Processing math: 100%

Your data matches 47 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000225
Mp00253: Decorated permutations permutationPermutations
Mp00060: Permutations Robinson-Schensted tableau shapeInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
St000225: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[3,+,1] => [3,2,1] => [1,1,1]
=> [1,1]
=> 0
[3,-,1] => [3,2,1] => [1,1,1]
=> [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
[-,4,-,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[2,1,4,3] => [2,1,4,3] => [2,2]
=> [2]
=> 0
[2,4,1,3] => [2,4,1,3] => [2,2]
=> [2]
=> 0
[2,4,+,1] => [2,4,3,1] => [2,1,1]
=> [1,1]
=> 0
[2,4,-,1] => [2,4,3,1] => [2,1,1]
=> [1,1]
=> 0
[3,1,4,2] => [3,1,4,2] => [2,2]
=> [2]
=> 0
[3,+,1,+] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 0
[3,-,1,+] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 0
[3,+,1,-] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 0
[3,-,1,-] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 0
[3,+,4,1] => [3,2,4,1] => [2,1,1]
=> [1,1]
=> 0
[3,-,4,1] => [3,2,4,1] => [2,1,1]
=> [1,1]
=> 0
[3,4,1,2] => [3,4,1,2] => [2,2]
=> [2]
=> 0
[3,4,2,1] => [3,4,2,1] => [2,1,1]
=> [1,1]
=> 0
[4,1,+,2] => [4,1,3,2] => [2,1,1]
=> [1,1]
=> 0
[4,1,-,2] => [4,1,3,2] => [2,1,1]
=> [1,1]
=> 0
[4,+,1,3] => [4,2,1,3] => [2,1,1]
=> [1,1]
=> 0
[4,-,1,3] => [4,2,1,3] => [2,1,1]
=> [1,1]
=> 0
[4,+,+,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 0
[4,-,+,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 0
[4,+,-,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 0
[4,-,-,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 0
[4,3,1,2] => [4,3,1,2] => [2,1,1]
=> [1,1]
=> 0
[4,3,2,1] => [4,3,2,1] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,+,5,+,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 0
[-,+,5,+,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 0
[+,-,5,+,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 0
[+,+,5,-,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 0
[-,-,5,+,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 0
[-,+,5,-,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 0
[+,-,5,-,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 0
[-,-,5,-,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 0
[+,3,2,5,4] => [1,3,2,5,4] => [3,2]
=> [2]
=> 0
[-,3,2,5,4] => [1,3,2,5,4] => [3,2]
=> [2]
=> 0
[+,3,5,2,4] => [1,3,5,2,4] => [3,2]
=> [2]
=> 0
[-,3,5,2,4] => [1,3,5,2,4] => [3,2]
=> [2]
=> 0
[+,3,5,+,2] => [1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 0
[-,3,5,+,2] => [1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 0
[+,3,5,-,2] => [1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 0
[-,3,5,-,2] => [1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 0
[+,4,2,5,3] => [1,4,2,5,3] => [3,2]
=> [2]
=> 0
[-,4,2,5,3] => [1,4,2,5,3] => [3,2]
=> [2]
=> 0
[+,4,+,2,+] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 0
[-,4,+,2,+] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 0
[+,4,-,2,+] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 0
Description
Difference between largest and smallest parts in a partition.
Matching statistic: St000386
Mp00253: Decorated permutations permutationPermutations
Mp00060: Permutations Robinson-Schensted tableau shapeInteger partitions
Mp00230: Integer partitions parallelogram polyominoDyck paths
St000386: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[3,+,1] => [3,2,1] => [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[3,-,1] => [3,2,1] => [1,1,1]
=> [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
[-,4,-,2] => [1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[2,1,4,3] => [2,1,4,3] => [2,2]
=> [1,1,1,0,0,0]
=> 0
[2,4,1,3] => [2,4,1,3] => [2,2]
=> [1,1,1,0,0,0]
=> 0
[2,4,+,1] => [2,4,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[2,4,-,1] => [2,4,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[3,1,4,2] => [3,1,4,2] => [2,2]
=> [1,1,1,0,0,0]
=> 0
[3,+,1,+] => [3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[3,-,1,+] => [3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[3,+,1,-] => [3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[3,-,1,-] => [3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[3,+,4,1] => [3,2,4,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[3,-,4,1] => [3,2,4,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[3,4,1,2] => [3,4,1,2] => [2,2]
=> [1,1,1,0,0,0]
=> 0
[3,4,2,1] => [3,4,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[4,1,+,2] => [4,1,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[4,1,-,2] => [4,1,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[4,+,1,3] => [4,2,1,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[4,-,1,3] => [4,2,1,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[4,+,+,1] => [4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[4,-,+,1] => [4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[4,+,-,1] => [4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[4,-,-,1] => [4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[4,3,1,2] => [4,3,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[4,3,2,1] => [4,3,2,1] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[+,+,5,+,3] => [1,2,5,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[-,+,5,+,3] => [1,2,5,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[+,-,5,+,3] => [1,2,5,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[+,+,5,-,3] => [1,2,5,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[-,-,5,+,3] => [1,2,5,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[-,+,5,-,3] => [1,2,5,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[+,-,5,-,3] => [1,2,5,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[-,-,5,-,3] => [1,2,5,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[+,3,2,5,4] => [1,3,2,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 0
[-,3,2,5,4] => [1,3,2,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 0
[+,3,5,2,4] => [1,3,5,2,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 0
[-,3,5,2,4] => [1,3,5,2,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 0
[+,3,5,+,2] => [1,3,5,4,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[-,3,5,+,2] => [1,3,5,4,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[+,3,5,-,2] => [1,3,5,4,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[-,3,5,-,2] => [1,3,5,4,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[+,4,2,5,3] => [1,4,2,5,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 0
[-,4,2,5,3] => [1,4,2,5,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 0
[+,4,+,2,+] => [1,4,3,2,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[-,4,+,2,+] => [1,4,3,2,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[+,4,-,2,+] => [1,4,3,2,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
Description
The number of factors DDU in a Dyck path.
