Your data matches 611 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
St001588: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> 0 = 1 - 1
[2]
=> 0 = 1 - 1
[1,1]
=> 0 = 1 - 1
[3]
=> 0 = 1 - 1
[2,1]
=> 1 = 2 - 1
[1,1,1]
=> 0 = 1 - 1
[4]
=> 0 = 1 - 1
[3,1]
=> 0 = 1 - 1
[2,2]
=> 0 = 1 - 1
[2,1,1]
=> 1 = 2 - 1
[1,1,1,1]
=> 0 = 1 - 1
[5]
=> 0 = 1 - 1
[4,1]
=> 1 = 2 - 1
[3,2]
=> 0 = 1 - 1
[3,1,1]
=> 0 = 1 - 1
[2,2,1]
=> 1 = 2 - 1
[2,1,1,1]
=> 1 = 2 - 1
[1,1,1,1,1]
=> 0 = 1 - 1
[6]
=> 0 = 1 - 1
[5,1]
=> 0 = 1 - 1
[4,2]
=> 0 = 1 - 1
[4,1,1]
=> 1 = 2 - 1
[3,3]
=> 0 = 1 - 1
[3,2,1]
=> 1 = 2 - 1
[3,1,1,1]
=> 0 = 1 - 1
[2,2,2]
=> 0 = 1 - 1
[2,2,1,1]
=> 1 = 2 - 1
[2,1,1,1,1]
=> 1 = 2 - 1
[1,1,1,1,1,1]
=> 0 = 1 - 1
Description
The number of distinct odd parts smaller than the largest even part in an integer partition.
Mp00317: Integer partitions odd partsBinary words
St000292: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> 1 => 0 = 1 - 1
[2]
=> 0 => 0 = 1 - 1
[1,1]
=> 11 => 0 = 1 - 1
[3]
=> 1 => 0 = 1 - 1
[2,1]
=> 01 => 1 = 2 - 1
[1,1,1]
=> 111 => 0 = 1 - 1
[4]
=> 0 => 0 = 1 - 1
[3,1]
=> 11 => 0 = 1 - 1
[2,2]
=> 00 => 0 = 1 - 1
[2,1,1]
=> 011 => 1 = 2 - 1
[1,1,1,1]
=> 1111 => 0 = 1 - 1
[5]
=> 1 => 0 = 1 - 1
[4,1]
=> 01 => 1 = 2 - 1
[3,2]
=> 10 => 0 = 1 - 1
[3,1,1]
=> 111 => 0 = 1 - 1
[2,2,1]
=> 001 => 1 = 2 - 1
[2,1,1,1]
=> 0111 => 1 = 2 - 1
[1,1,1,1,1]
=> 11111 => 0 = 1 - 1
[6]
=> 0 => 0 = 1 - 1
[5,1]
=> 11 => 0 = 1 - 1
[4,2]
=> 00 => 0 = 1 - 1
[4,1,1]
=> 011 => 1 = 2 - 1
[3,3]
=> 11 => 0 = 1 - 1
[3,2,1]
=> 101 => 1 = 2 - 1
[3,1,1,1]
=> 1111 => 0 = 1 - 1
[2,2,2]
=> 000 => 0 = 1 - 1
[2,2,1,1]
=> 0011 => 1 = 2 - 1
[2,1,1,1,1]
=> 01111 => 1 = 2 - 1
[1,1,1,1,1,1]
=> 111111 => 0 = 1 - 1
Description
The number of ascents of a binary word.
Mp00317: Integer partitions odd partsBinary words
St001355: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> 1 => 0 = 1 - 1
[2]
=> 0 => 0 = 1 - 1
[1,1]
=> 11 => 0 = 1 - 1
[3]
=> 1 => 0 = 1 - 1
[2,1]
=> 01 => 1 = 2 - 1
[1,1,1]
=> 111 => 0 = 1 - 1
[4]
=> 0 => 0 = 1 - 1
[3,1]
=> 11 => 0 = 1 - 1
[2,2]
=> 00 => 0 = 1 - 1
[2,1,1]
=> 011 => 1 = 2 - 1
[1,1,1,1]
=> 1111 => 0 = 1 - 1
[5]
=> 1 => 0 = 1 - 1
[4,1]
=> 01 => 1 = 2 - 1
[3,2]
=> 10 => 1 = 2 - 1
[3,1,1]
=> 111 => 0 = 1 - 1
[2,2,1]
=> 001 => 0 = 1 - 1
[2,1,1,1]
=> 0111 => 1 = 2 - 1
[1,1,1,1,1]
=> 11111 => 0 = 1 - 1
[6]
=> 0 => 0 = 1 - 1
[5,1]
=> 11 => 0 = 1 - 1
[4,2]
=> 00 => 0 = 1 - 1
[4,1,1]
=> 011 => 1 = 2 - 1
[3,3]
=> 11 => 0 = 1 - 1
[3,2,1]
=> 101 => 1 = 2 - 1
[3,1,1,1]
=> 1111 => 0 = 1 - 1
[2,2,2]
=> 000 => 0 = 1 - 1
[2,2,1,1]
=> 0011 => 1 = 2 - 1
[2,1,1,1,1]
=> 01111 => 1 = 2 - 1
[1,1,1,1,1,1]
=> 111111 => 0 = 1 - 1
Description
Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. Graphically, this is the number of returns to the main diagonal of the monotone lattice path of a binary word.
