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Your data matches 71 different statistics following compositions of up to 3 maps.
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Matching statistic: St000902
(load all 76 compositions to match this statistic)
(load all 76 compositions to match this statistic)
St000902: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1
[1,1] => 2
[2] => 1
[1,1,1] => 3
[1,2] => 1
[2,1] => 1
[3] => 1
[1,1,1,1] => 4
[1,1,2] => 1
[1,2,1] => 1
[1,3] => 1
[2,1,1] => 1
[2,2] => 2
[3,1] => 1
[4] => 1
[1,1,1,1,1] => 5
[1,1,1,2] => 1
[1,1,2,1] => 1
[1,1,3] => 1
[1,4] => 1
[2,1,1,1] => 1
[3,2] => 1
[4,1] => 1
[5] => 1
[1,1,1,1,1,1] => 6
[1,1,1,1,2] => 1
[1,2,2,1] => 2
[1,2,3] => 1
[1,4,1] => 1
[1,5] => 1
[2,1,1,1,1] => 1
[3,2,1] => 1
[3,3] => 2
[5,1] => 1
[6] => 1
[1,1,1,1,1,1,1] => 7
[1,6] => 1
[6,1] => 1
[7] => 1
[1,1,1,1,1,1,1,1] => 8
[8] => 1
[1,1,1,1,1,1,1,1,1] => 9
[9] => 1
Description
The minimal number of repetitions of an integer composition.
Matching statistic: St000900
(load all 30 compositions to match this statistic)
(load all 30 compositions to match this statistic)
Mp00315: Integer compositions —inverse Foata bijection⟶ Integer compositions
St000900: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000900: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1
[1,1] => [1,1] => 2
[2] => [2] => 1
[1,1,1] => [1,1,1] => 3
[1,2] => [1,2] => 1
[2,1] => [2,1] => 1
[3] => [3] => 1
[1,1,1,1] => [1,1,1,1] => 4
[1,1,2] => [1,1,2] => 1
[1,2,1] => [2,1,1] => 1
[1,3] => [1,3] => 1
[2,1,1] => [1,2,1] => 1
[2,2] => [2,2] => 2
[3,1] => [3,1] => 1
[4] => [4] => 1
[1,1,1,1,1] => [1,1,1,1,1] => 5
[1,1,1,2] => [1,1,1,2] => 1
[1,1,2,1] => [2,1,1,1] => 1
[1,1,3] => [1,1,3] => 1
[1,4] => [1,4] => 1
[2,1,1,1] => [1,1,2,1] => 1
[3,2] => [3,2] => 1
[4,1] => [4,1] => 1
[5] => [5] => 1
[1,1,1,1,1,1] => [1,1,1,1,1,1] => 6
[1,1,1,1,2] => [1,1,1,1,2] => 1
[1,2,2,1] => [2,2,1,1] => 2
[1,2,3] => [1,2,3] => 1
[1,4,1] => [4,1,1] => 1
[1,5] => [1,5] => 1
[2,1,1,1,1] => [1,1,1,2,1] => 1
[3,2,1] => [3,2,1] => 1
[3,3] => [3,3] => 2
[5,1] => [5,1] => 1
[6] => [6] => 1
[1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => 7
[1,6] => [1,6] => 1
[6,1] => [6,1] => 1
[7] => [7] => 1
[1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1] => 8
[8] => [8] => 1
[1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1] => 9
[9] => [9] => 1
Description
The minimal number of repetitions of a part in an integer composition.
This is the smallest letter in the word obtained by applying the delta morphism.
