 => 00110000001 => ? = 2 + 1
[2,6,7,8,10,12,1,3,4,5,9,11] => [10,2]
=> 100000000100 => 001000000001 => ? = 2 + 1
[2,6,7,8,11,12,1,3,4,5,9,10] => [12]
=> 1000000000000 => 0000000000001 => ? = 12 + 1
[2,6,7,9,10,12,1,3,4,5,8,11] => [8,2,2]
=> 10000001100 => 00110000001 => ? = 2 + 1
[2,6,7,9,11,12,1,3,4,5,8,10] => [10,2]
=> 100000000100 => 001000000001 => ? = 2 + 1
[2,6,7,10,11,12,1,3,4,5,8,9] => [12]
=> 1000000000000 => 0000000000001 => ? = 12 + 1
[2,6,8,9,10,12,1,3,4,5,7,11] => [6,2,2,2]
=> 1000011100 => 0011100001 => ? = 2 + 1
[2,6,8,9,11,12,1,3,4,5,7,10] => [8,2,2]
=> 10000001100 => 00110000001 => ? = 2 + 1
[2,6,8,10,11,12,1,3,4,5,7,9] => [10,2]
=> 100000000100 => 001000000001 => ? = 2 + 1
[2,6,9,10,11,12,1,3,4,5,7,8] => [12]
=> 1000000000000 => 0000000000001 => ? = 12 + 1
[2,7,9,10,11,12,1,3,4,5,6,8] => [9,3]
=> 10000001000 => 00010000001 => ? = 3 + 1
[2,8,9,10,11,12,1,3,4,5,6,7] => [12]
=> 1000000000000 => 0000000000001 => ? = 12 + 1
[3,4,6,8,10,12,1,2,5,7,9,11] => [9,3]
=> 10000001000 => 00010000001 => ? = 3 + 1
[3,4,6,9,10,12,1,2,5,7,8,11] => [12]
=> 1000000000000 => 0000000000001 => ? = 12 + 1
[3,4,6,10,11,12,1,2,5,7,8,9] => [12]
=> 1000000000000 => 0000000000001 => ? = 12 + 1
[3,4,7,9,10,12,1,2,5,6,8,11] => [9,3]
=> 10000001000 => 00010000001 => ? = 3 + 1
[3,4,7,10,11,12,1,2,5,6,8,9] => [9,3]
=> 10000001000 => 00010000001 => ? = 3 + 1
[3,4,8,9,10,12,1,2,5,6,7,11] => [12]
=> 1000000000000 => 0000000000001 => ? = 12 + 1
[3,4,8,9,11,12,1,2,5,6,7,10] => [9,3]
=> 10000001000 => 00010000001 => ? = 3 + 1
[3,4,8,10,11,12,1,2,5,6,7,9] => [12]
=> 1000000000000 => 0000000000001 => ? = 12 + 1
[3,5,6,8,10,12,1,2,4,7,9,11] => [12]
=> 1000000000000 => 0000000000001 => ? = 12 + 1
[3,5,6,9,10,12,1,2,4,7,8,11] => [10,2]
=> 100000000100 => 001000000001 => ? = 2 + 1
[3,5,7,8,10,12,1,2,4,6,9,11] => [9,3]
=> 10000001000 => 00010000001 => ? = 3 + 1
[3,5,8,9,10,12,1,2,4,6,7,11] => [10,2]
=> 100000000100 => 001000000001 => ? = 2 + 1
[3,5,9,10,11,12,1,2,4,6,7,8] => [12]
=> 1000000000000 => 0000000000001 => ? = 12 + 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: St000297
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 83% ●values known / values provided: 94%●distinct values known / distinct values provided: 83%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 83% ●values known / values provided: 94%●distinct values known / distinct values provided: 83%
Values
[1,2] => [1,1]
=> [2]
=> 100 => 1
[2,1] => [2]
=> [1,1]
=> 110 => 2
[1,2,3] => [1,1,1]
=> [3]
=> 1000 => 1
[1,3,2] => [2,1]
=> [2,1]
=> 1010 => 1
[2,1,3] => [2,1]
=> [2,1]
=> 1010 => 1
[2,3,1] => [3]
=> [1,1,1]
=> 1110 => 3
[3,1,2] => [3]
=> [1,1,1]
=> 1110 => 3
[3,2,1] => [2,1]
=> [2,1]
=> 1010 => 1
[1,2,3,4] => [1,1,1,1]
=> [4]
=> 10000 => 1
[1,2,4,3] => [2,1,1]
=> [3,1]
=> 10010 => 1
[1,3,2,4] => [2,1,1]
=> [3,1]
=> 10010 => 1
[1,3,4,2] => [3,1]
=> [2,1,1]
=> 10110 => 1
[1,4,2,3] => [3,1]
=> [2,1,1]
=> 10110 => 1
[1,4,3,2] => [2,1,1]
=> [3,1]
=> 10010 => 1
[2,1,3,4] => [2,1,1]
=> [3,1]
=> 10010 => 1
[2,1,4,3] => [2,2]
=> [2,2]
=> 1100 => 2
[2,3,1,4] => [3,1]
=> [2,1,1]
=> 10110 => 1
[2,3,4,1] => [4]
=> [1,1,1,1]
=> 11110 => 4
[2,4,1,3] => [4]
=> [1,1,1,1]
=> 11110 => 4
[2,4,3,1] => [3,1]
=> [2,1,1]
=> 10110 => 1
[3,1,2,4] => [3,1]
=> [2,1,1]
=> 10110 => 1
[3,1,4,2] => [4]
=> [1,1,1,1]
=> 11110 => 4
[3,2,1,4] => [2,1,1]
=> [3,1]
=> 10010 => 1
[3,2,4,1] => [3,1]
=> [2,1,1]
=> 10110 => 1
[3,4,1,2] => [2,2]
=> [2,2]
=> 1100 => 2
[3,4,2,1] => [4]
=> [1,1,1,1]
=> 11110 => 4
[4,1,2,3] => [4]
=> [1,1,1,1]
=> 11110 => 4
[4,1,3,2] => [3,1]
=> [2,1,1]
=> 10110 => 1
[4,2,1,3] => [3,1]
=> [2,1,1]
=> 10110 => 1
[4,2,3,1] => [2,1,1]
=> [3,1]
=> 10010 => 1
[4,3,1,2] => [4]
=> [1,1,1,1]
=> 11110 => 4
[4,3,2,1] => [2,2]
=> [2,2]
=> 1100 => 2
[1,2,3,4,5] => [1,1,1,1,1]
=> [5]
=> 100000 => 1
[1,2,3,5,4] => [2,1,1,1]
=> [4,1]
=> 100010 => 1
[1,2,4,3,5] => [2,1,1,1]
=> [4,1]
=> 100010 => 1
[1,2,4,5,3] => [3,1,1]
=> [3,1,1]
=> 100110 => 1
[1,2,5,3,4] => [3,1,1]
=> [3,1,1]
=> 100110 => 1
[1,2,5,4,3] => [2,1,1,1]
=> [4,1]
=> 100010 => 1
[1,3,2,4,5] => [2,1,1,1]
=> [4,1]
=> 100010 => 1
[1,3,2,5,4] => [2,2,1]
=> [3,2]
=> 10100 => 1
[1,3,4,2,5] => [3,1,1]
=> [3,1,1]
=> 100110 => 1
[1,3,4,5,2] => [4,1]
=> [2,1,1,1]
=> 101110 => 1
[1,3,5,2,4] => [4,1]
=> [2,1,1,1]
=> 101110 => 1
[1,3,5,4,2] => [3,1,1]
=> [3,1,1]
=> 100110 => 1
[1,4,2,3,5] => [3,1,1]
=> [3,1,1]
=> 100110 => 1
[1,4,2,5,3] => [4,1]
=> [2,1,1,1]
=> 101110 => 1
[1,4,3,2,5] => [2,1,1,1]
=> [4,1]
=> 100010 => 1
[1,4,3,5,2] => [3,1,1]
=> [3,1,1]
=> 100110 => 1
[1,4,5,2,3] => [2,2,1]
=> [3,2]
=> 10100 => 1
[1,4,5,3,2] => [4,1]
=> [2,1,1,1]
=> 101110 => 1
[1,2,3,4,5,6,7,8,9,11,10] => [2,1,1,1,1,1,1,1,1,1]
=> [10,1]
=> 100000000010 => ? = 1
[2,1,3,4,5,6,7,8,9,10,11] => [2,1,1,1,1,1,1,1,1,1]
=> [10,1]
=> 100000000010 => ? = 1
[11,1,2,3,4,5,6,7,8,9,10] => [11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> 111111111110 => ? = 11
[10,1,2,3,4,5,6,7,8,11,9] => [11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> 111111111110 => ? = 11
[11,1,2,3,4,5,6,7,10,8,9] => [11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> 111111111110 => ? = 11
[11,1,2,3,10,4,5,6,7,8,9] => [11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> 111111111110 => ? = 11
[10,1,2,11,3,4,5,6,7,8,9] => [11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> 111111111110 => ? = 11
[11,1,10,2,3,4,5,6,7,8,9] => [11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> 111111111110 => ? = 11
[2,3,4,5,6,7,8,9,10,11,1] => [11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> 111111111110 => ? = 11
[2,4,6,8,10,12,1,3,5,7,9,11] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> 1111111111110 => ? = 12
[2,4,6,8,11,12,1,3,5,7,9,10] => [9,3]
=> [2,2,2,1,1,1,1,1,1]
=> 11101111110 => ? = 3
[2,4,6,9,11,12,1,3,5,7,8,10] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> 1111111111110 => ? = 12
[2,4,7,9,10,12,1,3,5,6,8,11] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> 1111111111110 => ? = 12
[2,4,7,9,11,12,1,3,5,6,8,10] => [9,3]
=> [2,2,2,1,1,1,1,1,1]
=> 11101111110 => ? = 3
[2,4,7,10,11,12,1,3,5,6,8,9] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> 1111111111110 => ? = 12
[2,4,8,9,10,12,1,3,5,6,7,11] => [10,2]
=> [2,2,1,1,1,1,1,1,1,1]
=> 110111111110 => ? = 2
[2,4,8,9,11,12,1,3,5,6,7,10] => [7,3,2]
=> [3,3,2,1,1,1,1]
=> 1101011110 => ? = 2
[2,4,8,10,11,12,1,3,5,6,7,9] => [10,2]
=> [2,2,1,1,1,1,1,1,1,1]
=> 110111111110 => ? = 2
[2,4,9,10,11,12,1,3,5,6,7,8] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> 1111111111110 => ? = 12
[2,5,6,8,11,12,1,3,4,7,9,10] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> 1111111111110 => ? = 12
[2,5,6,9,11,12,1,3,4,7,8,10] => [10,2]
=> [2,2,1,1,1,1,1,1,1,1]
=> 110111111110 => ? = 2
[2,5,6,10,11,12,1,3,4,7,8,9] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> 1111111111110 => ? = 12
[2,5,7,8,10,12,1,3,4,6,9,11] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> 1111111111110 => ? = 12
[2,5,7,8,11,12,1,3,4,6,9,10] => [9,3]
=> [2,2,2,1,1,1,1,1,1]
=> 11101111110 => ? = 3
[2,5,7,9,10,12,1,3,4,6,8,11] => [10,2]
=> [2,2,1,1,1,1,1,1,1,1]
=> 110111111110 => ? = 2
[2,5,7,9,11,12,1,3,4,6,8,10] => [7,3,2]
=> [3,3,2,1,1,1,1]
=> 1101011110 => ? = 2
[2,5,8,9,10,12,1,3,4,6,7,11] => [8,2,2]
=> [3,3,1,1,1,1,1,1]
=> 11001111110 => ? = 2
[2,6,7,8,10,12,1,3,4,5,9,11] => [10,2]
=> [2,2,1,1,1,1,1,1,1,1]
=> 110111111110 => ? = 2
[2,6,7,8,11,12,1,3,4,5,9,10] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> 1111111111110 => ? = 12
[2,6,7,9,10,12,1,3,4,5,8,11] => [8,2,2]
=> [3,3,1,1,1,1,1,1]
=> 11001111110 => ? = 2
[2,6,7,9,11,12,1,3,4,5,8,10] => [10,2]
=> [2,2,1,1,1,1,1,1,1,1]
