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Your data matches 47 different statistics following compositions of up to 3 maps.
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Mp00166: Signed permutations even cycle typeInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
St000698: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2,3] => [1,1,1]
=> [1,1]
=> 1
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[1,2,3,-4] => [1,1,1]
=> [1,1]
=> 1
[1,2,-3,4] => [1,1,1]
=> [1,1]
=> 1
[1,-2,3,4] => [1,1,1]
=> [1,1]
=> 1
[-1,2,3,4] => [1,1,1]
=> [1,1]
=> 1
[1,2,4,3] => [2,1,1]
=> [1,1]
=> 1
[1,2,-4,-3] => [2,1,1]
=> [1,1]
=> 1
[1,3,2,4] => [2,1,1]
=> [1,1]
=> 1
[1,-3,-2,4] => [2,1,1]
=> [1,1]
=> 1
[1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[1,-4,3,-2] => [2,1,1]
=> [1,1]
=> 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> 1
[-2,-1,3,4] => [2,1,1]
=> [1,1]
=> 1
[2,1,4,3] => [2,2]
=> [2]
=> 1
[2,1,-4,-3] => [2,2]
=> [2]
=> 1
[-2,-1,4,3] => [2,2]
=> [2]
=> 1
[-2,-1,-4,-3] => [2,2]
=> [2]
=> 1
[3,2,1,4] => [2,1,1]
=> [1,1]
=> 1
[-3,2,-1,4] => [2,1,1]
=> [1,1]
=> 1
[3,4,1,2] => [2,2]
=> [2]
=> 1
[3,-4,1,-2] => [2,2]
=> [2]
=> 1
[-3,4,-1,2] => [2,2]
=> [2]
=> 1
[-3,-4,-1,-2] => [2,2]
=> [2]
=> 1
[4,2,3,1] => [2,1,1]
=> [1,1]
=> 1
[-4,2,3,-1] => [2,1,1]
=> [1,1]
=> 1
[4,3,2,1] => [2,2]
=> [2]
=> 1
[4,-3,-2,1] => [2,2]
=> [2]
=> 1
[-4,3,2,-1] => [2,2]
=> [2]
=> 1
[-4,-3,-2,-1] => [2,2]
=> [2]
=> 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 2
[1,2,3,4,-5] => [1,1,1,1]
=> [1,1,1]
=> 1
[1,2,3,-4,5] => [1,1,1,1]
=> [1,1,1]
=> 1
[1,2,3,-4,-5] => [1,1,1]
=> [1,1]
=> 1
[1,2,-3,4,5] => [1,1,1,1]
=> [1,1,1]
=> 1
[1,2,-3,4,-5] => [1,1,1]
=> [1,1]
=> 1
[1,2,-3,-4,5] => [1,1,1]
=> [1,1]
=> 1
[1,-2,3,4,5] => [1,1,1,1]
=> [1,1,1]
=> 1
[1,-2,3,4,-5] => [1,1,1]
=> [1,1]
=> 1
[1,-2,3,-4,5] => [1,1,1]
=> [1,1]
=> 1
[1,-2,-3,4,5] => [1,1,1]
=> [1,1]
=> 1
[-1,2,3,4,5] => [1,1,1,1]
=> [1,1,1]
=> 1
[-1,2,3,4,-5] => [1,1,1]
=> [1,1]
=> 1
[-1,2,3,-4,5] => [1,1,1]
=> [1,1]
=> 1
[-1,2,-3,4,5] => [1,1,1]
=> [1,1]
=> 1
[-1,-2,3,4,5] => [1,1,1]
=> [1,1]
=> 1
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,2,3,5,-4] => [1,1,1]
=> [1,1]
=> 1
[1,2,3,-5,4] => [1,1,1]
=> [1,1]
=> 1
[1,2,3,-5,-4] => [2,1,1,1]
=> [1,1,1]
=> 1
Description
The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. For any positive integer $k$, one associates a $k$-core to a partition by repeatedly removing all rim hooks of size $k$. This statistic counts the $2$-rim hooks that are removed in this process to obtain a $2$-core.