Matching statistic: St000940
Mp00253: Decorated permutations permutationPermutations
Mp00060: Permutations Robinson-Schensted tableau shapeInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
St000940: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[3,+,1] => [3,2,1] => [1,1,1]
=> [1,1]
=> 0
[3,-,1] => [3,2,1] => [1,1,1]
=> [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
[-,4,-,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[2,1,4,3] => [2,1,4,3] => [2,2]
=> [2]
=> 0
[2,4,1,3] => [2,4,1,3] => [2,2]
=> [2]
=> 0
[2,4,+,1] => [2,4,3,1] => [2,1,1]
=> [1,1]
=> 0
[2,4,-,1] => [2,4,3,1] => [2,1,1]
=> [1,1]
=> 0
[3,1,4,2] => [3,1,4,2] => [2,2]
=> [2]
=> 0
[3,+,1,+] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 0
[3,-,1,+] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 0
[3,+,1,-] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 0
[3,-,1,-] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 0
[3,+,4,1] => [3,2,4,1] => [2,1,1]
=> [1,1]
=> 0
[3,-,4,1] => [3,2,4,1] => [2,1,1]
=> [1,1]
=> 0
[3,4,1,2] => [3,4,1,2] => [2,2]
=> [2]
=> 0
[3,4,2,1] => [3,4,2,1] => [2,1,1]
=> [1,1]
=> 0
[4,1,+,2] => [4,1,3,2] => [2,1,1]
=> [1,1]
=> 0
[4,1,-,2] => [4,1,3,2] => [2,1,1]
=> [1,1]
=> 0
[4,+,1,3] => [4,2,1,3] => [2,1,1]
=> [1,1]
=> 0
[4,-,1,3] => [4,2,1,3] => [2,1,1]
=> [1,1]
=> 0
[4,+,+,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 0
[4,-,+,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 0
[4,+,-,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 0
[4,-,-,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 0
[4,3,1,2] => [4,3,1,2] => [2,1,1]
=> [1,1]
=> 0
[4,3,2,1] => [4,3,2,1] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,+,5,+,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 0
[-,+,5,+,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 0
[+,-,5,+,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 0
[+,+,5,-,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 0
[-,-,5,+,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 0
[-,+,5,-,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 0
[+,-,5,-,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 0
[-,-,5,-,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 0
[+,3,2,5,4] => [1,3,2,5,4] => [3,2]
=> [2]
=> 0
[-,3,2,5,4] => [1,3,2,5,4] => [3,2]
=> [2]
=> 0
[+,3,5,2,4] => [1,3,5,2,4] => [3,2]
=> [2]
=> 0
[-,3,5,2,4] => [1,3,5,2,4] => [3,2]
=> [2]
=> 0
[+,3,5,+,2] => [1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 0
[-,3,5,+,2] => [1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 0
[+,3,5,-,2] => [1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 0
[-,3,5,-,2] => [1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 0
[+,4,2,5,3] => [1,4,2,5,3] => [3,2]
=> [2]
=> 0
[-,4,2,5,3] => [1,4,2,5,3] => [3,2]
=> [2]
=> 0
[+,4,+,2,+] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 0
[-,4,+,2,+] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 0
[+,4,-,2,+] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 0
Description
The number of characters of the symmetric group whose value on the partition is zero. The maximal value for any given size is recorded in [2].
