Your data matches 375 different statistics following compositions of up to 3 maps.
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St001586: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> 0
[2]
=> 0
[1,1]
=> 0
[3]
=> 0
[2,1]
=> 1
[1,1,1]
=> 0
[4]
=> 0
[3,1]
=> 0
[2,2]
=> 0
[2,1,1]
=> 2
[1,1,1,1]
=> 0
[5]
=> 0
[4,1]
=> 1
[3,2]
=> 0
[3,1,1]
=> 0
[2,2,1]
=> 1
[2,1,1,1]
=> 3
[1,1,1,1,1]
=> 0
[6]
=> 0
[5,1]
=> 0
[4,2]
=> 0
[4,1,1]
=> 2
[3,3]
=> 0
[3,2,1]
=> 1
[3,1,1,1]
=> 0
[2,2,2]
=> 0
[2,2,1,1]
=> 2
[2,1,1,1,1]
=> 4
[1,1,1,1,1,1]
=> 0
[7]
=> 0
[6,1]
=> 1
[5,2]
=> 0
[5,1,1]
=> 0
[4,3]
=> 1
[4,2,1]
=> 1
[4,1,1,1]
=> 3
[3,3,1]
=> 0
[3,2,2]
=> 0
[3,2,1,1]
=> 2
[3,1,1,1,1]
=> 0
[2,2,2,1]
=> 1
[2,2,1,1,1]
=> 3
[2,1,1,1,1,1]
=> 5
[1,1,1,1,1,1,1]
=> 0
Description
The number of odd parts smaller than the largest even part in an integer partition.
Matching statistic: St000290
Mp00317: Integer partitions odd partsBinary words
Mp00104: Binary words reverseBinary words
St000290: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> 1 => 1 => 0
[2]
=> 0 => 0 => 0
[1,1]
=> 11 => 11 => 0
[3]
=> 1 => 1 => 0
[2,1]
=> 01 => 10 => 1
[1,1,1]
=> 111 => 111 => 0
[4]
=> 0 => 0 => 0
[3,1]
=> 11 => 11 => 0
[2,2]
=> 00 => 00 => 0
[2,1,1]
=> 011 => 110 => 2
[1,1,1,1]
=> 1111 => 1111 => 0
[5]
=> 1 => 1 => 0
[4,1]
=> 01 => 10 => 1
[3,2]
=> 10 => 01 => 0
[3,1,1]
=> 111 => 111 => 0
[2,2,1]
=> 001 => 100 => 1
[2,1,1,1]
=> 0111 => 1110 => 3
[1,1,1,1,1]
=> 11111 => 11111 => 0
[6]
=> 0 => 0 => 0
[5,1]
=> 11 => 11 => 0
[4,2]
=> 00 => 00 => 0
[4,1,1]
=> 011 => 110 => 2
[3,3]
=> 11 => 11 => 0
[3,2,1]
=> 101 => 101 => 1
[3,1,1,1]
=> 1111 => 1111 => 0
[2,2,2]
=> 000 => 000 => 0
[2,2,1,1]
=> 0011 => 1100 => 2
[2,1,1,1,1]
=> 01111 => 11110 => 4
[1,1,1,1,1,1]
=> 111111 => 111111 => 0
[7]
=> 1 => 1 => 0
[6,1]
=> 01 => 10 => 1
[5,2]
=> 10 => 01 => 0
[5,1,1]
=> 111 => 111 => 0
[4,3]
=> 01 => 10 => 1
[4,2,1]
=> 001 => 100 => 1
[4,1,1,1]
=> 0111 => 1110 => 3
[3,3,1]
=> 111 => 111 => 0
[3,2,2]
=> 100 => 001 => 0
[3,2,1,1]
=> 1011 => 1101 => 2
[3,1,1,1,1]
=> 11111 => 11111 => 0
[2,2,2,1]
=> 0001 => 1000 => 1
[2,2,1,1,1]
=> 00111 => 11100 => 3
[2,1,1,1,1,1]
=> 011111 => 111110 => 5
[1,1,1,1,1,1,1]
=> 1111111 => 1111111 => 0
Description
The major index of a binary word. This is the sum of the positions of descents, i.e., a one followed by a zero. For words of length $n$ with $a$ zeros, the generating function for the major index is the $q$-binomial coefficient $\binom{n}{a}_q$.