Matching statistic: St001232
Mp00166: Signed permutations even cycle typeInteger partitions
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00222: Dyck paths peaks-to-valleysDyck paths
St001232: Dyck paths ⟶ ℤResult quality: 41% values known / values provided: 41%distinct values known / distinct values provided: 50%
Values
[1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 1 + 1
[1,2,3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[1,2,-3,4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[1,-2,3,4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[-1,2,3,4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1 + 1
[1,2,-4,-3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1 + 1
[1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1 + 1
[1,-3,-2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1 + 1
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1 + 1
[1,-4,3,-2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1 + 1
[2,1,3,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1 + 1
[-2,-1,3,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1 + 1
[2,1,4,3] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[2,1,-4,-3] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[-2,-1,4,3] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[-2,-1,-4,-3] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1 + 1
[-3,2,-1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1 + 1
[3,4,1,2] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[3,-4,1,-2] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[-3,4,-1,2] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[-3,-4,-1,-2] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1 + 1
[-4,2,3,-1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1 + 1
[4,3,2,1] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[4,-3,-2,1] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[-4,3,2,-1] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[-4,-3,-2,-1] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 2 + 1
[1,2,3,4,-5] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 1 + 1
[1,2,3,-4,5] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 1 + 1
[1,2,3,-4,-5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[1,2,-3,4,5] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 1 + 1
[1,2,-3,4,-5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[1,2,-3,-4,5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[1,-2,3,4,5] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 1 + 1
[1,-2,3,4,-5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[1,-2,3,-4,5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[1,-2,-3,4,5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[-1,2,3,4,5] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 1 + 1
[-1,2,3,4,-5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[-1,2,3,-4,5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[-1,2,-3,4,5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[-1,-2,3,4,5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[1,2,3,5,4] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 1 + 1
[1,2,3,5,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[1,2,3,-5,4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[1,2,3,-5,-4] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 1 + 1
[1,2,-3,5,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1 + 1
[1,2,-3,-5,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1 + 1
[1,-2,3,5,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1 + 1
[1,-2,3,-5,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1 + 1
[-1,2,3,5,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1 + 1
[-1,2,3,-5,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1 + 1
[1,2,4,3,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 1 + 1
[1,2,4,3,-5] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1 + 1
[1,2,4,-3,5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[1,2,-4,3,5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[1,2,-4,-3,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 1 + 1
[1,2,-4,-3,-5] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1 + 1
[-1,3,2,5,4] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[-1,3,2,-5,-4] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[-1,-3,-2,5,4] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[-1,-3,-2,-5,-4] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[-1,4,5,2,3] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[-1,4,-5,2,-3] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[-1,-4,5,-2,3] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[-1,-4,-5,-2,-3] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[-1,5,4,3,2] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[-1,5,-4,-3,2] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[-1,-5,4,3,-2] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[-1,-5,-4,-3,-2] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[2,1,-3,5,4] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[2,1,-3,-5,-4] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[-2,-1,-3,5,4] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[-2,-1,-3,-5,-4] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[2,1,4,3,-5] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[2,1,-4,-3,-5] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[-2,-1,4,3,-5] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[-2,-1,-4,-3,-5] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[2,1,4,5,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[2,1,4,-5,-3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[2,1,-4,5,-3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[2,1,-4,-5,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[-2,-1,4,5,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[-2,-1,4,-5,-3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[-2,-1,-4,5,-3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[-2,-1,-4,-5,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[2,1,5,3,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[2,1,5,-3,-4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[2,1,-5,3,-4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[2,1,-5,-3,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[-2,-1,5,3,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[-2,-1,5,-3,-4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[-2,-1,-5,3,-4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[-2,-1,-5,-3,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[2,1,5,-4,3] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[2,1,-5,-4,-3] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000782