Matching statistic: St001124
Mp00253: Decorated permutations permutationPermutations
Mp00060: Permutations Robinson-Schensted tableau shapeInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
St001124: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[3,+,1] => [3,2,1] => [1,1,1]
=> [1,1]
=> 0
[3,-,1] => [3,2,1] => [1,1,1]
=> [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
[-,4,-,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[2,1,4,3] => [2,1,4,3] => [2,2]
=> [2]
=> 0
[2,4,1,3] => [2,4,1,3] => [2,2]
=> [2]
=> 0
[2,4,+,1] => [2,4,3,1] => [2,1,1]
=> [1,1]
=> 0
[2,4,-,1] => [2,4,3,1] => [2,1,1]
=> [1,1]
=> 0
[3,1,4,2] => [3,1,4,2] => [2,2]
=> [2]
=> 0
[3,+,1,+] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 0
[3,-,1,+] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 0
[3,+,1,-] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 0
[3,-,1,-] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 0
[3,+,4,1] => [3,2,4,1] => [2,1,1]
=> [1,1]
=> 0
[3,-,4,1] => [3,2,4,1] => [2,1,1]
=> [1,1]
=> 0
[3,4,1,2] => [3,4,1,2] => [2,2]
=> [2]
=> 0
[3,4,2,1] => [3,4,2,1] => [2,1,1]
=> [1,1]
=> 0
[4,1,+,2] => [4,1,3,2] => [2,1,1]
=> [1,1]
=> 0
[4,1,-,2] => [4,1,3,2] => [2,1,1]
=> [1,1]
=> 0
[4,+,1,3] => [4,2,1,3] => [2,1,1]
=> [1,1]
=> 0
[4,-,1,3] => [4,2,1,3] => [2,1,1]
=> [1,1]
=> 0
[4,+,+,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 0
[4,-,+,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 0
[4,+,-,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 0
[4,-,-,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 0
[4,3,1,2] => [4,3,1,2] => [2,1,1]
=> [1,1]
=> 0
[4,3,2,1] => [4,3,2,1] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,+,5,+,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 0
[-,+,5,+,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 0
[+,-,5,+,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 0
[+,+,5,-,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 0
[-,-,5,+,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 0
[-,+,5,-,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 0
[+,-,5,-,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 0
[-,-,5,-,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 0
[+,3,2,5,4] => [1,3,2,5,4] => [3,2]
=> [2]
=> 0
[-,3,2,5,4] => [1,3,2,5,4] => [3,2]
=> [2]
=> 0
[+,3,5,2,4] => [1,3,5,2,4] => [3,2]
=> [2]
=> 0
[-,3,5,2,4] => [1,3,5,2,4] => [3,2]
=> [2]
=> 0
[+,3,5,+,2] => [1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 0
[-,3,5,+,2] => [1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 0
[+,3,5,-,2] => [1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 0
[-,3,5,-,2] => [1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 0
[+,4,2,5,3] => [1,4,2,5,3] => [3,2]
=> [2]
=> 0
[-,4,2,5,3] => [1,4,2,5,3] => [3,2]
=> [2]
=> 0
[+,4,+,2,+] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 0
[-,4,+,2,+] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 0
[+,4,-,2,+] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 0
Description
The multiplicity of the standard representation in the Kronecker square corresponding to a partition. The Kronecker coefficient is the multiplicity gλμ,ν of the Specht module Sλ in SμSν: SμSν=λgλμ,νSλ This statistic records the Kronecker coefficient g(n1)1λ,λ, for λn>1. For n1 the statistic is undefined. It follows from [3, Prop.4.1] (or, slightly easier from [3, Thm.4.2]) that this is one less than [[St000159]], the number of distinct parts of the partition.
