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Your data matches 85 different statistics following compositions of up to 3 maps.
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Matching statistic: St000251
(load all 414 compositions to match this statistic)
(load all 414 compositions to match this statistic)
St000251: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> 1
{{1},{2}}
=> 0
{{1,2,3}}
=> 1
{{1,2},{3}}
=> 1
{{1,3},{2}}
=> 1
{{1},{2,3}}
=> 1
{{1},{2},{3}}
=> 0
{{1,2,3,4}}
=> 1
{{1,2,3},{4}}
=> 1
{{1,2,4},{3}}
=> 1
{{1,2},{3,4}}
=> 2
{{1,2},{3},{4}}
=> 1
{{1,3,4},{2}}
=> 1
{{1,3},{2,4}}
=> 2
{{1,3},{2},{4}}
=> 1
{{1,4},{2,3}}
=> 2
{{1},{2,3,4}}
=> 1
{{1},{2,3},{4}}
=> 1
{{1,4},{2},{3}}
=> 1
{{1},{2,4},{3}}
=> 1
{{1},{2},{3,4}}
=> 1
{{1},{2},{3},{4}}
=> 0
{{1,2,3,4,5}}
=> 1
{{1,2,3,4},{5}}
=> 1
{{1,2,3,5},{4}}
=> 1
{{1,2,3},{4,5}}
=> 2
{{1,2,3},{4},{5}}
=> 1
{{1,2,4,5},{3}}
=> 1
{{1,2,4},{3,5}}
=> 2
{{1,2,4},{3},{5}}
=> 1
{{1,2,5},{3,4}}
=> 2
{{1,2},{3,4,5}}
=> 2
{{1,2},{3,4},{5}}
=> 2
{{1,2,5},{3},{4}}
=> 1
{{1,2},{3,5},{4}}
=> 2
{{1,2},{3},{4,5}}
=> 2
{{1,2},{3},{4},{5}}
=> 1
{{1,3,4,5},{2}}
=> 1
{{1,3,4},{2,5}}
=> 2
{{1,3,4},{2},{5}}
=> 1
{{1,3,5},{2,4}}
=> 2
{{1,3},{2,4,5}}
=> 2
{{1,3},{2,4},{5}}
=> 2
{{1,3,5},{2},{4}}
=> 1
{{1,3},{2,5},{4}}
=> 2
{{1,3},{2},{4,5}}
=> 2
{{1,3},{2},{4},{5}}
=> 1
{{1,4,5},{2,3}}
=> 2
{{1,4},{2,3,5}}
=> 2
{{1,4},{2,3},{5}}
=> 2
Description
The number of nonsingleton blocks of a set partition.
Matching statistic: St000291
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000291: 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
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2] => 10 => 1
{{1},{2}}
=> [1,1] => 11 => 0
{{1,2,3}}
=> [3] => 100 => 1
{{1,2},{3}}
=> [2,1] => 101 => 1
{{1,3},{2}}
=> [2,1] => 101 => 1
{{1},{2,3}}
=> [1,2] => 110 => 1
{{1},{2},{3}}
=> [1,1,1] => 111 => 0
{{1,2,3,4}}
=> [4] => 1000 => 1
{{1,2,3},{4}}
=> [3,1] => 1001 => 1
{{1,2,4},{3}}
=> [3,1] => 1001 => 1
{{1,2},{3,4}}
=> [2,2] => 1010 => 2
{{1,2},{3},{4}}
=> [2,1,1] => 1011 => 1
{{1,3,4},{2}}
=> [3,1] => 1001 => 1
{{1,3},{2,4}}
=> [2,2] => 1010 => 2
{{1,3},{2},{4}}
=> [2,1,1] => 1011 => 1
{{1,4},{2,3}}
=> [2,2] => 1010 => 2
{{1},{2,3,4}}
=> [1,3] => 1100 => 1
{{1},{2,3},{4}}
=> [1,2,1] => 1101 => 1
{{1,4},{2},{3}}
=> [2,1,1] => 1011 => 1
{{1},{2,4},{3}}
=> [1,2,1] => 1101 => 1
{{1},{2},{3,4}}
=> [1,1,2] => 1110 => 1
{{1},{2},{3},{4}}
=> [1,1,1,1] => 1111 => 0
{{1,2,3,4,5}}
=> [5] => 10000 => 1
{{1,2,3,4},{5}}
=> [4,1] => 10001 => 1
{{1,2,3,5},{4}}
=> [4,1] => 10001 => 1
{{1,2,3},{4,5}}
=> [3,2] => 10010 => 2
{{1,2,3},{4},{5}}
=> [3,1,1] => 10011 => 1
{{1,2,4,5},{3}}
=> [4,1] => 10001 => 1
{{1,2,4},{3,5}}
=> [3,2] => 10010 => 2
{{1,2,4},{3},{5}}
=> [3,1,1] => 10011 => 1
{{1,2,5},{3,4}}
=> [3,2] => 10010 => 2
{{1,2},{3,4,5}}
=> [2,3] => 10100 => 2
{{1,2},{3,4},{5}}
=> [2,2,1] => 10101 => 2
{{1,2,5},{3},{4}}
=> [3,1,1] => 10011 => 1
{{1,2},{3,5},{4}}
=> [2,2,1] => 10101 => 2
{{1,2},{3},{4,5}}
=> [2,1,2] => 10110 => 2
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => 10111 => 1
{{1,3,4,5},{2}}
=> [4,1] => 10001 => 1
{{1,3,4},{2,5}}
=> [3,2] => 10010 => 2
{{1,3,4},{2},{5}}
=> [3,1,1] => 10011 => 1
{{1,3,5},{2,4}}
=> [3,2] => 10010 => 2
{{1,3},{2,4,5}}
=> [2,3] => 10100 => 2
{{1,3},{2,4},{5}}
=> [2,2,1] => 10101 => 2
{{1,3,5},{2},{4}}
=> [3,1,1] => 10011 => 1
{{1,3},{2,5},{4}}
=> [2,2,1] => 10101 => 2
{{1,3},{2},{4,5}}
=> [2,1,2] => 10110 => 2
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => 10111 => 1
{{1,4,5},{2,3}}
=> [3,2] => 10010 => 2
{{1,4},{2,3,5}}
=> [2,3] => 10100 => 2
{{1,4},{2,3},{5}}
=> [2,2,1] => 10101 => 2
Description
The number of descents of a binary word.