Mp00166: Signed permutations even cycle typeInteger partitions
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00146: Dyck paths to tunnel matchingPerfect matchings
St000782: Perfect matchings ⟶ ℤResult quality: 18% values known / values provided: 18%distinct values known / distinct values provided: 25%
Values
[1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 1
[1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> ? = 1
[1,2,3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 1
[1,2,-3,4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 1
[1,-2,3,4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 1
[-1,2,3,4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 1
[1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 1
[1,2,-4,-3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 1
[1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 1
[1,-3,-2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 1
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 1
[1,-4,3,-2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 1
[2,1,3,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 1
[-2,-1,3,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 1
[2,1,4,3] => [2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1
[2,1,-4,-3] => [2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1
[-2,-1,4,3] => [2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1
[-2,-1,-4,-3] => [2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1
[3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 1
[-3,2,-1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 1
[3,4,1,2] => [2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1
[3,-4,1,-2] => [2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1
[-3,4,-1,2] => [2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1
[-3,-4,-1,-2] => [2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1
[4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 1
[-4,2,3,-1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 1
[4,3,2,1] => [2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1
[4,-3,-2,1] => [2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1
[-4,3,2,-1] => [2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1
[-4,-3,-2,-1] => [2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9)]
=> ? = 2
[1,2,3,4,-5] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> ? = 1
[1,2,3,-4,5] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> ? = 1
[1,2,3,-4,-5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 1
[1,2,-3,4,5] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> ? = 1
[1,2,-3,4,-5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 1
[1,2,-3,-4,5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 1
[1,-2,3,4,5] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> ? = 1
[1,-2,3,4,-5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 1
[1,-2,3,-4,5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 1
[1,-2,-3,4,5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 1
[-1,2,3,4,5] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> ? = 1
[-1,2,3,4,-5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 1
[-1,2,3,-4,5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 1
[-1,2,-3,4,5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 1
[-1,-2,3,4,5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 1
[1,2,3,5,4] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> ? = 1
[1,2,3,5,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 1
[1,2,3,-5,4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 1
[1,2,3,-5,-4] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> ? = 1
[1,2,-3,5,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 1
[1,2,-3,-5,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 1
[1,-2,3,5,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 1
[1,-2,3,-5,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 1
[-1,2,3,5,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 1
[-1,2,3,-5,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 1
[1,2,4,3,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> ? = 1
[1,2,4,3,-5] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 1
[1,2,4,-3,5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 1
[1,2,-4,3,5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 1
[1,2,-4,-3,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> ? = 1
[1,2,-4,-3,-5] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 1
[1,-2,4,3,5] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 1
[1,-2,-4,-3,5] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 1
[-1,2,4,3,5] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 1
[-1,2,-4,-3,5] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 1
[1,2,4,5,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> ? = 1
[1,2,4,-5,-3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> ? = 1
[1,2,-4,5,-3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> ? = 1
[1,2,-4,-5,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> ? = 1
[1,2,5,3,4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> ? = 1
[1,2,5,-3,-4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> ? = 1
[1,2,-5,3,-4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> ? = 1
[1,2,-5,-3,4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> ? = 1
[1,2,5,4,3] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> ? = 1
[1,2,5,4,-3] => [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 1
[1,2,5,-4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 1
[1,2,-5,4,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 1
[1,2,-5,4,-3] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> ? = 1
[1,2,-5,-4,-3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 1
[1,-2,5,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 1
[1,-2,-5,4,-3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 1
[-1,2,5,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 1
[1,3,-2,4,5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 1
[1,-3,2,4,5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 1
[-1,3,2,5,4] => [2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1
[-1,3,2,-5,-4] => [2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1
[-1,-3,-2,5,4] => [2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1
[-1,-3,-2,-5,-4] => [2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1
[1,4,3,-2,5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 1
[1,-4,3,2,5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 1
[-1,4,5,2,3] => [2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1
[-1,4,-5,2,-3] => [2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1
[-1,-4,5,-2,3] => [2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1