Matching statistic: St001588
Mp00253: Decorated permutations permutationPermutations
Mp00060: Permutations Robinson-Schensted tableau shapeInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
St001588: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[3,+,1] => [3,2,1] => [1,1,1]
=> [1,1]
=> 0
[3,-,1] => [3,2,1] => [1,1,1]
=> [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
[-,4,-,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[2,1,4,3] => [2,1,4,3] => [2,2]
=> [2]
=> 0
[2,4,1,3] => [2,4,1,3] => [2,2]
=> [2]
=> 0
[2,4,+,1] => [2,4,3,1] => [2,1,1]
=> [1,1]
=> 0
[2,4,-,1] => [2,4,3,1] => [2,1,1]
=> [1,1]
=> 0
[3,1,4,2] => [3,1,4,2] => [2,2]
=> [2]
=> 0
[3,+,1,+] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 0
[3,-,1,+] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 0
[3,+,1,-] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 0
[3,-,1,-] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 0
[3,+,4,1] => [3,2,4,1] => [2,1,1]
=> [1,1]
=> 0
[3,-,4,1] => [3,2,4,1] => [2,1,1]
=> [1,1]
=> 0
[3,4,1,2] => [3,4,1,2] => [2,2]
=> [2]
=> 0
[3,4,2,1] => [3,4,2,1] => [2,1,1]
=> [1,1]
=> 0
[4,1,+,2] => [4,1,3,2] => [2,1,1]
=> [1,1]
=> 0
[4,1,-,2] => [4,1,3,2] => [2,1,1]
=> [1,1]
=> 0
[4,+,1,3] => [4,2,1,3] => [2,1,1]
=> [1,1]
=> 0
[4,-,1,3] => [4,2,1,3] => [2,1,1]
=> [1,1]
=> 0
[4,+,+,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 0
[4,-,+,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 0
[4,+,-,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 0
[4,-,-,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 0
[4,3,1,2] => [4,3,1,2] => [2,1,1]
=> [1,1]
=> 0
[4,3,2,1] => [4,3,2,1] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,+,5,+,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 0
[-,+,5,+,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 0
[+,-,5,+,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 0
[+,+,5,-,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 0
[-,-,5,+,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 0
[-,+,5,-,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 0
[+,-,5,-,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 0
[-,-,5,-,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 0
[+,3,2,5,4] => [1,3,2,5,4] => [3,2]
=> [2]
=> 0
[-,3,2,5,4] => [1,3,2,5,4] => [3,2]
=> [2]
=> 0
[+,3,5,2,4] => [1,3,5,2,4] => [3,2]
=> [2]
=> 0
[-,3,5,2,4] => [1,3,5,2,4] => [3,2]
=> [2]
=> 0
[+,3,5,+,2] => [1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 0
[-,3,5,+,2] => [1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 0
[+,3,5,-,2] => [1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 0
[-,3,5,-,2] => [1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 0
[+,4,2,5,3] => [1,4,2,5,3] => [3,2]
=> [2]
=> 0
[-,4,2,5,3] => [1,4,2,5,3] => [3,2]
=> [2]
=> 0
[+,4,+,2,+] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 0
[-,4,+,2,+] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 0
[+,4,-,2,+] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 0
Description
The number of distinct odd parts smaller than the largest even part in an integer partition.
Matching statistic: St000159
Mp00253: Decorated permutations permutationPermutations
Mp00060: Permutations Robinson-Schensted tableau shapeInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
St000159: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[3,+,1] => [3,2,1] => [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[3,-,1] => [3,2,1] => [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[+,4,+,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[-,4,+,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[+,4,-,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[-,4,-,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[2,1,4,3] => [2,1,4,3] => [2,2]
=> [2]
=> 1 = 0 + 1
[2,4,1,3] => [2,4,1,3] => [2,2]
=> [2]
=> 1 = 0 + 1
[2,4,+,1] => [2,4,3,1] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[2,4,-,1] => [2,4,3,1] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[3,1,4,2] => [3,1,4,2] => [2,2]
=> [2]
=> 1 = 0 + 1
[3,+,1,+] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[3,-,1,+] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[3,+,1,-] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[3,-,1,-] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[3,+,4,1] => [3,2,4,1] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[3,-,4,1] => [3,2,4,1] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[3,4,1,2] => [3,4,1,2] => [2,2]
=> [2]
=> 1 = 0 + 1
[3,4,2,1] => [3,4,2,1] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[4,1,+,2] => [4,1,3,2] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[4,1,-,2] => [4,1,3,2] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[4,+,1,3] => [4,2,1,3] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[4,-,1,3] => [4,2,1,3] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[4,+,+,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[4,-,+,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[4,+,-,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[4,-,-,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[4,3,1,2] => [4,3,1,2] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[4,3,2,1] => [4,3,2,1] => [1,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[+,+,5,+,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[-,+,5,+,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[+,-,5,+,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[+,+,5,-,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[-,-,5,+,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[-,+,5,-,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[+,-,5,-,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[-,-,5,-,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[+,3,2,5,4] => [1,3,2,5,4] => [3,2]
=> [2]
=> 1 = 0 + 1
[-,3,2,5,4] => [1,3,2,5,4] => [3,2]
=> [2]
=> 1 = 0 + 1
[+,3,5,2,4] => [1,3,5,2,4] => [3,2]
=> [2]
=> 1 = 0 + 1
[-,3,5,2,4] => [1,3,5,2,4] => [3,2]
=> [2]
=> 1 = 0 + 1
[+,3,5,+,2] => [1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[-,3,5,+,2] => [1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[+,3,5,-,2] => [1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[-,3,5,-,2] => [1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[+,4,2,5,3] => [1,4,2,5,3] => [3,2]
=> [2]
=> 1 = 0 + 1
[-,4,2,5,3] => [1,4,2,5,3] => [3,2]
=> [2]
=> 1 = 0 + 1
[+,4,+,2,+] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[-,4,+,2,+] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[+,4,-,2,+] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
Description
The number of distinct parts of the integer partition. This statistic is also the number of removeable cells of the partition, and the number of valleys of the Dyck path tracing the shape of the partition.