Mp00317: Integer partitions odd partsBinary words
Mp00178: Binary words to compositionInteger compositions
St000766: Integer compositions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> 1 => [1,1] => 0
[2]
=> 0 => [2] => 0
[1,1]
=> 11 => [1,1,1] => 0
[3]
=> 1 => [1,1] => 0
[2,1]
=> 01 => [2,1] => 1
[1,1,1]
=> 111 => [1,1,1,1] => 0
[4]
=> 0 => [2] => 0
[3,1]
=> 11 => [1,1,1] => 0
[2,2]
=> 00 => [3] => 0
[2,1,1]
=> 011 => [2,1,1] => 2
[1,1,1,1]
=> 1111 => [1,1,1,1,1] => 0
[5]
=> 1 => [1,1] => 0
[4,1]
=> 01 => [2,1] => 1
[3,2]
=> 10 => [1,2] => 0
[3,1,1]
=> 111 => [1,1,1,1] => 0
[2,2,1]
=> 001 => [3,1] => 1
[2,1,1,1]
=> 0111 => [2,1,1,1] => 3
[1,1,1,1,1]
=> 11111 => [1,1,1,1,1,1] => 0
[6]
=> 0 => [2] => 0
[5,1]
=> 11 => [1,1,1] => 0
[4,2]
=> 00 => [3] => 0
[4,1,1]
=> 011 => [2,1,1] => 2
[3,3]
=> 11 => [1,1,1] => 0
[3,2,1]
=> 101 => [1,2,1] => 1
[3,1,1,1]
=> 1111 => [1,1,1,1,1] => 0
[2,2,2]
=> 000 => [4] => 0
[2,2,1,1]
=> 0011 => [3,1,1] => 2
[2,1,1,1,1]
=> 01111 => [2,1,1,1,1] => 4
[1,1,1,1,1,1]
=> 111111 => [1,1,1,1,1,1,1] => 0
[7]
=> 1 => [1,1] => 0
[6,1]
=> 01 => [2,1] => 1
[5,2]
=> 10 => [1,2] => 0
[5,1,1]
=> 111 => [1,1,1,1] => 0
[4,3]
=> 01 => [2,1] => 1
[4,2,1]
=> 001 => [3,1] => 1
[4,1,1,1]
=> 0111 => [2,1,1,1] => 3
[3,3,1]
=> 111 => [1,1,1,1] => 0
[3,2,2]
=> 100 => [1,3] => 0
[3,2,1,1]
=> 1011 => [1,2,1,1] => 2
[3,1,1,1,1]
=> 11111 => [1,1,1,1,1,1] => 0
[2,2,2,1]
=> 0001 => [4,1] => 1
[2,2,1,1,1]
=> 00111 => [3,1,1,1] => 3
[2,1,1,1,1,1]
=> 011111 => [2,1,1,1,1,1] => 5
[1,1,1,1,1,1,1]
=> 1111111 => [1,1,1,1,1,1,1,1] => 0
Description
The number of inversions of an integer composition. This is the number of pairs $(i,j)$ such that $i < j$ and $c_i > c_j$.
