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Matching statistic: St000291
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(load all 6 compositions to match this statistic)
Mp00071: Permutations —descent 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] => [1] => 1 => 0 = 1 - 1
[1,2] => [2] => 10 => 1 = 2 - 1
[2,1] => [1,1] => 11 => 0 = 1 - 1
[1,2,3] => [3] => 100 => 1 = 2 - 1
[1,3,2] => [2,1] => 101 => 1 = 2 - 1
[2,1,3] => [1,2] => 110 => 1 = 2 - 1
[2,3,1] => [2,1] => 101 => 1 = 2 - 1
[3,1,2] => [1,2] => 110 => 1 = 2 - 1
[3,2,1] => [1,1,1] => 111 => 0 = 1 - 1
[1,2,3,4] => [4] => 1000 => 1 = 2 - 1
[1,2,4,3] => [3,1] => 1001 => 1 = 2 - 1
[1,3,2,4] => [2,2] => 1010 => 2 = 3 - 1
[1,3,4,2] => [3,1] => 1001 => 1 = 2 - 1
[1,4,2,3] => [2,2] => 1010 => 2 = 3 - 1
[1,4,3,2] => [2,1,1] => 1011 => 1 = 2 - 1
[2,1,3,4] => [1,3] => 1100 => 1 = 2 - 1
[2,1,4,3] => [1,2,1] => 1101 => 1 = 2 - 1
[2,3,1,4] => [2,2] => 1010 => 2 = 3 - 1
[2,3,4,1] => [3,1] => 1001 => 1 = 2 - 1
[2,4,1,3] => [2,2] => 1010 => 2 = 3 - 1
[2,4,3,1] => [2,1,1] => 1011 => 1 = 2 - 1
[3,1,2,4] => [1,3] => 1100 => 1 = 2 - 1
[3,1,4,2] => [1,2,1] => 1101 => 1 = 2 - 1
[3,2,1,4] => [1,1,2] => 1110 => 1 = 2 - 1
[3,2,4,1] => [1,2,1] => 1101 => 1 = 2 - 1
[3,4,1,2] => [2,2] => 1010 => 2 = 3 - 1
[3,4,2,1] => [2,1,1] => 1011 => 1 = 2 - 1
[4,1,2,3] => [1,3] => 1100 => 1 = 2 - 1
[4,1,3,2] => [1,2,1] => 1101 => 1 = 2 - 1
[4,2,1,3] => [1,1,2] => 1110 => 1 = 2 - 1
[4,2,3,1] => [1,2,1] => 1101 => 1 = 2 - 1
[4,3,1,2] => [1,1,2] => 1110 => 1 = 2 - 1
[4,3,2,1] => [1,1,1,1] => 1111 => 0 = 1 - 1
[1,2,3,4,5] => [5] => 10000 => 1 = 2 - 1
[1,2,3,5,4] => [4,1] => 10001 => 1 = 2 - 1
[1,2,4,3,5] => [3,2] => 10010 => 2 = 3 - 1
[1,2,4,5,3] => [4,1] => 10001 => 1 = 2 - 1
[1,2,5,3,4] => [3,2] => 10010 => 2 = 3 - 1
[1,2,5,4,3] => [3,1,1] => 10011 => 1 = 2 - 1
[1,3,2,4,5] => [2,3] => 10100 => 2 = 3 - 1
[1,3,2,5,4] => [2,2,1] => 10101 => 2 = 3 - 1
[1,3,4,2,5] => [3,2] => 10010 => 2 = 3 - 1
[1,3,4,5,2] => [4,1] => 10001 => 1 = 2 - 1
[1,3,5,2,4] => [3,2] => 10010 => 2 = 3 - 1
[1,3,5,4,2] => [3,1,1] => 10011 => 1 = 2 - 1
[1,4,2,3,5] => [2,3] => 10100 => 2 = 3 - 1
[1,4,2,5,3] => [2,2,1] => 10101 => 2 = 3 - 1
[1,4,3,2,5] => [2,1,2] => 10110 => 2 = 3 - 1
[1,4,3,5,2] => [2,2,1] => 10101 => 2 = 3 - 1
[1,4,5,2,3] => [3,2] => 10010 => 2 = 3 - 1
Description
The number of descents of a binary word.