=> 110111111110 => ? = 2
[2,6,7,10,11,12,1,3,4,5,8,9] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> 1111111111110 => ? = 12
[2,6,8,9,10,12,1,3,4,5,7,11] => [6,2,2,2]
=> [4,4,1,1,1,1]
=> 1100011110 => ? = 2
[2,6,8,9,11,12,1,3,4,5,7,10] => [8,2,2]
=> [3,3,1,1,1,1,1,1]
=> 11001111110 => ? = 2
[2,6,8,10,11,12,1,3,4,5,7,9] => [10,2]
=> [2,2,1,1,1,1,1,1,1,1]
=> 110111111110 => ? = 2
[2,6,9,10,11,12,1,3,4,5,7,8] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> 1111111111110 => ? = 12
[2,7,8,10,11,12,1,3,4,5,6,9] => [7,3,2]
=> [3,3,2,1,1,1,1]
=> 1101011110 => ? = 2
[2,7,9,10,11,12,1,3,4,5,6,8] => [9,3]
=> [2,2,2,1,1,1,1,1,1]
=> 11101111110 => ? = 3
[2,8,9,10,11,12,1,3,4,5,6,7] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> 1111111111110 => ? = 12
[3,4,6,8,10,12,1,2,5,7,9,11] => [9,3]
=> [2,2,2,1,1,1,1,1,1]
=> 11101111110 => ? = 3
[3,4,6,9,10,12,1,2,5,7,8,11] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> 1111111111110 => ? = 12
[3,4,6,10,11,12,1,2,5,7,8,9] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> 1111111111110 => ? = 12
[3,4,7,9,10,12,1,2,5,6,8,11] => [9,3]
=> [2,2,2,1,1,1,1,1,1]
=> 11101111110 => ? = 3
[3,4,7,10,11,12,1,2,5,6,8,9] => [9,3]
=> [2,2,2,1,1,1,1,1,1]
=> 11101111110 => ? = 3
[3,4,8,9,10,12,1,2,5,6,7,11] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> 1111111111110 => ? = 12
[3,4,8,9,11,12,1,2,5,6,7,10] => [9,3]
=> [2,2,2,1,1,1,1,1,1]
=> 11101111110 => ? = 3
[3,4,8,10,11,12,1,2,5,6,7,9] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> 1111111111110 => ? = 12
[3,5,6,8,10,12,1,2,4,7,9,11] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> 1111111111110 => ? = 12
[3,5,6,9,10,12,1,2,4,7,8,11] => [10,2]
=> [2,2,1,1,1,1,1,1,1,1]
=> 110111111110 => ? = 2
[3,5,6,10,11,12,1,2,4,7,8,9] => [8,4]
=> [2,2,2,2,1,1,1,1]
=> 1111011110 => ? = 4
Description
The number of leading ones in a binary word.
Matching statistic: St000733
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 83% ●values known / values provided: 93%●distinct values known / distinct values provided: 83%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 83% ●values known / values provided: 93%●distinct values known / distinct values provided: 83%
Values
[1,2] => [1,1]
=> [2]
=> [[1,2]]
=> 1
[2,1] => [2]
=> [1,1]
=> [[1],[2]]
=> 2
[1,2,3] => [1,1,1]
=> [3]
=> [[1,2,3]]
=> 1
[1,3,2] => [2,1]
=> [2,1]
=> [[1,3],[2]]
=> 1
[2,1,3] => [2,1]
=> [2,1]
=> [[1,3],[2]]
=> 1
[2,3,1] => [3]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3
[3,1,2] => [3]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3
[3,2,1] => [2,1]
=> [2,1]
=> [[1,3],[2]]
=> 1
[1,2,3,4] => [1,1,1,1]
=> [4]
=> [[1,2,3,4]]
=> 1
[1,2,4,3] => [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 1
[1,3,2,4] => [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 1
[1,3,4,2] => [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 1
[1,4,2,3] => [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 1
[1,4,3,2] => [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 1
[2,1,3,4] => [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 1
[2,1,4,3] => [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 2
[2,3,1,4] => [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 1
[2,3,4,1] => [4]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4
[2,4,1,3] => [4]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4
[2,4,3,1] => [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 1
[3,1,2,4] => [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 1
[3,1,4,2] => [4]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4
[3,2,1,4] => [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 1
[3,2,4,1] => [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 1
[3,4,1,2] => [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 2
[3,4,2,1] => [4]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4
[4,1,2,3] => [4]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4
[4,1,3,2] => [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 1
[4,2,1,3] => [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 1
[4,2,3,1] => [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 1
[4,3,1,2] => [4]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4
[4,3,2,1] => [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 2
[1,2,3,4,5] => [1,1,1,1,1]
=> [5]
=> [[1,2,3,4,5]]
=> 1
[1,2,3,5,4] => [2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 1
[1,2,4,3,5] => [2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 1
[1,2,4,5,3] => [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 1
[1,2,5,3,4] => [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 1
[1,2,5,4,3] => [2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 1
[1,3,2,4,5] => [2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 1
[1,3,2,5,4] => [2,2,1]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 1
[1,3,4,2,5] => [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 1
[1,3,4,5,2] => [4,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 1
[1,3,5,2,4] => [4,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 1
[1,3,5,4,2] => [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 1
[1,4,2,3,5] => [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 1
[1,4,2,5,3] => [4,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 1
[1,4,3,2,5] => [2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 1
[1,4,3,5,2] => [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 1
[1,4,5,2,3] => [2,2,1]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 1
[1,4,5,3,2] => [4,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 1
[1,2,3,4,5,6,7,8,9,11,10] => [2,1,1,1,1,1,1,1,1,1]
=> [10,1]
=> [[1,3,4,5,6,7,8,9,10,11],[2]]
=> ? = 1
[2,1,3,4,5,6,7,8,9,10,11] => [2,1,1,1,1,1,1,1,1,1]
=> [10,1]
=> [[1,3,4,5,6,7,8,9,10,11],[2]]
=> ? = 1
[11,1,2,3,4,5,6,7,8,9,10] => [11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 11
[10,1,2,3,4,5,6,7,8,11,9] => [11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 11
[11,1,2,3,4,5,6,7,10,8,9] => [11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 11
[11,1,2,3,4,5,6,10,7,8,9] => [9,2]
=> [2,2,1,1,1,1,1,1,1]
=> [[1,9],[2,11],[3],[4],[5],[6],[7],[8],[10]]
=> ? = 2
[10,1,2,3,4,5,11,6,7,8,9] => [8,3]
=> [2,2,2,1,1,1,1,1]
=> [[1,7],[2,9],[3,11],[4],[5],[6],[8],[10]]
=> ? = 3
[11,1,2,3,4,10,5,6,7,8,9] => [8,3]
=> [2,2,2,1,1,1,1,1]
=> [[1,7],[2,9],[3,11],[4],[5],[6],[8],[10]]
=> ? = 3
[11,1,2,3,10,4,5,6,7,8,9] => [11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 11
[10,1,2,11,3,4,5,6,7,8,9] => [11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 11
[11,1,10,2,3,4,5,6,7,8,9] => [11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 11
[11,10,1,2,3,4,5,6,7,8,9] => [6,5]
=> [2,2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? = 5
[10,9,8,7,6,5,4,3,2,1,11] => [2,2,2,2,2,1]
=> [6,5]
=> [[1,2,3,4,5,11],[6,7,8,9,10]]
=> ? = 1
[1,11,10,9,8,7,6,5,4,3,2] => [2,2,2,2,2,1]
=> [6,5]
=> [[1,2,3,4,5,11],[6,7,8,9,10]]
=> ? = 1
[2,3,4,5,6,7,8,9,10,11,1] => [11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 11
[2,4,6,8,10,12,1,3,5,7,9,11] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
[2,4,6,8,11,12,1,3,5,7,9,10] => [9,3]
=> [2,2,2,1,1,1,1,1,1]
=> [[1,8],[2,10],[3,12],[4],[5],[6],[7],[9],[11]]
=> ? = 3
[2,4,6,9,10,12,1,3,5,7,8,11] => [7,5]
=> [2,2,2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5,10],[7,12],[9],[11]]
=> ? = 5
[2,4,6,9,11,12,1,3,5,7,8,10] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
[2,4,6,10,11,12,1,3,5,7,8,9] => [7,5]
=> [2,2,2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5,10],[7,12],[9],[11]]
=> ? = 5
[2,4,7,8,11,12,1,3,5,6,9,10] => [6,3,3]
=> [3,3,3,1,1,1]