Matching statistic: St001485
Mp00317: Integer partitions odd partsBinary words
Mp00104: Binary words reverseBinary words
St001485: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> 1 => 1 => 0
[2]
=> 0 => 0 => 0
[1,1]
=> 11 => 11 => 0
[3]
=> 1 => 1 => 0
[2,1]
=> 01 => 10 => 1
[1,1,1]
=> 111 => 111 => 0
[4]
=> 0 => 0 => 0
[3,1]
=> 11 => 11 => 0
[2,2]
=> 00 => 00 => 0
[2,1,1]
=> 011 => 110 => 2
[1,1,1,1]
=> 1111 => 1111 => 0
[5]
=> 1 => 1 => 0
[4,1]
=> 01 => 10 => 1
[3,2]
=> 10 => 01 => 0
[3,1,1]
=> 111 => 111 => 0
[2,2,1]
=> 001 => 100 => 1
[2,1,1,1]
=> 0111 => 1110 => 3
[1,1,1,1,1]
=> 11111 => 11111 => 0
[6]
=> 0 => 0 => 0
[5,1]
=> 11 => 11 => 0
[4,2]
=> 00 => 00 => 0
[4,1,1]
=> 011 => 110 => 2
[3,3]
=> 11 => 11 => 0
[3,2,1]
=> 101 => 101 => 1
[3,1,1,1]
=> 1111 => 1111 => 0
[2,2,2]
=> 000 => 000 => 0
[2,2,1,1]
=> 0011 => 1100 => 2
[2,1,1,1,1]
=> 01111 => 11110 => 4
[1,1,1,1,1,1]
=> 111111 => 111111 => 0
[7]
=> 1 => 1 => 0
[6,1]
=> 01 => 10 => 1
[5,2]
=> 10 => 01 => 0
[5,1,1]
=> 111 => 111 => 0
[4,3]
=> 01 => 10 => 1
[4,2,1]
=> 001 => 100 => 1
[4,1,1,1]
=> 0111 => 1110 => 3
[3,3,1]
=> 111 => 111 => 0
[3,2,2]
=> 100 => 001 => 0
[3,2,1,1]
=> 1011 => 1101 => 2
[3,1,1,1,1]
=> 11111 => 11111 => 0
[2,2,2,1]
=> 0001 => 1000 => 1
[2,2,1,1,1]
=> 00111 => 11100 => 3
[2,1,1,1,1,1]
=> 011111 => 111110 => 5
[1,1,1,1,1,1,1]
=> 1111111 => 1111111 => 0
Description
The modular major index of a binary word. This is [[St000290]] modulo the length of the word.
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00102: Dyck paths rise compositionInteger compositions
Mp00041: Integer compositions conjugateInteger compositions
St000091: Integer compositions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1] => [1] => 0
[2]
=> [1,0,1,0]
=> [1,1] => [2] => 0
[1,1]
=> [1,1,0,0]
=> [2] => [1,1] => 0
[3]
=> [1,0,1,0,1,0]
=> [1,1,1] => [3] => 0
[2,1]
=> [1,0,1,1,0,0]
=> [1,2] => [1,2] => 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [2,1] => [2,1] => 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [4] => 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => [1,3] => 2
[2,2]
=> [1,1,1,0,0,0]
=> [3] => [1,1,1] => 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,2,1] => [2,2] => 0
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,1,1] => [3,1] => 0
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [5] => 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [1,4] => 3
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => [1,1,2] => 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [2,3] => 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,1] => [2,1,1] => 0
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [3,2] => 0
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => [4,1] => 0
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1] => [6] => 0
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,2] => [1,5] => 4
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [1,1,3] => 2
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,2,1] => [2,4] => 2
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [3,1] => [2,1,1] => 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [2,1,2] => 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,2,1,1] => [3,3] => 0
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [4] => [1,1,1,1] => 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,1] => [3,1,1] => 0
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,1,1,1] => [4,2] => 0
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1,1] => [5,1] => 0
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1] => [7] => 0
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,2] => [1,6] => 5
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,3] => [1,1,4] => 3
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,2,1] => [2,5] => 3
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [2,1,2] => 1
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,3,1] => [2,1,3] => 2
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,2,1,1] => [3,4] => 1
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1,1] => [3,1,1] => 0
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,1,1,2] => 1
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,3,1,1] => [3,1,2] => 1
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,2,1,1,1] => [4,3] => 0
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1] => [2,1,1,1] => 0
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,1,1,1] => [4,1,1] => 0
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,1,1,1,1] => [5,2] => 0
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1,1,1] => [6,1] => 0
Description
The descent variation of a composition. Defined in [1].
