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Your data matches 215 different statistics following compositions of up to 3 maps.
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Matching statistic: St001011
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
St001011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 0
[1,0,1,0]
=> 1
[1,1,0,0]
=> 0
[1,0,1,0,1,0]
=> 1
[1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0]
=> 1
[1,1,1,0,0,0]
=> 0
[1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> 1
Description
Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000291
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00100: Dyck paths —touch 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
[1,0]
=> [1] => 1 => 0
[1,0,1,0]
=> [1,1] => 11 => 0
[1,1,0,0]
=> [2] => 10 => 1
[1,0,1,0,1,0]
=> [1,1,1] => 111 => 0
[1,0,1,1,0,0]
=> [1,2] => 110 => 1
[1,1,0,0,1,0]
=> [2,1] => 101 => 1
[1,1,0,1,0,0]
=> [3] => 100 => 1
[1,1,1,0,0,0]
=> [3] => 100 => 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1111 => 0
[1,0,1,0,1,1,0,0]
=> [1,1,2] => 1110 => 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => 1101 => 1
[1,0,1,1,0,1,0,0]
=> [1,3] => 1100 => 1
[1,0,1,1,1,0,0,0]
=> [1,3] => 1100 => 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => 1011 => 1
[1,1,0,0,1,1,0,0]
=> [2,2] => 1010 => 2
[1,1,0,1,0,0,1,0]
=> [3,1] => 1001 => 1
[1,1,0,1,0,1,0,0]
=> [4] => 1000 => 1
[1,1,0,1,1,0,0,0]
=> [4] => 1000 => 1
[1,1,1,0,0,0,1,0]
=> [3,1] => 1001 => 1
[1,1,1,0,0,1,0,0]
=> [4] => 1000 => 1
[1,1,1,0,1,0,0,0]
=> [4] => 1000 => 1
[1,1,1,1,0,0,0,0]
=> [4] => 1000 => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => 11111 => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 11110 => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => 11101 => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => 11100 => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 11100 => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 11011 => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 11010 => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => 11001 => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => 11000 => 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => 11000 => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 11001 => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => 11000 => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => 11000 => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 11000 => 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => 10111 => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => 10110 => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => 10101 => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => 10100 => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 10100 => 2
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => 10011 => 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => 10010 => 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => 10001 => 1
[1,1,0,1,0,1,0,1,0,0]
=> [5] => 10000 => 1
[1,1,0,1,0,1,1,0,0,0]
=> [5] => 10000 => 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => 10001 => 1
[1,1,0,1,1,0,0,1,0,0]
=> [5] => 10000 => 1
[1,1,0,1,1,0,1,0,0,0]
=> [5] => 10000 => 1
[1,1,0,1,1,1,0,0,0,0]
=> [5] => 10000 => 1
Description
The number of descents of a binary word.
Matching statistic: St001280
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1]
=> 0
[1,0,1,0]
=> [1,1] => [1,1]
=> 0
[1,1,0,0]
=> [2] => [2]
=> 1
[1,0,1,0,1,0]
=> [1,1,1] => [1,1,1]
=> 0
[1,0,1,1,0,0]
=> [1,2] => [2,1]
=> 1
[1,1,0,0,1,0]
=> [2,1] => [2,1]
=> 1
[1,1,0,1,0,0]
=> [3] => [3]
=> 1
[1,1,1,0,0,0]
=> [3] => [3]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,1,1,1]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1,1]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,1,1]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,3] => [3,1]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,3] => [3,1]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [2,1,1]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,2] => [2,2]
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1] => [3,1]
=> 1
[1,1,0,1,0,1,0,0]
=> [4] => [4]
=> 1
[1,1,0,1,1,0,0,0]
=> [4] => [4]
=> 1
[1,1,1,0,0,0,1,0]
=> [3,1] => [3,1]
=> 1
[1,1,1,0,0,1,0,0]
=> [4] => [4]
=> 1
[1,1,1,0,1,0,0,0]
=> [4] => [4]
=> 1
[1,1,1,1,0,0,0,0]