=> [[1,5,6],[2,8,9],[3,11,12],[4],[7],[10]]
=> ? = 3
[2,4,7,9,10,12,1,3,5,6,8,11] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
[2,4,7,9,11,12,1,3,5,6,8,10] => [9,3]
=> [2,2,2,1,1,1,1,1,1]
=> [[1,8],[2,10],[3,12],[4],[5],[6],[7],[9],[11]]
=> ? = 3
[2,4,7,10,11,12,1,3,5,6,8,9] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
[2,4,8,9,10,12,1,3,5,6,7,11] => [10,2]
=> [2,2,1,1,1,1,1,1,1,1]
=> [[1,10],[2,12],[3],[4],[5],[6],[7],[8],[9],[11]]
=> ? = 2
[2,4,8,9,11,12,1,3,5,6,7,10] => [7,3,2]
=> [3,3,2,1,1,1,1]
=> [[1,6,9],[2,8,12],[3,11],[4],[5],[7],[10]]
=> ? = 2
[2,4,8,10,11,12,1,3,5,6,7,9] => [10,2]
=> [2,2,1,1,1,1,1,1,1,1]
=> [[1,10],[2,12],[3],[4],[5],[6],[7],[8],[9],[11]]
=> ? = 2
[2,4,9,10,11,12,1,3,5,6,7,8] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
[2,5,6,8,10,12,1,3,4,7,9,11] => [7,5]
=> [2,2,2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5,10],[7,12],[9],[11]]
=> ? = 5
[2,5,6,8,11,12,1,3,4,7,9,10] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
[2,5,6,9,10,12,1,3,4,7,8,11] => [5,5,2]
=> [3,3,2,2,2]
=> [[1,2,9],[3,4,12],[5,6],[7,8],[10,11]]
=> ? = 2
[2,5,6,9,11,12,1,3,4,7,8,10] => [10,2]
=> [2,2,1,1,1,1,1,1,1,1]
=> [[1,10],[2,12],[3],[4],[5],[6],[7],[8],[9],[11]]
=> ? = 2
[2,5,6,10,11,12,1,3,4,7,8,9] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
[2,5,7,8,10,12,1,3,4,6,9,11] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
[2,5,7,8,11,12,1,3,4,6,9,10] => [9,3]
=> [2,2,2,1,1,1,1,1,1]
=> [[1,8],[2,10],[3,12],[4],[5],[6],[7],[9],[11]]
=> ? = 3
[2,5,7,9,10,12,1,3,4,6,8,11] => [10,2]
=> [2,2,1,1,1,1,1,1,1,1]
=> [[1,10],[2,12],[3],[4],[5],[6],[7],[8],[9],[11]]
=> ? = 2
[2,5,7,9,11,12,1,3,4,6,8,10] => [7,3,2]
=> [3,3,2,1,1,1,1]
=> [[1,6,9],[2,8,12],[3,11],[4],[5],[7],[10]]
=> ? = 2
[2,5,7,10,11,12,1,3,4,6,8,9] => [7,5]
=> [2,2,2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5,10],[7,12],[9],[11]]
=> ? = 5
[2,5,8,9,10,12,1,3,4,6,7,11] => [8,2,2]
=> [3,3,1,1,1,1,1,1]
=> [[1,8,9],[2,11,12],[3],[4],[5],[6],[7],[10]]
=> ? = 2
[2,5,8,10,11,12,1,3,4,6,7,9] => [5,5,2]
=> [3,3,2,2,2]
=> [[1,2,9],[3,4,12],[5,6],[7,8],[10,11]]
=> ? = 2
[2,5,9,10,11,12,1,3,4,6,7,8] => [7,5]
=> [2,2,2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5,10],[7,12],[9],[11]]
=> ? = 5
[2,6,7,8,10,12,1,3,4,5,9,11] => [10,2]
=> [2,2,1,1,1,1,1,1,1,1]
=> [[1,10],[2,12],[3],[4],[5],[6],[7],[8],[9],[11]]
=> ? = 2
[2,6,7,8,11,12,1,3,4,5,9,10] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
[2,6,7,9,10,12,1,3,4,5,8,11] => [8,2,2]
=> [3,3,1,1,1,1,1,1]
=> [[1,8,9],[2,11,12],[3],[4],[5],[6],[7],[10]]
=> ? = 2
[2,6,7,9,11,12,1,3,4,5,8,10] => [10,2]
=> [2,2,1,1,1,1,1,1,1,1]
=> [[1,10],[2,12],[3],[4],[5],[6],[7],[8],[9],[11]]
=> ? = 2
[2,6,7,10,11,12,1,3,4,5,8,9] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
[2,6,8,9,10,12,1,3,4,5,7,11] => [6,2,2,2]
=> [4,4,1,1,1,1]
=> [[1,6,7,8],[2,10,11,12],[3],[4],[5],[9]]
=> ? = 2
[2,6,8,9,11,12,1,3,4,5,7,10] => [8,2,2]
=> [3,3,1,1,1,1,1,1]
=> [[1,8,9],[2,11,12],[3],[4],[5],[6],[7],[10]]
=> ? = 2
[2,6,8,10,11,12,1,3,4,5,7,9] => [10,2]
=> [2,2,1,1,1,1,1,1,1,1]
=> [[1,10],[2,12],[3],[4],[5],[6],[7],[8],[9],[11]]
=> ? = 2
[2,6,9,10,11,12,1,3,4,5,7,8] => [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
Description
The row containing the largest entry of a standard tableau.
Matching statistic: St000382
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 83% ●values known / values provided: 92%●distinct values known / distinct values provided: 83%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 83% ●values known / values provided: 92%●distinct values known / distinct values provided: 83%
Values
[1,2] => [1,1]
=> [[1],[2]]
=> [1,1] => 1
[2,1] => [2]
=> [[1,2]]
=> [2] => 2
[1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => 1
[1,3,2] => [2,1]
=> [[1,3],[2]]
=> [1,2] => 1
[2,1,3] => [2,1]
=> [[1,3],[2]]
=> [1,2] => 1
[2,3,1] => [3]
=> [[1,2,3]]
=> [3] => 3
[3,1,2] => [3]
=> [[1,2,3]]
=> [3] => 3
[3,2,1] => [2,1]
=> [[1,3],[2]]
=> [1,2] => 1
[1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => 1
[1,2,4,3] => [2,1,1]
=> [[1,4],[2],[3]]
=> [1,1,2] => 1
[1,3,2,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> [1,1,2] => 1
[1,3,4,2] => [3,1]
=> [[1,3,4],[2]]
=> [1,3] => 1
[1,4,2,3] => [3,1]
=> [[1,3,4],[2]]
=> [1,3] => 1
[1,4,3,2] => [2,1,1]
=> [[1,4],[2],[3]]
=> [1,1,2] => 1
[2,1,3,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> [1,1,2] => 1
[2,1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> [2,2] => 2
[2,3,1,4] => [3,1]
=> [[1,3,4],[2]]
=> [1,3] => 1
[2,3,4,1] => [4]
=> [[1,2,3,4]]
=> [4] => 4
[2,4,1,3] => [4]
=> [[1,2,3,4]]
=> [4] => 4
[2,4,3,1] => [3,1]
=> [[1,3,4],[2]]
=> [1,3] => 1
[3,1,2,4] => [3,1]
=> [[1,3,4],[2]]
=> [1,3] => 1
[3,1,4,2] => [4]
=> [[1,2,3,4]]
=> [4] => 4
[3,2,1,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> [1,1,2] => 1
[3,2,4,1] => [3,1]
=> [[1,3,4],[2]]
=> [1,3] => 1
[3,4,1,2] => [2,2]
=> [[1,2],[3,4]]
=> [2,2] => 2
[3,4,2,1] => [4]
=> [[1,2,3,4]]
=> [4] => 4
[4,1,2,3] => [4]
=> [[1,2,3,4]]
=> [4] => 4
[4,1,3,2] => [3,1]
=> [[1,3,4],[2]]
=> [1,3] => 1
[4,2,1,3] => [3,1]
=> [[1,3,4],[2]]
=> [1,3] => 1
[4,2,3,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> [1,1,2] => 1
[4,3,1,2] => [4]
=> [[1,2,3,4]]
=> [4] => 4
[4,3,2,1] => [2,2]
=> [[1,2],[3,4]]
=> [2,2] => 2
[1,2,3,4,5] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => 1
[1,2,3,5,4] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [1,1,1,2] => 1
[1,2,4,3,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [1,1,1,2] => 1
[1,2,4,5,3] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [1,1,3] => 1
[1,2,5,3,4] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [1,1,3] => 1
[1,2,5,4,3] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [1,1,1,2] => 1
[1,3,2,4,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [1,1,1,2] => 1
[1,3,2,5,4] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [1,2,2] => 1
[1,3,4,2,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [1,1,3] => 1
[1,3,4,5,2] => [4,1]
=> [[1,3,4,5],[2]]
=> [1,4] => 1
[1,3,5,2,4] => [4,1]
=> [[1,3,4,5],[2]]
=> [1,4] => 1
[1,3,5,4,2] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [1,1,3] => 1
[1,4,2,3,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [1,1,3] => 1
[1,4,2,5,3] => [4,1]
=> [[1,3,4,5],[2]]
=> [1,4] => 1
[1,4,3,2,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [1,1,1,2] => 1
[1,4,3,5,2] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [1,1,3] => 1
[1,4,5,2,3] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [1,2,2] => 1
[1,4,5,3,2] => [4,1]
=> [[1,3,4,5],[2]]
=> [1,4] => 1
[1,2,3,4,5,6,7,8,9,11,10] => [2,1,1,1,1,1,1,1,1,1]
=> [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? => ? = 1
[2,1,3,4,5,6,7,8,9,10,11] => [2,1,1,1,1,1,1,1,1,1]
=> [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? => ? = 1
[11,1,2,3,4,5,6,7,8,9,10] => [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? => ? = 11
[10,1,2,3,4,5,6,7,8,11,9] => [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? => ? = 11
[11,1,2,3,4,5,6,7,10,8,9] => [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? => ? = 11
[11,1,2,3,4,5,6,10,7,8,9] => [9,2]
=> [[1,2,5,6,7,8,9,10,11],[3,4]]
=> ? => ? = 2
[10,1,2,3,4,5,11,6,7,8,9] => [8,3]
=> [[1,2,3,7,8,9,10,11],[4,5,6]]
=> ? => ? = 3
[11,1,2,3,4,10,5,6,7,8,9] => [8,3]
=> [[1,2,3,7,8,9,10,11],[4,5,6]]
=> ? => ? = 3
[11,1,2,3,10,4,5,6,7,8,9] => [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? => ? = 11
[10,1,2,11,3,4,5,6,7,8,9] => [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? => ? = 11
[11,1,10,2,3,4,5,6,7,8,9] => [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? => ? = 11