Matching statistic: St000293
Mp00317: Integer partitions odd partsBinary words
Mp00104: Binary words reverseBinary words
Mp00096: Binary words Foata bijectionBinary words
St000293: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> 1 => 1 => 1 => 0
[2]
=> 0 => 0 => 0 => 0
[1,1]
=> 11 => 11 => 11 => 0
[3]
=> 1 => 1 => 1 => 0
[2,1]
=> 01 => 10 => 10 => 1
[1,1,1]
=> 111 => 111 => 111 => 0
[4]
=> 0 => 0 => 0 => 0
[3,1]
=> 11 => 11 => 11 => 0
[2,2]
=> 00 => 00 => 00 => 0
[2,1,1]
=> 011 => 110 => 110 => 2
[1,1,1,1]
=> 1111 => 1111 => 1111 => 0
[5]
=> 1 => 1 => 1 => 0
[4,1]
=> 01 => 10 => 10 => 1
[3,2]
=> 10 => 01 => 01 => 0
[3,1,1]
=> 111 => 111 => 111 => 0
[2,2,1]
=> 001 => 100 => 010 => 1
[2,1,1,1]
=> 0111 => 1110 => 1110 => 3
[1,1,1,1,1]
=> 11111 => 11111 => 11111 => 0
[6]
=> 0 => 0 => 0 => 0
[5,1]
=> 11 => 11 => 11 => 0
[4,2]
=> 00 => 00 => 00 => 0
[4,1,1]
=> 011 => 110 => 110 => 2
[3,3]
=> 11 => 11 => 11 => 0
[3,2,1]
=> 101 => 101 => 101 => 1
[3,1,1,1]
=> 1111 => 1111 => 1111 => 0
[2,2,2]
=> 000 => 000 => 000 => 0
[2,2,1,1]
=> 0011 => 1100 => 0110 => 2
[2,1,1,1,1]
=> 01111 => 11110 => 11110 => 4
[1,1,1,1,1,1]
=> 111111 => 111111 => 111111 => 0
[7]
=> 1 => 1 => 1 => 0
[6,1]
=> 01 => 10 => 10 => 1
[5,2]
=> 10 => 01 => 01 => 0
[5,1,1]
=> 111 => 111 => 111 => 0
[4,3]
=> 01 => 10 => 10 => 1
[4,2,1]
=> 001 => 100 => 010 => 1
[4,1,1,1]
=> 0111 => 1110 => 1110 => 3
[3,3,1]
=> 111 => 111 => 111 => 0
[3,2,2]
=> 100 => 001 => 001 => 0
[3,2,1,1]
=> 1011 => 1101 => 1101 => 2
[3,1,1,1,1]
=> 11111 => 11111 => 11111 => 0
[2,2,2,1]
=> 0001 => 1000 => 0010 => 1
[2,2,1,1,1]
=> 00111 => 11100 => 01110 => 3
[2,1,1,1,1,1]
=> 011111 => 111110 => 111110 => 5
[1,1,1,1,1,1,1]
=> 1111111 => 1111111 => 1111111 => 0
Description
The number of inversions of a binary word.