=> [4] => [4]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,1,1,1,1]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [2,1,1,1]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,1,1,1]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [3,1,1]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [3,1,1]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [2,1,1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [2,2,1]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [3,1,1]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [4,1]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [4,1]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [3,1,1]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [4,1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [4,1]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [2,1,1,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [2,2,1]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,2,1]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [3,2]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [3,2]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [3,1,1]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [3,2]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [4,1]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [5] => [5]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [5] => [5]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [4,1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [5] => [5]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [5] => [5]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [5] => [5]
=> 1
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St000292
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00100: Dyck paths —touch 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
[1,0]
=> [1] => 1 => 1 => 0
[1,0,1,0]
=> [1,1] => 11 => 11 => 0
[1,1,0,0]
=> [2] => 10 => 01 => 1
[1,0,1,0,1,0]
=> [1,1,1] => 111 => 111 => 0
[1,0,1,1,0,0]
=> [1,2] => 110 => 011 => 1
[1,1,0,0,1,0]
=> [2,1] => 101 => 101 => 1
[1,1,0,1,0,0]
=> [3] => 100 => 001 => 1
[1,1,1,0,0,0]
=> [3] => 100 => 001 => 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1111 => 1111 => 0
[1,0,1,0,1,1,0,0]
=> [1,1,2] => 1110 => 0111 => 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => 1101 => 1011 => 1
[1,0,1,1,0,1,0,0]
=> [1,3] => 1100 => 0011 => 1
[1,0,1,1,1,0,0,0]
=> [1,3] => 1100 => 0011 => 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => 1011 => 1101 => 1
[1,1,0,0,1,1,0,0]
=> [2,2] => 1010 => 0101 => 2
[1,1,0,1,0,0,1,0]
=> [3,1] => 1001 => 1001 => 1
[1,1,0,1,0,1,0,0]
=> [4] => 1000 => 0001 => 1
[1,1,0,1,1,0,0,0]
=> [4] => 1000 => 0001 => 1
[1,1,1,0,0,0,1,0]
=> [3,1] => 1001 => 1001 => 1
[1,1,1,0,0,1,0,0]
=> [4] => 1000 => 0001 => 1
[1,1,1,0,1,0,0,0]
=> [4] => 1000 => 0001 => 1
[1,1,1,1,0,0,0,0]
=> [4] => 1000 => 0001 => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => 11111 => 11111 => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 11110 => 01111 => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => 11101 => 10111 => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => 11100 => 00111 => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 11100 => 00111 => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 11011 => 11011 => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 11010 => 01011 => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => 11001 => 10011 => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => 11000 => 00011 => 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => 11000 => 00011 => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 11001 => 10011 => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => 11000 => 00011 => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => 11000 => 00011 => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 11000 => 00011 => 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => 10111 => 11101 => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => 10110 => 01101 => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => 10101 => 10101 => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => 10100 => 00101 => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 10100 => 00101 => 2
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => 10011 => 11001 => 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => 10010 => 01001 => 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => 10001 => 10001 => 1
[1,1,0,1,0,1,0,1,0,0]
=> [5] => 10000 => 00001 => 1
[1,1,0,1,0,1,1,0,0,0]
=> [5] => 10000 => 00001 => 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => 10001 => 10001 => 1
[1,1,0,1,1,0,0,1,0,0]
=> [5] => 10000 => 00001 => 1
[1,1,0,1,1,0,1,0,0,0]
=> [5] => 10000 => 00001 => 1
[1,1,0,1,1,1,0,0,0,0]
=> [5] => 10000 => 00001 => 1
Description
The number of ascents of a binary word.
Matching statistic: St000340
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000340: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000340: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1,0]
=> 0
[1,0,1,0]
=> [2,1] => [1,2] => [1,0,1,0]
=> 0
[1,1,0,0]
=> [1,2] => [2,1] => [1,1,0,0]
=> 1
[1,0,1,0,1,0]
=> [3,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> 0
[1,0,1,1,0,0]
=> [2,3,1] => [3,1,2] => [1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0]
=> [3,1,2] => [1,3,2] => [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [2,1,3] => [3,2,1] => [1,1,1,0,0,0]
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [2,3,1] => [1,1,0,1,0,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> 1
Description
The number of non-final maximal constant sub-paths of length greater than one.
