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Your data matches 19 different statistics following compositions of up to 3 maps.
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Matching statistic: St000931
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000931: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000931: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+,+] => [1,2] => [1,1]
=> [1,1,0,0]
=> 0
[-,+] => [1,2] => [1,1]
=> [1,1,0,0]
=> 0
[+,-] => [1,2] => [1,1]
=> [1,1,0,0]
=> 0
[-,-] => [1,2] => [1,1]
=> [1,1,0,0]
=> 0
[+,+,+] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[-,+,+] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[+,-,+] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[+,+,-] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[-,-,+] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[-,+,-] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[+,-,-] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[-,-,-] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[+,3,2] => [1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> 0
[-,3,2] => [1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> 0
[2,1,+] => [2,1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 0
[2,1,-] => [2,1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 0
[3,+,1] => [3,2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 0
[3,-,1] => [3,2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 0
[+,+,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[-,+,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[+,-,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[+,+,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[+,+,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[-,-,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[-,+,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[-,+,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[+,-,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[+,-,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[+,+,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[-,-,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[-,-,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[-,+,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[+,-,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[-,-,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[+,+,4,3] => [1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[-,+,4,3] => [1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[+,-,4,3] => [1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[-,-,4,3] => [1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[+,3,2,+] => [1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[-,3,2,+] => [1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[+,3,2,-] => [1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[-,3,2,-] => [1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[+,3,4,2] => [1,3,4,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[-,3,4,2] => [1,3,4,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[+,4,2,3] => [1,4,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[-,4,2,3] => [1,4,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[+,4,+,2] => [1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[-,4,+,2] => [1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[+,4,-,2] => [1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[-,4,-,2] => [1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
Description
The number of occurrences of the pattern UUU in a Dyck path.
The number of Dyck paths with statistic value 0 are counted by the Motzkin numbers [1].
Matching statistic: St001251
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001251: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001251: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+,+] => [1,2] => [1,1]
=> [1]
=> 0
[-,+] => [1,2] => [1,1]
=> [1]
=> 0
[+,-] => [1,2] => [1,1]
=> [1]
=> 0
[-,-] => [1,2] => [1,1]
=> [1]
=> 0
[+,+,+] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[-,+,+] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[+,-,+] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[+,+,-] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[-,-,+] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[-,+,-] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[+,-,-] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[-,-,-] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[+,3,2] => [1,3,2] => [2,1]
=> [1]
=> 0
[-,3,2] => [1,3,2] => [2,1]
=> [1]
=> 0
[2,1,+] => [2,1,3] => [2,1]
=> [1]
=> 0
[2,1,-] => [2,1,3] => [2,1]
=> [1]
=> 0
[3,+,1] => [3,2,1] => [2,1]
=> [1]
=> 0
[3,-,1] => [3,2,1] => [2,1]
=> [1]
=> 0
[+,+,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,+,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,-,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,+,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,+,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,-,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,+,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,+,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,-,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,-,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,+,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,-,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,-,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,+,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,-,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,-,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,+,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[-,+,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[+,-,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[-,-,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[+,3,2,+] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[-,3,2,+] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[+,3,2,-] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[-,3,2,-] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[+,3,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> 0
[-,3,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> 0
[+,4,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> 0
[-,4,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> 0
[+,4,+,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[-,4,+,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[+,4,-,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[-,4,-,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
Description
The number of parts of a partition that are not congruent 1 modulo 3.