Mp00317: Integer partitions odd partsBinary words
Mp00178: Binary words to compositionInteger compositions
St000760: Integer compositions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> 1 => [1,1] => 1
[2]
=> 0 => [2] => 1
[1,1]
=> 11 => [1,1,1] => 1
[3]
=> 1 => [1,1] => 1
[2,1]
=> 01 => [2,1] => 2
[1,1,1]
=> 111 => [1,1,1,1] => 1
[4]
=> 0 => [2] => 1
[3,1]
=> 11 => [1,1,1] => 1
[2,2]
=> 00 => [3] => 1
[2,1,1]
=> 011 => [2,1,1] => 2
[1,1,1,1]
=> 1111 => [1,1,1,1,1] => 1
[5]
=> 1 => [1,1] => 1
[4,1]
=> 01 => [2,1] => 2
[3,2]
=> 10 => [1,2] => 1
[3,1,1]
=> 111 => [1,1,1,1] => 1
[2,2,1]
=> 001 => [3,1] => 2
[2,1,1,1]
=> 0111 => [2,1,1,1] => 2
[1,1,1,1,1]
=> 11111 => [1,1,1,1,1,1] => 1
[6]
=> 0 => [2] => 1
[5,1]
=> 11 => [1,1,1] => 1
[4,2]
=> 00 => [3] => 1
[4,1,1]
=> 011 => [2,1,1] => 2
[3,3]
=> 11 => [1,1,1] => 1
[3,2,1]
=> 101 => [1,2,1] => 2
[3,1,1,1]
=> 1111 => [1,1,1,1,1] => 1
[2,2,2]
=> 000 => [4] => 1
[2,2,1,1]
=> 0011 => [3,1,1] => 2
[2,1,1,1,1]
=> 01111 => [2,1,1,1,1] => 2
[1,1,1,1,1,1]
=> 111111 => [1,1,1,1,1,1,1] => 1
Description
The length of the longest strictly decreasing subsequence of parts of an integer composition. By the Greene-Kleitman theorem, regarding the composition as a word, this is the length of the partition associated by the Robinson-Schensted-Knuth correspondence.
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00032: Dyck paths inverse zeta mapDyck paths
St001732: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1,0]
=> 1
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 1
Description
The number of peaks visible from the left. This is, the number of left-to-right maxima of the heights of the peaks of a Dyck path.
Mp00317: Integer partitions odd partsBinary words
Mp00104: Binary words reverseBinary 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
[2]
=> 0 => 0 => 0 = 1 - 1
[1,1]
=> 11 => 11 => 0 = 1 - 1
[3]
=> 1 => 1 => 0 = 1 - 1
[2,1]
=> 01 => 10 => 1 = 2 - 1
[1,1,1]
=> 111 => 111 => 0 = 1 - 1
[4]
=> 0 => 0 => 0 = 1 - 1
[3,1]
=> 11 => 11 => 0 = 1 - 1
[2,2]
=> 00 => 00 => 0 = 1 - 1
[2,1,1]
=> 011 => 110 => 1 = 2 - 1
[1,1,1,1]
=> 1111 => 1111 => 0 = 1 - 1
[5]
=> 1 => 1 => 0 = 1 - 1
[4,1]
=> 01 => 10 => 1 = 2 - 1
[3,2]
=> 10 => 01 => 0 = 1 - 1
[3,1,1]
=> 111 => 111 => 0 = 1 - 1
[2,2,1]
=> 001 => 100 => 1 = 2 - 1
[2,1,1,1]
=> 0111 => 1110 => 1 = 2 - 1
[1,1,1,1,1]
=> 11111 => 11111 => 0 = 1 - 1
[6]
=> 0 => 0 => 0 = 1 - 1
[5,1]
=> 11 => 11 => 0 = 1 - 1
[4,2]
=> 00 => 00 => 0 = 1 - 1
[4,1,1]
=> 011 => 110 => 1 = 2 - 1
[3,3]
=> 11 => 11 => 0 = 1 - 1
[3,2,1]
=> 101 => 101 => 1 = 2 - 1
[3,1,1,1]
=> 1111 => 1111 => 0 = 1 - 1
[2,2,2]
=> 000 => 000 => 0 = 1 - 1
[2,2,1,1]
=> 0011 => 1100 => 1 = 2 - 1
[2,1,1,1,1]
=> 01111 => 11110 => 1 = 2 - 1
[1,1,1,1,1,1]
=> 111111 => 111111 => 0 = 1 - 1
Description
The number of descents of a binary word.