[11,10,1,2,3,4,5,6,7,8,9] => [6,5]
=> [[1,2,3,4,5,11],[6,7,8,9,10]]
=> ? => ? = 5
[10,9,8,7,6,5,4,3,2,1,11] => [2,2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? = 1
[1,11,10,9,8,7,6,5,4,3,2] => [2,2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? = 1
[2,3,4,5,6,7,8,9,10,11,1] => [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? => ? = 11
[2,4,6,8,10,12,1,3,5,7,9,11] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [12] => ? = 12
[2,4,6,8,11,12,1,3,5,7,9,10] => [9,3]
=> [[1,2,3,7,8,9,10,11,12],[4,5,6]]
=> ? => ? = 3
[2,4,6,9,10,12,1,3,5,7,8,11] => [7,5]
=> [[1,2,3,4,5,11,12],[6,7,8,9,10]]
=> ? => ? = 5
[2,4,6,9,11,12,1,3,5,7,8,10] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [12] => ? = 12
[2,4,6,10,11,12,1,3,5,7,8,9] => [7,5]
=> [[1,2,3,4,5,11,12],[6,7,8,9,10]]
=> ? => ? = 5
[2,4,7,8,11,12,1,3,5,6,9,10] => [6,3,3]
=> [[1,2,3,10,11,12],[4,5,6],[7,8,9]]
=> ? => ? = 3
[2,4,7,9,10,12,1,3,5,6,8,11] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [12] => ? = 12
[2,4,7,9,11,12,1,3,5,6,8,10] => [9,3]
=> [[1,2,3,7,8,9,10,11,12],[4,5,6]]
=> ? => ? = 3
[2,4,7,10,11,12,1,3,5,6,8,9] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [12] => ? = 12
[2,4,8,9,10,12,1,3,5,6,7,11] => [10,2]
=> [[1,2,5,6,7,8,9,10,11,12],[3,4]]
=> ? => ? = 2
[2,4,8,9,11,12,1,3,5,6,7,10] => [7,3,2]
=> [[1,2,5,9,10,11,12],[3,4,8],[6,7]]
=> ? => ? = 2
[2,4,8,10,11,12,1,3,5,6,7,9] => [10,2]
=> [[1,2,5,6,7,8,9,10,11,12],[3,4]]
=> ? => ? = 2
[2,4,9,10,11,12,1,3,5,6,7,8] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [12] => ? = 12
[2,5,6,8,10,12,1,3,4,7,9,11] => [7,5]
=> [[1,2,3,4,5,11,12],[6,7,8,9,10]]
=> ? => ? = 5
[2,5,6,8,11,12,1,3,4,7,9,10] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [12] => ? = 12
[2,5,6,9,10,12,1,3,4,7,8,11] => [5,5,2]
=> [[1,2,5,6,7],[3,4,10,11,12],[8,9]]
=> ? => ? = 2
[2,5,6,9,11,12,1,3,4,7,8,10] => [10,2]
=> [[1,2,5,6,7,8,9,10,11,12],[3,4]]
=> ? => ? = 2
[2,5,6,10,11,12,1,3,4,7,8,9] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [12] => ? = 12
[2,5,7,8,10,12,1,3,4,6,9,11] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [12] => ? = 12
[2,5,7,8,11,12,1,3,4,6,9,10] => [9,3]
=> [[1,2,3,7,8,9,10,11,12],[4,5,6]]
=> ? => ? = 3
[2,5,7,9,10,12,1,3,4,6,8,11] => [10,2]
=> [[1,2,5,6,7,8,9,10,11,12],[3,4]]
=> ? => ? = 2
[2,5,7,9,11,12,1,3,4,6,8,10] => [7,3,2]
=> [[1,2,5,9,10,11,12],[3,4,8],[6,7]]
=> ? => ? = 2
[2,5,7,10,11,12,1,3,4,6,8,9] => [7,5]
=> [[1,2,3,4,5,11,12],[6,7,8,9,10]]
=> ? => ? = 5
[2,5,8,9,10,12,1,3,4,6,7,11] => [8,2,2]
=> [[1,2,7,8,9,10,11,12],[3,4],[5,6]]
=> ? => ? = 2
[2,5,8,9,11,12,1,3,4,6,7,10] => [5,3,2,2]
=> [[1,2,7,11,12],[3,4,10],[5,6],[8,9]]
=> [2,2,3,5] => ? = 2
[2,5,8,10,11,12,1,3,4,6,7,9] => [5,5,2]
=> [[1,2,5,6,7],[3,4,10,11,12],[8,9]]
=> ? => ? = 2
[2,5,9,10,11,12,1,3,4,6,7,8] => [7,5]
=> [[1,2,3,4,5,11,12],[6,7,8,9,10]]
=> ? => ? = 5
[2,6,7,8,10,12,1,3,4,5,9,11] => [10,2]
=> [[1,2,5,6,7,8,9,10,11,12],[3,4]]
=> ? => ? = 2
[2,6,7,8,11,12,1,3,4,5,9,10] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [12] => ? = 12
[2,6,7,9,10,12,1,3,4,5,8,11] => [8,2,2]
=> [[1,2,7,8,9,10,11,12],[3,4],[5,6]]
=> ? => ? = 2
[2,6,7,9,11,12,1,3,4,5,8,10] => [10,2]
=> [[1,2,5,6,7,8,9,10,11,12],[3,4]]
=> ? => ? = 2
[2,6,7,10,11,12,1,3,4,5,8,9] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [12] => ? = 12
[2,6,8,9,10,12,1,3,4,5,7,11] => [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ? => ? = 2
[2,6,8,9,11,12,1,3,4,5,7,10] => [8,2,2]
=> [[1,2,7,8,9,10,11,12],[3,4],[5,6]]
=> ? => ? = 2
[2,6,8,10,11,12,1,3,4,5,7,9] => [10,2]
=> [[1,2,5,6,7,8,9,10,11,12],[3,4]]
=> ? => ? = 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)
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 83% ●values known / values provided: 91%●distinct values known / distinct values provided: 83%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 83% ●values known / values provided: 91%●distinct values known / distinct values provided: 83%
Values
[1,2] => [1,1]
=> 110 => [2,1] => 1
[2,1] => [2]
=> 100 => [1,2] => 2
[1,2,3] => [1,1,1]
=> 1110 => [3,1] => 1
[1,3,2] => [2,1]
=> 1010 => [1,1,1,1] => 1
[2,1,3] => [2,1]
=> 1010 => [1,1,1,1] => 1
[2,3,1] => [3]
=> 1000 => [1,3] => 3
[3,1,2] => [3]
=> 1000 => [1,3] => 3
[3,2,1] => [2,1]
=> 1010 => [1,1,1,1] => 1
[1,2,3,4] => [1,1,1,1]
=> 11110 => [4,1] => 1
[1,2,4,3] => [2,1,1]
=> 10110 => [1,1,2,1] => 1
[1,3,2,4] => [2,1,1]
=> 10110 => [1,1,2,1] => 1
[1,3,4,2] => [3,1]
=> 10010 => [1,2,1,1] => 1
[1,4,2,3] => [3,1]
=> 10010 => [1,2,1,1] => 1
[1,4,3,2] => [2,1,1]
=> 10110 => [1,1,2,1] => 1
[2,1,3,4] => [2,1,1]
=> 10110 => [1,1,2,1] => 1
[2,1,4,3] => [2,2]
=> 1100 => [2,2] => 2
[2,3,1,4] => [3,1]
=> 10010 => [1,2,1,1] => 1
[2,3,4,1] => [4]
=> 10000 => [1,4] => 4
[2,4,1,3] => [4]
=> 10000 => [1,4] => 4
[2,4,3,1] => [3,1]
=> 10010 => [1,2,1,1] => 1
[3,1,2,4] => [3,1]
=> 10010 => [1,2,1,1] => 1
[3,1,4,2] => [4]
=> 10000 => [1,4] => 4
[3,2,1,4] => [2,1,1]
=> 10110 => [1,1,2,1] => 1
[3,2,4,1] => [3,1]
=> 10010 => [1,2,1,1] => 1
[3,4,1,2] => [2,2]
=> 1100 => [2,2] => 2
[3,4,2,1] => [4]
=> 10000 => [1,4] => 4
[4,1,2,3] => [4]
=> 10000 => [1,4] => 4
[4,1,3,2] => [3,1]
=> 10010 => [1,2,1,1] => 1
[4,2,1,3] => [3,1]
=> 10010 => [1,2,1,1] => 1
[4,2,3,1] => [2,1,1]
=> 10110 => [1,1,2,1] => 1
[4,3,1,2] => [4]
=> 10000 => [1,4] => 4
[4,3,2,1] => [2,2]
=> 1100 => [2,2] => 2
[1,2,3,4,5] => [1,1,1,1,1]
=> 111110 => [5,1] => 1
[1,2,3,5,4] => [2,1,1,1]
=> 101110 => [1,1,3,1] => 1
[1,2,4,3,5] => [2,1,1,1]
=> 101110 => [1,1,3,1] => 1
[1,2,4,5,3] => [3,1,1]
=> 100110 => [1,2,2,1] => 1
[1,2,5,3,4] => [3,1,1]
=> 100110 => [1,2,2,1] => 1
[1,2,5,4,3] => [2,1,1,1]
=> 101110 => [1,1,3,1] => 1
[1,3,2,4,5] => [2,1,1,1]
=> 101110 => [1,1,3,1] => 1
[1,3,2,5,4] => [2,2,1]
=> 11010 => [2,1,1,1] => 1
[1,3,4,2,5] => [3,1,1]
=> 100110 => [1,2,2,1] => 1
[1,3,4,5,2] => [4,1]
=> 100010 => [1,3,1,1] => 1
[1,3,5,2,4] => [4,1]
=> 100010 => [1,3,1,1] => 1
[1,3,5,4,2] => [3,1,1]
=> 100110 => [1,2,2,1] => 1
[1,4,2,3,5] => [3,1,1]
=> 100110 => [1,2,2,1] => 1
[1,4,2,5,3] => [4,1]
=> 100010 => [1,3,1,1] => 1
[1,4,3,2,5] => [2,1,1,1]
=> 101110 => [1,1,3,1] => 1
[1,4,3,5,2] => [3,1,1]
=> 100110 => [1,2,2,1] => 1
[1,4,5,2,3] => [2,2,1]
=> 11010 => [2,1,1,1] => 1
[1,4,5,3,2] => [4,1]
=> 100010 => [1,3,1,1] => 1
[2,1,4,3,6,5,8,7,10,9,12,11] => [2,2,2,2,2,2]
=> 11111100 => [6,2] => ? = 2
[6,5,4,3,2,1,12,11,10,9,8,7] => [2,2,2,2,2,2]
=> 11111100 => [6,2] => ? = 2
[12,11,10,9,8,7,6,5,4,3,2,1] => [2,2,2,2,2,2]
=> 11111100 => [6,2] => ? = 2
[1,2,3,4,5,6,7,8,9,11,10] => [2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? => ? = 1
[2,1,3,4,5,6,7,8,9,10,11] => [2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? => ? = 1
[11,1,2,3,4,5,6,7,8,9,10] => [11]
=> 100000000000 => ? => ? = 11
[10,1,2,3,4,5,6,7,8,11,9] => [11]
=> 100000000000 => ? => ? = 11
[11,1,2,3,4,5,6,7,10,8,9] => [11]
=> 100000000000 => ? => ? = 11
[11,1,2,3,4,5,6,10,7,8,9] => [9,2]
=> 10000000100 => [1,7,1,2] => ? = 2
[10,1,2,3,4,5,11,6,7,8,9] => [8,3]
=> 1000001000 => ? => ? = 3
[11,1,2,3,4,10,5,6,7,8,9] => [8,3]
=> 1000001000 => ? => ? = 3
[11,1,2,3,10,4,5,6,7,8,9] => [11]
=> 100000000000 => ? => ? = 11
[10,1,2,11,3,4,5,6,7,8,9] => [11]
=> 100000000000 => ? => ? = 11
[11,1,10,2,3,4,5,6,7,8,9] => [11]
=> 100000000000 => ? => ? = 11
[11,10,1,2,3,4,5,6,7,8,9] => [6,5]
=> 10100000 => [1,1,1,5] => ? = 5
[2,3,4,5,6,7,8,9,10,11,1] => [11]
=> 100000000000 => ? => ? = 11
[2,4,6,8,10,12,1,3,5,7,9,11] => [12]
=> 1000000000000 => ? => ? = 12
[2,4,6,8,11,12,1,3,5,7,9,10] => [9,3]
=> 10000001000 => ? => ? = 3
[2,4,6,9,10,12,1,3,5,7,8,11] => [7,5]
=> 100100000 => [1,2,1,5] => ? = 5
[2,4,6,9,11,12,1,3,5,7,8,10] => [12]