[-1,-4,-5,-2,-3] => [2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1
[1,5,3,4,-2] => [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 1
[1,-5,3,4,2] => [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 1
[-1,5,4,3,2] => [2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1
[-1,5,-4,-3,2] => [2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1
[-1,-5,4,3,-2] => [2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1
Description
The indicator function of whether a given perfect matching is an L & P matching. An L&P matching is built inductively as follows: starting with either a single edge, or a hairpin $([1,3],[2,4])$, insert a noncrossing matching or inflate an edge by a ladder, that is, a number of nested edges. The number of L&P matchings is (see [thm. 1, 2]) $$\frac{1}{2} \cdot 4^{n} + \frac{1}{n + 1}{2 \, n \choose n} - {2 \, n + 1 \choose n} + {2 \, n - 1 \choose n - 1}$$
Mp00166: Signed permutations even cycle typeInteger partitions
Mp00044: Integer partitions conjugateInteger partitions
Mp00095: Integer partitions to binary wordBinary words
St001491: Binary words ⟶ ℤResult quality: 18% values known / values provided: 18%distinct values known / distinct values provided: 25%
Values
[1,2,3] => [1,1,1]
=> [3]
=> 1000 => 1
[1,2,3,4] => [1,1,1,1]
=> [4]
=> 10000 => ? = 1
[1,2,3,-4] => [1,1,1]
=> [3]
=> 1000 => 1
[1,2,-3,4] => [1,1,1]
=> [3]
=> 1000 => 1
[1,-2,3,4] => [1,1,1]
=> [3]
=> 1000 => 1
[-1,2,3,4] => [1,1,1]
=> [3]
=> 1000 => 1
[1,2,4,3] => [2,1,1]
=> [3,1]
=> 10010 => ? = 1
[1,2,-4,-3] => [2,1,1]
=> [3,1]
=> 10010 => ? = 1
[1,3,2,4] => [2,1,1]
=> [3,1]
=> 10010 => ? = 1
[1,-3,-2,4] => [2,1,1]
=> [3,1]
=> 10010 => ? = 1
[1,4,3,2] => [2,1,1]
=> [3,1]
=> 10010 => ? = 1
[1,-4,3,-2] => [2,1,1]
=> [3,1]
=> 10010 => ? = 1
[2,1,3,4] => [2,1,1]
=> [3,1]
=> 10010 => ? = 1
[-2,-1,3,4] => [2,1,1]
=> [3,1]
=> 10010 => ? = 1
[2,1,4,3] => [2,2]
=> [2,2]
=> 1100 => 1
[2,1,-4,-3] => [2,2]
=> [2,2]
=> 1100 => 1
[-2,-1,4,3] => [2,2]
=> [2,2]
=> 1100 => 1
[-2,-1,-4,-3] => [2,2]
=> [2,2]
=> 1100 => 1
[3,2,1,4] => [2,1,1]
=> [3,1]
=> 10010 => ? = 1
[-3,2,-1,4] => [2,1,1]
=> [3,1]
=> 10010 => ? = 1
[3,4,1,2] => [2,2]
=> [2,2]
=> 1100 => 1
[3,-4,1,-2] => [2,2]
=> [2,2]
=> 1100 => 1
[-3,4,-1,2] => [2,2]
=> [2,2]
=> 1100 => 1
[-3,-4,-1,-2] => [2,2]
=> [2,2]
=> 1100 => 1
[4,2,3,1] => [2,1,1]
=> [3,1]
=> 10010 => ? = 1
[-4,2,3,-1] => [2,1,1]
=> [3,1]
=> 10010 => ? = 1
[4,3,2,1] => [2,2]
=> [2,2]
=> 1100 => 1
[4,-3,-2,1] => [2,2]
=> [2,2]
=> 1100 => 1
[-4,3,2,-1] => [2,2]
=> [2,2]
=> 1100 => 1
[-4,-3,-2,-1] => [2,2]
=> [2,2]
=> 1100 => 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [5]
=> 100000 => ? = 2
[1,2,3,4,-5] => [1,1,1,1]
=> [4]
=> 10000 => ? = 1
[1,2,3,-4,5] => [1,1,1,1]
=> [4]
=> 10000 => ? = 1
[1,2,3,-4,-5] => [1,1,1]
=> [3]
=> 1000 => 1
[1,2,-3,4,5] => [1,1,1,1]
=> [4]
=> 10000 => ? = 1
[1,2,-3,4,-5] => [1,1,1]
=> [3]
=> 1000 => 1
[1,2,-3,-4,5] => [1,1,1]
=> [3]
=> 1000 => 1
[1,-2,3,4,5] => [1,1,1,1]
=> [4]
=> 10000 => ? = 1
[1,-2,3,4,-5] => [1,1,1]
=> [3]
=> 1000 => 1
[1,-2,3,-4,5] => [1,1,1]
=> [3]
=> 1000 => 1
[1,-2,-3,4,5] => [1,1,1]
=> [3]
=> 1000 => 1
[-1,2,3,4,5] => [1,1,1,1]
=> [4]
=> 10000 => ? = 1
[-1,2,3,4,-5] => [1,1,1]
=> [3]
=> 1000 => 1
[-1,2,3,-4,5] => [1,1,1]
=> [3]
=> 1000 => 1
[-1,2,-3,4,5] => [1,1,1]
=> [3]
=> 1000 => 1
[-1,-2,3,4,5] => [1,1,1]
=> [3]
=> 1000 => 1
[1,2,3,5,4] => [2,1,1,1]
=> [4,1]
=> 100010 => ? = 1
[1,2,3,5,-4] => [1,1,1]
=> [3]
=> 1000 => 1
[1,2,3,-5,4] => [1,1,1]
=> [3]
=> 1000 => 1
[1,2,3,-5,-4] => [2,1,1,1]
=> [4,1]
=> 100010 => ? = 1
[1,2,-3,5,4] => [2,1,1]
=> [3,1]
=> 10010 => ? = 1
[1,2,-3,-5,-4] => [2,1,1]
=> [3,1]
=> 10010 => ? = 1
[1,-2,3,5,4] => [2,1,1]
=> [3,1]
=> 10010 => ? = 1
[1,-2,3,-5,-4] => [2,1,1]
=> [3,1]
=> 10010 => ? = 1
[-1,2,3,5,4] => [2,1,1]
=> [3,1]
=> 10010 => ? = 1
[-1,2,3,-5,-4] => [2,1,1]
=> [3,1]
=> 10010 => ? = 1
[1,2,4,3,5] => [2,1,1,1]
=> [4,1]
=> 100010 => ? = 1
[1,2,4,3,-5] => [2,1,1]
=> [3,1]
=> 10010 => ? = 1
[1,2,4,-3,5] => [1,1,1]
=> [3]
=> 1000 => 1
[1,2,-4,3,5] => [1,1,1]
=> [3]
=> 1000 => 1
[1,2,-4,-3,5] => [2,1,1,1]
=> [4,1]
=> 100010 => ? = 1
[1,2,-4,-3,-5] => [2,1,1]
=> [3,1]
=> 10010 => ? = 1
[1,-2,4,3,5] => [2,1,1]
=> [3,1]
=> 10010 => ? = 1
[1,-2,-4,-3,5] => [2,1,1]
=> [3,1]
=> 10010 => ? = 1
[-1,2,4,3,5] => [2,1,1]
=> [3,1]
=> 10010 => ? = 1
[-1,2,-4,-3,5] => [2,1,1]
=> [3,1]
=> 10010 => ? = 1
[1,2,4,5,3] => [3,1,1]
=> [3,1,1]
=> 100110 => ? = 1
[1,2,4,-5,-3] => [3,1,1]
=> [3,1,1]
=> 100110 => ? = 1
[1,2,-4,5,-3] => [3,1,1]
=> [3,1,1]
=> 100110 => ? = 1
[1,2,-4,-5,3] => [3,1,1]
=> [3,1,1]
=> 100110 => ? = 1
[1,2,5,3,4] => [3,1,1]
=> [3,1,1]
=> 100110 => ? = 1
[1,2,5,-3,-4] => [3,1,1]
=> [3,1,1]
=> 100110 => ? = 1
[1,2,-5,3,-4] => [3,1,1]
=> [3,1,1]
=> 100110 => ? = 1
[1,2,-5,-3,4] => [3,1,1]
=> [3,1,1]
=> 100110 => ? = 1
[1,2,5,4,3] => [2,1,1,1]
=> [4,1]
=> 100010 => ? = 1
[1,2,5,4,-3] => [1,1,1]
=> [3]
=> 1000 => 1
[1,2,5,-4,3] => [2,1,1]
=> [3,1]
=> 10010 => ? = 1
[1,2,-5,4,3] => [1,1,1]
=> [3]
=> 1000 => 1
[1,2,-5,4,-3] => [2,1,1,1]
=> [4,1]
=> 100010 => ? = 1
[1,2,-5,-4,-3] => [2,1,1]
=> [3,1]
=> 10010 => ? = 1
[1,-2,5,4,3] => [2,1,1]
=> [3,1]
=> 10010 => ? = 1
[1,-2,-5,4,-3] => [2,1,1]
=> [3,1]
=> 10010 => ? = 1
[-1,2,5,4,3] => [2,1,1]
=> [3,1]
=> 10010 => ? = 1
[1,3,-2,4,5] => [1,1,1]
=> [3]
=> 1000 => 1
[1,-3,2,4,5] => [1,1,1]
=> [3]
=> 1000 => 1
[-1,3,2,5,4] => [2,2]
=> [2,2]
=> 1100 => 1
[-1,3,2,-5,-4] => [2,2]
=> [2,2]
=> 1100 => 1
[-1,-3,-2,5,4] => [2,2]
=> [2,2]
=> 1100 => 1
[-1,-3,-2,-5,-4] => [2,2]
=> [2,2]
=> 1100 => 1
[1,4,3,-2,5] => [1,1,1]
=> [3]
=> 1000 => 1
[1,-4,3,2,5] => [1,1,1]
=> [3]
=> 1000 => 1
[-1,4,5,2,3] => [2,2]
=> [2,2]
=> 1100 => 1
[-1,4,-5,2,-3] => [2,2]
=> [2,2]
=> 1100 => 1
[-1,-4,5,-2,3] => [2,2]
=> [2,2]
=> 1100 => 1
[-1,-4,-5,-2,-3] => [2,2]
=> [2,2]
=> 1100 => 1
[1,5,3,4,-2] => [1,1,1]
=> [3]
=> 1000 => 1
[1,-5,3,4,2] => [1,1,1]
=> [3]
=> 1000 => 1
[-1,5,4,3,2] => [2,2]
=> [2,2]
=> 1100 => 1
[-1,5,-4,-3,2] => [2,2]
=> [2,2]
=> 1100 => 1
[-1,-5,4,3,-2] => [2,2]
=> [2,2]
=> 1100 => 1
Description
The number of indecomposable projective-injective modules in the algebra corresponding to a subset. Let $A_n=K[x]/(x^n)$. We associate to a nonempty subset S of an (n-1)-set the module $M_S$, which is the direct sum of $A_n$-modules with indecomposable non-projective direct summands of dimension $i$ when $i$ is in $S$ (note that such modules have vector space dimension at most n-1). Then the corresponding algebra associated to S is the stable endomorphism ring of $M_S$. We decode the subset as a binary word so that for example the subset $S=\{1,3 \} $ of $\{1,2,3 \}$ is decoded as 101.