Mp00317: Integer partitions odd partsBinary words
Mp00136: Binary words rotate back-to-frontBinary words
St000875: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> 1 => 1 => 0 = 1 - 1
[2]
=> 0 => 0 => 0 = 1 - 1
[1,1]
=> 11 => 11 => 0 = 1 - 1
[3]
=> 1 => 1 => 0 = 1 - 1
[2,1]
=> 01 => 10 => 1 = 2 - 1
[1,1,1]
=> 111 => 111 => 0 = 1 - 1
[4]
=> 0 => 0 => 0 = 1 - 1
[3,1]
=> 11 => 11 => 0 = 1 - 1
[2,2]
=> 00 => 00 => 0 = 1 - 1
[2,1,1]
=> 011 => 101 => 1 = 2 - 1
[1,1,1,1]
=> 1111 => 1111 => 0 = 1 - 1
[5]
=> 1 => 1 => 0 = 1 - 1
[4,1]
=> 01 => 10 => 1 = 2 - 1
[3,2]
=> 10 => 01 => 0 = 1 - 1
[3,1,1]
=> 111 => 111 => 0 = 1 - 1
[2,2,1]
=> 001 => 100 => 1 = 2 - 1
[2,1,1,1]
=> 0111 => 1011 => 1 = 2 - 1
[1,1,1,1,1]
=> 11111 => 11111 => 0 = 1 - 1
[6]
=> 0 => 0 => 0 = 1 - 1
[5,1]
=> 11 => 11 => 0 = 1 - 1
[4,2]
=> 00 => 00 => 0 = 1 - 1
[4,1,1]
=> 011 => 101 => 1 = 2 - 1
[3,3]
=> 11 => 11 => 0 = 1 - 1
[3,2,1]
=> 101 => 110 => 1 = 2 - 1
[3,1,1,1]
=> 1111 => 1111 => 0 = 1 - 1
[2,2,2]
=> 000 => 000 => 0 = 1 - 1
[2,2,1,1]
=> 0011 => 1001 => 1 = 2 - 1
[2,1,1,1,1]
=> 01111 => 10111 => 1 = 2 - 1
[1,1,1,1,1,1]
=> 111111 => 111111 => 0 = 1 - 1
Description
The semilength of the longest Dyck word in the Catalan factorisation of a binary word. Every binary word can be written in a unique way as $(\mathcal D 0)^\ell \mathcal D (1 \mathcal D)^m$, where $\mathcal D$ is the set of Dyck words. This is the Catalan factorisation, see [1, sec.9.1.2]. This statistic records the semilength of the longest Dyck word in this factorisation.