Matching statistic: St000297
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00040: Integer compositions —to partition⟶ 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%
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]
=> 10 => 1
[1,1] => [1,1]
=> 110 => 2
[2] => [2]
=> 100 => 1
[1,1,1] => [1,1,1]
=> 1110 => 3
[1,2] => [2,1]
=> 1010 => 1
[2,1] => [2,1]
=> 1010 => 1
[3] => [3]
=> 1000 => 1
[1,1,1,1] => [1,1,1,1]
=> 11110 => 4
[1,1,2] => [2,1,1]
=> 10110 => 1
[1,2,1] => [2,1,1]
=> 10110 => 1
[1,3] => [3,1]
=> 10010 => 1
[2,1,1] => [2,1,1]
=> 10110 => 1
[2,2] => [2,2]
=> 1100 => 2
[3,1] => [3,1]
=> 10010 => 1
[4] => [4]
=> 10000 => 1
[1,1,1,1,1] => [1,1,1,1,1]
=> 111110 => 5
[1,1,1,2] => [2,1,1,1]
=> 101110 => 1
[1,1,2,1] => [2,1,1,1]
=> 101110 => 1
[1,1,3] => [3,1,1]
=> 100110 => 1
[1,4] => [4,1]
=> 100010 => 1
[2,1,1,1] => [2,1,1,1]
=> 101110 => 1
[3,2] => [3,2]
=> 10100 => 1
[4,1] => [4,1]
=> 100010 => 1
[5] => [5]
=> 100000 => 1
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> 1111110 => 6
[1,1,1,1,2] => [2,1,1,1,1]
=> 1011110 => 1
[1,2,2,1] => [2,2,1,1]
=> 110110 => 2
[1,2,3] => [3,2,1]
=> 101010 => 1
[1,4,1] => [4,1,1]
=> 1000110 => 1
[1,5] => [5,1]
=> 1000010 => 1
[2,1,1,1,1] => [2,1,1,1,1]
=> 1011110 => 1
[3,2,1] => [3,2,1]
=> 101010 => 1
[3,3] => [3,3]
=> 11000 => 2
[5,1] => [5,1]
=> 1000010 => 1
[6] => [6]
=> 1000000 => 1
[1,1,1,1,1,1,1] => [1,1,1,1,1,1,1]
=> 11111110 => 7
[1,6] => [6,1]
=> 10000010 => 1
[6,1] => [6,1]
=> 10000010 => 1
[7] => [7]
=> 10000000 => 1
[1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1]
=> 111111110 => 8
[8] => [8]
=> 100000000 => 1
[1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1]
=> 1111111110 => 9
[9] => [9]
=> 1000000000 => 1
Description
The number of leading ones in a binary word.
Matching statistic: St000627
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00314: Integer compositions —Foata bijection⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000627: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00094: Integer compositions —to binary word⟶ Binary words
St000627: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1 => 1
[1,1] => [1,1] => 11 => 2
[2] => [2] => 10 => 1
[1,1,1] => [1,1,1] => 111 => 3
[1,2] => [1,2] => 110 => 1
[2,1] => [2,1] => 101 => 1
[3] => [3] => 100 => 1
[1,1,1,1] => [1,1,1,1] => 1111 => 4
[1,1,2] => [1,1,2] => 1110 => 1
[1,2,1] => [2,1,1] => 1011 => 1
[1,3] => [1,3] => 1100 => 1
[2,1,1] => [1,2,1] => 1101 => 1
[2,2] => [2,2] => 1010 => 2
[3,1] => [3,1] => 1001 => 1
[4] => [4] => 1000 => 1
[1,1,1,1,1] => [1,1,1,1,1] => 11111 => 5
[1,1,1,2] => [1,1,1,2] => 11110 => 1
[1,1,2,1] => [2,1,1,1] => 10111 => 1
[1,1,3] => [1,1,3] => 11100 => 1
[1,4] => [1,4] => 11000 => 1
[2,1,1,1] => [1,1,2,1] => 11101 => 1
[3,2] => [3,2] => 10010 => 1
[4,1] => [4,1] => 10001 => 1
[5] => [5] => 10000 => 1
[1,1,1,1,1,1] => [1,1,1,1,1,1] => 111111 => 6
[1,1,1,1,2] => [1,1,1,1,2] => 111110 => 1
[1,2,2,1] => [2,1,2,1] => 101101 => 2
[1,2,3] => [1,2,3] => 110100 => 1
[1,4,1] => [4,1,1] => 100011 => 1
[1,5] => [1,5] => 110000 => 1
[2,1,1,1,1] => [1,1,1,2,1] => 111101 => 1
[3,2,1] => [3,2,1] => 100101 => 1
[3,3] => [3,3] => 100100 => 2
[5,1] => [5,1] => 100001 => 1
[6] => [6] => 100000 => 1
[1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => 1111111 => 7
[1,6] => [1,6] => 1100000 => 1
[6,1] => [6,1] => 1000001 => 1
[7] => [7] => 1000000 => 1
[1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1] => 11111111 => 8
[8] => [8] => 10000000 => 1
[1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1] => 111111111 => 9
[9] => [9] => 100000000 => 1
Description
The exponent of a binary word.
This is the largest number $e$ such that $w$ is the concatenation of $e$ identical factors. This statistic is also called '''frequency'''.