Matching statistic: St001280
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+,+] => [1,2] => [1,1]
=> [1]
=> 0
[-,+] => [1,2] => [1,1]
=> [1]
=> 0
[+,-] => [1,2] => [1,1]
=> [1]
=> 0
[-,-] => [1,2] => [1,1]
=> [1]
=> 0
[+,+,+] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[-,+,+] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[+,-,+] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[+,+,-] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[-,-,+] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[-,+,-] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[+,-,-] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[-,-,-] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[+,3,2] => [1,3,2] => [2,1]
=> [1]
=> 0
[-,3,2] => [1,3,2] => [2,1]
=> [1]
=> 0
[2,1,+] => [2,1,3] => [2,1]
=> [1]
=> 0
[2,1,-] => [2,1,3] => [2,1]
=> [1]
=> 0
[3,+,1] => [3,2,1] => [2,1]
=> [1]
=> 0
[3,-,1] => [3,2,1] => [2,1]
=> [1]
=> 0
[+,+,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,+,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,-,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,+,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,+,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,-,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,+,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,+,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,-,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,-,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,+,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,-,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,-,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,+,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,-,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,-,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,+,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[-,+,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[+,-,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[-,-,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[+,3,2,+] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[-,3,2,+] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[+,3,2,-] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[-,3,2,-] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[+,3,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> 0
[-,3,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> 0
[+,4,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> 0
[-,4,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> 0
[+,4,+,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[-,4,+,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[+,4,-,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[-,4,-,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St000024
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+,+] => [1,2] => [1,1]
=> [1,1,0,0]
=> 1 = 0 + 1
[-,+] => [1,2] => [1,1]
=> [1,1,0,0]
=> 1 = 0 + 1
[+,-] => [1,2] => [1,1]
=> [1,1,0,0]
=> 1 = 0 + 1
[-,-] => [1,2] => [1,1]
=> [1,1,0,0]
=> 1 = 0 + 1
[+,+,+] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[-,+,+] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[+,-,+] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[+,+,-] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[-,-,+] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[-,+,-] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[+,-,-] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[-,-,-] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[+,3,2] => [1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[-,3,2] => [1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[2,1,+] => [2,1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[2,1,-] => [2,1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[3,+,1] => [3,2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[3,-,1] => [3,2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[+,+,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[-,+,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[+,-,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[+,+,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[+,+,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[-,-,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[-,+,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[-,+,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[+,-,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[+,-,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[+,+,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[-,-,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[-,-,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[-,+,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[+,-,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[-,-,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[+,+,4,3] => [1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[-,+,4,3] => [1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[+,-,4,3] => [1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[-,-,4,3] => [1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[+,3,2,+] => [1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[-,3,2,+] => [1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[+,3,2,-] => [1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[-,3,2,-] => [1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[+,3,4,2] => [1,3,4,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[-,3,4,2] => [1,3,4,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[+,4,2,3] => [1,4,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[-,4,2,3] => [1,4,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[+,4,+,2] => [1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[-,4,+,2] => [1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[+,4,-,2] => [1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[-,4,-,2] => [1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
Description
The number of double up and double down steps of a Dyck path.
In other words, this is the number of double rises (and, equivalently, the number of double falls) of a Dyck path.
Matching statistic: St000442
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000442: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000442: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+,+] => [1,2] => [1,1]
=> [1,1,0,0]
=> 1 = 0 + 1
[-,+] => [1,2] => [1,1]
=> [1,1,0,0]
=> 1 = 0 + 1
[+,-] => [1,2] => [1,1]
=> [1,1,0,0]
=> 1 = 0 + 1
[-,-] => [1,2] => [1,1]
=> [1,1,0,0]
=> 1 = 0 + 1
[+,+,+] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[-,+,+] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[+,-,+] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[+,+,-] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[-,-,+] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[-,+,-] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[+,-,-] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[-,-,-] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[+,3,2] => [1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[-,3,2] => [1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[2,1,+] => [2,1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[2,1,-] => [2,1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[3,+,1] => [3,2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[3,-,1] => [3,2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[+,+,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[-,+,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[+,-,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[+,+,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[+,+,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[-,-,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[-,+,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[-,+,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[+,-,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[+,-,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[+,+,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[-,-,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[-,-,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[-,+,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[+,-,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[-,-,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[+,+,4,3] => [1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[-,+,4,3] => [1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[+,-,4,3] => [1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[-,-,4,3] => [1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[+,3,2,+] => [1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[-,3,2,+] => [1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[+,3,2,-] => [1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[-,3,2,-] => [1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[+,3,4,2] => [1,3,4,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[-,3,4,2] => [1,3,4,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[+,4,2,3] => [1,4,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[-,4,2,3] => [1,4,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[+,4,+,2] => [1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[-,4,+,2] => [1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[+,4,-,2] => [1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[-,4,-,2] => [1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
Description
The maximal area to the right of an up step of a Dyck path.