Matching statistic: St000292
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000292: 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
Mp00104: Binary words —reverse⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2] => 10 => 01 => 1
{{1},{2}}
=> [1,1] => 11 => 11 => 0
{{1,2,3}}
=> [3] => 100 => 001 => 1
{{1,2},{3}}
=> [2,1] => 101 => 101 => 1
{{1,3},{2}}
=> [2,1] => 101 => 101 => 1
{{1},{2,3}}
=> [1,2] => 110 => 011 => 1
{{1},{2},{3}}
=> [1,1,1] => 111 => 111 => 0
{{1,2,3,4}}
=> [4] => 1000 => 0001 => 1
{{1,2,3},{4}}
=> [3,1] => 1001 => 1001 => 1
{{1,2,4},{3}}
=> [3,1] => 1001 => 1001 => 1
{{1,2},{3,4}}
=> [2,2] => 1010 => 0101 => 2
{{1,2},{3},{4}}
=> [2,1,1] => 1011 => 1101 => 1
{{1,3,4},{2}}
=> [3,1] => 1001 => 1001 => 1
{{1,3},{2,4}}
=> [2,2] => 1010 => 0101 => 2
{{1,3},{2},{4}}
=> [2,1,1] => 1011 => 1101 => 1
{{1,4},{2,3}}
=> [2,2] => 1010 => 0101 => 2
{{1},{2,3,4}}
=> [1,3] => 1100 => 0011 => 1
{{1},{2,3},{4}}
=> [1,2,1] => 1101 => 1011 => 1
{{1,4},{2},{3}}
=> [2,1,1] => 1011 => 1101 => 1
{{1},{2,4},{3}}
=> [1,2,1] => 1101 => 1011 => 1
{{1},{2},{3,4}}
=> [1,1,2] => 1110 => 0111 => 1
{{1},{2},{3},{4}}
=> [1,1,1,1] => 1111 => 1111 => 0
{{1,2,3,4,5}}
=> [5] => 10000 => 00001 => 1
{{1,2,3,4},{5}}
=> [4,1] => 10001 => 10001 => 1
{{1,2,3,5},{4}}
=> [4,1] => 10001 => 10001 => 1
{{1,2,3},{4,5}}
=> [3,2] => 10010 => 01001 => 2
{{1,2,3},{4},{5}}
=> [3,1,1] => 10011 => 11001 => 1
{{1,2,4,5},{3}}
=> [4,1] => 10001 => 10001 => 1
{{1,2,4},{3,5}}
=> [3,2] => 10010 => 01001 => 2
{{1,2,4},{3},{5}}
=> [3,1,1] => 10011 => 11001 => 1
{{1,2,5},{3,4}}
=> [3,2] => 10010 => 01001 => 2
{{1,2},{3,4,5}}
=> [2,3] => 10100 => 00101 => 2
{{1,2},{3,4},{5}}
=> [2,2,1] => 10101 => 10101 => 2
{{1,2,5},{3},{4}}
=> [3,1,1] => 10011 => 11001 => 1
{{1,2},{3,5},{4}}
=> [2,2,1] => 10101 => 10101 => 2
{{1,2},{3},{4,5}}
=> [2,1,2] => 10110 => 01101 => 2
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => 10111 => 11101 => 1
{{1,3,4,5},{2}}
=> [4,1] => 10001 => 10001 => 1
{{1,3,4},{2,5}}
=> [3,2] => 10010 => 01001 => 2
{{1,3,4},{2},{5}}
=> [3,1,1] => 10011 => 11001 => 1
{{1,3,5},{2,4}}
=> [3,2] => 10010 => 01001 => 2
{{1,3},{2,4,5}}
=> [2,3] => 10100 => 00101 => 2
{{1,3},{2,4},{5}}
=> [2,2,1] => 10101 => 10101 => 2
{{1,3,5},{2},{4}}
=> [3,1,1] => 10011 => 11001 => 1
{{1,3},{2,5},{4}}
=> [2,2,1] => 10101 => 10101 => 2
{{1,3},{2},{4,5}}
=> [2,1,2] => 10110 => 01101 => 2
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => 10111 => 11101 => 1
{{1,4,5},{2,3}}
=> [3,2] => 10010 => 01001 => 2
{{1,4},{2,3,5}}
=> [2,3] => 10100 => 00101 => 2
{{1,4},{2,3},{5}}
=> [2,2,1] => 10101 => 10101 => 2
Description
The number of ascents of a binary word.