Matching statistic: St001011
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> 0 = 1 - 1
[1,2] => [2] => [1,1,0,0]
=> 0 = 1 - 1
[2,1] => [1,1] => [1,0,1,0]
=> 1 = 2 - 1
[1,2,3] => [3] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> 1 = 2 - 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> 1 = 2 - 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> 1 = 2 - 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
=> 1 = 2 - 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 1 = 2 - 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1 = 2 - 1
Description
Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001280
Mp00071: Permutations —descent 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] => [1] => [1]
=> 0 = 1 - 1
[1,2] => [2] => [2]
=> 1 = 2 - 1
[2,1] => [1,1] => [1,1]
=> 0 = 1 - 1
[1,2,3] => [3] => [3]
=> 1 = 2 - 1
[1,3,2] => [2,1] => [2,1]
=> 1 = 2 - 1
[2,1,3] => [1,2] => [2,1]
=> 1 = 2 - 1
[2,3,1] => [2,1] => [2,1]
=> 1 = 2 - 1
[3,1,2] => [1,2] => [2,1]
=> 1 = 2 - 1
[3,2,1] => [1,1,1] => [1,1,1]
=> 0 = 1 - 1
[1,2,3,4] => [4] => [4]
=> 1 = 2 - 1
[1,2,4,3] => [3,1] => [3,1]
=> 1 = 2 - 1
[1,3,2,4] => [2,2] => [2,2]
=> 2 = 3 - 1
[1,3,4,2] => [3,1] => [3,1]
=> 1 = 2 - 1
[1,4,2,3] => [2,2] => [2,2]
=> 2 = 3 - 1
[1,4,3,2] => [2,1,1] => [2,1,1]
=> 1 = 2 - 1
[2,1,3,4] => [1,3] => [3,1]
=> 1 = 2 - 1
[2,1,4,3] => [1,2,1] => [2,1,1]
=> 1 = 2 - 1
[2,3,1,4] => [2,2] => [2,2]
=> 2 = 3 - 1
[2,3,4,1] => [3,1] => [3,1]
=> 1 = 2 - 1
[2,4,1,3] => [2,2] => [2,2]
=> 2 = 3 - 1
[2,4,3,1] => [2,1,1] => [2,1,1]
=> 1 = 2 - 1
[3,1,2,4] => [1,3] => [3,1]
=> 1 = 2 - 1
[3,1,4,2] => [1,2,1] => [2,1,1]
=> 1 = 2 - 1
[3,2,1,4] => [1,1,2] => [2,1,1]
=> 1 = 2 - 1
[3,2,4,1] => [1,2,1] => [2,1,1]
=> 1 = 2 - 1
[3,4,1,2] => [2,2] => [2,2]
=> 2 = 3 - 1
[3,4,2,1] => [2,1,1] => [2,1,1]
=> 1 = 2 - 1
[4,1,2,3] => [1,3] => [3,1]
=> 1 = 2 - 1
[4,1,3,2] => [1,2,1] => [2,1,1]
=> 1 = 2 - 1
[4,2,1,3] => [1,1,2] => [2,1,1]
=> 1 = 2 - 1
[4,2,3,1] => [1,2,1] => [2,1,1]
=> 1 = 2 - 1
[4,3,1,2] => [1,1,2] => [2,1,1]
=> 1 = 2 - 1
[4,3,2,1] => [1,1,1,1] => [1,1,1,1]
=> 0 = 1 - 1
[1,2,3,4,5] => [5] => [5]
=> 1 = 2 - 1
[1,2,3,5,4] => [4,1] => [4,1]
=> 1 = 2 - 1
[1,2,4,3,5] => [3,2] => [3,2]
=> 2 = 3 - 1
[1,2,4,5,3] => [4,1] => [4,1]
=> 1 = 2 - 1
[1,2,5,3,4] => [3,2] => [3,2]
=> 2 = 3 - 1
[1,2,5,4,3] => [3,1,1] => [3,1,1]
=> 1 = 2 - 1
[1,3,2,4,5] => [2,3] => [3,2]
=> 2 = 3 - 1
[1,3,2,5,4] => [2,2,1] => [2,2,1]
=> 2 = 3 - 1
[1,3,4,2,5] => [3,2] => [3,2]
=> 2 = 3 - 1
[1,3,4,5,2] => [4,1] => [4,1]
=> 1 = 2 - 1
[1,3,5,2,4] => [3,2] => [3,2]
=> 2 = 3 - 1
[1,3,5,4,2] => [3,1,1] => [3,1,1]
=> 1 = 2 - 1
[1,4,2,3,5] => [2,3] => [3,2]
=> 2 = 3 - 1
[1,4,2,5,3] => [2,2,1] => [2,2,1]
=> 2 = 3 - 1
[1,4,3,2,5] => [2,1,2] => [2,2,1]
=> 2 = 3 - 1
[1,4,3,5,2] => [2,2,1] => [2,2,1]
=> 2 = 3 - 1
[1,4,5,2,3] => [3,2] => [3,2]
=> 2 = 3 - 1
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St000010