=> 1000000000000 => ? => ? = 12
[2,4,6,10,11,12,1,3,5,7,8,9] => [7,5]
=> 100100000 => [1,2,1,5] => ? = 5
[2,4,7,8,10,12,1,3,5,6,9,11] => [6,6]
=> 11000000 => [2,6] => ? = 6
[2,4,7,8,11,12,1,3,5,6,9,10] => [6,3,3]
=> 100011000 => [1,3,2,3] => ? = 3
[2,4,7,9,10,12,1,3,5,6,8,11] => [12]
=> 1000000000000 => ? => ? = 12
[2,4,7,9,11,12,1,3,5,6,8,10] => [9,3]
=> 10000001000 => ? => ? = 3
[2,4,7,10,11,12,1,3,5,6,8,9] => [12]
=> 1000000000000 => ? => ? = 12
[2,4,8,9,10,12,1,3,5,6,7,11] => [10,2]
=> 100000000100 => ? => ? = 2
[2,4,8,9,11,12,1,3,5,6,7,10] => [7,3,2]
=> 1000010100 => ? => ? = 2
[2,4,8,10,11,12,1,3,5,6,7,9] => [10,2]
=> 100000000100 => ? => ? = 2
[2,4,9,10,11,12,1,3,5,6,7,8] => [12]
=> 1000000000000 => ? => ? = 12
[2,5,6,8,10,12,1,3,4,7,9,11] => [7,5]
=> 100100000 => [1,2,1,5] => ? = 5
[2,5,6,8,11,12,1,3,4,7,9,10] => [12]
=> 1000000000000 => ? => ? = 12
[2,5,6,9,11,12,1,3,4,7,8,10] => [10,2]
=> 100000000100 => ? => ? = 2
[2,5,6,10,11,12,1,3,4,7,8,9] => [12]
=> 1000000000000 => ? => ? = 12
[2,5,7,8,10,12,1,3,4,6,9,11] => [12]
=> 1000000000000 => ? => ? = 12
[2,5,7,8,11,12,1,3,4,6,9,10] => [9,3]
=> 10000001000 => ? => ? = 3
[2,5,7,9,10,12,1,3,4,6,8,11] => [10,2]
=> 100000000100 => ? => ? = 2
[2,5,7,9,11,12,1,3,4,6,8,10] => [7,3,2]
=> 1000010100 => ? => ? = 2
[2,5,7,10,11,12,1,3,4,6,8,9] => [7,5]
=> 100100000 => [1,2,1,5] => ? = 5
[2,5,8,9,10,12,1,3,4,6,7,11] => [8,2,2]
=> 10000001100 => ? => ? = 2
[2,5,8,9,11,12,1,3,4,6,7,10] => [5,3,2,2]
=> 100101100 => [1,2,1,1,2,2] => ? = 2
[2,5,9,10,11,12,1,3,4,6,7,8] => [7,5]
=> 100100000 => [1,2,1,5] => ? = 5
[2,6,7,8,10,12,1,3,4,5,9,11] => [10,2]
=> 100000000100 => ? => ? = 2
[2,6,7,8,11,12,1,3,4,5,9,10] => [12]
=> 1000000000000 => ? => ? = 12
[2,6,7,9,10,12,1,3,4,5,8,11] => [8,2,2]
=> 10000001100 => ? => ? = 2
[2,6,7,9,11,12,1,3,4,5,8,10] => [10,2]
=> 100000000100 => ? => ? = 2
[2,6,7,10,11,12,1,3,4,5,8,9] => [12]
=> 1000000000000 => ? => ? = 12
[2,6,8,9,10,12,1,3,4,5,7,11] => [6,2,2,2]
=> 1000011100 => ? => ? = 2
[2,6,8,9,11,12,1,3,4,5,7,10] => [8,2,2]
=> 10000001100 => ? => ? = 2
[2,6,8,10,11,12,1,3,4,5,7,9] => [10,2]
=> 100000000100 => ? => ? = 2
Description
The last part of an integer composition.
Matching statistic: St000745
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 83% ●values known / values provided: 91%●distinct values known / distinct values provided: 83%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 83% ●values known / values provided: 91%●distinct values known / distinct values provided: 83%
Values
[1,2] => [1,1]
=> [[1],[2]]
=> [[1,2]]
=> 1
[2,1] => [2]
=> [[1,2]]
=> [[1],[2]]
=> 2
[1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> 1
[1,3,2] => [2,1]
=> [[1,3],[2]]
=> [[1,2],[3]]
=> 1
[2,1,3] => [2,1]
=> [[1,3],[2]]
=> [[1,2],[3]]
=> 1
[2,3,1] => [3]
=> [[1,2,3]]
=> [[1],[2],[3]]
=> 3
[3,1,2] => [3]
=> [[1,2,3]]
=> [[1],[2],[3]]
=> 3
[3,2,1] => [2,1]
=> [[1,3],[2]]
=> [[1,2],[3]]
=> 1
[1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> 1
[1,2,4,3] => [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 1
[1,3,2,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 1
[1,3,4,2] => [3,1]
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 1
[1,4,2,3] => [3,1]
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 1
[1,4,3,2] => [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 1
[2,1,3,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 1
[2,1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 2
[2,3,1,4] => [3,1]
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 1
[2,3,4,1] => [4]
=> [[1,2,3,4]]
=> [[1],[2],[3],[4]]
=> 4
[2,4,1,3] => [4]
=> [[1,2,3,4]]
=> [[1],[2],[3],[4]]
=> 4
[2,4,3,1] => [3,1]
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 1
[3,1,2,4] => [3,1]
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 1
[3,1,4,2] => [4]
=> [[1,2,3,4]]
=> [[1],[2],[3],[4]]
=> 4
[3,2,1,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 1
[3,2,4,1] => [3,1]
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 1
[3,4,1,2] => [2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 2
[3,4,2,1] => [4]
=> [[1,2,3,4]]
=> [[1],[2],[3],[4]]
=> 4
[4,1,2,3] => [4]
=> [[1,2,3,4]]
=> [[1],[2],[3],[4]]
=> 4
[4,1,3,2] => [3,1]
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 1
[4,2,1,3] => [3,1]
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 1
[4,2,3,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 1
[4,3,1,2] => [4]
=> [[1,2,3,4]]
=> [[1],[2],[3],[4]]
=> 4
[4,3,2,1] => [2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 2
[1,2,3,4,5] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [[1,2,3,4,5]]
=> 1
[1,2,3,5,4] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> 1
[1,2,4,3,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> 1
[1,2,4,5,3] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> 1
[1,2,5,3,4] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> 1
[1,2,5,4,3] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> 1
[1,3,2,4,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> 1
[1,3,2,5,4] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [[1,2,4],[3,5]]
=> 1
[1,3,4,2,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> 1
[1,3,4,5,2] => [4,1]
=> [[1,3,4,5],[2]]
=> [[1,2],[3],[4],[5]]
=> 1
[1,3,5,2,4] => [4,1]
=> [[1,3,4,5],[2]]
=> [[1,2],[3],[4],[5]]
=> 1
[1,3,5,4,2] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> 1
[1,4,2,3,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> 1
[1,4,2,5,3] => [4,1]
=> [[1,3,4,5],[2]]
=> [[1,2],[3],[4],[5]]
=> 1
[1,4,3,2,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> 1
[1,4,3,5,2] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> 1
[1,4,5,2,3] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [[1,2,4],[3,5]]
=> 1
[1,4,5,3,2] => [4,1]
=> [[1,3,4,5],[2]]
=> [[1,2],[3],[4],[5]]
=> 1
[1,2,3,4,5,6,7,8,9,11,10] => [2,1,1,1,1,1,1,1,1,1]
=> [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> [[1,2,3,4,5,6,7,8,9,10],[11]]
=> ? = 1
[2,1,3,4,5,6,7,8,9,10,11] => [2,1,1,1,1,1,1,1,1,1]
=> [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> [[1,2,3,4,5,6,7,8,9,10],[11]]
=> ? = 1
[11,1,2,3,4,5,6,7,8,9,10] => [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 11
[10,1,2,3,4,5,6,7,8,11,9] => [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 11
[11,1,2,3,4,5,6,7,10,8,9] => [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 11
[11,1,2,3,4,5,6,10,7,8,9] => [9,2]
=> [[1,2,5,6,7,8,9,10,11],[3,4]]
=> [[1,3],[2,4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 2
[10,1,2,3,4,5,11,6,7,8,9] => [8,3]
=> [[1,2,3,7,8,9,10,11],[4,5,6]]
=> [[1,4],[2,5],[3,6],[7],[8],[9],[10],[11]]
=> ? = 3
[11,1,2,3,4,10,5,6,7,8,9] => [8,3]
=> [[1,2,3,7,8,9,10,11],[4,5,6]]
=> [[1,4],[2,5],[3,6],[7],[8],[9],[10],[11]]
=> ? = 3
[11,1,2,3,10,4,5,6,7,8,9] => [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 11
[10,1,2,11,3,4,5,6,7,8,9] => [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 11
[11,1,10,2,3,4,5,6,7,8,9] => [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 11
[11,10,1,2,3,4,5,6,7,8,9] => [6,5]
=> [[1,2,3,4,5,11],[6,7,8,9,10]]
=> [[1,6],[2,7],[3,8],[4,9],[5,10],[11]]
=> ? = 5
[10,9,8,7,6,5,4,3,2,1,11] => [2,2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8,11],[10]]
=> [[1,2,4,6,8,10],[3,5,7,9,11]]
=> ? = 1
[1,11,10,9,8,7,6,5,4,3,2] => [2,2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8,11],[10]]
=> [[1,2,4,6,8,10],[3,5,7,9,11]]
=> ? = 1
[2,3,4,5,6,7,8,9,10,11,1] => [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 11
[2,4,6,8,10,12,1,3,5,7,9,11] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
[2,4,6,8,11,12,1,3,5,7,9,10] => [9,3]
=> [[1,2,3,7,8,9,10,11,12],[4,5,6]]
=> ?