Matching statistic: St000769
Mp00317: Integer partitions odd partsBinary words
Mp00178: Binary words to compositionInteger compositions
Mp00315: Integer compositions inverse Foata bijectionInteger compositions
St000769: Integer compositions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> 1 => [1,1] => [1,1] => 0
[2]
=> 0 => [2] => [2] => 0
[1,1]
=> 11 => [1,1,1] => [1,1,1] => 0
[3]
=> 1 => [1,1] => [1,1] => 0
[2,1]
=> 01 => [2,1] => [2,1] => 1
[1,1,1]
=> 111 => [1,1,1,1] => [1,1,1,1] => 0
[4]
=> 0 => [2] => [2] => 0
[3,1]
=> 11 => [1,1,1] => [1,1,1] => 0
[2,2]
=> 00 => [3] => [3] => 0
[2,1,1]
=> 011 => [2,1,1] => [1,2,1] => 2
[1,1,1,1]
=> 1111 => [1,1,1,1,1] => [1,1,1,1,1] => 0
[5]
=> 1 => [1,1] => [1,1] => 0
[4,1]
=> 01 => [2,1] => [2,1] => 1
[3,2]
=> 10 => [1,2] => [1,2] => 0
[3,1,1]
=> 111 => [1,1,1,1] => [1,1,1,1] => 0
[2,2,1]
=> 001 => [3,1] => [3,1] => 1
[2,1,1,1]
=> 0111 => [2,1,1,1] => [1,1,2,1] => 3
[1,1,1,1,1]
=> 11111 => [1,1,1,1,1,1] => [1,1,1,1,1,1] => 0
[6]
=> 0 => [2] => [2] => 0
[5,1]
=> 11 => [1,1,1] => [1,1,1] => 0
[4,2]
=> 00 => [3] => [3] => 0
[4,1,1]
=> 011 => [2,1,1] => [1,2,1] => 2
[3,3]
=> 11 => [1,1,1] => [1,1,1] => 0
[3,2,1]
=> 101 => [1,2,1] => [2,1,1] => 1
[3,1,1,1]
=> 1111 => [1,1,1,1,1] => [1,1,1,1,1] => 0
[2,2,2]
=> 000 => [4] => [4] => 0
[2,2,1,1]
=> 0011 => [3,1,1] => [1,3,1] => 2
[2,1,1,1,1]
=> 01111 => [2,1,1,1,1] => [1,1,1,2,1] => 4
[1,1,1,1,1,1]
=> 111111 => [1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => 0
[7]
=> 1 => [1,1] => [1,1] => 0
[6,1]
=> 01 => [2,1] => [2,1] => 1
[5,2]
=> 10 => [1,2] => [1,2] => 0
[5,1,1]
=> 111 => [1,1,1,1] => [1,1,1,1] => 0
[4,3]
=> 01 => [2,1] => [2,1] => 1
[4,2,1]
=> 001 => [3,1] => [3,1] => 1
[4,1,1,1]
=> 0111 => [2,1,1,1] => [1,1,2,1] => 3
[3,3,1]
=> 111 => [1,1,1,1] => [1,1,1,1] => 0
[3,2,2]
=> 100 => [1,3] => [1,3] => 0
[3,2,1,1]
=> 1011 => [1,2,1,1] => [1,2,1,1] => 2
[3,1,1,1,1]
=> 11111 => [1,1,1,1,1,1] => [1,1,1,1,1,1] => 0
[2,2,2,1]
=> 0001 => [4,1] => [4,1] => 1
[2,2,1,1,1]
=> 00111 => [3,1,1,1] => [1,1,3,1] => 3
[2,1,1,1,1,1]
=> 011111 => [2,1,1,1,1,1] => [1,1,1,1,2,1] => 5
[1,1,1,1,1,1,1]
=> 1111111 => [1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1] => 0
Description
The major index of a composition regarded as a word. This is the sum of the positions of the descents of the composition. For the statistic which interprets the composition as a descent set, see [[St000008]].
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00132: Dyck paths switch returns and last double riseDyck paths
Mp00143: Dyck paths inverse promotionDyck paths
St001172: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1,0]
=> [1,0]
=> 0
[2]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 0
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,1,0,0,0]
=> 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 0
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 0
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 0
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 4
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 0
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 0
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 0
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 0
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 2
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 5
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 3
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> 0
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 1
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> 0
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 0
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> 0
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> 1
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> 1
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> 2
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> 3
Description
The number of 1-rises at odd height of a Dyck path.