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00308: Integer partitions —Bulgarian solitaire⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00308: Integer partitions —Bulgarian solitaire⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1]
=> [1]
=> 1
[1,2] => [2] => [2]
=> [1,1]
=> 2
[2,1] => [1,1] => [1,1]
=> [2]
=> 1
[1,2,3] => [3] => [3]
=> [2,1]
=> 2
[1,3,2] => [2,1] => [2,1]
=> [2,1]
=> 2
[2,1,3] => [1,2] => [2,1]
=> [2,1]
=> 2
[2,3,1] => [2,1] => [2,1]
=> [2,1]
=> 2
[3,1,2] => [1,2] => [2,1]
=> [2,1]
=> 2
[3,2,1] => [1,1,1] => [1,1,1]
=> [3]
=> 1
[1,2,3,4] => [4] => [4]
=> [3,1]
=> 2
[1,2,4,3] => [3,1] => [3,1]
=> [2,2]
=> 2
[1,3,2,4] => [2,2] => [2,2]
=> [2,1,1]
=> 3
[1,3,4,2] => [3,1] => [3,1]
=> [2,2]
=> 2
[1,4,2,3] => [2,2] => [2,2]
=> [2,1,1]
=> 3
[1,4,3,2] => [2,1,1] => [2,1,1]
=> [3,1]
=> 2
[2,1,3,4] => [1,3] => [3,1]
=> [2,2]
=> 2
[2,1,4,3] => [1,2,1] => [2,1,1]
=> [3,1]
=> 2
[2,3,1,4] => [2,2] => [2,2]
=> [2,1,1]
=> 3
[2,3,4,1] => [3,1] => [3,1]
=> [2,2]
=> 2
[2,4,1,3] => [2,2] => [2,2]
=> [2,1,1]
=> 3
[2,4,3,1] => [2,1,1] => [2,1,1]
=> [3,1]
=> 2
[3,1,2,4] => [1,3] => [3,1]
=> [2,2]
=> 2
[3,1,4,2] => [1,2,1] => [2,1,1]
=> [3,1]
=> 2
[3,2,1,4] => [1,1,2] => [2,1,1]
=> [3,1]
=> 2
[3,2,4,1] => [1,2,1] => [2,1,1]
=> [3,1]
=> 2
[3,4,1,2] => [2,2] => [2,2]
=> [2,1,1]
=> 3
[3,4,2,1] => [2,1,1] => [2,1,1]
=> [3,1]
=> 2
[4,1,2,3] => [1,3] => [3,1]
=> [2,2]
=> 2
[4,1,3,2] => [1,2,1] => [2,1,1]
=> [3,1]
=> 2
[4,2,1,3] => [1,1,2] => [2,1,1]
=> [3,1]
=> 2
[4,2,3,1] => [1,2,1] => [2,1,1]
=> [3,1]
=> 2
[4,3,1,2] => [1,1,2] => [2,1,1]
=> [3,1]
=> 2
[4,3,2,1] => [1,1,1,1] => [1,1,1,1]
=> [4]
=> 1
[1,2,3,4,5] => [5] => [5]
=> [4,1]
=> 2
[1,2,3,5,4] => [4,1] => [4,1]
=> [3,2]
=> 2
[1,2,4,3,5] => [3,2] => [3,2]
=> [2,2,1]
=> 3
[1,2,4,5,3] => [4,1] => [4,1]
=> [3,2]
=> 2
[1,2,5,3,4] => [3,2] => [3,2]
=> [2,2,1]
=> 3
[1,2,5,4,3] => [3,1,1] => [3,1,1]
=> [3,2]
=> 2
[1,3,2,4,5] => [2,3] => [3,2]
=> [2,2,1]
=> 3
[1,3,2,5,4] => [2,2,1] => [2,2,1]
=> [3,1,1]
=> 3
[1,3,4,2,5] => [3,2] => [3,2]
=> [2,2,1]
=> 3
[1,3,4,5,2] => [4,1] => [4,1]
=> [3,2]
=> 2
[1,3,5,2,4] => [3,2] => [3,2]
=> [2,2,1]
=> 3
[1,3,5,4,2] => [3,1,1] => [3,1,1]
=> [3,2]
=> 2
[1,4,2,3,5] => [2,3] => [3,2]
=> [2,2,1]
=> 3
[1,4,2,5,3] => [2,2,1] => [2,2,1]
=> [3,1,1]
=> 3
[1,4,3,2,5] => [2,1,2] => [2,2,1]
=> [3,1,1]
=> 3
[1,4,3,5,2] => [2,2,1] => [2,2,1]
=> [3,1,1]
=> 3
[1,4,5,2,3] => [3,2] => [3,2]
=> [2,2,1]
=> 3
Description
The length of the partition.
Matching statistic: St000013
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => {{1}}
=> [1]
=> [1,0]
=> 1
[1,2] => {{1},{2}}
=> [1,1]
=> [1,1,0,0]
=> 2
[2,1] => {{1,2}}
=> [2]
=> [1,0,1,0]
=> 1
[1,2,3] => {{1},{2},{3}}
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,3,2] => {{1},{2,3}}
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[2,1,3] => {{1,2},{3}}
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[2,3,1] => {{1,2,3}}