Matching statistic: St000390
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000390: 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
Mp00105: Binary words —complement⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2] => 10 => 01 => 1
{{1},{2}}
=> [1,1] => 11 => 00 => 0
{{1,2,3}}
=> [3] => 100 => 011 => 1
{{1,2},{3}}
=> [2,1] => 101 => 010 => 1
{{1,3},{2}}
=> [2,1] => 101 => 010 => 1
{{1},{2,3}}
=> [1,2] => 110 => 001 => 1
{{1},{2},{3}}
=> [1,1,1] => 111 => 000 => 0
{{1,2,3,4}}
=> [4] => 1000 => 0111 => 1
{{1,2,3},{4}}
=> [3,1] => 1001 => 0110 => 1
{{1,2,4},{3}}
=> [3,1] => 1001 => 0110 => 1
{{1,2},{3,4}}
=> [2,2] => 1010 => 0101 => 2
{{1,2},{3},{4}}
=> [2,1,1] => 1011 => 0100 => 1
{{1,3,4},{2}}
=> [3,1] => 1001 => 0110 => 1
{{1,3},{2,4}}
=> [2,2] => 1010 => 0101 => 2
{{1,3},{2},{4}}
=> [2,1,1] => 1011 => 0100 => 1
{{1,4},{2,3}}
=> [2,2] => 1010 => 0101 => 2
{{1},{2,3,4}}
=> [1,3] => 1100 => 0011 => 1
{{1},{2,3},{4}}
=> [1,2,1] => 1101 => 0010 => 1
{{1,4},{2},{3}}
=> [2,1,1] => 1011 => 0100 => 1
{{1},{2,4},{3}}
=> [1,2,1] => 1101 => 0010 => 1
{{1},{2},{3,4}}
=> [1,1,2] => 1110 => 0001 => 1
{{1},{2},{3},{4}}
=> [1,1,1,1] => 1111 => 0000 => 0
{{1,2,3,4,5}}
=> [5] => 10000 => 01111 => 1
{{1,2,3,4},{5}}
=> [4,1] => 10001 => 01110 => 1
{{1,2,3,5},{4}}
=> [4,1] => 10001 => 01110 => 1
{{1,2,3},{4,5}}
=> [3,2] => 10010 => 01101 => 2
{{1,2,3},{4},{5}}
=> [3,1,1] => 10011 => 01100 => 1
{{1,2,4,5},{3}}
=> [4,1] => 10001 => 01110 => 1
{{1,2,4},{3,5}}
=> [3,2] => 10010 => 01101 => 2
{{1,2,4},{3},{5}}
=> [3,1,1] => 10011 => 01100 => 1
{{1,2,5},{3,4}}
=> [3,2] => 10010 => 01101 => 2
{{1,2},{3,4,5}}
=> [2,3] => 10100 => 01011 => 2
{{1,2},{3,4},{5}}
=> [2,2,1] => 10101 => 01010 => 2
{{1,2,5},{3},{4}}
=> [3,1,1] => 10011 => 01100 => 1
{{1,2},{3,5},{4}}
=> [2,2,1] => 10101 => 01010 => 2
{{1,2},{3},{4,5}}
=> [2,1,2] => 10110 => 01001 => 2
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => 10111 => 01000 => 1
{{1,3,4,5},{2}}
=> [4,1] => 10001 => 01110 => 1
{{1,3,4},{2,5}}
=> [3,2] => 10010 => 01101 => 2
{{1,3,4},{2},{5}}
=> [3,1,1] => 10011 => 01100 => 1
{{1,3,5},{2,4}}
=> [3,2] => 10010 => 01101 => 2
{{1,3},{2,4,5}}
=> [2,3] => 10100 => 01011 => 2
{{1,3},{2,4},{5}}
=> [2,2,1] => 10101 => 01010 => 2
{{1,3,5},{2},{4}}
=> [3,1,1] => 10011 => 01100 => 1
{{1,3},{2,5},{4}}
=> [2,2,1] => 10101 => 01010 => 2
{{1,3},{2},{4,5}}
=> [2,1,2] => 10110 => 01001 => 2
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => 10111 => 01000 => 1
{{1,4,5},{2,3}}
=> [3,2] => 10010 => 01101 => 2
{{1,4},{2,3,5}}
=> [2,3] => 10100 => 01011 => 2
{{1,4},{2,3},{5}}
=> [2,2,1] => 10101 => 01010 => 2
Description
The number of runs of ones in a binary word.
Matching statistic: St000288
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2,1] => [2,1] => 1 => 1
{{1},{2}}
=> [1,2] => [1,2] => 0 => 0
{{1,2,3}}
=> [2,3,1] => [3,1,2] => 10 => 1
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => 10 => 1
{{1,3},{2}}
=> [3,2,1] => [2,3,1] => 01 => 1
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => 01 => 1
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => 00 => 0
{{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => 100 => 1
{{1,2,3},{4}}
=> [2,3,1,4] => [3,1,2,4] => 100 => 1
{{1,2,4},{3}}
=> [2,4,3,1] => [3,4,1,2] => 010 => 1
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => 101 => 2
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => 100 => 1
{{1,3,4},{2}}
=> [3,2,4,1] => [2,4,1,3] => 010 => 1
{{1,3},{2,4}}
=> [3,4,1,2] => [3,1,4,2] => 101 => 2
{{1,3},{2},{4}}
=> [3,2,1,4] => [2,3,1,4] => 010 => 1
{{1,4},{2,3}}
=> [4,3,2,1] => [3,2,4,1] => 101 => 2
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,2,3] => 010 => 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => 010 => 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,3,4,1] => 001 => 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,3,4,2] => 001 => 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => 001 => 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => 000 => 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [5,1,2,3,4] => 1000 => 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [4,1,2,3,5] => 1000 => 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [4,5,1,2,3] => 0100 => 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [3,1,2,5,4] => 1001 => 2
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [3,1,2,4,5] => 1000 => 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [3,5,1,2,4] => 0100 => 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [4,1,2,5,3] => 1001 => 2
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [3,4,1,2,5] => 0100 => 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [4,3,5,1,2] => 1010 => 2
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,5,3,4] => 1010 => 2
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,1,4,3,5] => 1010 => 2
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [3,4,5,1,2] => 0010 => 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,4,5,3] => 1001 => 2
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => 1001 => 2
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => 1000 => 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [2,5,1,3,4] => 0100 => 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [4,1,3,5,2] => 1001 => 2
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [2,4,1,3,5] => 0100 => 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [4,2,5,1,3] => 1010 => 2
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,1,5,2,4] => 1010 => 2
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [3,1,4,2,5] => 1010 => 2
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [2,4,5,1,3] => 0010 => 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [3,1,4,5,2] => 1001 => 2
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [2,3,1,5,4] => 0101 => 2
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [2,3,1,4,5] => 0100 => 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [3,2,5,1,4] => 1010 => 2
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [4,1,5,2,3] => 1010 => 2
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [3,2,4,1,5] => 1010 => 2
{{1},{2,3,4,5,6,7},{8}}
=> ? => ? => ? => ? = 1
{{1,2,5,6,7,8},{3,4}}
=> [2,5,4,3,6,7,8,1] => [4,3,8,1,2,5,6,7] => ? => ? = 2
{{1,2,4,6,7,8},{3,5}}
=> [2,4,5,6,3,7,8,1] => [5,3,8,1,2,4,6,7] => ? => ? = 2
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000389
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000389: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000389: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2,1] => [2,1] => 1 => 1
{{1},{2}}
=> [1,2] => [1,2] => 0 => 0
{{1,2,3}}
=> [2,3,1] => [3,1,2] => 10 => 1
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => 10 => 1
{{1,3},{2}}
=> [3,2,1] => [2,3,1] => 01 => 1
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => 01 => 1
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => 00 => 0
{{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => 100 => 1
{{1,2,3},{4}}
=> [2,3,1,4] => [3,1,2,4] => 100 => 1
{{1,2,4},{3}}
=> [2,4,3,1] => [3,4,1,2] => 010 => 1