Matching statistic: St001092
Mp00202: Integer partitions first row removalInteger partitions
Mp00044: Integer partitions conjugateInteger partitions
St001092: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> []
=> []
=> 0 = 1 - 1
[2]
=> []
=> []
=> 0 = 1 - 1
[1,1]
=> [1]
=> [1]
=> 0 = 1 - 1
[3]
=> []
=> []
=> 0 = 1 - 1
[2,1]
=> [1]
=> [1]
=> 0 = 1 - 1
[1,1,1]
=> [1,1]
=> [2]
=> 1 = 2 - 1
[4]
=> []
=> []
=> 0 = 1 - 1
[3,1]
=> [1]
=> [1]
=> 0 = 1 - 1
[2,2]
=> [2]
=> [1,1]
=> 0 = 1 - 1
[2,1,1]
=> [1,1]
=> [2]
=> 1 = 2 - 1
[1,1,1,1]
=> [1,1,1]
=> [3]
=> 0 = 1 - 1
[5]
=> []
=> []
=> 0 = 1 - 1
[4,1]
=> [1]
=> [1]
=> 0 = 1 - 1
[3,2]
=> [2]
=> [1,1]
=> 0 = 1 - 1
[3,1,1]
=> [1,1]
=> [2]
=> 1 = 2 - 1
[2,2,1]
=> [2,1]
=> [2,1]
=> 1 = 2 - 1
[2,1,1,1]
=> [1,1,1]
=> [3]
=> 0 = 1 - 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 1 = 2 - 1
[6]
=> []
=> []
=> 0 = 1 - 1
[5,1]
=> [1]
=> [1]
=> 0 = 1 - 1
[4,2]
=> [2]
=> [1,1]
=> 0 = 1 - 1
[4,1,1]
=> [1,1]
=> [2]
=> 1 = 2 - 1
[3,3]
=> [3]
=> [1,1,1]
=> 0 = 1 - 1
[3,2,1]
=> [2,1]
=> [2,1]
=> 1 = 2 - 1
[3,1,1,1]
=> [1,1,1]
=> [3]
=> 0 = 1 - 1
[2,2,2]
=> [2,2]
=> [2,2]
=> 1 = 2 - 1
[2,2,1,1]
=> [2,1,1]
=> [3,1]
=> 0 = 1 - 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 1 = 2 - 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> 0 = 1 - 1
Description
The number of distinct even parts of a partition. See Section 3.3.1 of [1].
Mp00317: Integer partitions odd partsBinary words
Mp00136: Binary words rotate back-to-frontBinary words
St001421: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> 1 => 1 => 0 = 1 - 1
[2]
=> 0 => 0 => 0 = 1 - 1
[1,1]
=> 11 => 11 => 0 = 1 - 1
[3]
=> 1 => 1 => 0 = 1 - 1
[2,1]
=> 01 => 10 => 1 = 2 - 1
[1,1,1]
=> 111 => 111 => 0 = 1 - 1
[4]
=> 0 => 0 => 0 = 1 - 1
[3,1]
=> 11 => 11 => 0 = 1 - 1
[2,2]
=> 00 => 00 => 0 = 1 - 1
[2,1,1]
=> 011 => 101 => 1 = 2 - 1
[1,1,1,1]
=> 1111 => 1111 => 0 = 1 - 1
[5]
=> 1 => 1 => 0 = 1 - 1
[4,1]
=> 01 => 10 => 1 = 2 - 1
[3,2]
=> 10 => 01 => 0 = 1 - 1
[3,1,1]
=> 111 => 111 => 0 = 1 - 1
[2,2,1]
=> 001 => 100 => 1 = 2 - 1
[2,1,1,1]
=> 0111 => 1011 => 1 = 2 - 1
[1,1,1,1,1]
=> 11111 => 11111 => 0 = 1 - 1
[6]
=> 0 => 0 => 0 = 1 - 1
[5,1]
=> 11 => 11 => 0 = 1 - 1
[4,2]
=> 00 => 00 => 0 = 1 - 1
[4,1,1]
=> 011 => 101 => 1 = 2 - 1
[3,3]
=> 11 => 11 => 0 = 1 - 1
[3,2,1]
=> 101 => 110 => 1 = 2 - 1
[3,1,1,1]
=> 1111 => 1111 => 0 = 1 - 1
[2,2,2]
=> 000 => 000 => 0 = 1 - 1
[2,2,1,1]
=> 0011 => 1001 => 1 = 2 - 1
[2,1,1,1,1]
=> 01111 => 10111 => 1 = 2 - 1
[1,1,1,1,1,1]
=> 111111 => 111111 => 0 = 1 - 1
Description
Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word.
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00129: Dyck paths to 321-avoiding permutation (Billey-Jockusch-Stanley)Permutations
Mp00223: Permutations runsortPermutations
St000007: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1] => [1] => 1
[2]
=> [1,0,1,0]
=> [2,1] => [1,2] => 1
[1,1]
=> [1,1,0,0]
=> [1,2] => [1,2] => 1
[3]
=> [1,0,1,0,1,0]
=> [2,3,1] => [1,2,3] => 1
[2,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => [1,3,2] => 2
[1,1,1]
=> [1,1,0,1,0,0]
=> [3,1,2] => [1,2,3] => 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [1,2,3,4] => 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [1,4,2,3] => 2
[2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [1,3,2,4] => 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [1,2,3,4] => 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [1,5,2,3,4] => 2
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [1,3,4,2] => 2
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [1,4,2,3,5] => 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,2,3] => 2
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [1,3,2,4,5] => 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [1,2,3,4,5] => 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,1] => [1,2,3,4,5,6] => 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,1,6] => [1,6,2,3,4,5] => 2
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [1,4,5,2,3] => 2
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,3,4,6,1,5] => [1,5,2,3,4,6] => 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [1,2,3,4] => 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [1,5,2,3,4] => 2
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,3,5,6,1,4] => [1,4,2,3,5,6] => 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => [1,4,5,2,3] => 2
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,4,5,6,1,3] => [1,3,2,4,5,6] => 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,5,6,1,2] => [1,2,3,4,5,6] => 1
Description
The number of saliances of the permutation. A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern $([1], {(1,1)})$, i.e., the upper right quadrant is shaded, see [1].