Matching statistic: St001207
Mp00166: Signed permutations even cycle typeInteger partitions
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00201: Dyck paths RingelPermutations
St001207: Permutations ⟶ ℤResult quality: 18% values known / values provided: 18%distinct values known / distinct values provided: 25%
Values
[1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 3 = 1 + 2
[1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => ? = 1 + 2
[1,2,3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 3 = 1 + 2
[1,2,-3,4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 3 = 1 + 2
[1,-2,3,4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 3 = 1 + 2
[-1,2,3,4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 3 = 1 + 2
[1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 1 + 2
[1,2,-4,-3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 1 + 2
[1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 1 + 2
[1,-3,-2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 1 + 2
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 1 + 2
[1,-4,3,-2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 1 + 2
[2,1,3,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 1 + 2
[-2,-1,3,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 1 + 2
[2,1,4,3] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 1 + 2
[2,1,-4,-3] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 1 + 2
[-2,-1,4,3] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 1 + 2
[-2,-1,-4,-3] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 1 + 2
[3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 1 + 2
[-3,2,-1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 1 + 2
[3,4,1,2] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 1 + 2
[3,-4,1,-2] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 1 + 2
[-3,4,-1,2] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 1 + 2
[-3,-4,-1,-2] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 1 + 2
[4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 1 + 2
[-4,2,3,-1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 1 + 2
[4,3,2,1] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 1 + 2
[4,-3,-2,1] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 1 + 2
[-4,3,2,-1] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 1 + 2
[-4,-3,-2,-1] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 1 + 2
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => ? = 2 + 2
[1,2,3,4,-5] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => ? = 1 + 2
[1,2,3,-4,5] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => ? = 1 + 2
[1,2,3,-4,-5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 3 = 1 + 2
[1,2,-3,4,5] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => ? = 1 + 2
[1,2,-3,4,-5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 3 = 1 + 2
[1,2,-3,-4,5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 3 = 1 + 2
[1,-2,3,4,5] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => ? = 1 + 2
[1,-2,3,4,-5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 3 = 1 + 2
[1,-2,3,-4,5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 3 = 1 + 2
[1,-2,-3,4,5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 3 = 1 + 2
[-1,2,3,4,5] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => ? = 1 + 2
[-1,2,3,4,-5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 3 = 1 + 2
[-1,2,3,-4,5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 3 = 1 + 2
[-1,2,-3,4,5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 3 = 1 + 2
[-1,-2,3,4,5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 3 = 1 + 2
[1,2,3,5,4] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ? = 1 + 2
[1,2,3,5,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 3 = 1 + 2
[1,2,3,-5,4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 3 = 1 + 2
[1,2,3,-5,-4] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ? = 1 + 2
[1,2,-3,5,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 1 + 2
[1,2,-3,-5,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 1 + 2
[1,-2,3,5,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 1 + 2
[1,-2,3,-5,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 1 + 2
[-1,2,3,5,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 1 + 2
[-1,2,3,-5,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 1 + 2
[1,2,4,3,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ? = 1 + 2
[1,2,4,3,-5] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 1 + 2
[1,2,4,-3,5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 3 = 1 + 2
[1,2,-4,3,5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 3 = 1 + 2
[1,2,-4,-3,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ? = 1 + 2
[1,2,-4,-3,-5] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 1 + 2
[1,-2,4,3,5] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 1 + 2
[1,-2,-4,-3,5] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 1 + 2
[-1,2,4,3,5] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 1 + 2
[-1,2,-4,-3,5] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 1 + 2
[1,2,4,5,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 1 + 2
[1,2,4,-5,-3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 1 + 2
[1,2,-4,5,-3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 1 + 2
[1,2,-4,-5,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 1 + 2
[1,2,5,3,4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 1 + 2
[1,2,5,-3,-4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 1 + 2
[1,2,-5,3,-4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 1 + 2
[1,2,-5,-3,4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 1 + 2
[1,2,5,4,3] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ? = 1 + 2
[1,2,5,4,-3] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 3 = 1 + 2
[1,2,5,-4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 1 + 2
[1,2,-5,4,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 3 = 1 + 2
[1,2,-5,4,-3] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ? = 1 + 2
[1,2,-5,-4,-3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 1 + 2
[1,-2,5,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 1 + 2
[1,-2,-5,4,-3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 1 + 2
[-1,2,5,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 1 + 2
[1,3,-2,4,5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 3 = 1 + 2
[1,-3,2,4,5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 3 = 1 + 2
[-1,3,2,5,4] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 1 + 2
[-1,3,2,-5,-4] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 1 + 2
[-1,-3,-2,5,4] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 1 + 2
[-1,-3,-2,-5,-4] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 1 + 2
[1,4,3,-2,5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 3 = 1 + 2
[1,-4,3,2,5] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 3 = 1 + 2
[-1,4,5,2,3] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 1 + 2
[-1,4,-5,2,-3] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 1 + 2
[-1,-4,5,-2,3] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 1 + 2
[-1,-4,-5,-2,-3] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 1 + 2
[1,5,3,4,-2] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 3 = 1 + 2
[1,-5,3,4,2] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 3 = 1 + 2
[-1,5,4,3,2] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 1 + 2
[-1,5,-4,-3,2] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 1 + 2
[-1,-5,4,3,-2] => [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 1 + 2
Description
The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
Mp00166: Signed permutations even cycle typeInteger partitions
Mp00042: Integer partitions initial tableauStandard tableaux