Matching statistic: St000340
Mp00253: Decorated permutations permutationPermutations
Mp00060: Permutations Robinson-Schensted tableau shapeInteger partitions
Mp00230: Integer partitions parallelogram polyominoDyck paths
St000340: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[3,+,1] => [3,2,1] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[3,-,1] => [3,2,1] => [1,1,1]
=> [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
[-,4,-,2] => [1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[2,1,4,3] => [2,1,4,3] => [2,2]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[2,4,1,3] => [2,4,1,3] => [2,2]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[2,4,+,1] => [2,4,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[2,4,-,1] => [2,4,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[3,1,4,2] => [3,1,4,2] => [2,2]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[3,+,1,+] => [3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[3,-,1,+] => [3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[3,+,1,-] => [3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[3,-,1,-] => [3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[3,+,4,1] => [3,2,4,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[3,-,4,1] => [3,2,4,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[3,4,1,2] => [3,4,1,2] => [2,2]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[3,4,2,1] => [3,4,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[4,1,+,2] => [4,1,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[4,1,-,2] => [4,1,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[4,+,1,3] => [4,2,1,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[4,-,1,3] => [4,2,1,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[4,+,+,1] => [4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[4,-,+,1] => [4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[4,+,-,1] => [4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[4,-,-,1] => [4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[4,3,1,2] => [4,3,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[4,3,2,1] => [4,3,2,1] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[+,+,5,+,3] => [1,2,5,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[-,+,5,+,3] => [1,2,5,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[+,-,5,+,3] => [1,2,5,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[+,+,5,-,3] => [1,2,5,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[-,-,5,+,3] => [1,2,5,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[-,+,5,-,3] => [1,2,5,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[+,-,5,-,3] => [1,2,5,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[-,-,5,-,3] => [1,2,5,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[+,3,2,5,4] => [1,3,2,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[-,3,2,5,4] => [1,3,2,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[+,3,5,2,4] => [1,3,5,2,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[-,3,5,2,4] => [1,3,5,2,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[+,3,5,+,2] => [1,3,5,4,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[-,3,5,+,2] => [1,3,5,4,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[+,3,5,-,2] => [1,3,5,4,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[-,3,5,-,2] => [1,3,5,4,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[+,4,2,5,3] => [1,4,2,5,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[-,4,2,5,3] => [1,4,2,5,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[+,4,+,2,+] => [1,4,3,2,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[-,4,+,2,+] => [1,4,3,2,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[+,4,-,2,+] => [1,4,3,2,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
Description
The number of non-final maximal constant sub-paths of length greater than one. This is the total number of occurrences of the patterns 110 and 001.
Matching statistic: St001128
Mp00253: Decorated permutations permutationPermutations
Mp00060: Permutations Robinson-Schensted tableau shapeInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
St001128: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[3,+,1] => [3,2,1] => [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[3,-,1] => [3,2,1] => [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[+,4,+,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[-,4,+,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[+,4,-,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[-,4,-,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[2,1,4,3] => [2,1,4,3] => [2,2]
=> [2]
=> 1 = 0 + 1
[2,4,1,3] => [2,4,1,3] => [2,2]
=> [2]
=> 1 = 0 + 1
[2,4,+,1] => [2,4,3,1] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[2,4,-,1] => [2,4,3,1] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[3,1,4,2] => [3,1,4,2] => [2,2]
=> [2]
=> 1 = 0 + 1
[3,+,1,+] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[3,-,1,+] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[3,+,1,-] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[3,-,1,-] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[3,+,4,1] => [3,2,4,1] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[3,-,4,1] => [3,2,4,1] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[3,4,1,2] => [3,4,1,2] => [2,2]
=> [2]
=> 1 = 0 + 1
[3,4,2,1] => [3,4,2,1] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[4,1,+,2] => [4,1,3,2] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[4,1,-,2] => [4,1,3,2] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[4,+,1,3] => [4,2,1,3] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[4,-,1,3] => [4,2,1,3] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[4,+,+,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[4,-,+,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[4,+,-,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[4,-,-,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[4,3,1,2] => [4,3,1,2] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[4,3,2,1] => [4,3,2,1] => [1,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[+,+,5,+,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[-,+,5,+,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[+,-,5,+,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[+,+,5,-,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[-,-,5,+,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[-,+,5,-,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[+,-,5,-,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[-,-,5,-,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[+,3,2,5,4] => [1,3,2,5,4] => [3,2]
=> [2]
=> 1 = 0 + 1
[-,3,2,5,4] => [1,3,2,5,4] => [3,2]
=> [2]
=> 1 = 0 + 1
[+,3,5,2,4] => [1,3,5,2,4] => [3,2]
=> [2]
=> 1 = 0 + 1
[-,3,5,2,4] => [1,3,5,2,4] => [3,2]
=> [2]
=> 1 = 0 + 1
[+,3,5,+,2] => [1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[-,3,5,+,2] => [1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[+,3,5,-,2] => [1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[-,3,5,-,2] => [1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[+,4,2,5,3] => [1,4,2,5,3] => [3,2]
=> [2]
=> 1 = 0 + 1
[-,4,2,5,3] => [1,4,2,5,3] => [3,2]
=> [2]
=> 1 = 0 + 1
[+,4,+,2,+] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[-,4,+,2,+] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[+,4,-,2,+] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
Description
The exponens consonantiae of a partition. This is the quotient of the least common multiple and the greatest common divior of the parts of the partiton. See [1, Caput sextum, §19-§22].