The following 601 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000390The number of runs of ones in a binary word. St000628The balance of a binary word. St000659The number of rises of length at least 2 of a Dyck path. St000662The staircase size of the code of a permutation. St000701The protection number of a binary tree. St000758The length of the longest staircase fitting into an integer composition. St000764The number of strong records in an integer composition. St000765The number of weak records in an integer composition. St000767The number of runs in an integer composition. St000847The number of standard Young tableaux whose descent set is the binary word. St000903The number of different parts of an integer composition. St000920The logarithmic height of a Dyck path. St001151The number of blocks with odd minimum. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St000317The cycle descent number of a permutation. St000366The number of double descents of a permutation. St000386The number of factors DDU in a Dyck path. St000534The number of 2-rises of a permutation. St000648The number of 2-excedences of a permutation. St000658The number of rises of length 2 of a Dyck path. St000660The number of rises of length at least 3 of a Dyck path. St000661The number of rises of length 3 of a Dyck path. St000663The number of right floats of a permutation. St000710The number of big deficiencies of a permutation. St000761The number of ascents in an integer composition. St000768The number of peaks in an integer composition. St000769The major index of a composition regarded as a word. St000779The tier of a permutation. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001115The number of even descents of a permutation. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001673The degree of asymmetry of an integer composition. St001728The number of invisible descents of a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St000035The number of left outer peaks of a permutation. St000630The length of the shortest palindromic decomposition of a binary word. St000886The number of permutations with the same antidiagonal sums. St000931The number of occurrences of the pattern UUU in a Dyck path. St000360The number of occurrences of the pattern 32-1. St000389The number of runs of ones of odd length in a binary word. St000486The number of cycles of length at least 3 of a permutation. St000538The number of even inversions of a permutation. St000664The number of right ropes of a permutation. St000836The number of descents of distance 2 of a permutation. St000871The number of very big ascents of a permutation. St000872The number of very big descents of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000444The length of the maximal rise of a Dyck path. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000335The difference of lower and upper interactions. St000352The Elizalde-Pak rank of a permutation. St000640The rank of the largest boolean interval in a poset. 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. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St001114The number of odd descents 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. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. 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. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001737The number of descents of type 2 in a permutation. St001928The number of non-overlapping descents in a permutation. St000252The number of nodes of degree 3 of a binary tree. St000338The number of pixed points of a permutation. St000649The number of 3-excedences of a permutation. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001204Call 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 special CNakayama algebra. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001549The number of restricted non-inversions between exceedances. St001552The number of inversions between excedances and fixed points of a permutation. St000298The order dimension or Dushnik-Miller dimension of a poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001487The number of inner corners of a skew partition. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St000617The number of global maxima of a Dyck path. St000633The size of the automorphism group of a poset. St001399The distinguishing number of a poset. St000850The number of 1/2-balanced pairs in a poset. St000993The multiplicity of the largest part of an integer partition. St001162The minimum jump of a permutation. St000842The breadth of a permutation. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000783The side length of the largest staircase partition fitting into a partition. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001432The order dimension of the partition. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001530The depth of a Dyck path. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St000025The number of initial rises of a Dyck path. St000053The number of valleys of the Dyck path. St000054The first entry of the permutation. St000092The number of outer peaks of a permutation. St000124The cardinality of the preimage of the Simion-Schmidt map. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000184The size of the centralizer of any permutation of given cycle type. St000306The bounce count of a Dyck path. St000331The number of upper interactions of a Dyck path. St000353The number of inner valleys of a permutation. St000354The number of recoils of a permutation. St000396The register function (or Horton-Strahler number) of a binary tree. St000531The leading coefficient of the rook polynomial of an integer partition. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000542The number of left-to-right-minima of a permutation. St000619The number of cyclic descents of a permutation. St000679The pruning number of an ordered tree. St000759The smallest missing part in an integer partition. St000862The number of parts of the shifted shape of a permutation. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St001006Number of simple modules with projective dimension equal to the global dimension of 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. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. 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. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001202Call 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 special CNakayama 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 $eA$. 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 $K[x]/(x^n)$. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. 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. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. 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. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001420Half the length of a longest factor which is its own reverse-complement 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. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001498The normalised height of a Nakayama algebra with magnitude 1. St001499The number of indecomposable projective-injective modules of a magnitude 1 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. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001665The number of pure excedances of a permutation. St001729The number of visible descents of a permutation. St001733The number of weak left to right maxima of a Dyck path. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St001884The number of borders of a binary word. St001913The number of preimages of an integer partition in Bulgarian solitaire. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000488The number of cycles of a permutation of length at most 2. St000516The number of stretching pairs of a permutation. St000552The number of cut vertices of a graph. St000650The number of 3-rises of a permutation. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001513The number of nested exceedences of a permutation. St001520The number of strict 3-descents. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001556The number of inversions of the third entry of a permutation. St001577The minimal number of edges to add or remove to make a graph a cograph. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001715The number of non-records in a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000259The diameter of a connected graph. St001045The number of leaves in the subtree not containing one in the decreasing labelled binary unordered tree associated with the perfect matching. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001569The maximal modular displacement of a permutation. St001597The Frobenius rank of a skew partition. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000706The product of the factorials of the multiplicities of an integer partition. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001568The smallest positive integer that does not appear twice in the partition. St001712The number of natural descents of a standard Young tableau. 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 $eA$. St001722The number of minimal chains with small intervals between a binary word and the top element. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001128The exponens consonantiae of a partition. St000060The greater neighbor of the maximum. St000260The radius of a connected graph. St000402Half the size of the symmetry class of a permutation. St000487The length of the shortest cycle of a permutation. St000568The hook number of a binary tree. St000678The number of up steps after the last double rise of a Dyck path. St000910The number of maximal chains of minimal length in a poset. St000990The first ascent of a permutation. St001052The length of the exterior of a permutation. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001220The width of a permutation. St001095The number of non-isomorphic posets with precisely one further covering relation. St001964The interval resolution global dimension of a poset. St000781The number of proper colouring schemes of a Ferrers diagram. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001780The order of promotion on the set of standard tableaux of given shape. St001820The size of the image of the pop stack sorting operator. 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. St001933The largest multiplicity of a part in an integer partition. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001846The number of elements which do not have a complement in the lattice. St000454The largest eigenvalue of a graph if it is integral. St001330The hat guessing number of a graph. St001811The Castelnuovo-Mumford regularity of a permutation. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000618The number of self-evacuating tableaux of given shape. St000667The greatest common divisor of the parts of the partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001389The number of partitions of the same length below the given integer partition. St001571The Cartan determinant of the integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001924The number of cells in an integer partition whose arm and leg length coincide. St000003The number of standard Young tableaux of the partition. St000005The bounce statistic of a Dyck path. St000006The dinv of a Dyck path. St000010The length of the partition. St000026The position of the first return of a Dyck path. St000049The number of set partitions whose sorted block sizes correspond to the partition. St000075The orbit size of a standard tableau under promotion. St000120The number of left tunnels of a Dyck path. St000147The largest part of an integer partition. St000159The number of distinct parts of the integer partition. St000160The multiplicity of the smallest part of a partition. St000182The number of permutations whose cycle type is the given integer partition. St000183The side length of the Durfee square of an integer partition. St000212The number of standard Young tableaux for an integer partition such that no two consecutive entries appear in the same row. St000275Number of permutations whose sorted list of non zero multiplicities of the Lehmer code is the given partition. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000321The number of integer partitions of n that are dominated by an integer partition. St000326The position of the first one in a binary word after appending a 1 at the end. St000345The number of refinements of a partition. St000346The number of coarsenings of a partition. St000378The diagonal inversion number of an integer partition. St000443The number of long tunnels of a Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000517The Kreweras number of an integer partition. St000529The number of permutations whose descent word is the given binary word. St000543The size of the conjugacy class of a binary word. St000626The minimal period of a binary word. St000627The exponent of a binary word. St000655The length of the minimal rise of a Dyck path. St000674The number of hills of a Dyck path. St000675The number of centered multitunnels of a Dyck path. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000691The number of changes of a binary word. St000705The number of semistandard tableaux on a given integer partition of n with maximal entry n. St000734The last entry in the first row of a standard tableau. St000814The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions. St000897The number of different multiplicities of parts of an integer partition. St000913The number of ways to refine the partition into singletons. St000935The number of ordered refinements of an integer partition. St000947The major index east count of a Dyck path. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St000983The length of the longest alternating subword. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001129The product of the squares of the parts of a partition. St001161The major index north count of a Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows: St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001256Number of simple reflexive modules that are 2-stable reflexive. St001274The number of indecomposable injective modules with projective dimension equal to two. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001313The number of Dyck paths above the lattice path given by a binary word. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001365The number of lattice paths of the same length weakly above the path given by a binary word. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001462The number of factors of a standard tableaux under concatenation. St001471The magnitude of a Dyck path. St001490The number of connected components of a skew partition. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001523The degree of symmetry of a Dyck path. St001527The cyclic permutation representation number of an integer partition. St001595The number of standard Young tableaux of the skew partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. 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. St001809The index of the step at the first peak of maximal height in a Dyck path. St001814The number of partitions interlacing the given partition. St001838The number of nonempty primitive factors of a binary word. St001885The number of binary words with the same proper border set. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St001955The number of natural descents for set-valued two row standard Young tableaux. St001435The number of missing boxes in the first row. St000181The number of connected components of the Hasse diagram for the poset. St000902 The minimal number of repetitions of an integer composition. St001247The number of parts of a partition that are not congruent 2 modulo 3. 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. St001890The maximum magnitude of the Möbius function of a poset. St000297The number of leading ones in a binary word. St000392The length of the longest run of ones in a binary word. St000982The length of the longest constant subword. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001423The number of distinct cubes in a binary word. St001372The length of a longest cyclic run of ones of a binary word. St000527The width of the poset. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000455The second largest eigenvalue of a graph if it is integral. St000045The number of linear extensions of a binary tree. St000460The hook length of the last cell along the main diagonal of an integer partition. St000741The Colin de Verdière graph invariant. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001360The number of covering relations in Young's lattice below a partition. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001763The Hurwitz number of an integer partition. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St000068The number of minimal elements in a poset. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000456The monochromatic index of a connected graph. St000451The length of the longest pattern of the form k 1 2. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000782The indicator function of whether a given perfect matching is an L & P matching. St000939The number of characters of the symmetric group whose value on the partition is positive. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St000284The Plancherel distribution on integer partitions. St000668The least common multiple of the parts of the partition. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000933The number of multipartitions of sizes given by an integer partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000735The last entry on the main diagonal of a standard tableau. St000817The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions. St000818The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. St000937The number of positive values of the symmetric group character corresponding to the partition. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000058The order of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001344The neighbouring number of a permutation. St001896The number of right descents of a signed permutations. St000090The variation of a composition. St000091The descent variation of a composition. St000217The number of occurrences of the pattern 312 in a permutation. St000233The number of nestings of a set partition. St000358The number of occurrences of the pattern 31-2. St000365The number of double ascents of a permutation. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St001438The number of missing boxes of a skew partition. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001705The number of occurrences of the pattern 2413 in a permutation. St001857The number of edges in the reduced word graph of a signed permutation. St001866The nesting alignments of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000929The constant term of the character polynomial of an integer partition. St000022The number of fixed points of a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000546The number of global descents of a permutation. St000731The number of double exceedences of a permutation. St001488The number of corners of a skew partition. St001624The breadth of a lattice. 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. St001713The difference of the first and last value in the first row of the Gelfand-Tsetlin pattern. St001875The number of simple modules with projective dimension at most 1. St000021The number of descents of a permutation. St000023The number of inner peaks of a permutation. St000056The decomposition (or block) number of a permutation. St000154The sum of the descent bottoms of a permutation. St000210Minimum over maximum difference of elements in cycles. St000234The number of global ascents of a permutation. St000253The crossing number of a set partition. St000563The number of overlapping pairs of blocks of a set partition. St000570The Edelman-Greene number of a permutation. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000601The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, (2,3) are consecutive in a block. St000646The number of big ascents of a permutation. St000654The first descent of a permutation. St000681The Grundy value of Chomp on Ferrers diagrams. St000694The number of affine bounded permutations that project to a given permutation. St000729The minimal arc length of a set partition. St000732The number of double deficiencies of a permutation. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St000864The number of circled entries of the shifted recording tableau of a permutation. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St000925The number of topologically connected components of a set partition. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001388The number of non-attacking neighbors of a permutation. St001413Half the length of the longest even length palindromic prefix of a binary word. St001461The number of topologically connected components of the chord diagram of a permutation. St001469The holeyness of a permutation. St001470The cyclic holeyness of a permutation. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001781The interlacing number of a set partition. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St001806The upper middle entry of a permutation. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001889The size of the connectivity set of a signed permutation. St000039The number of crossings of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000084The number of subtrees. St000099The number of valleys of a permutation, including the boundary. St000105The number of blocks in the set partition. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000219The number of occurrences of the pattern 231 in a permutation. St000221The number of strong fixed points of a permutation. St000236The number of cyclical small weak excedances. St000239The number of small weak excedances. St000241The number of cyclical small excedances. St000247The number of singleton blocks of a set partition. St000248The number of anti-singletons of a set partition. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000251The number of nonsingleton blocks of a set partition. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000295The length of the border of a binary word. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000328The maximum number of child nodes in a tree. St000355The number of occurrences of the pattern 21-3. St000367The number of simsun double descents of a permutation. St000406The number of occurrences of the pattern 3241 in a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000462The major index minus the number of excedences of a permutation. St000470The number of runs in a permutation. St000485The length of the longest cycle of a permutation. St000496The rcs statistic of a set partition. St000500Eigenvalues of the random-to-random operator acting on the regular representation. St000502The number of successions of a set partitions. St000504The cardinality of the first block of a set partition. St000557The number of occurrences of the pattern {{1},{2},{3}} in a set partition. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000573The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton and 2 a maximal element. St000574The number of occurrences of the pattern {{1},{2}} such that 1 is a minimal and 2 a maximal element. St000575The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element and 2 a singleton. St000576The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal and 2 a minimal element. St000578The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton. St000580The number of occurrences of the pattern {{1},{2},{3}} such that 2 is minimal, 3 is maximal. St000583The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1, 2 are maximal. St000584The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal, 3 is maximal. St000587The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. St000590The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is maximal, (2,3) are consecutive in a block. St000591The number of occurrences of the pattern {{1},{2},{3}} such that 2 is maximal. St000592The number of occurrences of the pattern {{1},{2},{3}} such that 1 is maximal. St000593The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal. St000596The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is maximal. St000603The number of occurrences of the pattern {{1},{2},{3}} such that 2,3 are minimal. St000604The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 is maximal. St000608The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal, 3 is maximal. St000615The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal. St000623The number of occurrences of the pattern 52341 in a permutation. St000666The number of right tethers of a permutation. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000750The number of occurrences of the pattern 4213 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000943The number of spots the most unlucky car had to go further in a parking function. St000961The shifted major index of a permutation. St000962The 3-shifted major index of a permutation. St000963The 2-shifted major index of a permutation. St000989The number of final rises of a permutation. St000991The number of right-to-left minima of a permutation. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001062The maximal size of a block of a set partition. St001075The minimal size of a block of a set partition. St001082The number of boxed occurrences of 123 in a permutation. St001130The number of two successive successions in a permutation. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001301The first Betti number of the order complex associated with the poset. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001371The length of the longest Yamanouchi prefix of a binary word. St001381The fertility of a permutation. St001396Number of triples of incomparable elements in a finite poset. St001402The number of separators in a permutation. St001403The number of vertical separators in a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001537The number of cyclic crossings of a permutation. St001550The number of inversions between exceedances where the greater exceedance is linked. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St001735The number of permutations with the same set of runs. St001741The largest integer such that all patterns of this size are contained in the permutation. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001801Half the number of preimage-image pairs of different parity in a permutation. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001847The number of occurrences of the pattern 1432 in a permutation. St001850The number of Hecke atoms of a permutation. St001851The number of Hecke atoms of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St001903The number of fixed points of a parking function. St001905The number of preferred parking spots in a parking function less than the index of the car. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000495The number of inversions of distance at most 2 of a permutation. St000638The number of up-down runs of a permutation. St000824The sum of the number of descents and the number of recoils of a permutation. St000831The number of indices that are either descents or recoils. St001424The number of distinct squares in a binary word. St001472The permanent of the Coxeter matrix of the poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000632The jump number of the poset. St001635The trace of the square of the Coxeter matrix of the incidence algebra of a poset. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000307The number of rowmotion orbits of a poset. St000907The number of maximal antichains of minimal length in a poset. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St000524The number of posets with the same order polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000717The number of ordinal summands of a poset. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000102The charge of a semistandard tableau. St001948The number of augmented double ascents of a permutation.