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00143: Dyck paths inverse promotionDyck paths
Mp00024: Dyck paths to 321-avoiding permutationPermutations
St001810: Permutations ⟶ ℤResult quality: 91% values known / values provided: 91%distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,2] => 0
[2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => 0
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 0
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,3,2] => 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 0
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => 0
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => 2
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,6,1,2,3,4] => 0
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => 0
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 0
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,3,4,2] => 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,2,3,5,4] => 3
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => 0
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [6,7,1,2,3,4,5] => ? = 0
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [4,6,1,2,3,5] => 0
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => 0
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => 0
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 0
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,2,5,3,4] => 2
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,2,4,5,3] => 2
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,2,3,4,6,5] => 4
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7] => 0
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [7,8,1,2,3,4,5,6] => ? ∊ {0,1,1}
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0,1,1,0,0]
=> [5,7,1,2,3,4,6] => ? ∊ {0,1,1}
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [4,5,1,2,3,6] => 0
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [3,6,1,2,4,5] => 0
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => 0
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => 0
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,5,2,3,4] => 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => 0
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => 0
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,2,4,3,5] => 2
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,2,3,6,4,5] => 3
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,2,3,5,6,4] => 3
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,2,3,4,5,7,6] => 5
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7,8] => ? ∊ {0,1,1}
Description
The number of fixed points of a permutation smaller than its largest moved point.
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00024: Dyck paths to 321-avoiding permutationPermutations
Mp00149: Permutations Lehmer code rotationPermutations
St000367: Permutations ⟶ ℤResult quality: 83% values known / values provided: 89%distinct values known / distinct values provided: 83%
Values
[1]
=> [1,0,1,0]
=> [2,1] => [1,2] => 0
[2]
=> [1,1,0,0,1,0]
=> [3,1,2] => [1,3,2] => 0
[1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => [3,1,2] => 0
[3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,3,4,2] => 0
[2,1]
=> [1,0,1,0,1,0]
=> [2,1,3] => [3,2,1] => 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [3,4,1,2] => 0
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [1,3,4,5,2] => 0
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [4,2,3,1] => 0
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,1,3,2] => 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [3,4,2,1] => 2
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [3,4,5,1,2] => 0
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => [1,3,4,5,6,2] => 0
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,3,5] => [5,2,3,4,1] => 0
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [4,2,1,3] => 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [3,2,4,1] => 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [3,1,4,2] => 0
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [3,4,5,2,1] => 3
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [3,4,5,6,1,2] => 0
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => [1,3,4,5,6,7,2] => 0
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1,2,3,4,6] => [6,2,3,4,5,1] => 0
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => [5,2,3,1,4] => 0
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [4,2,3,5,1] => 0
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [5,1,3,4,2] => 0
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [3,2,1,4] => 2
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [3,4,2,5,1] => 2
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [4,5,1,3,2] => 0
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [3,4,1,5,2] => 0
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,3,4,5,1,6] => [3,4,5,6,2,1] => 4
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [3,4,5,6,7,1,2] => ? = 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,7] => [1,3,4,5,6,7,8,2] => ? ∊ {0,1,1,5}
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,7] => [7,2,3,4,5,6,1] => ? ∊ {0,1,1,5}
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,1,2,3,6,4] => [6,2,3,4,1,5] => 0
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5,6] => [5,2,3,4,6,1] => 0
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [5,2,1,4,3] => 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [4,2,3,1,5] => 0
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [3,2,4,5,1] => 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [4,1,3,5,2] => 0
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [4,5,2,1,3] => 2
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [3,4,2,1,5] => 3
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,3,4,1,5,6] => [3,4,5,2,6,1] => 3
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [3,5,1,4,2] => 0
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,3,4,6,1,5] => [3,4,5,1,6,2] => 0
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [2,3,4,5,6,1,7] => [3,4,5,6,7,2,1] => ? ∊ {0,1,1,5}
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => [3,4,5,6,7,8,1,2] => ? ∊ {0,1,1,5}
Description
The number of simsun double descents of a permutation. The restriction of a permutation $\pi$ to $[k] = \{1,\ldots,k\}$ is given in one-line notation by the subword of $\pi$ of letters in $[k]$. A simsun double descent of a permutation $\pi$ is a double descent of any restriction of $\pi$ to $[1,\ldots,k]$ for some $k$. (Note here that the same double descent can appear in multiple restrictions!)