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => 101 => 2
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => 100 => 1
{{1,3,4},{2}}
=> [3,2,4,1] => [2,4,1,3] => 010 => 1
{{1,3},{2,4}}
=> [3,4,1,2] => [3,1,4,2] => 101 => 2
{{1,3},{2},{4}}
=> [3,2,1,4] => [2,3,1,4] => 010 => 1
{{1,4},{2,3}}
=> [4,3,2,1] => [3,2,4,1] => 101 => 2
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,2,3] => 010 => 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => 010 => 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,3,4,1] => 001 => 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,3,4,2] => 001 => 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => 001 => 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => 000 => 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [5,1,2,3,4] => 1000 => 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [4,1,2,3,5] => 1000 => 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [4,5,1,2,3] => 0100 => 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [3,1,2,5,4] => 1001 => 2
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [3,1,2,4,5] => 1000 => 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [3,5,1,2,4] => 0100 => 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [4,1,2,5,3] => 1001 => 2
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [3,4,1,2,5] => 0100 => 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [4,3,5,1,2] => 1010 => 2
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,5,3,4] => 1010 => 2
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,1,4,3,5] => 1010 => 2
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [3,4,5,1,2] => 0010 => 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,4,5,3] => 1001 => 2
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => 1001 => 2
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => 1000 => 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [2,5,1,3,4] => 0100 => 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [4,1,3,5,2] => 1001 => 2
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [2,4,1,3,5] => 0100 => 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [4,2,5,1,3] => 1010 => 2
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,1,5,2,4] => 1010 => 2
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [3,1,4,2,5] => 1010 => 2
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [2,4,5,1,3] => 0010 => 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [3,1,4,5,2] => 1001 => 2
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [2,3,1,5,4] => 0101 => 2
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [2,3,1,4,5] => 0100 => 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [3,2,5,1,4] => 1010 => 2
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [4,1,5,2,3] => 1010 => 2
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [3,2,4,1,5] => 1010 => 2
{{1},{2,3,4,5,6,7},{8}}
=> ? => ? => ? => ? = 1
{{1,2,5,6,7,8},{3,4}}
=> [2,5,4,3,6,7,8,1] => [4,3,8,1,2,5,6,7] => ? => ? = 2
{{1,2,4,6,7,8},{3,5}}
=> [2,4,5,6,3,7,8,1] => [5,3,8,1,2,4,6,7] => ? => ? = 2
Description
The number of runs of ones of odd length in a binary word.
Matching statistic: St000157
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2,1] => [2,1] => [[1],[2]]
=> 1
{{1},{2}}
=> [1,2] => [1,2] => [[1,2]]
=> 0
{{1,2,3}}
=> [2,3,1] => [3,1,2] => [[1,3],[2]]
=> 1
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => [[1,3],[2]]
=> 1
{{1,3},{2}}
=> [3,2,1] => [2,3,1] => [[1,2],[3]]
=> 1
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => [[1,2],[3]]
=> 1
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => [[1,2,3]]
=> 0
{{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => [[1,3,4],[2]]
=> 1
{{1,2,3},{4}}
=> [2,3,1,4] => [3,1,2,4] => [[1,3,4],[2]]
=> 1
{{1,2,4},{3}}
=> [2,4,3,1] => [3,4,1,2] => [[1,2],[3,4]]
=> 1
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => [[1,3],[2,4]]
=> 2
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => [[1,3,4],[2]]
=> 1
{{1,3,4},{2}}
=> [3,2,4,1] => [2,4,1,3] => [[1,2],[3,4]]
=> 1
{{1,3},{2,4}}
=> [3,4,1,2] => [3,1,4,2] => [[1,3],[2,4]]
=> 2
{{1,3},{2},{4}}
=> [3,2,1,4] => [2,3,1,4] => [[1,2,4],[3]]
=> 1
{{1,4},{2,3}}
=> [4,3,2,1] => [3,2,4,1] => [[1,3],[2],[4]]
=> 2
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,2,3] => [[1,2,4],[3]]
=> 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => [[1,2,4],[3]]
=> 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,3,4,1] => [[1,2,3],[4]]
=> 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,3,4,2] => [[1,2,3],[4]]
=> 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => [[1,2,3],[4]]
=> 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
=> 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [5,1,2,3,4] => [[1,3,4,5],[2]]
=> 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [4,1,2,3,5] => [[1,3,4,5],[2]]
=> 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [4,5,1,2,3] => [[1,2,5],[3,4]]
=> 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [3,1,2,5,4] => [[1,3,4],[2,5]]
=> 2
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [3,1,2,4,5] => [[1,3,4,5],[2]]
=> 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [3,5,1,2,4] => [[1,2,5],[3,4]]
=> 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [4,1,2,5,3] => [[1,3,4],[2,5]]
=> 2
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [3,4,1,2,5] => [[1,2,5],[3,4]]
=> 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [4,3,5,1,2] => [[1,3],[2,5],[4]]
=> 2
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,5,3,4] => [[1,3,5],[2,4]]
=> 2
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,1,4,3,5] => [[1,3,5],[2,4]]
=> 2
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [3,4,5,1,2] => [[1,2,3],[4,5]]
=> 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,4,5,3] => [[1,3,4],[2,5]]
=> 2
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => [[1,3,4],[2,5]]
=> 2
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => [[1,3,4,5],[2]]
=> 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [2,5,1,3,4] => [[1,2,5],[3,4]]
=> 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [4,1,3,5,2] => [[1,3,4],[2],[5]]
=> 2
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [2,4,1,3,5] => [[1,2,5],[3,4]]
=> 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [4,2,5,1,3] => [[1,3],[2,5],[4]]
=> 2
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,1,5,2,4] => [[1,3,5],[2,4]]
=> 2
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [3,1,4,2,5] => [[1,3,5],[2,4]]
=> 2
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [2,4,5,1,3] => [[1,2,3],[4,5]]
=> 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [3,1,4,5,2] => [[1,3,4],[2,5]]
=> 2
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [2,3,1,5,4] => [[1,2,4],[3,5]]
=> 2
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [2,3,1,4,5] => [[1,2,4,5],[3]]
=> 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [3,2,5,1,4] => [[1,3],[2,5],[4]]
=> 2
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [4,1,5,2,3] => [[1,3,5],[2,4]]
=> 2
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [3,2,4,1,5] => [[1,3,5],[2],[4]]
=> 2
{{1},{2,3,4,5,6,7},{8}}
=> ? => ? => ?