=> ? = 3
[2,4,6,9,10,12,1,3,5,7,8,11] => [7,5]
=> [[1,2,3,4,5,11,12],[6,7,8,9,10]]
=> ?
=> ? = 5
[2,4,6,9,11,12,1,3,5,7,8,10] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
[2,4,6,10,11,12,1,3,5,7,8,9] => [7,5]
=> [[1,2,3,4,5,11,12],[6,7,8,9,10]]
=> ?
=> ? = 5
[2,4,7,8,11,12,1,3,5,6,9,10] => [6,3,3]
=> [[1,2,3,10,11,12],[4,5,6],[7,8,9]]
=> ?
=> ? = 3
[2,4,7,9,10,12,1,3,5,6,8,11] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
[2,4,7,9,11,12,1,3,5,6,8,10] => [9,3]
=> [[1,2,3,7,8,9,10,11,12],[4,5,6]]
=> ?
=> ? = 3
[2,4,7,10,11,12,1,3,5,6,8,9] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
[2,4,8,9,10,12,1,3,5,6,7,11] => [10,2]
=> [[1,2,5,6,7,8,9,10,11,12],[3,4]]
=> [[1,3],[2,4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 2
[2,4,8,9,11,12,1,3,5,6,7,10] => [7,3,2]
=> [[1,2,5,9,10,11,12],[3,4,8],[6,7]]
=> ?
=> ? = 2
[2,4,8,10,11,12,1,3,5,6,7,9] => [10,2]
=> [[1,2,5,6,7,8,9,10,11,12],[3,4]]
=> [[1,3],[2,4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 2
[2,4,9,10,11,12,1,3,5,6,7,8] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
[2,5,6,8,10,12,1,3,4,7,9,11] => [7,5]
=> [[1,2,3,4,5,11,12],[6,7,8,9,10]]
=> ?
=> ? = 5
[2,5,6,8,11,12,1,3,4,7,9,10] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
[2,5,6,9,10,12,1,3,4,7,8,11] => [5,5,2]
=> [[1,2,5,6,7],[3,4,10,11,12],[8,9]]
=> [[1,3,8],[2,4,9],[5,10],[6,11],[7,12]]
=> ? = 2
[2,5,6,9,11,12,1,3,4,7,8,10] => [10,2]
=> [[1,2,5,6,7,8,9,10,11,12],[3,4]]
=> [[1,3],[2,4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 2
[2,5,6,10,11,12,1,3,4,7,8,9] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
[2,5,7,8,10,12,1,3,4,6,9,11] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
[2,5,7,8,11,12,1,3,4,6,9,10] => [9,3]
=> [[1,2,3,7,8,9,10,11,12],[4,5,6]]
=> ?
=> ? = 3
[2,5,7,9,10,12,1,3,4,6,8,11] => [10,2]
=> [[1,2,5,6,7,8,9,10,11,12],[3,4]]
=> [[1,3],[2,4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 2
[2,5,7,9,11,12,1,3,4,6,8,10] => [7,3,2]
=> [[1,2,5,9,10,11,12],[3,4,8],[6,7]]
=> ?
=> ? = 2
[2,5,7,10,11,12,1,3,4,6,8,9] => [7,5]
=> [[1,2,3,4,5,11,12],[6,7,8,9,10]]
=> ?
=> ? = 5
[2,5,8,9,10,12,1,3,4,6,7,11] => [8,2,2]
=> [[1,2,7,8,9,10,11,12],[3,4],[5,6]]
=> ?
=> ? = 2
[2,5,8,9,11,12,1,3,4,6,7,10] => [5,3,2,2]
=> [[1,2,7,11,12],[3,4,10],[5,6],[8,9]]
=> [[1,3,5,8],[2,4,6,9],[7,10],[11],[12]]
=> ? = 2
[2,5,8,10,11,12,1,3,4,6,7,9] => [5,5,2]
=> [[1,2,5,6,7],[3,4,10,11,12],[8,9]]
=> [[1,3,8],[2,4,9],[5,10],[6,11],[7,12]]
=> ? = 2
[2,5,9,10,11,12,1,3,4,6,7,8] => [7,5]
=> [[1,2,3,4,5,11,12],[6,7,8,9,10]]
=> ?
=> ? = 5
[2,6,7,8,10,12,1,3,4,5,9,11] => [10,2]
=> [[1,2,5,6,7,8,9,10,11,12],[3,4]]
=> [[1,3],[2,4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 2
[2,6,7,8,11,12,1,3,4,5,9,10] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
[2,6,7,9,10,12,1,3,4,5,8,11] => [8,2,2]
=> [[1,2,7,8,9,10,11,12],[3,4],[5,6]]
=> ?
=> ? = 2
[2,6,7,9,11,12,1,3,4,5,8,10] => [10,2]
=> [[1,2,5,6,7,8,9,10,11,12],[3,4]]
=> [[1,3],[2,4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 2
[2,6,7,10,11,12,1,3,4,5,8,9] => [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
[2,6,8,9,10,12,1,3,4,5,7,11] => [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ?
=> ? = 2
[2,6,8,9,11,12,1,3,4,5,7,10] => [8,2,2]
=> [[1,2,7,8,9,10,11,12],[3,4],[5,6]]
=> ?
=> ? = 2
[2,6,8,10,11,12,1,3,4,5,7,9] => [10,2]
=> [[1,2,5,6,7,8,9,10,11,12],[3,4]]
=> [[1,3],[2,4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 2
Description
The index of the last row whose first entry is the row number in a standard Young tableau.
Matching statistic: St001038
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001038: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 88%●distinct values known / distinct values provided: 67%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001038: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 88%●distinct values known / distinct values provided: 67%
Values
[1,2] => [1,1]
=> [1,1,0,0]
=> 1
[2,1] => [2]
=> [1,0,1,0]
=> 2
[1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,3,1] => [3]
=> [1,0,1,0,1,0]
=> 3
[3,1,2] => [3]
=> [1,0,1,0,1,0]
=> 3
[3,2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,3,4,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,4,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,1,3,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,1,4,3] => [2,2]
=> [1,1,1,0,0,0]
=> 2
[2,3,1,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[2,3,4,1] => [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[2,4,1,3] => [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[2,4,3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[3,1,2,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[3,1,4,2] => [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[3,2,4,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[3,4,1,2] => [2,2]
=> [1,1,1,0,0,0]
=> 2
[3,4,2,1] => [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[4,1,2,3] => [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[4,1,3,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[4,2,1,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[4,3,1,2] => [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[4,3,2,1] => [2,2]
=> [1,1,1,0,0,0]
=> 2
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,2,3,5,4] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,2,4,3,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,2,4,5,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,2,5,3,4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,2,5,4,3] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,3,2,4,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,3,2,5,4] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,3,4,2,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,3,4,5,2] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,3,5,2,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,3,5,4,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,4,2,3,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,4,2,5,3] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,4,3,2,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,4,3,5,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,4,5,2,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,4,5,3,2] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,2,3,4,5,6,7,8] => [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
[9,1,2,3,4,5,6,7,8] => [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[8,1,2,3,4,5,6,9,7] => [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[7,1,2,3,4,5,9,6,8] => [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[9,1,2,3,4,5,8,6,7] => [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[7,1,2,3,4,5,8,9,6] => [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[6,1,2,3,4,9,5,7,8] => [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[6,1,2,3,4,8,5,9,7] => [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[9,1,2,3,4,7,5,6,8] => [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[8,1,2,3,4,7,5,9,6] => [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[6,1,2,3,4,7,9,5,8] => [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[6,1,2,3,4,9,8,5,7] => [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[9,1,2,3,4,7,8,5,6] => [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[6,1,2,3,4,7,8,9,5] => [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[5,1,2,3,9,4,6,7,8] => [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[5,1,2,3,8,4,6,9,7] => [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[9,1,2,3,6,4,5,7,8] => [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[5,1,2,3,6,9,8,4,7] => [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[5,1,2,3,6,7,8,9,4] => [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[4,1,2,8,3,5,6,9,7] => [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[4,1,2,9,3,5,8,6,7] => [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[4,1,2,5,9,3,6,7,8] => [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[4,1,2,5,6,7,9,3,8] => [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[4,1,2,9,6,7,8,3,5] => [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[9,1,2,5,6,7,8,3,4] => [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[4,1,2,5,6,7,8,9,3] => [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[3,1,9,2,4,5,6,7,8] => [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[3,1,8,2,4,5,6,9,7] => [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[3,1,9,2,4,5,8,6,7] => [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[3,1,5,2,7,4,9,6,8] => [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[8,1,4,2,3,5,6,9,7] => [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[9,1,4,2,3,5,8,6,7] => [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[6,1,4,2,3,7,8,9,5] => [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[9,1,8,2,3,4,5,6,7] => [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[3,1,4,9,2,5,6,7,8] => [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[3,1,4,9,2,5,8,6,7] => [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[9,1,4,5,2,3,6,7,8] => [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[7,1,4,5,2,3,8,9,6] => [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[7,1,9,6,2,3,4,5,8] => [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[9,1,8,6,2,3,4,5,7] => [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[7,1,8,9,2,3,4,5,6] => [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[3,1,8,9,7,2,4,5,6] => [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[8,1,4,5,6,2,3,9,7] => [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[8,1,4,9,7,2,3,5,6] => [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[8,1,9,7,6,2,3,4,5] => [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[7,1,4,5,6,2,8,9,3] => [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[3,1,4,5,6,9,2,7,8] => [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[3,1,4,5,6,8,2,9,7] => [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[9,1,4,5,6,7,2,3,8] => [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[8,1,4,5,6,7,2,9,3] => [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
Description
The minimal height of a column in the parallelogram polyomino associated with the Dyck path.