Matching statistic: St000013
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+,+] => [1,2] => [1,1]
=> [1,1,0,0]
=> 2 = 0 + 2
[-,+] => [1,2] => [1,1]
=> [1,1,0,0]
=> 2 = 0 + 2
[+,-] => [1,2] => [1,1]
=> [1,1,0,0]
=> 2 = 0 + 2
[-,-] => [1,2] => [1,1]
=> [1,1,0,0]
=> 2 = 0 + 2
[+,+,+] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 2 = 0 + 2
[-,+,+] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 2 = 0 + 2
[+,-,+] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 2 = 0 + 2
[+,+,-] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 2 = 0 + 2
[-,-,+] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 2 = 0 + 2
[-,+,-] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 2 = 0 + 2
[+,-,-] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 2 = 0 + 2
[-,-,-] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 2 = 0 + 2
[+,3,2] => [1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> 2 = 0 + 2
[-,3,2] => [1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> 2 = 0 + 2
[2,1,+] => [2,1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 2 = 0 + 2
[2,1,-] => [2,1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 2 = 0 + 2
[3,+,1] => [3,2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 2 = 0 + 2
[3,-,1] => [3,2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 2 = 0 + 2
[+,+,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[-,+,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[+,-,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[+,+,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[+,+,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[-,-,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[-,+,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[-,+,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[+,-,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[+,-,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[+,+,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[-,-,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[-,-,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[-,+,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[+,-,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[-,-,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[+,+,4,3] => [1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
[-,+,4,3] => [1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
[+,-,4,3] => [1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
[-,-,4,3] => [1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
[+,3,2,+] => [1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
[-,3,2,+] => [1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
[+,3,2,-] => [1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
[-,3,2,-] => [1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
[+,3,4,2] => [1,3,4,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 0 + 2
[-,3,4,2] => [1,3,4,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 0 + 2
[+,4,2,3] => [1,4,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 0 + 2
[-,4,2,3] => [1,4,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 0 + 2
[+,4,+,2] => [1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
[-,4,+,2] => [1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
[+,4,-,2] => [1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
[-,4,-,2] => [1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
Description
The height of a Dyck path.
The height of a Dyck path $D$ of semilength $n$ is defined as the maximal height of a peak of $D$. The height of $D$ at position $i$ is the number of up-steps minus the number of down-steps before position $i$.
Matching statistic: St000443
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000443: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000443: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+,+] => [1,2] => [1,1]
=> [1,1,0,0]
=> 2 = 0 + 2
[-,+] => [1,2] => [1,1]
=> [1,1,0,0]
=> 2 = 0 + 2
[+,-] => [1,2] => [1,1]
=> [1,1,0,0]
=> 2 = 0 + 2
[-,-] => [1,2] => [1,1]
=> [1,1,0,0]
=> 2 = 0 + 2
[+,+,+] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 2 = 0 + 2
[-,+,+] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 2 = 0 + 2
[+,-,+] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 2 = 0 + 2
[+,+,-] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 2 = 0 + 2
[-,-,+] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 2 = 0 + 2
[-,+,-] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 2 = 0 + 2
[+,-,-] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 2 = 0 + 2
[-,-,-] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 2 = 0 + 2
[+,3,2] => [1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> 2 = 0 + 2
[-,3,2] => [1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> 2 = 0 + 2
[2,1,+] => [2,1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 2 = 0 + 2
[2,1,-] => [2,1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 2 = 0 + 2
[3,+,1] => [3,2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 2 = 0 + 2
[3,-,1] => [3,2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 2 = 0 + 2
[+,+,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[-,+,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[+,-,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[+,+,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[+,+,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[-,-,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[-,+,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[-,+,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[+,-,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[+,-,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[+,+,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[-,-,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[-,-,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[-,+,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[+,-,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[-,-,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[+,+,4,3] => [1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
[-,+,4,3] => [1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
[+,-,4,3] => [1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
[-,-,4,3] => [1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
[+,3,2,+] => [1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
[-,3,2,+] => [1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
[+,3,2,-] => [1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
[-,3,2,-] => [1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
[+,3,4,2] => [1,3,4,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 0 + 2
[-,3,4,2] => [1,3,4,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 0 + 2
[+,4,2,3] => [1,4,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 0 + 2
[-,4,2,3] => [1,4,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 0 + 2
[+,4,+,2] => [1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
[-,4,+,2] => [1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
[+,4,-,2] => [1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
[-,4,-,2] => [1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
Description
The number of long tunnels of a Dyck path.