=> ? = 1
{{1,2,5,6,7,8},{3,4}}
=> [2,5,4,3,6,7,8,1] => [4,3,8,1,2,5,6,7] => ?
=> ? = 2
{{1,3,4,6,7,8},{2,5}}
=> [3,5,4,6,2,7,8,1] => [5,2,8,1,3,4,6,7] => ?
=> ? = 2
{{1,2,4,6,7,8},{3,5}}
=> [2,4,5,6,3,7,8,1] => [5,3,8,1,2,4,6,7] => ?
=> ? = 2
Description
The number of descents of a standard tableau.
Entry $i$ of a standard Young tableau is a descent if $i+1$ appears in a row below the row of $i$.
Matching statistic: St001280
(load all 30 compositions to match this statistic)
(load all 30 compositions to match this statistic)
Mp00176: Set partitions —rotate decreasing⟶ Set partitions
Mp00164: Set partitions —Chen Deng Du Stanley Yan⟶ Set partitions
Mp00079: Set partitions —shape⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00164: Set partitions —Chen Deng Du Stanley Yan⟶ Set partitions
Mp00079: Set partitions —shape⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> {{1,2}}
=> {{1,2}}
=> [2]
=> 1
{{1},{2}}
=> {{1},{2}}
=> {{1},{2}}
=> [1,1]
=> 0
{{1,2,3}}
=> {{1,2,3}}
=> {{1,2,3}}
=> [3]
=> 1
{{1,2},{3}}
=> {{1,3},{2}}
=> {{1,3},{2}}
=> [2,1]
=> 1
{{1,3},{2}}
=> {{1},{2,3}}
=> {{1},{2,3}}
=> [2,1]
=> 1
{{1},{2,3}}
=> {{1,2},{3}}
=> {{1,2},{3}}
=> [2,1]
=> 1
{{1},{2},{3}}
=> {{1},{2},{3}}
=> {{1},{2},{3}}
=> [1,1,1]
=> 0
{{1,2,3,4}}
=> {{1,2,3,4}}
=> {{1,2,3,4}}
=> [4]
=> 1
{{1,2,3},{4}}
=> {{1,2,4},{3}}
=> {{1,2,4},{3}}
=> [3,1]
=> 1
{{1,2,4},{3}}
=> {{1,3,4},{2}}
=> {{1,3,4},{2}}
=> [3,1]
=> 1
{{1,2},{3,4}}
=> {{1,4},{2,3}}
=> {{1,3},{2,4}}
=> [2,2]
=> 2
{{1,2},{3},{4}}
=> {{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> [2,1,1]
=> 1
{{1,3,4},{2}}
=> {{1},{2,3,4}}
=> {{1},{2,3,4}}
=> [3,1]
=> 1
{{1,3},{2,4}}
=> {{1,3},{2,4}}
=> {{1,4},{2,3}}
=> [2,2]
=> 2
{{1,3},{2},{4}}
=> {{1},{2,4},{3}}
=> {{1},{2,4},{3}}
=> [2,1,1]
=> 1
{{1,4},{2,3}}
=> {{1,2},{3,4}}
=> {{1,2},{3,4}}
=> [2,2]
=> 2
{{1},{2,3,4}}
=> {{1,2,3},{4}}
=> {{1,2,3},{4}}
=> [3,1]
=> 1
{{1},{2,3},{4}}
=> {{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> [2,1,1]
=> 1
{{1,4},{2},{3}}
=> {{1},{2},{3,4}}
=> {{1},{2},{3,4}}
=> [2,1,1]
=> 1
{{1},{2,4},{3}}
=> {{1,3},{2},{4}}
=> {{1,3},{2},{4}}
=> [2,1,1]
=> 1
{{1},{2},{3,4}}
=> {{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> [2,1,1]
=> 1
{{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> [1,1,1,1]
=> 0
{{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> [5]
=> 1
{{1,2,3,4},{5}}
=> {{1,2,3,5},{4}}
=> {{1,2,3,5},{4}}
=> [4,1]
=> 1
{{1,2,3,5},{4}}