The following 365 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001552The number of inversions between excedances and fixed points of a permutation. St000356The number of occurrences of the pattern 13-2. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000732The number of double deficiencies of a permutation. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St000383The last part of an integer composition. St000366The number of double descents of a permutation. St001130The number of two successive successions in a permutation. St001705The number of occurrences of the pattern 2413 in a permutation. St001847The number of occurrences of the pattern 1432 in a permutation. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001557The number of inversions of the second entry of a permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001235The global dimension of the corresponding Comp-Nakayama algebra. St000225Difference between largest and smallest parts in a partition. St000377The dinv defect of an integer partition. St000931The number of occurrences of the pattern UUU in a Dyck path. St000944The 3-degree of an integer partition. St001091The number of parts in an integer partition whose next smaller part has the same size. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001175The size of a partition minus the hook length of the base cell. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001095The number of non-isomorphic posets with precisely one further covering relation. St001964The interval resolution global dimension of a poset. St000455The second largest eigenvalue of a graph if it is integral. St001498The normalised height of a Nakayama algebra with magnitude 1. St000260The radius of a connected graph. St001330The hat guessing number of a graph. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St001820The size of the image of the pop stack sorting operator. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000674The number of hills of a Dyck path. St000929The constant term of the character polynomial of an integer partition. St000941The number of characters of the symmetric group whose value on the partition is even. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The 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 vertex labelled trees. St001099The 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 leaf labelled binary trees. St001100The 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 leaf labelled trees. 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. St001176The size of a partition minus its first part. 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. St001280The number of parts of an integer partition that are at least two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001541The Gini index of an integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001846The number of elements which do not have a complement in the lattice. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St000181The number of connected components of the Hasse diagram for the poset. St000454The largest eigenvalue of a graph if it is integral. St001490The number of connected components of a skew partition. St001645The pebbling number of a connected graph. St001890The maximum magnitude of the Möbius function of a poset. St001556The number of inversions of the third entry of a permutation. St000741The Colin de Verdière graph invariant. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000488The number of cycles of a permutation of length at most 2. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. 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. St001730The number of times the path corresponding to a binary word crosses the base line. St001948The number of augmented double ascents of a permutation. St000456The monochromatic index of a connected graph. St000902 The minimal number of repetitions of an integer composition. St001884The number of borders of a binary word. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000058The order of a permutation. St001651The Frankl number of a lattice. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000068The number of minimal elements in a poset. St000534The number of 2-rises of a permutation. St000451The length of the longest pattern of the form k 1 2. St000842The breadth of a permutation. 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. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000679The pruning number of an ordered tree. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St000090The variation of a composition. St000217The number of occurrences of the pattern 312 in a permutation. St000233The number of nestings of a set partition. St000338The number of pixed points of a permutation. St000358The number of occurrences of the pattern 31-2. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000650The number of 3-rises of a permutation. St000664The number of right ropes of a permutation. St000681The Grundy value of Chomp on Ferrers diagrams. St000709The number of occurrences of 14-2-3 or 14-3-2. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000779The tier of a permutation. St000937The number of positive values of the symmetric group character corresponding to the partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001811The Castelnuovo-Mumford regularity of 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. St001162The minimum jump of a permutation. St001344The neighbouring number of a permutation. St001487The number of inner corners of a skew partition. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001896The number of right descents of a signed permutations. St000527The width of the poset. St001738The minimal order of a graph which is not an induced subgraph of the given graph. 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. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000731The number of double exceedences of a permutation. St000871The number of very big ascents of a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000035The number of left outer peaks of a permutation. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000862The number of parts of the shifted shape 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. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000039The number of crossings of a permutation. 