=> [3]
=> [1,0,1,0,1,0]
=> 1
[3,1,2] => {{1,3},{2}}
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[3,2,1] => {{1,3},{2}}
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,2,3,4] => {{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2
[1,2,4,3] => {{1},{2},{3,4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,3,2,4] => {{1},{2,3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,3,4,2] => {{1},{2,3,4}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,4,2,3] => {{1},{2,4},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,4,3,2] => {{1},{2,4},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,1,3,4] => {{1,2},{3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,1,4,3] => {{1,2},{3,4}}
=> [2,2]
=> [1,1,1,0,0,0]
=> 3
[2,3,1,4] => {{1,2,3},{4}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[2,3,4,1] => {{1,2,3,4}}
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 1
[2,4,1,3] => {{1,2,4},{3}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[2,4,3,1] => {{1,2,4},{3}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[3,1,2,4] => {{1,3},{2},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[3,1,4,2] => {{1,3,4},{2}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[3,2,1,4] => {{1,3},{2},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[3,2,4,1] => {{1,3,4},{2}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[3,4,1,2] => {{1,3},{2,4}}
=> [2,2]
=> [1,1,1,0,0,0]
=> 3
[3,4,2,1] => {{1,3},{2,4}}
=> [2,2]
=> [1,1,1,0,0,0]
=> 3
[4,1,2,3] => {{1,4},{2},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[4,1,3,2] => {{1,4},{2},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[4,2,1,3] => {{1,4},{2},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[4,2,3,1] => {{1,4},{2},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[4,3,1,2] => {{1,4},{2,3}}
=> [2,2]
=> [1,1,1,0,0,0]
=> 3
[4,3,2,1] => {{1,4},{2,3}}
=> [2,2]
=> [1,1,1,0,0,0]
=> 3
[1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,2,4,5,3] => {{1},{2},{3,4,5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[1,2,5,3,4] => {{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,3,2,5,4] => {{1},{2,3},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3
[1,3,4,2,5] => {{1},{2,3,4},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[1,3,4,5,2] => {{1},{2,3,4,5}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2
[1,3,5,2,4] => {{1},{2,3,5},{4}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[1,3,5,4,2] => {{1},{2,3,5},{4}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[1,4,2,3,5] => {{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,4,2,5,3] => {{1},{2,4,5},{3}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,4,3,5,2] => {{1},{2,4,5},{3}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[1,4,5,2,3] => {{1},{2,4},{3,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3
Description
The height of a Dyck path.