=> {{1,2,4,5},{3}}
=> {{1,2,4,5},{3}}
=> [4,1]
=> 1
{{1,2,3},{4,5}}
=> {{1,2,5},{3,4}}
=> {{1,2,4},{3,5}}
=> [3,2]
=> 2
{{1,2,3},{4},{5}}
=> {{1,2,5},{3},{4}}
=> {{1,2,5},{3},{4}}
=> [3,1,1]
=> 1
{{1,2,4,5},{3}}
=> {{1,3,4,5},{2}}
=> {{1,3,4,5},{2}}
=> [4,1]
=> 1
{{1,2,4},{3,5}}
=> {{1,3,5},{2,4}}
=> {{1,5},{2,3,4}}
=> [3,2]
=> 2
{{1,2,4},{3},{5}}
=> {{1,3,5},{2},{4}}
=> {{1,3,5},{2},{4}}
=> [3,1,1]
=> 1
{{1,2,5},{3,4}}
=> {{1,4,5},{2,3}}
=> {{1,3},{2,4,5}}
=> [3,2]
=> 2
{{1,2},{3,4,5}}
=> {{1,5},{2,3,4}}
=> {{1,3,5},{2,4}}
=> [3,2]
=> 2
{{1,2},{3,4},{5}}
=> {{1,5},{2,3},{4}}
=> {{1,3},{2,5},{4}}
=> [2,2,1]
=> 2
{{1,2,5},{3},{4}}
=> {{1,4,5},{2},{3}}
=> {{1,4,5},{2},{3}}
=> [3,1,1]
=> 1
{{1,2},{3,5},{4}}
=> {{1,5},{2,4},{3}}
=> {{1,4},{2,5},{3}}
=> [2,2,1]
=> 2
{{1,2},{3},{4,5}}
=> {{1,5},{2},{3,4}}
=> {{1,4},{2},{3,5}}
=> [2,2,1]
=> 2
{{1,2},{3},{4},{5}}
=> {{1,5},{2},{3},{4}}
=> {{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> 1
{{1,3,4,5},{2}}
=> {{1},{2,3,4,5}}
=> {{1},{2,3,4,5}}
=> [4,1]
=> 1
{{1,3,4},{2,5}}
=> {{1,4},{2,3,5}}
=> {{1,3,4},{2,5}}
=> [3,2]
=> 2
{{1,3,4},{2},{5}}
=> {{1},{2,3,5},{4}}
=> {{1},{2,3,5},{4}}
=> [3,1,1]
=> 1
{{1,3,5},{2,4}}
=> {{1,3},{2,4,5}}
=> {{1,4,5},{2,3}}
=> [3,2]
=> 2
{{1,3},{2,4,5}}
=> {{1,3,4},{2,5}}
=> {{1,4},{2,3,5}}
=> [3,2]
=> 2
{{1,3},{2,4},{5}}
=> {{1,3},{2,5},{4}}
=> {{1,5},{2,3},{4}}
=> [2,2,1]
=> 2
{{1,3,5},{2},{4}}
=> {{1},{2,4,5},{3}}
=> {{1},{2,4,5},{3}}
=> [3,1,1]
=> 1
{{1,3},{2,5},{4}}
=> {{1,4},{2,5},{3}}
=> {{1,5},{2,4},{3}}
=> [2,2,1]
=> 2
{{1,3},{2},{4,5}}
=> {{1},{2,5},{3,4}}
=> {{1},{2,4},{3,5}}
=> [2,2,1]
=> 2
{{1,3},{2},{4},{5}}
=> {{1},{2,5},{3},{4}}
=> {{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> 1
{{1,4,5},{2,3}}
=> {{1,2},{3,4,5}}
=> {{1,2},{3,4,5}}
=> [3,2]
=> 2
{{1,4},{2,3,5}}
=> {{1,2,4},{3,5}}
=> {{1,2,5},{3,4}}
=> [3,2]
=> 2
{{1,4},{2,3},{5}}
=> {{1,2},{3,5},{4}}
=> {{1,2},{3,5},{4}}
=> [2,2,1]
=> 2
{{1},{2},{3,4,5,6,7,8}}
=> {{1},{2,3,4,5,6,7},{8}}
=> {{1},{2,3,4,5,6,7},{8}}
=> ?
=> ? = 1
{{1},{2,3,5,6,7,8},{4}}
=> {{1,2,4,5,6,7},{3},{8}}
=> {{1,2,4,5,6,7},{3},{8}}
=> ?
=> ? = 1
{{1,2,3,4,5,6},{7,8}}
=> {{1,2,3,4,5,8},{6,7}}
=> {{1,2,3,4,5,7},{6,8}}
=> ?
=> ? = 2
{{1,2,3,4,7,8},{5,6}}
=> {{1,2,3,6,7,8},{4,5}}
=> {{1,2,3,5},{4,6,7,8}}
=> ?
=> ? = 2
{{1,3,4,5,6,7},{2,8}}
=> {{1,7},{2,3,4,5,6,8}}
=> {{1,3,5,8},{2,4,6,7}}
=> ?
=> ? = 2
{{1,2,4,5,6,7},{3,8}}
=> {{1,3,4,5,6,8},{2,7}}
=> {{1,4,6,7},{2,3,5,8}}
=> ?