Matching statistic: St000318
Mp00253: Decorated permutations permutationPermutations
Mp00060: Permutations Robinson-Schensted tableau shapeInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
St000318: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[3,+,1] => [3,2,1] => [1,1,1]
=> [1,1]
=> 2 = 0 + 2
[3,-,1] => [3,2,1] => [1,1,1]
=> [1,1]
=> 2 = 0 + 2
[+,4,+,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 2 = 0 + 2
[-,4,+,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 2 = 0 + 2
[+,4,-,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 2 = 0 + 2
[-,4,-,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 2 = 0 + 2
[2,1,4,3] => [2,1,4,3] => [2,2]
=> [2]
=> 2 = 0 + 2
[2,4,1,3] => [2,4,1,3] => [2,2]
=> [2]
=> 2 = 0 + 2
[2,4,+,1] => [2,4,3,1] => [2,1,1]
=> [1,1]
=> 2 = 0 + 2
[2,4,-,1] => [2,4,3,1] => [2,1,1]
=> [1,1]
=> 2 = 0 + 2
[3,1,4,2] => [3,1,4,2] => [2,2]
=> [2]
=> 2 = 0 + 2
[3,+,1,+] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 2 = 0 + 2
[3,-,1,+] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 2 = 0 + 2
[3,+,1,-] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 2 = 0 + 2
[3,-,1,-] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 2 = 0 + 2
[3,+,4,1] => [3,2,4,1] => [2,1,1]
=> [1,1]
=> 2 = 0 + 2
[3,-,4,1] => [3,2,4,1] => [2,1,1]
=> [1,1]
=> 2 = 0 + 2
[3,4,1,2] => [3,4,1,2] => [2,2]
=> [2]
=> 2 = 0 + 2
[3,4,2,1] => [3,4,2,1] => [2,1,1]
=> [1,1]
=> 2 = 0 + 2
[4,1,+,2] => [4,1,3,2] => [2,1,1]
=> [1,1]
=> 2 = 0 + 2
[4,1,-,2] => [4,1,3,2] => [2,1,1]
=> [1,1]
=> 2 = 0 + 2
[4,+,1,3] => [4,2,1,3] => [2,1,1]
=> [1,1]
=> 2 = 0 + 2
[4,-,1,3] => [4,2,1,3] => [2,1,1]
=> [1,1]
=> 2 = 0 + 2
[4,+,+,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 2 = 0 + 2
[4,-,+,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 2 = 0 + 2
[4,+,-,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 2 = 0 + 2
[4,-,-,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 2 = 0 + 2
[4,3,1,2] => [4,3,1,2] => [2,1,1]
=> [1,1]
=> 2 = 0 + 2
[4,3,2,1] => [4,3,2,1] => [1,1,1,1]
=> [1,1,1]
=> 2 = 0 + 2
[+,+,5,+,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 2 = 0 + 2
[-,+,5,+,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 2 = 0 + 2
[+,-,5,+,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 2 = 0 + 2
[+,+,5,-,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 2 = 0 + 2
[-,-,5,+,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 2 = 0 + 2
[-,+,5,-,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 2 = 0 + 2
[+,-,5,-,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 2 = 0 + 2
[-,-,5,-,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 2 = 0 + 2
[+,3,2,5,4] => [1,3,2,5,4] => [3,2]
=> [2]
=> 2 = 0 + 2
[-,3,2,5,4] => [1,3,2,5,4] => [3,2]
=> [2]
=> 2 = 0 + 2
[+,3,5,2,4] => [1,3,5,2,4] => [3,2]
=> [2]
=> 2 = 0 + 2
[-,3,5,2,4] => [1,3,5,2,4] => [3,2]
=> [2]
=> 2 = 0 + 2
[+,3,5,+,2] => [1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 2 = 0 + 2
[-,3,5,+,2] => [1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 2 = 0 + 2
[+,3,5,-,2] => [1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 2 = 0 + 2
[-,3,5,-,2] => [1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 2 = 0 + 2
[+,4,2,5,3] => [1,4,2,5,3] => [3,2]
=> [2]
=> 2 = 0 + 2
[-,4,2,5,3] => [1,4,2,5,3] => [3,2]
=> [2]
=> 2 = 0 + 2
[+,4,+,2,+] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 2 = 0 + 2
[-,4,+,2,+] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 2 = 0 + 2
[+,4,-,2,+] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 2 = 0 + 2
Description
The number of addable cells of the Ferrers diagram of an integer partition.