The height of a Dyck path $D$ of semilength $n$ is defined as the maximal height of a peak of $D$. The height of $D$ at position $i$ is the number of up-steps minus the number of down-steps before position $i$.
Matching statistic: St001007
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001007: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001007: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => {{1}}
=> [1]
=> [1,0]
=> 1
[1,2] => {{1},{2}}
=> [1,1]
=> [1,1,0,0]
=> 2
[2,1] => {{1,2}}
=> [2]
=> [1,0,1,0]
=> 1
[1,2,3] => {{1},{2},{3}}
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,3,2] => {{1},{2,3}}
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[2,1,3] => {{1,2},{3}}
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[2,3,1] => {{1,2,3}}
=> [3]
=> [1,0,1,0,1,0]
=> 1
[3,1,2] => {{1,3},{2}}
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[3,2,1] => {{1,3},{2}}
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,2,3,4] => {{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2
[1,2,4,3] => {{1},{2},{3,4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,3,2,4] => {{1},{2,3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,3,4,2] => {{1},{2,3,4}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,4,2,3] => {{1},{2,4},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,4,3,2] => {{1},{2,4},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,1,3,4] => {{1,2},{3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,1,4,3] => {{1,2},{3,4}}
=> [2,2]
=> [1,1,1,0,0,0]
=> 3
[2,3,1,4] => {{1,2,3},{4}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[2,3,4,1] => {{1,2,3,4}}
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 1
[2,4,1,3] => {{1,2,4},{3}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[2,4,3,1] => {{1,2,4},{3}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[3,1,2,4] => {{1,3},{2},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[3,1,4,2] => {{1,3,4},{2}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[3,2,1,4] => {{1,3},{2},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[3,2,4,1] => {{1,3,4},{2}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[3,4,1,2] => {{1,3},{2,4}}
=> [2,2]
=> [1,1,1,0,0,0]
=> 3
[3,4,2,1] => {{1,3},{2,4}}
=> [2,2]
=> [1,1,1,0,0,0]
=> 3
[4,1,2,3] => {{1,4},{2},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[4,1,3,2] => {{1,4},{2},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[4,2,1,3] => {{1,4},{2},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[4,2,3,1] => {{1,4},{2},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[4,3,1,2] => {{1,4},{2,3}}
=> [2,2]
=> [1,1,1,0,0,0]
=> 3
[4,3,2,1] => {{1,4},{2,3}}
=> [2,2]
=> [1,1,1,0,0,0]
=> 3
[1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,2,4,5,3] => {{1},{2},{3,4,5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[1,2,5,3,4] => {{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,3,2,5,4] => {{1},{2,3},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3
[1,3,4,2,5] => {{1},{2,3,4},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[1,3,4,5,2] => {{1},{2,3,4,5}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2
[1,3,5,2,4] => {{1},{2,3,5},{4}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[1,3,5,4,2] => {{1},{2,3,5},{4}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[1,4,2,3,5] => {{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,4,2,5,3] => {{1},{2,4,5},{3}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,4,3,5,2] => {{1},{2,4,5},{3}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[1,4,5,2,3] => {{1},{2,4},{3,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3
Description
Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001068
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001068: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001068: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> [1,0]
=> 1
[1,2] => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 2
[2,1] => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 1
[1,2,3] => [3] => [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2
[3,1,2] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3
Description
Number of torsionless simple modules in the corresponding Nakayama algebra.