This is the total number of occurrences of the patterns $110$ and $001$.
Matching statistic: St000390
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00100: Dyck paths —touch 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
[1,0]
=> [1] => 1 => 0 => 0
[1,0,1,0]
=> [1,1] => 11 => 00 => 0
[1,1,0,0]
=> [2] => 10 => 01 => 1
[1,0,1,0,1,0]
=> [1,1,1] => 111 => 000 => 0
[1,0,1,1,0,0]
=> [1,2] => 110 => 001 => 1
[1,1,0,0,1,0]
=> [2,1] => 101 => 010 => 1
[1,1,0,1,0,0]
=> [3] => 100 => 011 => 1
[1,1,1,0,0,0]
=> [3] => 100 => 011 => 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1111 => 0000 => 0
[1,0,1,0,1,1,0,0]
=> [1,1,2] => 1110 => 0001 => 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => 1101 => 0010 => 1
[1,0,1,1,0,1,0,0]
=> [1,3] => 1100 => 0011 => 1
[1,0,1,1,1,0,0,0]
=> [1,3] => 1100 => 0011 => 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => 1011 => 0100 => 1
[1,1,0,0,1,1,0,0]
=> [2,2] => 1010 => 0101 => 2
[1,1,0,1,0,0,1,0]
=> [3,1] => 1001 => 0110 => 1
[1,1,0,1,0,1,0,0]
=> [4] => 1000 => 0111 => 1
[1,1,0,1,1,0,0,0]
=> [4] => 1000 => 0111 => 1
[1,1,1,0,0,0,1,0]
=> [3,1] => 1001 => 0110 => 1
[1,1,1,0,0,1,0,0]
=> [4] => 1000 => 0111 => 1
[1,1,1,0,1,0,0,0]
=> [4] => 1000 => 0111 => 1
[1,1,1,1,0,0,0,0]
=> [4] => 1000 => 0111 => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => 11111 => 00000 => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 11110 => 00001 => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => 11101 => 00010 => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => 11100 => 00011 => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 11100 => 00011 => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 11011 => 00100 => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 11010 => 00101 => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => 11001 => 00110 => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => 11000 => 00111 => 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => 11000 => 00111 => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 11001 => 00110 => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => 11000 => 00111 => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => 11000 => 00111 => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 11000 => 00111 => 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => 10111 => 01000 => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => 10110 => 01001 => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => 10101 => 01010 => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => 10100 => 01011 => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 10100 => 01011 => 2
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => 10011 => 01100 => 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => 10010 => 01101 => 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => 10001 => 01110 => 1
[1,1,0,1,0,1,0,1,0,0]
=> [5] => 10000 => 01111 => 1
[1,1,0,1,0,1,1,0,0,0]
=> [5] => 10000 => 01111 => 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => 10001 => 01110 => 1
[1,1,0,1,1,0,0,1,0,0]
=> [5] => 10000 => 01111 => 1
[1,1,0,1,1,0,1,0,0,0]
=> [5] => 10000 => 01111 => 1
[1,1,0,1,1,1,0,0,0,0]
=> [5] => 10000 => 01111 => 1
Description
The number of runs of ones in a binary word.
Matching statistic: St001333
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001333: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001333: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => ([],1)
=> 0
[1,0,1,0]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[1,1,0,0]
=> [1,2] => [1,2] => ([],2)
=> 0
[1,0,1,0,1,0]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> 1
[1,0,1,1,0,0]
=> [2,3,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,1,0,0,1,0]
=> [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[1,1,0,1,0,0]
=> [1,3,2] => [1,3,2] => ([(1,2)],3)
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => ([],3)
=> 0
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> 1
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> 1
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> 1
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 1
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [5,2,4,1,3] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [5,2,1,4,3] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [5,3,1,4,2] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [5,1,3,4,2] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [5,1,3,2,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
Description
The cardinality of a minimal edge-isolating set of a graph.
Let $\mathcal F$ be a set of graphs. A set of vertices $S$ is $\mathcal F$-isolating, if the subgraph induced by the vertices in the complement of the closed neighbourhood of $S$ does not contain any graph in $\mathcal F$.