Matching statistic: St000657
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St000657: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St000657: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 1
[1,1] => [1,1] => [2] => 2
[2] => [2] => [1] => 1
[1,1,1] => [1,1,1] => [3] => 3
[1,2] => [2,1] => [1,1] => 1
[2,1] => [1,2] => [1,1] => 1
[3] => [3] => [1] => 1
[1,1,1,1] => [1,1,1,1] => [4] => 4
[1,1,2] => [2,1,1] => [1,2] => 1
[1,2,1] => [1,1,2] => [2,1] => 1
[1,3] => [3,1] => [1,1] => 1
[2,1,1] => [1,2,1] => [1,1,1] => 1
[2,2] => [2,2] => [2] => 2
[3,1] => [1,3] => [1,1] => 1
[4] => [4] => [1] => 1
[1,1,1,1,1] => [1,1,1,1,1] => [5] => 5
[1,1,1,2] => [2,1,1,1] => [1,3] => 1
[1,1,2,1] => [1,1,1,2] => [3,1] => 1
[1,1,3] => [3,1,1] => [1,2] => 1
[1,4] => [4,1] => [1,1] => 1
[2,1,1,1] => [1,2,1,1] => [1,1,2] => 1
[3,2] => [2,3] => [1,1] => 1
[4,1] => [1,4] => [1,1] => 1
[5] => [5] => [1] => 1
[1,1,1,1,1,1] => [1,1,1,1,1,1] => [6] => 6
[1,1,1,1,2] => [2,1,1,1,1] => [1,4] => 1
[1,2,2,1] => [1,1,2,2] => [2,2] => 2
[1,2,3] => [3,1,2] => [1,1,1] => 1
[1,4,1] => [1,1,4] => [2,1] => 1
[1,5] => [5,1] => [1,1] => 1
[2,1,1,1,1] => [1,2,1,1,1] => [1,1,3] => 1
[3,2,1] => [1,3,2] => [1,1,1] => 1
[3,3] => [3,3] => [2] => 2
[5,1] => [1,5] => [1,1] => 1
[6] => [6] => [1] => 1
[1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => [7] => 7
[1,6] => [6,1] => [1,1] => 1
[6,1] => [1,6] => [1,1] => 1
[7] => [7] => [1] => 1
[1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1] => [8] => 8
[8] => [8] => [1] => 1
[1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1] => [9] => 9
[9] => [9] => [1] => 1
Description
The smallest part of an integer composition.
Matching statistic: St000667
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000667: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000667: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [1]
=> 1
[1,1] => [1,1]
=> [2]
=> 2
[2] => [2]
=> [1,1]
=> 1
[1,1,1] => [1,1,1]
=> [3]
=> 3
[1,2] => [2,1]
=> [2,1]
=> 1
[2,1] => [2,1]
=> [2,1]
=> 1
[3] => [3]
=> [1,1,1]
=> 1
[1,1,1,1] => [1,1,1,1]
=> [4]
=> 4
[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]
=> 1
[1,1,1,1,1] => [1,1,1,1,1]
=> [5]
=> 5
[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
[1,4] => [4,1]
=> [2,1,1,1]
=> 1
[2,1,1,1] => [2,1,1,1]
=> [4,1]
=> 1
[3,2] => [3,2]
=> [2,2,1]
=> 1
[4,1] => [4,1]
=> [2,1,1,1]
=> 1
[5] => [5]
=> [1,1,1,1,1]
=> 1
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> [6]
=> 6
[1,1,1,1,2] => [2,1,1,1,1]
=> [5,1]
=> 1
[1,2,2,1] => [2,2,1,1]
=> [4,2]
=> 2
[1,2,3] => [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
[3,2,1] => [3,2,1]
=> [3,2,1]
=> 1
[3,3] => [3,3]
=> [2,2,2]
=> 2
[5,1] => [5,1]
=> [2,1,1,1,1]
=> 1
[6] => [6]
=> [1,1,1,1,1,1]
=> 1
[1,1,1,1,1,1,1] => [1,1,1,1,1,1,1]
=> [7]
=> 7
[1,6] => [6,1]
=> [2,1,1,1,1,1]
=> 1
[6,1] => [6,1]
=> [2,1,1,1,1,1]
=> 1
[7] => [7]
=> [1,1,1,1,1,1,1]
=> 1
[1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1]
=> [8]
=> 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]
=> [9]
=> 9
[9] => [9]
=> [1,1,1,1,1,1,1,1,1]
=> 1
Description
The greatest common divisor of the parts of the partition.