Matching statistic: St000657
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000657: Integer compositions ⟶ ℤResult quality: 75% ●values known / values provided: 84%●distinct values known / distinct values provided: 75%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000657: Integer compositions ⟶ ℤResult quality: 75% ●values known / values provided: 84%●distinct values known / distinct values provided: 75%
Values
[1,2] => [1,1]
=> [[1],[2]]
=> [1,1] => 1
[2,1] => [2]
=> [[1,2]]
=> [2] => 2
[1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => 1
[1,3,2] => [2,1]
=> [[1,2],[3]]
=> [2,1] => 1
[2,1,3] => [2,1]
=> [[1,2],[3]]
=> [2,1] => 1
[2,3,1] => [3]
=> [[1,2,3]]
=> [3] => 3
[3,1,2] => [3]
=> [[1,2,3]]
=> [3] => 3
[3,2,1] => [2,1]
=> [[1,2],[3]]
=> [2,1] => 1
[1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => 1
[1,2,4,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> [2,1,1] => 1
[1,3,2,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [2,1,1] => 1
[1,3,4,2] => [3,1]
=> [[1,2,3],[4]]
=> [3,1] => 1
[1,4,2,3] => [3,1]
=> [[1,2,3],[4]]
=> [3,1] => 1
[1,4,3,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> [2,1,1] => 1
[2,1,3,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [2,1,1] => 1
[2,1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> [2,2] => 2
[2,3,1,4] => [3,1]
=> [[1,2,3],[4]]
=> [3,1] => 1
[2,3,4,1] => [4]
=> [[1,2,3,4]]
=> [4] => 4
[2,4,1,3] => [4]
=> [[1,2,3,4]]
=> [4] => 4
[2,4,3,1] => [3,1]
=> [[1,2,3],[4]]
=> [3,1] => 1
[3,1,2,4] => [3,1]
=> [[1,2,3],[4]]
=> [3,1] => 1
[3,1,4,2] => [4]
=> [[1,2,3,4]]
=> [4] => 4
[3,2,1,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [2,1,1] => 1
[3,2,4,1] => [3,1]
=> [[1,2,3],[4]]
=> [3,1] => 1
[3,4,1,2] => [2,2]
=> [[1,2],[3,4]]
=> [2,2] => 2
[3,4,2,1] => [4]
=> [[1,2,3,4]]
=> [4] => 4
[4,1,2,3] => [4]
=> [[1,2,3,4]]
=> [4] => 4
[4,1,3,2] => [3,1]
=> [[1,2,3],[4]]
=> [3,1] => 1
[4,2,1,3] => [3,1]
=> [[1,2,3],[4]]
=> [3,1] => 1
[4,2,3,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> [2,1,1] => 1
[4,3,1,2] => [4]
=> [[1,2,3,4]]
=> [4] => 4
[4,3,2,1] => [2,2]
=> [[1,2],[3,4]]
=> [2,2] => 2
[1,2,3,4,5] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => 1
[1,2,3,5,4] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1] => 1
[1,2,4,3,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1] => 1
[1,2,4,5,3] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [3,1,1] => 1
[1,2,5,3,4] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [3,1,1] => 1
[1,2,5,4,3] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1] => 1
[1,3,2,4,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1] => 1
[1,3,2,5,4] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [2,2,1] => 1
[1,3,4,2,5] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [3,1,1] => 1
[1,3,4,5,2] => [4,1]
=> [[1,2,3,4],[5]]
=> [4,1] => 1
[1,3,5,2,4] => [4,1]
=> [[1,2,3,4],[5]]
=> [4,1] => 1
[1,3,5,4,2] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [3,1,1] => 1
[1,4,2,3,5] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [3,1,1] => 1
[1,4,2,5,3] => [4,1]
=> [[1,2,3,4],[5]]
=> [4,1] => 1
[1,4,3,2,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1] => 1
[1,4,3,5,2] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [3,1,1] => 1
[1,4,5,2,3] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [2,2,1] => 1
[1,4,5,3,2] => [4,1]
=> [[1,2,3,4],[5]]
=> [4,1] => 1
[2,1,4,3,6,5,8,7,10,9] => [2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [2,2,2,2,2] => ? = 2
[3,4,1,2,6,5,8,7,10,9] => [2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [2,2,2,2,2] => ? = 2
[4,3,2,1,6,5,8,7,10,9] => [2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [2,2,2,2,2] => ? = 2
[6,5,4,3,2,1,8,7,10,9] => [2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [2,2,2,2,2] => ? = 2
[8,9,5,6,3,4,10,1,2,7] => [2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [2,2,2,2,2] => ? = 2
[4,3,2,1,8,7,6,5,10,9] => [2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [2,2,2,2,2] => ? = 2
[8,7,6,5,4,3,2,1,10,9] => [2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [2,2,2,2,2] => ? = 2
[9,7,6,5,4,3,2,10,1,8] => [2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [2,2,2,2,2] => ? = 2
[10,8,9,6,7,4,5,2,3,1] => [2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [2,2,2,2,2] => ? = 2
[6,7,8,9,10,1,2,3,4,5] => [2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [2,2,2,2,2] => ? = 2
[7,3,2,8,9,10,1,4,5,6] => [2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [2,2,2,2,2] => ? = 2
[8,5,4,3,2,9,10,1,6,7] => [2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [2,2,2,2,2] => ? = 2
[6,5,4,3,2,1,10,9,8,7] => [2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [2,2,2,2,2] => ? = 2
[9,7,5,10,3,8,2,6,1,4] => [2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [2,2,2,2,2] => ? = 2
[4,3,2,1,10,9,8,7,6,5] => [2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [2,2,2,2,2] => ? = 2
[2,1,10,9,8,7,6,5,4,3] => [2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [2,2,2,2,2] => ? = 2
[9,10,8,7,6,5,4,3,1,2] => [2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [2,2,2,2,2] => ? = 2
[10,9,8,7,6,5,4,3,2,1] => [2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [2,2,2,2,2] => ? = 2
[3,4,2,1,6,5,8,7,10,9] => [4,2,2,2]
=> [[1,2,3,4],[5,6],[7,8],[9,10]]
=> [4,2,2,2] => ? = 2
[3,5,2,6,4,1,8,7,10,9] => [6,2,2]
=> [[1,2,3,4,5,6],[7,8],[9,10]]
=> [6,2,2] => ? = 2
[3,5,2,7,4,8,6,1,10,9] => [8,2]
=> [[1,2,3,4,5,6,7,8],[9,10]]
=> [8,2] => ? = 2
[3,5,2,7,4,9,6,10,8,1] => [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> [10] => ? = 10
[2,1,5,6,4,3,8,7,10,9] => [4,2,2,2]
=> [[1,2,3,4],[5,6],[7,8],[9,10]]
=> [4,2,2,2] => ? = 2
[4,5,6,3,2,1,8,7,10,9] => [4,2,2,2]
=> [[1,2,3,4],[5,6],[7,8],[9,10]]
=> [4,2,2,2] => ? = 2
[4,5,7,3,2,8,6,1,10,9] => [6,2,2]
=> [[1,2,3,4,5,6],[7,8],[9,10]]
=> [6,2,2] => ? = 2
[4,5,7,3,2,9,6,10,8,1] => [8,2]
=> [[1,2,3,4,5,6,7,8],[9,10]]
=> [8,2] => ? = 2
[2,1,5,7,4,8,6,3,10,9] => [6,2,2]
=> [[1,2,3,4,5,6],[7,8],[9,10]]
=> [6,2,2] => ? = 2
[4,6,7,3,8,5,2,1,10,9] => [8,2]
=> [[1,2,3,4,5,6,7,8],[9,10]]
=> [8,2] => ? = 2
[4,6,7,3,9,5,2,10,8,1] => [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> [10] => ? = 10
[2,1,5,7,4,9,6,10,8,3] => [8,2]
=> [[1,2,3,4,5,6,7,8],[9,10]]
=> [8,2] => ? = 2
[4,6,8,3,9,5,10,7,2,1] => [6,4]
=> [[1,2,3,4,5,6],[7,8,9,10]]
=> [6,4] => ? = 4
[2,1,4,3,7,8,6,5,10,9] => [4,2,2,2]
=> [[1,2,3,4],[5,6],[7,8],[9,10]]
=> [4,2,2,2] => ? = 2
[3,4,2,1,7,8,6,5,10,9] => [4,4,2]
=> [[1,2,3,4],[5,6,7,8],[9,10]]
=> [4,4,2] => ? = 2
[3,6,2,7,8,5,4,1,10,9] => [6,2,2]
=> [[1,2,3,4,5,6],[7,8],[9,10]]
=> [6,2,2] => ? = 2
[3,6,2,7,9,5,4,10,8,1] => [8,2]
=> [[1,2,3,4,5,6,7,8],[9,10]]
=> [8,2] => ? = 2
[2,1,6,7,8,5,4,3,10,9] => [4,2,2,2]
=> [[1,2,3,4],[5,6],[7,8],[9,10]]
=> [4,2,2,2] => ? = 2
[5,6,7,8,4,3,2,1,10,9] => [4,4,2]
=> [[1,2,3,4],[5,6,7,8],[9,10]]
=> [4,4,2] => ? = 2
[5,6,7,9,4,3,2,10,8,1] => [6,4]
=> [[1,2,3,4,5,6],[7,8,9,10]]
=> [6,4] => ? = 4
[2,1,6,7,9,5,4,10,8,3] => [6,2,2]
=> [[1,2,3,4,5,6],[7,8],[9,10]]
=> [6,2,2] => ? = 2
[5,6,8,9,4,3,10,7,2,1] => [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> [10] => ? = 10
[2,1,4,3,7,9,6,10,8,5] => [6,2,2]
=> [[1,2,3,4,5,6],[7,8],[9,10]]
=> [6,2,2] => ? = 2
[3,4,2,1,7,9,6,10,8,5] => [6,4]
=> [[1,2,3,4,5,6],[7,8,9,10]]
=> [6,4] => ? = 4
[3,6,2,8,9,5,10,7,4,1] => [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> [10] => ? = 10
[2,1,6,8,9,5,10,7,4,3] => [8,2]
=> [[1,2,3,4,5,6,7,8],[9,10]]
=> [8,2] => ? = 2
[5,7,8,9,4,10,6,3,2,1] => [8,2]
=> [[1,2,3,4,5,6,7,8],[9,10]]
=> [8,2] => ? = 2
[6,7,8,9,10,1,4,5,3,2] => [8,2]
=> [[1,2,3,4,5,6,7,8],[9,10]]
=> [8,2] => ? = 2
[6,7,8,9,10,1,4,5,2,3] => [4,4,2]
=> [[1,2,3,4],[5,6,7,8],[9,10]]
=> [4,4,2] => ? = 2
[6,7,8,9,10,1,3,4,5,2] => [8,2]
=> [[1,2,3,4,5,6,7,8],[9,10]]
=> [8,2] => ? = 2
[6,7,8,9,10,2,3,4,1,5] => [8,2]
=> [[1,2,3,4,5,6,7,8],[9,10]]
=> [8,2] => ? = 2
[2,1,4,3,6,5,9,10,8,7] => [4,2,2,2]
=> [[1,2,3,4],[5,6],[7,8],[9,10]]
=> [4,2,2,2] => ? = 2
Description
The smallest part of an integer composition.