Matching statistic: St001203
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001203: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001203: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> [1,0]
=> 1
[1,2] => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 2
[2,1] => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 1
[1,2,3] => [3] => [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2
[3,1,2] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3
Description
We 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:
In the list $L$ delete the first entry $c_0$ and substract from all other entries $n-1$ and then append the last element 1 (this was suggested by Christian Stump). The result is a Kupisch series of an LNakayama algebra.
Example:
[5,6,6,6,6] goes into [2,2,2,2,1].
Now associate to the CNakayama algebra with the above properties the Dyck path corresponding to the Kupisch series of the LNakayama algebra.
The statistic return the global dimension of the CNakayama algebra divided by 2.
Matching statistic: St001471
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001471: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001471: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> [1,0]
=> 1
[1,2] => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 2
[2,1] => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 1
[1,2,3] => [3] => [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2
[3,1,2] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3
Description
The magnitude of a Dyck path.
The magnitude of a finite dimensional algebra with invertible Cartan matrix C is defined as the sum of all entries of the inverse of C.
We define the magnitude of a Dyck path as the magnitude of the corresponding LNakayama algebra.
Matching statistic: St001674
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001674: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001674: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
=> ([],1)
=> 1
[1,2] => [2] => ([],2)
=> ([],1)
=> 1
[2,1] => [1,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,2,3] => [3] => ([],3)
=> ([],1)
=> 1
[1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
[2,1,3] => [1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 2
[2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
[3,1,2] => [1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 2
[3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[1,2,3,4] => [4] => ([],4)
=> ([],1)
=> 1
[1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
[1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2
[1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
[1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2
[1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[2,1,3,4] => [1,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> 2
[2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2
[2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
[2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2
[2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[3,1,2,4] => [1,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> 2
[3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2
[3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[4,1,2,3] => [1,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> 2
[4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,2,3,4,5] => [5] => ([],5)
=> ([],1)
=> 1
[1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
[1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
[1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
[1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
[1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,4,3,2,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[1,4,3,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
Description
The number of vertices of the largest induced star graph in the graph.
The following 97 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000024The number of double up and double down steps of a Dyck path. St000053The number of valleys of the Dyck path. St000292The number of ascents of a binary word. St000340The number of non-final maximal constant sub-paths of length greater than one. St000390The number of runs of ones in a binary word. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001028Number of simple modules with injective dimension equal to the dominant dimension 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. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001512The minimum rank of a graph. St000251The number of nonsingleton blocks of a set partition. St000659The number of rises of length at least 2 of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St000658The number of rises of length 2 of a Dyck path. St000919The number of maximal left branches of a binary tree. St001354The number of series nodes in the modular decomposition of a graph. 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. St000299The number of nonisomorphic vertex-induced subtrees. 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. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St000306The bounce count of a Dyck path. St000470The number of runs in a 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. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001928The number of non-overlapping descents in a permutation. St000354The number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000670The reversal length of a permutation. St000834The number of right outer peaks of a permutation. St001726The number of visible inversions of a permutation. St000264The girth of a graph, which is not a tree. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000035The number of left outer peaks of a permutation. St000665The number of rafts of a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. 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. 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. St000015The number of peaks of a Dyck path. St000325The width of the tree associated to a permutation. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St000021The number of descents of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000238The number of indices that are not small weak excedances. St000316The number of non-left-to-right-maxima of a permutation. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. 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 $eA$. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001874Lusztig's a-function for the symmetric group. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. 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 $eA$. St000454The largest eigenvalue of a graph if it is integral. 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 $eA$. St001875The number of simple modules with projective dimension at most 1. St001060The distinguishing index of a graph. St000256The number of parts from which one can substract 2 and still get an integer partition. St001330The hat guessing number of a graph. St000455The second largest eigenvalue of a graph if it is integral. St001720The minimal length of a chain of small intervals in a lattice. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000023The number of inner peaks of a permutation. St000353The number of inner valleys of a permutation. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000779The tier of a permutation. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St001935The number of ascents in a parking function. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001597The Frobenius rank of a skew partition. St000862The number of parts of the shifted shape of a permutation. St001624The breadth of a lattice.
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