This statistic returns the cardinality of the smallest isolating set when $\mathcal F$ contains only the graph with one edge.
Matching statistic: St000010
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1]
=> 1 = 0 + 1
[1,0,1,0]
=> [1,2] => [1,2] => [1,1]
=> 2 = 1 + 1
[1,1,0,0]
=> [2,1] => [2,1] => [2]
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,3,2] => [2,1]
=> 2 = 1 + 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => [2,1]
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [2,1]
=> 2 = 1 + 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,3,1] => [2,1]
=> 2 = 1 + 1
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => [3]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,4,3,2] => [3,1]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,4,3,2] => [3,1]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,4,3,2] => [3,1]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => [3,1]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,3,2] => [3,1]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,4,3] => [2,2]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => [2,2]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,4,1,3] => [2,1,1]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,4,3,1] => [3,1]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,4,3,1] => [3,1]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => [3,1]
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => [3,1]
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,4,2,1] => [3,1]
=> 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => [4]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,5,4,3,2] => [4,1]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,5,4,3,2] => [4,1]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,5,4,3,2] => [4,1]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,5,4,3,2] => [4,1]
=> 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,5,4,3,2] => [4,1]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,5,4,3,2] => [4,1]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,5,4,3,2] => [4,1]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,5,4,3,2] => [4,1]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,4,3,2] => [4,1]
=> 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,5,4,3,2] => [4,1]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,5,4,3,2] => [4,1]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,5,4,3,2] => [4,1]
=> 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,5,4,3,2] => [4,1]
=> 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,5,4,3,2] => [4,1]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,5,4,3] => [3,2]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,5,4,3] => [3,2]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,5,4,3] => [3,2]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => [3,2]
=> 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,5,4,3] => [3,2]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,5,1,4,3] => [3,1,1]
=> 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,5,1,4,3] => [3,1,1]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,5,4,1,3] => [3,1,1]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,5,4,3,1] => [4,1]
=> 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [2,5,4,3,1] => [4,1]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [2,5,4,1,3] => [3,1,1]
=> 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [2,5,4,3,1] => [4,1]
=> 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [2,5,4,3,1] => [4,1]
=> 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,5,4,3,1] => [4,1]
=> 2 = 1 + 1
Description
The length of the partition.
Matching statistic: St000507
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [[1]]
=> 1 = 0 + 1
[1,0,1,0]
=> [1,2] => [1,2] => [[1,2]]
=> 2 = 1 + 1
[1,1,0,0]
=> [2,1] => [2,1] => [[1],[2]]
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,3,2] => [[1,2],[3]]
=> 2 = 1 + 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => [[1,2],[3]]
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [[1,3],[2]]
=> 2 = 1 + 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,3,1] => [[1,2],[3]]
=> 2 = 1 + 1
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => [[1],[2],[3]]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,4,3] => [[1,3],[2,4]]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => [[1,3],[2,4]]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,4,1,3] => [[1,2],[3,4]]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,4,3,1] => [[1,2],[3],[4]]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,4,3,1] => [[1,2],[3],[4]]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => [[1,4],[2],[3]]
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => [[1,3],[2],[4]]
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,4,2,1] => [[1,2],[3],[4]]
=> 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,5,4,3] => [[1,3],[2,4],[5]]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,5,4,3] => [[1,3],[2,4],[5]]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,5,4,3] => [[1,3],[2,4],[5]]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => [[1,3],[2,4],[5]]
=> 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,5,4,3] => [[1,3],[2,4],[5]]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,5,1,4,3] => [[1,2],[3,4],[5]]
=> 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,5,1,4,3] => [[1,2],[3,4],[5]]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,5,4,1,3] => [[1,2],[3,5],[4]]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,5,4,3,1] => [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [2,5,4,3,1] => [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [2,5,4,1,3] => [[1,2],[3,5],[4]]
=> 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [2,5,4,3,1] => [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [2,5,4,3,1] => [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,5,4,3,1] => [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
Description
The number of ascents of a standard tableau.
Entry $i$ of a standard Young tableau is an '''ascent''' if $i+1$ appears to the right or above $i$ in the tableau (with respect to the English notation for tableaux).