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
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [[1]]
=> 1
[1,1] => [1,1]
=> [[1],[2]]
=> 2
[2] => [2]
=> [[1,2]]
=> 1
[1,1,1] => [1,1,1]
=> [[1],[2],[3]]
=> 3
[1,2] => [2,1]
=> [[1,3],[2]]
=> 1
[2,1] => [2,1]
=> [[1,3],[2]]
=> 1
[3] => [3]
=> [[1,2,3]]
=> 1
[1,1,1,1] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4
[1,1,2] => [2,1,1]
=> [[1,4],[2],[3]]
=> 1
[1,2,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> 1
[1,3] => [3,1]
=> [[1,3,4],[2]]
=> 1
[2,1,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> 1
[2,2] => [2,2]
=> [[1,2],[3,4]]
=> 2
[3,1] => [3,1]
=> [[1,3,4],[2]]
=> 1
[4] => [4]
=> [[1,2,3,4]]
=> 1
[1,1,1,1,1] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 5
[1,1,1,2] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 1
[1,1,2,1] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 1
[1,1,3] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 1
[1,4] => [4,1]
=> [[1,3,4,5],[2]]
=> 1
[2,1,1,1] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 1
[3,2] => [3,2]
=> [[1,2,5],[3,4]]
=> 1
[4,1] => [4,1]
=> [[1,3,4,5],[2]]
=> 1
[5] => [5]
=> [[1,2,3,4,5]]
=> 1
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> 6
[1,1,1,1,2] => [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> 1
[1,2,2,1] => [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> 2
[1,2,3] => [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> 1
[1,4,1] => [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> 1
[1,5] => [5,1]
=> [[1,3,4,5,6],[2]]
=> 1
[2,1,1,1,1] => [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> 1
[3,2,1] => [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> 1
[3,3] => [3,3]
=> [[1,2,3],[4,5,6]]
=> 2
[5,1] => [5,1]
=> [[1,3,4,5,6],[2]]
=> 1
[6] => [6]
=> [[1,2,3,4,5,6]]
=> 1
[1,1,1,1,1,1,1] => [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> 7
[1,6] => [6,1]
=> [[1,3,4,5,6,7],[2]]
=> 1
[6,1] => [6,1]
=> [[1,3,4,5,6,7],[2]]
=> 1
[7] => [7]
=> [[1,2,3,4,5,6,7]]
=> 1
[1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> 8
[8] => [8]
=> [[1,2,3,4,5,6,7,8]]
=> 1
[1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> 9
[9] => [9]
=> [[1,2,3,4,5,6,7,8,9]]
=> 1
Description
The row containing the largest entry of a standard tableau.
Matching statistic: St000883
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000883: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000883: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => 1
[1,1] => [1,0,1,0]
=> [2,1] => 2
[2] => [1,1,0,0]
=> [1,2] => 1
[1,1,1] => [1,0,1,0,1,0]
=> [3,2,1] => 3
[1,2] => [1,0,1,1,0,0]
=> [2,3,1] => 1
[2,1] => [1,1,0,0,1,0]
=> [3,1,2] => 1
[3] => [1,1,1,0,0,0]
=> [1,2,3] => 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 4
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 1
[1,3] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2
[3,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
[4] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => 5
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => 6
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,2,1] => 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,5,2,3,1] => 2
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,5,6,2,3,1] => 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,5,1] => 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1,2] => 1
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [6,4,5,1,2,3] => 1
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [4,5,6,1,2,3] => 2
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 1
[6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => 1
[1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,2,1] => 7
[1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 1
[6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => 1
[7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7] => 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]
=> [8,7,6,5,4,3,2,1] => 8
[8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7,8] => 1
[1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [9,8,7,6,5,4,3,2,1] => 9
[9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7,8,9] => 1
Description
The number of longest increasing subsequences of a permutation.