A long tunnel of a Dyck path is a longest sequence of consecutive usual tunnels, i.e., a longest sequence of tunnels where the end point of one is the starting point of the next. See [1] for the definition of tunnels.
Matching statistic: St000444
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000444: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000444: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+,+] => [1,2] => [1,1]
=> [1,1,0,0]
=> 2 = 0 + 2
[-,+] => [1,2] => [1,1]
=> [1,1,0,0]
=> 2 = 0 + 2
[+,-] => [1,2] => [1,1]
=> [1,1,0,0]
=> 2 = 0 + 2
[-,-] => [1,2] => [1,1]
=> [1,1,0,0]
=> 2 = 0 + 2
[+,+,+] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 2 = 0 + 2
[-,+,+] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 2 = 0 + 2
[+,-,+] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 2 = 0 + 2
[+,+,-] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 2 = 0 + 2
[-,-,+] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 2 = 0 + 2
[-,+,-] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 2 = 0 + 2
[+,-,-] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 2 = 0 + 2
[-,-,-] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 2 = 0 + 2
[+,3,2] => [1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> 2 = 0 + 2
[-,3,2] => [1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> 2 = 0 + 2
[2,1,+] => [2,1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 2 = 0 + 2
[2,1,-] => [2,1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 2 = 0 + 2
[3,+,1] => [3,2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 2 = 0 + 2
[3,-,1] => [3,2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 2 = 0 + 2
[+,+,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[-,+,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[+,-,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[+,+,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[+,+,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[-,-,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[-,+,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[-,+,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[+,-,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[+,-,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[+,+,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[-,-,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[-,-,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[-,+,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[+,-,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[-,-,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[+,+,4,3] => [1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
[-,+,4,3] => [1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
[+,-,4,3] => [1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
[-,-,4,3] => [1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
[+,3,2,+] => [1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
[-,3,2,+] => [1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
[+,3,2,-] => [1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
[-,3,2,-] => [1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
[+,3,4,2] => [1,3,4,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 0 + 2
[-,3,4,2] => [1,3,4,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 0 + 2
[+,4,2,3] => [1,4,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 0 + 2
[-,4,2,3] => [1,4,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 0 + 2
[+,4,+,2] => [1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
[-,4,+,2] => [1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
[+,4,-,2] => [1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
[-,4,-,2] => [1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
Description
The length of the maximal rise of a Dyck path.