=> ? = 2
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St000010
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00164: Set partitions —Chen Deng Du Stanley Yan⟶ Set partitions
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00308: Integer partitions —Bulgarian solitaire⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00308: Integer partitions —Bulgarian solitaire⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> {{1,2}}
=> [2]
=> [1,1]
=> 2 = 1 + 1
{{1},{2}}
=> {{1},{2}}
=> [1,1]
=> [2]
=> 1 = 0 + 1
{{1,2,3}}
=> {{1,2,3}}
=> [3]
=> [2,1]
=> 2 = 1 + 1
{{1,2},{3}}
=> {{1,2},{3}}
=> [2,1]
=> [2,1]
=> 2 = 1 + 1
{{1,3},{2}}
=> {{1,3},{2}}
=> [2,1]
=> [2,1]
=> 2 = 1 + 1
{{1},{2,3}}
=> {{1},{2,3}}
=> [2,1]
=> [2,1]
=> 2 = 1 + 1
{{1},{2},{3}}
=> {{1},{2},{3}}
=> [1,1,1]
=> [3]
=> 1 = 0 + 1
{{1,2,3,4}}
=> {{1,2,3,4}}
=> [4]
=> [3,1]
=> 2 = 1 + 1
{{1,2,3},{4}}
=> {{1,2,3},{4}}
=> [3,1]
=> [2,2]
=> 2 = 1 + 1
{{1,2,4},{3}}
=> {{1,2,4},{3}}
=> [3,1]
=> [2,2]
=> 2 = 1 + 1
{{1,2},{3,4}}
=> {{1,2},{3,4}}
=> [2,2]
=> [2,1,1]
=> 3 = 2 + 1
{{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> [2,1,1]
=> [3,1]
=> 2 = 1 + 1
{{1,3,4},{2}}
=> {{1,3,4},{2}}
=> [3,1]
=> [2,2]
=> 2 = 1 + 1
{{1,3},{2,4}}
=> {{1,4},{2,3}}
=> [2,2]
=> [2,1,1]
=> 3 = 2 + 1
{{1,3},{2},{4}}
=> {{1,3},{2},{4}}
=> [2,1,1]
=> [3,1]
=> 2 = 1 + 1
{{1,4},{2,3}}
=> {{1,3},{2,4}}
=> [2,2]
=> [2,1,1]
=> 3 = 2 + 1
{{1},{2,3,4}}
=> {{1},{2,3,4}}
=> [3,1]
=> [2,2]
=> 2 = 1 + 1
{{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> [2,1,1]
=> [3,1]
=> 2 = 1 + 1
{{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> [2,1,1]
=> [3,1]
=> 2 = 1 + 1
{{1},{2,4},{3}}
=> {{1},{2,4},{3}}
=> [2,1,1]
=> [3,1]
=> 2 = 1 + 1
{{1},{2},{3,4}}
=> {{1},{2},{3,4}}
=> [2,1,1]
=> [3,1]
=> 2 = 1 + 1
{{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> [1,1,1,1]
=> [4]
=> 1 = 0 + 1
{{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> [5]
=> [4,1]
=> 2 = 1 + 1
{{1,2,3,4},{5}}
=> {{1,2,3,4},{5}}
=> [4,1]
=> [3,2]
=> 2 = 1 + 1
{{1,2,3,5},{4}}
=> {{1,2,3,5},{4}}
=> [4,1]
=> [3,2]
=> 2 = 1 + 1
{{1,2,3},{4,5}}
=> {{1,2,3},{4,5}}
=> [3,2]
=> [2,2,1]
=> 3 = 2 + 1
{{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> [3,1,1]
=> [3,2]
=> 2 = 1 + 1
{{1,2,4,5},{3}}
=> {{1,2,4,5},{3}}
=> [4,1]
=> [3,2]
=> 2 = 1 + 1
{{1,2,4},{3,5}}
=> {{1,2,5},{3,4}}
=> [3,2]
=> [2,2,1]
=> 3 = 2 + 1
{{1,2,4},{3},{5}}
=> {{1,2,4},{3},{5}}
=> [3,1,1]
=> [3,2]
=> 2 = 1 + 1
{{1,2,5},{3,4}}
=> {{1,2,4},{3,5}}
=> [3,2]
=> [2,2,1]
=> 3 = 2 + 1
{{1,2},{3,4,5}}
=> {{1,2},{3,4,5}}
=> [3,2]
=> [2,2,1]
=> 3 = 2 + 1
{{1,2},{3,4},{5}}
=> {{1,2},{3,4},{5}}
=> [2,2,1]
=> [3,1,1]
=> 3 = 2 + 1
{{1,2,5},{3},{4}}
=> {{1,2,5},{3},{4}}
=> [3,1,1]
=> [3,2]
=> 2 = 1 + 1
{{1,2},{3,5},{4}}
=> {{1,2},{3,5},{4}}
=> [2,2,1]
=> [3,1,1]
=> 3 = 2 + 1
{{1,2},{3},{4,5}}
=> {{1,2},{3},{4,5}}
=> [2,2,1]
=> [3,1,1]
=> 3 = 2 + 1
{{1,2},{3},{4},{5}}
=> {{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [4,1]
=> 2 = 1 + 1
{{1,3,4,5},{2}}
=> {{1,3,4,5},{2}}
=> [4,1]
=> [3,2]
=> 2 = 1 + 1
{{1,3,4},{2,5}}
=> {{1,4},{2,3,5}}
=> [3,2]
=> [2,2,1]
=> 3 = 2 + 1
{{1,3,4},{2},{5}}
=> {{1,3,4},{2},{5}}
=> [3,1,1]
=> [3,2]
=> 2 = 1 + 1
{{1,3,5},{2,4}}
=> {{1,5},{2,3,4}}
=> [3,2]
=> [2,2,1]
=> 3 = 2 + 1
{{1,3},{2,4,5}}
=> {{1,4,5},{2,3}}
=> [3,2]
=> [2,2,1]
=> 3 = 2 + 1
{{1,3},{2,4},{5}}
=> {{1,4},{2,3},{5}}
=> [2,2,1]
=> [3,1,1]
=> 3 = 2 + 1
{{1,3,5},{2},{4}}
=> {{1,3,5},{2},{4}}
=> [3,1,1]
=> [3,2]
=> 2 = 1 + 1
{{1,3},{2,5},{4}}
=> {{1,5},{2,3},{4}}
=> [2,2,1]
=> [3,1,1]
=> 3 = 2 + 1
{{1,3},{2},{4,5}}
=> {{1,3},{2},{4,5}}
=> [2,2,1]
=> [3,1,1]
=> 3 = 2 + 1
{{1,3},{2},{4},{5}}
=> {{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [4,1]
=> 2 = 1 + 1
{{1,4,5},{2,3}}
=> {{1,3},{2,4,5}}
=> [3,2]
=> [2,2,1]
=> 3 = 2 + 1
{{1,4},{2,3,5}}
=> {{1,3,4},{2,5}}
=> [3,2]
=> [2,2,1]
=> 3 = 2 + 1
{{1,4},{2,3},{5}}
=> {{1,3},{2,4},{5}}
=> [2,2,1]
=> [3,1,1]
=> 3 = 2 + 1
{{1},{2,3,5,6,7,8},{4}}
=> {{1},{2,3,5,6,7,8},{4}}
=> ?