Matching statistic: St001803
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St001803: Standard tableaux ⟶ ℤResult quality: 67% ●values known / values provided: 79%●distinct values known / distinct values provided: 67%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St001803: Standard tableaux ⟶ ℤResult quality: 67% ●values known / values provided: 79%●distinct values known / distinct values provided: 67%
Values
[1,2] => [1,1]
=> [2]
=> [[1,2]]
=> 0 = 1 - 1
[2,1] => [2]
=> [1,1]
=> [[1],[2]]
=> 1 = 2 - 1
[1,2,3] => [1,1,1]
=> [3]
=> [[1,2,3]]
=> 0 = 1 - 1
[1,3,2] => [2,1]
=> [2,1]
=> [[1,3],[2]]
=> 0 = 1 - 1
[2,1,3] => [2,1]
=> [2,1]
=> [[1,3],[2]]
=> 0 = 1 - 1
[2,3,1] => [3]
=> [1,1,1]
=> [[1],[2],[3]]
=> 2 = 3 - 1
[3,1,2] => [3]
=> [1,1,1]
=> [[1],[2],[3]]
=> 2 = 3 - 1
[3,2,1] => [2,1]
=> [2,1]
=> [[1,3],[2]]
=> 0 = 1 - 1
[1,2,3,4] => [1,1,1,1]
=> [4]
=> [[1,2,3,4]]
=> 0 = 1 - 1
[1,2,4,3] => [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 0 = 1 - 1
[1,3,2,4] => [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 0 = 1 - 1
[1,3,4,2] => [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 0 = 1 - 1
[1,4,2,3] => [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 0 = 1 - 1
[1,4,3,2] => [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 0 = 1 - 1
[2,1,3,4] => [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 0 = 1 - 1
[2,1,4,3] => [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 1 = 2 - 1
[2,3,1,4] => [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 0 = 1 - 1
[2,3,4,1] => [4]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 3 = 4 - 1
[2,4,1,3] => [4]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 3 = 4 - 1
[2,4,3,1] => [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 0 = 1 - 1
[3,1,2,4] => [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 0 = 1 - 1
[3,1,4,2] => [4]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 3 = 4 - 1
[3,2,1,4] => [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 0 = 1 - 1
[3,2,4,1] => [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 0 = 1 - 1
[3,4,1,2] => [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 1 = 2 - 1
[3,4,2,1] => [4]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 3 = 4 - 1
[4,1,2,3] => [4]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 3 = 4 - 1
[4,1,3,2] => [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 0 = 1 - 1
[4,2,1,3] => [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 0 = 1 - 1
[4,2,3,1] => [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 0 = 1 - 1
[4,3,1,2] => [4]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 3 = 4 - 1
[4,3,2,1] => [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 1 = 2 - 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [5]
=> [[1,2,3,4,5]]
=> 0 = 1 - 1
[1,2,3,5,4] => [2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 0 = 1 - 1
[1,2,4,3,5] => [2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 0 = 1 - 1
[1,2,4,5,3] => [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 0 = 1 - 1
[1,2,5,3,4] => [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 0 = 1 - 1
[1,2,5,4,3] => [2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 0 = 1 - 1
[1,3,2,4,5] => [2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 0 = 1 - 1
[1,3,2,5,4] => [2,2,1]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 0 = 1 - 1
[1,3,4,2,5] => [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 0 = 1 - 1
[1,3,4,5,2] => [4,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 0 = 1 - 1
[1,3,5,2,4] => [4,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 0 = 1 - 1
[1,3,5,4,2] => [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 0 = 1 - 1
[1,4,2,3,5] => [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 0 = 1 - 1
[1,4,2,5,3] => [4,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 0 = 1 - 1
[1,4,3,2,5] => [2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 0 = 1 - 1
[1,4,3,5,2] => [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 0 = 1 - 1
[1,4,5,2,3] => [2,2,1]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 0 = 1 - 1
[1,4,5,3,2] => [4,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 0 = 1 - 1
[2,1,4,3,6,5,8,7,10,9] => [2,2,2,2,2]
=> [5,5]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> ? = 2 - 1
[3,4,1,2,6,5,8,7,10,9] => [2,2,2,2,2]
=> [5,5]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> ? = 2 - 1
[4,3,2,1,6,5,8,7,10,9] => [2,2,2,2,2]
=> [5,5]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> ? = 2 - 1
[6,5,4,3,2,1,8,7,10,9] => [2,2,2,2,2]
=> [5,5]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> ? = 2 - 1
[8,9,5,6,3,4,10,1,2,7] => [2,2,2,2,2]
=> [5,5]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> ? = 2 - 1
[4,3,2,1,8,7,6,5,10,9] => [2,2,2,2,2]
=> [5,5]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> ? = 2 - 1
[8,7,6,5,4,3,2,1,10,9] => [2,2,2,2,2]
=> [5,5]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> ? = 2 - 1
[9,7,6,5,4,3,2,10,1,8] => [2,2,2,2,2]
=> [5,5]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> ? = 2 - 1
[10,8,9,6,7,4,5,2,3,1] => [2,2,2,2,2]
=> [5,5]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> ? = 2 - 1
[6,7,8,9,10,1,2,3,4,5] => [2,2,2,2,2]
=> [5,5]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> ? = 2 - 1
[7,3,2,8,9,10,1,4,5,6] => [2,2,2,2,2]
=> [5,5]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> ? = 2 - 1
[8,5,4,3,2,9,10,1,6,7] => [2,2,2,2,2]
=> [5,5]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> ? = 2 - 1
[6,5,4,3,2,1,10,9,8,7] => [2,2,2,2,2]
=> [5,5]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> ? = 2 - 1
[9,7,5,10,3,8,2,6,1,4] => [2,2,2,2,2]
=> [5,5]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> ? = 2 - 1
[4,3,2,1,10,9,8,7,6,5] => [2,2,2,2,2]
=> [5,5]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> ? = 2 - 1
[2,1,10,9,8,7,6,5,4,3] => [2,2,2,2,2]
=> [5,5]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> ? = 2 - 1
[9,10,8,7,6,5,4,3,1,2] => [2,2,2,2,2]
=> [5,5]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> ? = 2 - 1
[10,9,8,7,6,5,4,3,2,1] => [2,2,2,2,2]
=> [5,5]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> ? = 2 - 1
[9,1,2,3,4,5,6,7,8] => [9]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 9 - 1
[8,1,2,3,4,5,6,9,7] => [9]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 9 - 1
[7,1,2,3,4,5,9,6,8] => [9]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 9 - 1
[9,1,2,3,4,5,8,6,7] => [9]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 9 - 1
[7,1,2,3,4,5,8,9,6] => [9]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 9 - 1
[6,1,2,3,4,9,5,7,8] => [9]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 9 - 1
[6,1,2,3,4,8,5,9,7] => [9]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 9 - 1
[9,1,2,3,4,7,5,6,8] => [9]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 9 - 1
[9,1,2,3,4,8,5,6,7] => [7,2]
=> [2,2,1,1,1,1,1]
=> [[1,7],[2,9],[3],[4],[5],[6],[8]]
=> ? = 2 - 1
[8,1,2,3,4,7,5,9,6] => [9]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 9 - 1
[6,1,2,3,4,7,9,5,8] => [9]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 9 - 1
[6,1,2,3,4,9,8,5,7] => [9]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 9 - 1
[9,1,2,3,4,7,8,5,6] => [9]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 9 - 1
[6,1,2,3,4,7,8,9,5] => [9]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 9 - 1
[5,1,2,3,9,4,6,7,8] => [9]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 9 - 1
[5,1,2,3,8,4,6,9,7] => [9]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 9 - 1
[9,1,2,3,6,4,5,7,8] => [9]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 9 - 1
[8,1,2,3,9,4,5,6,7] => [6,3]
=> [2,2,2,1,1,1]
=> [[1,5],[2,7],[3,9],[4],[6],[8]]
=> ? = 3 - 1
[5,1,2,3,9,8,4,6,7] => [7,2]
=> [2,2,1,1,1,1,1]
=> [[1,7],[2,9],[3],[4],[5],[6],[8]]
=> ? = 2 - 1
[9,1,2,3,6,8,4,5,7] => [6,3]
=> [2,2,2,1,1,1]
=> [[1,5],[2,7],[3,9],[4],[6],[8]]
=> ? = 3 - 1
[5,1,2,3,6,9,8,4,7] => [9]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 9 - 1
[5,1,2,3,6,7,8,9,4] => [9]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 9 - 1
[4,1,2,8,3,5,6,9,7] => [9]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 9 - 1
[4,1,2,9,3,5,8,6,7] => [9]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 9 - 1
[4,1,2,9,3,8,5,6,7] => [7,2]
=> [2,2,1,1,1,1,1]
=> [[1,7],[2,9],[3],[4],[5],[6],[8]]
=> ? = 2 - 1
[9,1,2,8,3,4,5,6,7] => [6,3]
=> [2,2,2,1,1,1]
=> [[1,5],[2,7],[3,9],[4],[6],[8]]
=> ? = 3 - 1
[7,1,2,8,3,4,5,9,6] => [5,4]
=> [2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8]]
=> ? = 4 - 1
[4,1,2,5,9,3,6,7,8] => [9]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 9 - 1
[4,1,2,8,9,3,5,6,7] => [6,3]
=> [2,2,2,1,1,1]
=> [[1,5],[2,7],[3,9],[4],[6],[8]]
=> ? = 3 - 1
[8,1,2,5,9,3,4,6,7] => [5,4]
=> [2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8]]
=> ? = 4 - 1
[8,1,2,9,6,3,4,5,7] => [6,3]
=> [2,2,2,1,1,1]
=> [[1,5],[2,7],[3,9],[4],[6],[8]]
=> ? = 3 - 1
[4,1,2,5,9,8,3,6,7] => [7,2]
=> [2,2,1,1,1,1,1]
=> [[1,7],[2,9],[3],[4],[5],[6],[8]]
=> ? = 2 - 1
Description
The maximal overlap of the cylindrical tableau associated with a tableau.
A cylindrical tableau associated with a standard Young tableau $T$ is the skew row-strict tableau obtained by gluing two copies of $T$ such that the inner shape is a rectangle.
The overlap, recorded in this statistic, equals $\max_C\big(2\ell(T) - \ell(C)\big)$, where $\ell$ denotes the number of rows of a tableau and the maximum is taken over all cylindrical tableaux.
In particular, the statistic equals $0$, if and only if the last entry of the first row is larger than or equal to the first entry of the last row. Moreover, the statistic attains its maximal value, the number of rows of the tableau minus 1, if and only if the tableau consists of a single column.
The following 11 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. St000990The first ascent of a permutation. St001075The minimal size of a block of a set partition. 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. St000090The variation of a composition. St000260The radius of a connected graph. St000487The length of the shortest cycle of a permutation. St000314The number of left-to-right-maxima of a permutation. St001236The dominant dimension of the corresponding Comp-Nakayama algebra.
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