Matching statistic: St001007
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001007: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001007: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+,+] => [1,2] => [1,1]
=> [1,1,0,0]
=> 2 = 0 + 2
[-,+] => [1,2] => [1,1]
=> [1,1,0,0]
=> 2 = 0 + 2
[+,-] => [1,2] => [1,1]
=> [1,1,0,0]
=> 2 = 0 + 2
[-,-] => [1,2] => [1,1]
=> [1,1,0,0]
=> 2 = 0 + 2
[+,+,+] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 2 = 0 + 2
[-,+,+] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 2 = 0 + 2
[+,-,+] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 2 = 0 + 2
[+,+,-] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 2 = 0 + 2
[-,-,+] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 2 = 0 + 2
[-,+,-] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 2 = 0 + 2
[+,-,-] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 2 = 0 + 2
[-,-,-] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 2 = 0 + 2
[+,3,2] => [1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> 2 = 0 + 2
[-,3,2] => [1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> 2 = 0 + 2
[2,1,+] => [2,1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 2 = 0 + 2
[2,1,-] => [2,1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 2 = 0 + 2
[3,+,1] => [3,2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 2 = 0 + 2
[3,-,1] => [3,2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 2 = 0 + 2
[+,+,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[-,+,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[+,-,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[+,+,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[+,+,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[-,-,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[-,+,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[-,+,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[+,-,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[+,-,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[+,+,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[-,-,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[-,-,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[-,+,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[+,-,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[-,-,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[+,+,4,3] => [1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
[-,+,4,3] => [1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
[+,-,4,3] => [1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
[-,-,4,3] => [1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
[+,3,2,+] => [1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
[-,3,2,+] => [1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
[+,3,2,-] => [1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
[-,3,2,-] => [1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
[+,3,4,2] => [1,3,4,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 0 + 2
[-,3,4,2] => [1,3,4,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 0 + 2
[+,4,2,3] => [1,4,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 0 + 2
[-,4,2,3] => [1,4,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 0 + 2
[+,4,+,2] => [1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
[-,4,+,2] => [1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
[+,4,-,2] => [1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
[-,4,-,2] => [1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
Description
Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001187
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001187: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001187: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+,+] => [1,2] => [1,1]
=> [1,1,0,0]
=> 2 = 0 + 2
[-,+] => [1,2] => [1,1]
=> [1,1,0,0]
=> 2 = 0 + 2
[+,-] => [1,2] => [1,1]
=> [1,1,0,0]
=> 2 = 0 + 2
[-,-] => [1,2] => [1,1]
=> [1,1,0,0]
=> 2 = 0 + 2
[+,+,+] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 2 = 0 + 2
[-,+,+] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 2 = 0 + 2
[+,-,+] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 2 = 0 + 2
[+,+,-] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 2 = 0 + 2
[-,-,+] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 2 = 0 + 2
[-,+,-] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 2 = 0 + 2
[+,-,-] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 2 = 0 + 2
[-,-,-] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 2 = 0 + 2
[+,3,2] => [1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> 2 = 0 + 2
[-,3,2] => [1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> 2 = 0 + 2
[2,1,+] => [2,1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 2 = 0 + 2
[2,1,-] => [2,1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 2 = 0 + 2
[3,+,1] => [3,2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 2 = 0 + 2
[3,-,1] => [3,2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 2 = 0 + 2
[+,+,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[-,+,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[+,-,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[+,+,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[+,+,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[-,-,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[-,+,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[-,+,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[+,-,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[+,-,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[+,+,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[-,-,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[-,-,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[-,+,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[+,-,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[-,-,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 0 + 2
[+,+,4,3] => [1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
[-,+,4,3] => [1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
[+,-,4,3] => [1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
[-,-,4,3] => [1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
[+,3,2,+] => [1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
[-,3,2,+] => [1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
[+,3,2,-] => [1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
[-,3,2,-] => [1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
[+,3,4,2] => [1,3,4,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 0 + 2
[-,3,4,2] => [1,3,4,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 0 + 2
[+,4,2,3] => [1,4,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 0 + 2
[-,4,2,3] => [1,4,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 0 + 2
[+,4,+,2] => [1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
[-,4,+,2] => [1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
[+,4,-,2] => [1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
[-,4,-,2] => [1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
Description
The number of simple modules with grade at least one in the corresponding Nakayama algebra.
The following 9 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001964The interval resolution global dimension of a poset. St000260The radius of a connected graph. St000456The monochromatic index of a connected graph. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. 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.
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