Mp00253: Decorated permutations permutationPermutations
Mp00254: Permutations Inverse fireworks mapPermutations
Mp00170: Permutations to signed permutationSigned permutations
St001771: Signed permutations ⟶ ℤResult quality: 6% values known / values provided: 6%distinct values known / distinct values provided: 50%
Values
[3,+,1] => [3,2,1] => [3,2,1] => [3,2,1] => 0
[3,-,1] => [3,2,1] => [3,2,1] => [3,2,1] => 0
[+,4,+,2] => [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 0
[-,4,+,2] => [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 0
[+,4,-,2] => [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 0
[-,4,-,2] => [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 0
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[2,4,1,3] => [2,4,1,3] => [2,4,1,3] => [2,4,1,3] => 0
[2,4,+,1] => [2,4,3,1] => [1,4,3,2] => [1,4,3,2] => 0
[2,4,-,1] => [2,4,3,1] => [1,4,3,2] => [1,4,3,2] => 0
[3,1,4,2] => [3,1,4,2] => [2,1,4,3] => [2,1,4,3] => 0
[3,+,1,+] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 0
[3,-,1,+] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 0
[3,+,1,-] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 0
[3,-,1,-] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 0
[3,+,4,1] => [3,2,4,1] => [2,1,4,3] => [2,1,4,3] => 0
[3,-,4,1] => [3,2,4,1] => [2,1,4,3] => [2,1,4,3] => 0
[3,4,1,2] => [3,4,1,2] => [2,4,1,3] => [2,4,1,3] => 0
[3,4,2,1] => [3,4,2,1] => [1,4,3,2] => [1,4,3,2] => 0
[4,1,+,2] => [4,1,3,2] => [4,1,3,2] => [4,1,3,2] => 0
[4,1,-,2] => [4,1,3,2] => [4,1,3,2] => [4,1,3,2] => 0
[4,+,1,3] => [4,2,1,3] => [4,2,1,3] => [4,2,1,3] => 0
[4,-,1,3] => [4,2,1,3] => [4,2,1,3] => [4,2,1,3] => 0
[4,+,+,1] => [4,2,3,1] => [4,1,3,2] => [4,1,3,2] => 0
[4,-,+,1] => [4,2,3,1] => [4,1,3,2] => [4,1,3,2] => 0
[4,+,-,1] => [4,2,3,1] => [4,1,3,2] => [4,1,3,2] => 0
[4,-,-,1] => [4,2,3,1] => [4,1,3,2] => [4,1,3,2] => 0
[4,3,1,2] => [4,3,1,2] => [4,3,1,2] => [4,3,1,2] => 0
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0
[+,+,5,+,3] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[-,+,5,+,3] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[+,-,5,+,3] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[+,+,5,-,3] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[-,-,5,+,3] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[-,+,5,-,3] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[+,-,5,-,3] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[-,-,5,-,3] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[+,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[-,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[+,3,5,2,4] => [1,3,5,2,4] => [1,3,5,2,4] => [1,3,5,2,4] => 0
[-,3,5,2,4] => [1,3,5,2,4] => [1,3,5,2,4] => [1,3,5,2,4] => 0
[+,3,5,+,2] => [1,3,5,4,2] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[-,3,5,+,2] => [1,3,5,4,2] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[+,3,5,-,2] => [1,3,5,4,2] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[-,3,5,-,2] => [1,3,5,4,2] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[+,4,2,5,3] => [1,4,2,5,3] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[-,4,2,5,3] => [1,4,2,5,3] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[+,4,+,2,+] => [1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 0
[-,4,+,2,+] => [1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 0
[+,4,-,2,+] => [1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 0
[2,1,+,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
[2,1,-,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
[2,1,4,3,+] => [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 0
[2,1,4,3,-] => [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 0
[2,1,4,5,3] => [2,1,4,5,3] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
[2,1,5,3,4] => [2,1,5,3,4] => [2,1,5,3,4] => [2,1,5,3,4] => ? = 0
[2,1,5,+,3] => [2,1,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1
[2,1,5,-,3] => [2,1,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1
[2,4,1,3,+] => [2,4,1,3,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 0
[2,4,1,3,-] => [2,4,1,3,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 0
[2,5,1,3,4] => [2,5,1,3,4] => [2,5,1,3,4] => [2,5,1,3,4] => ? = 0
[2,5,1,+,3] => [2,5,1,4,3] => [2,5,1,4,3] => [2,5,1,4,3] => ? = 1
[2,5,1,-,3] => [2,5,1,4,3] => [2,5,1,4,3] => [2,5,1,4,3] => ? = 1
[2,5,4,1,3] => [2,5,4,1,3] => [2,5,4,1,3] => [2,5,4,1,3] => ? = 1
[3,1,2,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => ? = 0