Matching statistic: St000007
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [[1]]
=> [1] => 1
[1,1] => [1,1]
=> [[1],[2]]
=> [2,1] => 2
[2] => [2]
=> [[1,2]]
=> [1,2] => 1
[1,1,1] => [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 3
[1,2] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 1
[2,1] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 1
[3] => [3]
=> [[1,2,3]]
=> [1,2,3] => 1
[1,1,1,1] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 4
[1,1,2] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 1
[1,2,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 1
[1,3] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 1
[2,1,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 1
[2,2] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[3,1] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 1
[4] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1
[1,1,1,1,1] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 5
[1,1,1,2] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 1
[1,1,2,1] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 1
[1,1,3] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 1
[1,4] => [4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 1
[2,1,1,1] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 1
[3,2] => [3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => 1
[4,1] => [4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 1
[5] => [5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 1
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => 6
[1,1,1,1,2] => [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => 1
[1,2,2,1] => [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => 2
[1,2,3] => [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => 1
[1,4,1] => [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => 1
[1,5] => [5,1]
=> [[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => 1
[2,1,1,1,1] => [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => 1
[3,2,1] => [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => 1
[3,3] => [3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => 2
[5,1] => [5,1]
=> [[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => 1
[6] => [6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => 1
[1,1,1,1,1,1,1] => [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => 7
[1,6] => [6,1]
=> [[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => 1
[6,1] => [6,1]
=> [[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => 1
[7] => [7]
=> [[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => 1
[1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,2,1] => 8
[8] => [8]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => 1
[1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> [9,8,7,6,5,4,3,2,1] => 9
[9] => [9]
=> [[1,2,3,4,5,6,7,8,9]]
=> [1,2,3,4,5,6,7,8,9] => 1
Description
The number of saliances of the permutation.
A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern $([1], {(1,1)})$, i.e., the upper right quadrant is shaded, see [1].
Matching statistic: St000382
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> 10 => [1,1] => 1
[1,1] => [1,1]
=> 110 => [2,1] => 2
[2] => [2]
=> 100 => [1,2] => 1
[1,1,1] => [1,1,1]
=> 1110 => [3,1] => 3
[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] => 1
[1,1,1,1] => [1,1,1,1]
=> 11110 => [4,1] => 4
[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] => 1
[1,1,1,1,1] => [1,1,1,1,1]
=> 111110 => [5,1] => 5
[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
[1,4] => [4,1]
=> 100010 => [1,3,1,1] => 1
[2,1,1,1] => [2,1,1,1]
=> 101110 => [1,1,3,1] => 1
[3,2] => [3,2]
=> 10100 => [1,1,1,2] => 1
[4,1] => [4,1]
=> 100010 => [1,3,1,1] => 1
[5] => [5]
=> 100000 => [1,5] => 1
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> 1111110 => [6,1] => 6
[1,1,1,1,2] => [2,1,1,1,1]
=> 1011110 => [1,1,4,1] => 1
[1,2,2,1] => [2,2,1,1]
=> 110110 => [2,1,2,1] => 2
[1,2,3] => [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
[3,2,1] => [3,2,1]
=> 101010 => [1,1,1,1,1,1] => 1
[3,3] => [3,3]
=> 11000 => [2,3] => 2
[5,1] => [5,1]
=> 1000010 => [1,4,1,1] => 1
[6] => [6]
=> 1000000 => [1,6] => 1
[1,1,1,1,1,1,1] => [1,1,1,1,1,1,1]
=> 11111110 => [7,1] => 7
[1,6] => [6,1]
=> 10000010 => [1,5,1,1] => 1
[6,1] => [6,1]
=> 10000010 => [1,5,1,1] => 1
[7] => [7]
=> 10000000 => [1,7] => 1
[1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1]
=> 111111110 => [8,1] => 8
[8] => [8]
=> 100000000 => [1,8] => 1
[1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1]
=> 1111111110 => [9,1] => 9
[9] => [9]
=> 1000000000 => [1,9] => 1
Description
The first part of an integer composition.
The following 61 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000383The last part of an integer composition. St000617The number of global maxima of a Dyck path. St000655The length of the minimal rise of a Dyck path. St000765The number of weak records in an integer composition. St000326The position of the first one in a binary word after appending a 1 at the end. St000993The multiplicity of the largest part of an integer partition. St000909The number of maximal chains of maximal size in a poset. St000439The position of the first down step of a Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000911The number of maximal antichains of maximal size in a poset. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001481The minimal height of a peak of a Dyck path. St001733The number of weak left to right maxima of a Dyck path. St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St000363The number of minimal vertex covers of a graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue 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. St000990The first ascent of a permutation. St000654The first descent of a permutation. St000061The number of nodes on the left branch of a binary tree. St001571The Cartan determinant of the integer partition. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000379The number of Hamiltonian cycles in a graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000455The second largest eigenvalue of a graph if it is integral. St000699The toughness times the least common multiple of 1,. St001884The number of borders of a binary word. St001330The hat guessing number of a graph. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000907The number of maximal antichains of minimal length in a poset. St000456The monochromatic index of a connected graph. St000806The semiperimeter of the associated bargraph. St000650The number of 3-rises of a permutation. St000264The girth of a graph, which is not a tree. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001645The pebbling number of a connected graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
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