Matching statistic: St000251
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000251: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000251: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => {{1}}
=> ? = 0
[1,0,1,0]
=> [2,1] => [2,1] => {{1,2}}
=> 1
[1,1,0,0]
=> [1,2] => [1,2] => {{1},{2}}
=> 0
[1,0,1,0,1,0]
=> [3,2,1] => [3,1,2] => {{1,2,3}}
=> 1
[1,0,1,1,0,0]
=> [2,3,1] => [3,2,1] => {{1,3},{2}}
=> 1
[1,1,0,0,1,0]
=> [3,1,2] => [2,3,1] => {{1,2,3}}
=> 1
[1,1,0,1,0,0]
=> [2,1,3] => [2,1,3] => {{1,2},{3}}
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => {{1},{2},{3}}
=> 0
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4,1,2,3] => {{1,2,3,4}}
=> 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [4,1,3,2] => {{1,2,4},{3}}
=> 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [4,3,1,2] => {{1,2,3,4}}
=> 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [4,3,2,1] => {{1,4},{2,3}}
=> 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,2,3,1] => {{1,4},{2},{3}}
=> 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [2,4,1,3] => {{1,2,3,4}}
=> 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,4,3,1] => {{1,2,4},{3}}
=> 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [3,1,4,2] => {{1,2,3,4}}
=> 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,1,2,4] => {{1,2,3},{4}}
=> 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [3,2,1,4] => {{1,3},{2},{4}}
=> 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,3,4,1] => {{1,2,3,4}}
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [2,3,1,4] => {{1,2,3},{4}}
=> 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => {{1,2},{3},{4}}
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [5,1,2,3,4] => {{1,2,3,4,5}}
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [5,1,2,4,3] => {{1,2,3,5},{4}}
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [5,1,4,2,3] => {{1,2,3,4,5}}
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [5,1,4,3,2] => {{1,2,5},{3,4}}
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [5,1,3,4,2] => {{1,2,5},{3},{4}}
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [5,3,1,2,4] => {{1,2,3,4,5}}
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [5,3,1,4,2] => {{1,2,3,5},{4}}
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [5,4,2,1,3] => {{1,2,3,4,5}}
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [5,4,2,3,1] => {{1,5},{2,3,4}}
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [5,4,3,2,1] => {{1,5},{2,4},{3}}
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [5,3,4,1,2] => {{1,2,3,4,5}}
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [5,3,4,2,1] => {{1,5},{2,3,4}}
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [5,3,2,4,1] => {{1,5},{2,3},{4}}
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => {{1,5},{2},{3},{4}}
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [2,5,1,3,4] => {{1,2,3,4,5}}
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [2,5,1,4,3] => {{1,2,3,5},{4}}
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [2,5,4,1,3] => {{1,2,3,4,5}}
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [2,5,4,3,1] => {{1,2,5},{3,4}}
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [2,5,3,4,1] => {{1,2,5},{3},{4}}
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [3,1,5,2,4] => {{1,2,3,4,5}}
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [3,1,5,4,2] => {{1,2,3,5},{4}}
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [4,1,2,5,3] => {{1,2,3,4,5}}
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [4,1,2,3,5] => {{1,2,3,4},{5}}
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [4,1,3,2,5] => {{1,2,4},{3},{5}}
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [4,3,1,5,2] => {{1,2,3,4,5}}
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [4,3,1,2,5] => {{1,2,3,4},{5}}
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [4,3,2,1,5] => {{1,4},{2,3},{5}}
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => {{1,4},{2},{3},{5}}
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [2,3,5,1,4] => {{1,2,3,4,5}}
=> 1
Description
The number of nonsingleton blocks of a set partition.