Mp00082: Standard tableaux to Gelfand-Tsetlin patternGelfand-Tsetlin patterns
St001713: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 6% values known / values provided: 6%distinct values known / distinct values provided: 25%
Values
[1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 1 - 1
[1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 1 - 1
[1,2,3,-4] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 1 - 1
[1,2,-3,4] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 1 - 1
[1,-2,3,4] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 1 - 1
[-1,2,3,4] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 1 - 1
[1,2,4,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 1 - 1
[1,2,-4,-3] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 1 - 1
[1,3,2,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 1 - 1
[1,-3,-2,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 1 - 1
[1,4,3,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 1 - 1
[1,-4,3,-2] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 1 - 1
[2,1,3,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 1 - 1
[-2,-1,3,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 1 - 1
[2,1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 1 - 1
[2,1,-4,-3] => [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 1 - 1
[-2,-1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 1 - 1
[-2,-1,-4,-3] => [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 1 - 1
[3,2,1,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 1 - 1
[-3,2,-1,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 1 - 1
[3,4,1,2] => [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 1 - 1
[3,-4,1,-2] => [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 1 - 1
[-3,4,-1,2] => [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 1 - 1
[-3,-4,-1,-2] => [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 1 - 1
[4,2,3,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 1 - 1
[-4,2,3,-1] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 1 - 1
[4,3,2,1] => [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 1 - 1
[4,-3,-2,1] => [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 1 - 1
[-4,3,2,-1] => [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 1 - 1
[-4,-3,-2,-1] => [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 1 - 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 2 - 1
[1,2,3,4,-5] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 1 - 1
[1,2,3,-4,5] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 1 - 1
[1,2,3,-4,-5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 1 - 1
[1,2,-3,4,5] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 1 - 1
[1,2,-3,4,-5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 1 - 1
[1,2,-3,-4,5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 1 - 1
[1,-2,3,4,5] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 1 - 1
[1,-2,3,4,-5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 1 - 1
[1,-2,3,-4,5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 1 - 1
[1,-2,-3,4,5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 1 - 1
[-1,2,3,4,5] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? = 1 - 1
[-1,2,3,4,-5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 1 - 1
[-1,2,3,-4,5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 1 - 1
[-1,2,-3,4,5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 1 - 1
[-1,-2,3,4,5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 1 - 1
[1,2,3,5,4] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[2,1,1,1,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 1 - 1
[1,2,3,5,-4] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 1 - 1
[1,2,3,-5,4] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 1 - 1
[1,2,3,-5,-4] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[2,1,1,1,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 1 - 1
[1,2,-3,5,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 1 - 1
[1,2,-3,-5,-4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 1 - 1
[1,-2,3,5,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 1 - 1
[1,-2,3,-5,-4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 1 - 1
[-1,2,3,5,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 1 - 1
[-1,2,3,-5,-4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 1 - 1
[1,2,4,3,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[2,1,1,1,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 1 - 1
[1,2,4,3,-5] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 1 - 1
[1,2,4,-3,5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 1 - 1
[1,2,-4,3,5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 1 - 1
[1,2,-4,-3,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[2,1,1,1,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 1 - 1
[1,2,-4,-3,-5] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 1 - 1
[1,-2,4,3,5] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 1 - 1
[1,-2,-4,-3,5] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 1 - 1
[-1,2,4,3,5] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 1 - 1
[-1,2,-4,-3,5] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 1 - 1
[1,2,4,5,3] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> ? = 1 - 1
[1,2,4,-5,-3] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> ? = 1 - 1
[1,2,-4,5,-3] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> ? = 1 - 1
[1,2,5,4,-3] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 1 - 1
[1,2,-5,4,3] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 1 - 1
[1,3,-2,4,5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 1 - 1
[1,-3,2,4,5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 1 - 1
[1,4,3,-2,5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 1 - 1
[1,-4,3,2,5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 1 - 1
[1,5,3,4,-2] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 1 - 1
[1,-5,3,4,2] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 1 - 1
[2,-1,3,4,5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 1 - 1
[-2,1,3,4,5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 1 - 1
[3,2,-1,4,5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 1 - 1
[-3,2,1,4,5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 1 - 1
[4,2,3,-1,5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 1 - 1
[-4,2,3,1,5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 1 - 1
[5,2,3,4,-1] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 1 - 1
[-5,2,3,4,1] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 1 - 1
[1,2,3,6,4,-5] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 1 - 1
[1,2,3,5,6,-4] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 1 - 1
[1,2,5,4,6,-3] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 1 - 1
[1,2,6,3,5,-4] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 1 - 1
[1,2,4,6,5,-3] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 1 - 1
[1,5,3,4,6,-2] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 1 - 1
[1,4,3,6,5,-2] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 1 - 1
[1,6,2,4,5,-3] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 1 - 1
[1,3,6,4,5,-2] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 1 - 1
[5,2,3,4,6,-1] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 1 - 1
[4,2,3,6,5,-1] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 1 - 1
[3,2,6,4,5,-1] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 1 - 1
[6,1,3,4,5,-2] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 1 - 1
[2,6,3,4,5,-1] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 1 - 1
[-3,-2,1,4,5,6] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0 = 1 - 1
Description
The difference of the first and last value in the first row of the Gelfand-Tsetlin pattern.