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. St000234The number of global ascents of a permutation. St000247The number of singleton 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. St000317The cycle descent number of a permutation. St000355The number of occurrences of the pattern 21-3. St000360The number of occurrences of the pattern 32-1. St000365The number of double ascents of a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. 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. St000486The number of cycles of length at least 3 of a permutation. St000496The rcs statistic of a set partition. St000500Eigenvalues of the random-to-random operator acting on the regular representation. St000516The number of stretching pairs of a permutation. 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. St000563The number of overlapping pairs of blocks of a set partition. St000573The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton 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. St000649The number of 3-excedences of a permutation. St000666The number of right tethers of a permutation. 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. 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. St000850The number of 1/2-balanced pairs in a poset. St000872The number of very big descents of a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. 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. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001060The distinguishing index of a graph. St001061The number of indices that are both descents and recoils of a permutation. St001082The number of boxed occurrences of 123 in a permutation. 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. St001470The cyclic holeyness of a permutation. St001513The number of nested exceedences of a permutation. St001537The number of cyclic crossings of a permutation. St001549The number of restricted non-inversions between exceedances. St001550The number of inversions between exceedances where the greater exceedance is linked. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. 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. St001715The number of non-records in a permutation. St001728The number of invisible descents of a permutation. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001781The interlacing number of a set partition. St001803The maximal overlap of the cylindrical tableau associated with a tableau. 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. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. 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. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000210Minimum over maximum difference of elements in cycles. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000253The crossing number of a set partition. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000353The number of inner valleys of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. 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. St000633The size of the automorphism group of a poset. St000646The number of big ascents of a permutation. St000654The first descent of a permutation. St000663The number of right floats of a permutation. St000694The number of affine bounded permutations that project to a given permutation. St000729The minimal arc length of a set partition. 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. St000886The number of permutations with the same antidiagonal sums. 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. St000990The first ascent of a permutation. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. 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 $K[x]/(x^n)$. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001388The number of non-attacking neighbors of a permutation. St001399The distinguishing number of a poset. 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. St001462The number of factors of a standard tableaux under concatenation. St001469The holeyness of a permutation. St001472The permanent of the Coxeter matrix of the poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001652The length of a longest interval of consecutive numbers. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001662The length of the longest factor of consecutive numbers in a permutation. St001665The number of pure excedances of a permutation. St001712The number of natural descents of a standard Young tableau. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. 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. St001928The number of non-overlapping descents in a permutation. St000062The length of the longest increasing subsequence of the permutation. St000075The orbit size of a standard tableau under promotion. St000084The number of subtrees. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000105The number of blocks in the set partition. St000236The number of cyclical small weak excedances. St000239The number of small weak excedances. St000241The number of cyclical small excedances. 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. 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. St000354The number of recoils of a permutation. St000470The number of runs in a permutation. St000485The length of the longest cycle of a permutation. St000487The length of the shortest cycle of a permutation. St000502The number of successions of a set partitions. St000504The cardinality of the first block of a set partition. St000542The number of left-to-right-minima of a permutation. St000574The number of occurrences of the pattern {{1},{2}} such that 1 is a minimal and 2 a maximal element. St000619The number of cyclic descents of a permutation. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St000836The number of descents of distance 2 of a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. 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. St001062The maximal size of a block of a set partition. St001075The minimal size of a block of a set partition. St001114The number of odd descents of a permutation. St001151The number of blocks with odd minimum. 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 $eA$. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001569The maximal modular displacement of 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. St001801Half the number of preimage-image pairs of different parity in a permutation. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001905The number of preferred parking spots in a parking function less than the index of the car. 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. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path.