The following 205 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000254The nesting number of a set partition. St000659The number of rises of length at least 2 of a Dyck path. St000919The number of maximal left branches of a binary tree. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001613The binary logarithm of the size of the center of a lattice. St001340The cardinality of a minimal non-edge isolating set of a graph. St000258The burning number of a graph. St001732The number of peaks visible from the left. St000035The number of left outer peaks of a permutation. St000884The number of isolated descents of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000374The number of exclusive right-to-left minima of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000912The number of maximal antichains in a poset. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000260The radius of a connected graph. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001432The order dimension of the partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001933The largest multiplicity of a part in an integer partition. St000245The number of ascents of a permutation. St000834The number of right outer peaks of a permutation. St000454The largest eigenvalue of a graph if it is integral. St000024The number of double up and double down steps of a Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St000658The number of rises of length 2 of a Dyck path. St000672The number of minimal elements in Bruhat order not less than the permutation. St001354The number of series nodes in the modular decomposition of a graph. St000386The number of factors DDU in a Dyck path. St000706The product of the factorials of the multiplicities of an integer partition. St000993The multiplicity of the largest part of an integer partition. St000053The number of valleys of the Dyck path. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000306The bounce count of a Dyck path. St000331The number of upper interactions of a Dyck path. St000627The exponent of a binary word. St000628The balance of a binary word. St000655The length of the minimal rise of a Dyck path. St000675The number of centered multitunnels of a Dyck path. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the 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. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001481The minimal height of a peak of a Dyck path. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001722The number of minimal chains with small intervals between a binary word and the top element. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St001884The number of borders of a binary word. St000702The number of weak deficiencies of a permutation. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001665The number of pure excedances of a permutation. St000781The number of proper colouring schemes of a Ferrers diagram. St001597The Frobenius rank of a skew partition. St001737The number of descents of type 2 in a permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000741The Colin de Verdière graph invariant. St001489The maximum of the number of descents and the number of inverse descents. St001729The number of visible descents of a permutation. St001928The number of non-overlapping descents in a permutation. St000470The number of runs in a permutation. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001568The smallest positive integer that does not appear twice in the partition. St000764The number of strong records in an integer composition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000640The rank of the largest boolean interval in a poset. St000914The sum of the values of the Möbius function of a poset. St000100The number of linear extensions of a poset. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000618The number of self-evacuating tableaux of given shape. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000934The 2-degree of an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St001128The exponens consonantiae of a partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001564The value of the forgotten symmetric functions when all variables set to 1. St001593This is the number of standard Young tableaux of the given shifted shape. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St001890The maximum magnitude of the Möbius function of a poset. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001298The number of repeated entries in the Lehmer code of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000021The number of descents of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000456The monochromatic index of a connected graph. 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. 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. St000062The length of the longest increasing subsequence of the permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000308The height of the tree associated to a permutation. St000325The width of the tree associated to a permutation. St000443The number of long tunnels of a Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000083The number of left oriented leafs of a binary tree except the first one. St001344The neighbouring number of a permutation. St000007The number of saliances of the permutation. St001330The hat guessing number of a graph. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001162The minimum jump of a permutation. St000570The Edelman-Greene number of a permutation. St000871The number of very big ascents of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St000259The diameter of a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000352The Elizalde-Pak rank of a permutation. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000542The number of left-to-right-minima of a permutation. St000356The number of occurrences of the pattern 13-2. St000486The number of cycles of length at least 3 of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St000455The second largest eigenvalue of a graph if it is integral. St001877Number of indecomposable injective modules with projective dimension 2. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000647The number of big descents of a permutation. St001644The dimension of a graph. St000181The number of connected components of the Hasse diagram for the poset. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St000023The number of inner peaks of a permutation. St000779The tier of a 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. St001728The number of invisible descents of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000099The number of valleys of a permutation, including the boundary. St001235The global dimension of the corresponding Comp-Nakayama algebra. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000862The number of parts of the shifted shape of a permutation. St000633The size of the automorphism group of a poset. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000660The number of rises of length at least 3 of a Dyck path. St001104The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. 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 $eA$. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001624The breadth of a lattice. 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 $eA$. St000632The jump number of the poset. St000807The sum of the heights of the valleys of the associated bargraph. St001335The cardinality of a minimal cycle-isolating set of a graph. St001394The genus of a permutation. St001413Half the length of the longest even length palindromic prefix of a binary word. St001423The number of distinct cubes in a binary word. St001470The cyclic holeyness of a permutation. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001935The number of ascents in a parking function. St001960The number of descents of a permutation minus one if its first entry is not one. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000298The order dimension or Dushnik-Miller dimension of a poset. St000758The length of the longest staircase fitting into an integer composition. St000805The number of peaks of the associated bargraph. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001741The largest integer such that all patterns of this size are contained in the permutation. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St000028The number of stack-sorts needed to sort a permutation.
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