Mp00245: Signed permutations standardizePermutations
Mp00305: Permutations parking functionParking functions
St000188: Parking functions ⟶ ℤResult quality: 3% values known / values provided: 3%distinct values known / distinct values provided: 25%
Values
[1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,3,-4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,-3,4] => [1,2,4,3] => [1,2,4,3] => 0 = 1 - 1
[1,-2,3,4] => [1,4,2,3] => [1,4,2,3] => 0 = 1 - 1
[-1,2,3,4] => [4,1,2,3] => [4,1,2,3] => 0 = 1 - 1
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0 = 1 - 1
[1,2,-4,-3] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0 = 1 - 1
[1,-3,-2,4] => [1,3,4,2] => [1,3,4,2] => 0 = 1 - 1
[1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 0 = 1 - 1
[1,-4,3,-2] => [1,3,2,4] => [1,3,2,4] => 0 = 1 - 1
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0 = 1 - 1
[-2,-1,3,4] => [3,4,1,2] => [3,4,1,2] => 0 = 1 - 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0 = 1 - 1
[2,1,-4,-3] => [2,1,3,4] => [2,1,3,4] => 0 = 1 - 1
[-2,-1,4,3] => [3,4,2,1] => [3,4,2,1] => 0 = 1 - 1
[-2,-1,-4,-3] => [3,4,1,2] => [3,4,1,2] => 0 = 1 - 1
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 0 = 1 - 1
[-3,2,-1,4] => [3,1,4,2] => [3,1,4,2] => 0 = 1 - 1
[3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 0 = 1 - 1
[3,-4,1,-2] => [2,3,1,4] => [2,3,1,4] => 0 = 1 - 1
[-3,4,-1,2] => [3,2,4,1] => [3,2,4,1] => 0 = 1 - 1
[-3,-4,-1,-2] => [2,1,4,3] => [2,1,4,3] => 0 = 1 - 1
[4,2,3,1] => [4,2,3,1] => [4,2,3,1] => 0 = 1 - 1
[-4,2,3,-1] => [3,1,2,4] => [3,1,2,4] => 0 = 1 - 1
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
[4,-3,-2,1] => [2,3,4,1] => [2,3,4,1] => 0 = 1 - 1
[-4,3,2,-1] => [3,2,1,4] => [3,2,1,4] => 0 = 1 - 1
[-4,-3,-2,-1] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 2 - 1
[1,2,3,4,-5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 1 - 1
[1,2,3,-4,5] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 1 - 1
[1,2,3,-4,-5] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 1 - 1
[1,2,-3,4,5] => [1,2,5,3,4] => [1,2,5,3,4] => ? = 1 - 1
[1,2,-3,4,-5] => [1,2,5,3,4] => [1,2,5,3,4] => ? = 1 - 1
[1,2,-3,-4,5] => [1,2,5,4,3] => [1,2,5,4,3] => ? = 1 - 1
[1,-2,3,4,5] => [1,5,2,3,4] => [1,5,2,3,4] => ? = 1 - 1
[1,-2,3,4,-5] => [1,5,2,3,4] => [1,5,2,3,4] => ? = 1 - 1
[1,-2,3,-4,5] => [1,5,2,4,3] => [1,5,2,4,3] => ? = 1 - 1
[1,-2,-3,4,5] => [1,5,4,2,3] => [1,5,4,2,3] => ? = 1 - 1
[-1,2,3,4,5] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 1 - 1
[-1,2,3,4,-5] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 1 - 1
[-1,2,3,-4,5] => [5,1,2,4,3] => [5,1,2,4,3] => ? = 1 - 1
[-1,2,-3,4,5] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 1 - 1
[-1,-2,3,4,5] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 1 - 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 1 - 1
[1,2,3,5,-4] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 1 - 1
[1,2,3,-5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 1 - 1
[1,2,3,-5,-4] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 1 - 1
[1,2,-3,5,4] => [1,2,5,4,3] => [1,2,5,4,3] => ? = 1 - 1
[1,2,-3,-5,-4] => [1,2,5,3,4] => [1,2,5,3,4] => ? = 1 - 1
[1,-2,3,5,4] => [1,5,2,4,3] => [1,5,2,4,3] => ? = 1 - 1
[1,-2,3,-5,-4] => [1,5,2,3,4] => [1,5,2,3,4] => ? = 1 - 1
[-1,2,3,5,4] => [5,1,2,4,3] => [5,1,2,4,3] => ? = 1 - 1
[-1,2,3,-5,-4] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 1 - 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ? = 1 - 1
[1,2,4,3,-5] => [1,2,4,3,5] => [1,2,4,3,5] => ? = 1 - 1
[1,2,4,-3,5] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 1 - 1
[1,2,-4,3,5] => [1,2,5,3,4] => [1,2,5,3,4] => ? = 1 - 1
[1,2,-4,-3,5] => [1,2,4,5,3] => [1,2,4,5,3] => ? = 1 - 1
[1,2,-4,-3,-5] => [1,2,4,5,3] => [1,2,4,5,3] => ? = 1 - 1
[1,-2,4,3,5] => [1,5,3,2,4] => [1,5,3,2,4] => ? = 1 - 1
[1,-2,-4,-3,5] => [1,5,3,4,2] => [1,5,3,4,2] => ? = 1 - 1
[-1,2,4,3,5] => [5,1,3,2,4] => [5,1,3,2,4] => ? = 1 - 1
[-1,2,-4,-3,5] => [5,1,3,4,2] => [5,1,3,4,2] => ? = 1 - 1
[1,2,4,5,3] => [1,2,4,5,3] => [1,2,4,5,3] => ? = 1 - 1
[1,2,4,-5,-3] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 1 - 1
[1,2,-4,5,-3] => [1,2,4,3,5] => [1,2,4,3,5] => ? = 1 - 1
[1,2,-4,-5,3] => [1,2,5,4,3] => [1,2,5,4,3] => ? = 1 - 1
[1,2,5,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => ? = 1 - 1
[1,2,5,-3,-4] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 1 - 1
[1,2,-5,3,-4] => [1,2,4,3,5] => [1,2,4,3,5] => ? = 1 - 1
[1,2,-5,-3,4] => [1,2,4,5,3] => [1,2,4,5,3] => ? = 1 - 1
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => ? = 1 - 1
[1,2,5,4,-3] => [1,2,4,3,5] => [1,2,4,3,5] => ? = 1 - 1
[1,2,5,-4,3] => [1,2,4,5,3] => [1,2,4,5,3] => ? = 1 - 1
[1,2,-5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => ? = 1 - 1
[1,2,-5,4,-3] => [1,2,4,3,5] => [1,2,4,3,5] => ? = 1 - 1
[1,2,-5,-4,-3] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 1 - 1
Description
The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. The area Dyck path corresponding to a parking function for a parking function $p_1,\ldots,p_n$ of length $n$, is given by $\binom{n+1}{2} - \sum_i p_i$. The total displacement of a parking function $ p \in PF_n $ is defined by $$ \operatorname{disp}(p) := \sum_{i=1}^{n} d_i, $$ where the displacement vector $ d := (d_1, d_2, \ldots, d_n) $ and $ d_i := \pi^{-1}(i) - p_i $ for all $ i \in [n] $, such that each $ d_i $ is the positive difference between the actual spot a car parks and its preferred spot.