[3,1,4,2,+] => [3,1,4,2,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 0
[3,1,4,2,-] => [3,1,4,2,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 0
[3,1,4,5,2] => [3,1,4,5,2] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
[3,1,5,2,4] => [3,1,5,2,4] => [3,1,5,2,4] => [3,1,5,2,4] => ? = 0
[3,1,5,+,2] => [3,1,5,4,2] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1
[3,1,5,-,2] => [3,1,5,4,2] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1
[3,+,1,+,+] => [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 0
[3,-,1,+,+] => [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 0
[3,+,1,-,+] => [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 0
[3,+,1,+,-] => [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 0
[3,-,1,-,+] => [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 0
[3,-,1,+,-] => [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 0
[3,+,1,-,-] => [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 0
[3,-,1,-,-] => [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 0
[3,+,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 1
[3,-,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 1
[3,+,4,1,+] => [3,2,4,1,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 0
[3,-,4,1,+] => [3,2,4,1,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 0
[3,+,4,1,-] => [3,2,4,1,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 0
[3,-,4,1,-] => [3,2,4,1,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 0
[3,+,4,5,1] => [3,2,4,5,1] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
[3,-,4,5,1] => [3,2,4,5,1] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
[3,+,5,1,4] => [3,2,5,1,4] => [3,2,5,1,4] => [3,2,5,1,4] => ? = 1
[3,-,5,1,4] => [3,2,5,1,4] => [3,2,5,1,4] => [3,2,5,1,4] => ? = 1
[3,+,5,+,1] => [3,2,5,4,1] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1
[3,-,5,+,1] => [3,2,5,4,1] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1
[3,+,5,-,1] => [3,2,5,4,1] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1
[3,-,5,-,1] => [3,2,5,4,1] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1
[3,4,1,2,+] => [3,4,1,2,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 0
[3,4,1,2,-] => [3,4,1,2,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 0
[3,5,1,2,4] => [3,5,1,2,4] => [3,5,1,2,4] => [3,5,1,2,4] => ? = 0
[3,5,1,+,2] => [3,5,1,4,2] => [2,5,1,4,3] => [2,5,1,4,3] => ? = 1
[3,5,1,-,2] => [3,5,1,4,2] => [2,5,1,4,3] => [2,5,1,4,3] => ? = 1
[3,5,2,1,4] => [3,5,2,1,4] => [3,5,2,1,4] => [3,5,2,1,4] => ? = 1
[3,5,2,+,1] => [3,5,2,4,1] => [2,5,1,4,3] => [2,5,1,4,3] => ? = 1
Description
The number of occurrences of the signed pattern 1-2 in a signed permutation. This is the number of pairs 1i<jn such that 0<π(i)<π(j).
The following 37 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St000068The number of minimal elements in a poset. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001863The number of weak excedances of a signed permutation. St001864The number of excedances of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001889The size of the connectivity set of a signed permutation. St001866The nesting alignments of a signed permutation. St001893The flag descent of a signed permutation. St001845The number of join irreducibles minus the rank of a lattice. St001862The number of crossings of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001896The number of right descents of a signed permutations. St001892The flag excedance statistic of a signed permutation. St001301The first Betti number of the order complex associated with the poset. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. 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. St000943The number of spots the most unlucky car had to go further in a parking function. St001946The number of descents in a parking function. St001371The length of the longest Yamanouchi prefix of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. St001851The number of Hecke atoms of a signed permutation. St001927Sparre Andersen's number of positives of a signed 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]/(xn). St001768The number of reduced words of a signed permutation. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000298The order dimension or Dushnik-Miller dimension of a poset. St000640The rank of the largest boolean interval in a poset. St000907The number of maximal antichains of minimal length in a poset. St001490The number of connected components of a skew partition.