=> ?
=> ? = 1 + 1
{{1},{2,3,4,5,6,7},{8}}
=> {{1},{2,3,4,5,6,7},{8}}
=> ?
=> ?
=> ? = 1 + 1
{{1,4,5,6,7,8},{2},{3}}
=> {{1,4,5,6,7,8},{2},{3}}
=> ?
=> ?
=> ? = 1 + 1
{{1,3,5,6,7,8},{2},{4}}
=> {{1,3,5,6,7,8},{2},{4}}
=> ?
=> ?
=> ? = 1 + 1
{{1,2,3,6,7,8},{4,5}}
=> {{1,2,3,5},{4,6,7,8}}
=> ?
=> ?
=> ? = 2 + 1
{{1,2,3,4,5,8},{6,7}}
=> {{1,2,3,4,5,7},{6,8}}
=> ?
=> ?
=> ? = 2 + 1
{{1,3,4,5,6,8},{2,7}}
=> {{1,4,6,7},{2,3,5,8}}
=> ?
=> ?
=> ? = 2 + 1
{{1,7},{2,3,4,5,6,8}}
=> {{1,3,5,8},{2,4,6,7}}
=> ?
=> ?
=> ? = 2 + 1
Description
The length of the partition.
Matching statistic: St000659
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000659: Dyck paths ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000659: Dyck paths ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2] => [2] => [1,1,0,0]
=> 1
{{1},{2}}
=> [1,1] => [1,1] => [1,0,1,0]
=> 0
{{1,2,3}}
=> [3] => [3] => [1,1,1,0,0,0]
=> 1
{{1,2},{3}}
=> [2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
{{1,3},{2}}
=> [2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
{{1},{2,3}}
=> [1,2] => [2,1] => [1,1,0,0,1,0]
=> 1
{{1},{2},{3}}
=> [1,1,1] => [1,1,1] => [1,0,1,0,1,0]
=> 0
{{1,2,3,4}}
=> [4] => [4] => [1,1,1,1,0,0,0,0]
=> 1
{{1,2,3},{4}}
=> [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
{{1,2,4},{3}}
=> [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
{{1,2},{3,4}}
=> [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
{{1,2},{3},{4}}
=> [2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
{{1,3,4},{2}}
=> [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
{{1,3},{2,4}}
=> [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
{{1,3},{2},{4}}
=> [2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
{{1,4},{2,3}}
=> [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
{{1},{2,3,4}}
=> [1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
{{1},{2,3},{4}}
=> [1,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
{{1,4},{2},{3}}
=> [2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
{{1},{2,4},{3}}
=> [1,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
{{1},{2},{3,4}}
=> [1,1,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
{{1},{2},{3},{4}}
=> [1,1,1,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
{{1,2,3,4,5}}
=> [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
{{1,2,3,4},{5}}
=> [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,2,3,5},{4}}
=> [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,2,3},{4,5}}
=> [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
{{1,2,4,5},{3}}
=> [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,2,4},{3,5}}
=> [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
{{1,2,5},{3,4}}
=> [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
{{1,2},{3,4,5}}
=> [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
{{1,2},{3},{4,5}}
=> [2,1,2] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
{{1,3,4,5},{2}}
=> [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,3,4},{2,5}}
=> [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
{{1,3,5},{2,4}}
=> [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
{{1,3},{2,4,5}}
=> [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
{{1,3},{2},{4,5}}
=> [2,1,2] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
{{1,4,5},{2,3}}
=> [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
{{1,4},{2,3,5}}
=> [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
{{1,4},{2,3},{5}}
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
{{1},{2},{3,4,5,6,7,8}}
=> [1,1,6] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
{{1},{2,3,4,5,6,7,8}}
=> [1,7] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
{{1,2},{3,4,5,6,7,8}}
=> [2,6] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 2
{{1,8},{2,3,4,5,6,7}}
=> [2,6] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 2
{{1,2,3,4,5,6,7,8}}
=> [8] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
{{1,3},{2,4,5,6,7,8}}
=> [2,6] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 2
{{1,4},{2,3,5,6,7,8}}
=> [2,6] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 2
{{1,5},{2,3,4,6,7,8}}
=> [2,6] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 2
{{1,6},{2,3,4,5,7,8}}
=> [2,6] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 2
{{1,7},{2,3,4,5,6,8}}
=> [2,6] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 2
Description
The number of rises of length at least 2 of a Dyck path.
The following 75 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000147The largest part of an integer partition. St000665The number of rafts of a permutation. St000834The number of right outer peaks of a permutation. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000919The number of maximal left branches of a binary tree. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St000053The number of valleys of the Dyck path. St000211The rank of the set partition. St000558The number of occurrences of the pattern {{1,2}} in a set partition. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St001471The magnitude of a Dyck path. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St000668The least common multiple of the parts of the partition. St001354The number of series nodes in the modular decomposition of a graph. St000306The bounce count of a Dyck path. St000884The number of isolated descents of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001737The number of descents of type 2 in a permutation. St001928The number of non-overlapping descents in a permutation. St000470The number of runs in a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000670The reversal length of a permutation. St000662The staircase size of the code of a permutation. St000035The number of left outer peaks of a permutation. St000703The number of deficiencies of a permutation. St000354The number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001665The number of pure excedances of a permutation. St001726The number of visible inversions of a permutation. St001729The number of visible descents of a permutation. St000245The number of ascents of a permutation. St000702The number of weak deficiencies of a permutation. St001427The number of descents of a signed permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000021The number of descents of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000325The width of the tree associated to a permutation. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001874Lusztig's a-function for the symmetric group. St000015The number of peaks of a Dyck path. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000023The number of inner peaks of a permutation. St000710The number of big deficiencies of a permutation. St000779The tier of a permutation. St000099The number of valleys of a permutation, including the boundary. 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 $eA$. St000455The second largest eigenvalue of a graph if it is integral. St001864The number of excedances of a signed permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St001946The number of descents in a parking function.
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