Mp00245: Signed permutations standardizePermutations
Mp00305: Permutations parking functionParking functions
St000195: Parking functions ⟶ ℤResult quality: 3% values known / values provided: 3%distinct values known / distinct values provided: 25%
Values
[1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,3,-4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,-3,4] => [1,2,4,3] => [1,2,4,3] => 0 = 1 - 1
[1,-2,3,4] => [1,4,2,3] => [1,4,2,3] => 0 = 1 - 1
[-1,2,3,4] => [4,1,2,3] => [4,1,2,3] => 0 = 1 - 1
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0 = 1 - 1
[1,2,-4,-3] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0 = 1 - 1
[1,-3,-2,4] => [1,3,4,2] => [1,3,4,2] => 0 = 1 - 1
[1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 0 = 1 - 1
[1,-4,3,-2] => [1,3,2,4] => [1,3,2,4] => 0 = 1 - 1
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0 = 1 - 1
[-2,-1,3,4] => [3,4,1,2] => [3,4,1,2] => 0 = 1 - 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0 = 1 - 1
[2,1,-4,-3] => [2,1,3,4] => [2,1,3,4] => 0 = 1 - 1
[-2,-1,4,3] => [3,4,2,1] => [3,4,2,1] => 0 = 1 - 1
[-2,-1,-4,-3] => [3,4,1,2] => [3,4,1,2] => 0 = 1 - 1
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 0 = 1 - 1
[-3,2,-1,4] => [3,1,4,2] => [3,1,4,2] => 0 = 1 - 1
[3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 0 = 1 - 1
[3,-4,1,-2] => [2,3,1,4] => [2,3,1,4] => 0 = 1 - 1
[-3,4,-1,2] => [3,2,4,1] => [3,2,4,1] => 0 = 1 - 1
[-3,-4,-1,-2] => [2,1,4,3] => [2,1,4,3] => 0 = 1 - 1
[4,2,3,1] => [4,2,3,1] => [4,2,3,1] => 0 = 1 - 1
[-4,2,3,-1] => [3,1,2,4] => [3,1,2,4] => 0 = 1 - 1
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
[4,-3,-2,1] => [2,3,4,1] => [2,3,4,1] => 0 = 1 - 1
[-4,3,2,-1] => [3,2,1,4] => [3,2,1,4] => 0 = 1 - 1
[-4,-3,-2,-1] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 2 - 1
[1,2,3,4,-5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 1 - 1
[1,2,3,-4,5] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 1 - 1
[1,2,3,-4,-5] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 1 - 1
[1,2,-3,4,5] => [1,2,5,3,4] => [1,2,5,3,4] => ? = 1 - 1
[1,2,-3,4,-5] => [1,2,5,3,4] => [1,2,5,3,4] => ? = 1 - 1
[1,2,-3,-4,5] => [1,2,5,4,3] => [1,2,5,4,3] => ? = 1 - 1
[1,-2,3,4,5] => [1,5,2,3,4] => [1,5,2,3,4] => ? = 1 - 1
[1,-2,3,4,-5] => [1,5,2,3,4] => [1,5,2,3,4] => ? = 1 - 1
[1,-2,3,-4,5] => [1,5,2,4,3] => [1,5,2,4,3] => ? = 1 - 1
[1,-2,-3,4,5] => [1,5,4,2,3] => [1,5,4,2,3] => ? = 1 - 1
[-1,2,3,4,5] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 1 - 1
[-1,2,3,4,-5] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 1 - 1
[-1,2,3,-4,5] => [5,1,2,4,3] => [5,1,2,4,3] => ? = 1 - 1
[-1,2,-3,4,5] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 1 - 1
[-1,-2,3,4,5] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 1 - 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 1 - 1
[1,2,3,5,-4] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 1 - 1
[1,2,3,-5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 1 - 1
[1,2,3,-5,-4] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 1 - 1
[1,2,-3,5,4] => [1,2,5,4,3] => [1,2,5,4,3] => ? = 1 - 1
[1,2,-3,-5,-4] => [1,2,5,3,4] => [1,2,5,3,4] => ? = 1 - 1
[1,-2,3,5,4] => [1,5,2,4,3] => [1,5,2,4,3] => ? = 1 - 1
[1,-2,3,-5,-4] => [1,5,2,3,4] => [1,5,2,3,4] => ? = 1 - 1
[-1,2,3,5,4] => [5,1,2,4,3] => [5,1,2,4,3] => ? = 1 - 1
[-1,2,3,-5,-4] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 1 - 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ? = 1 - 1
[1,2,4,3,-5] => [1,2,4,3,5] => [1,2,4,3,5] => ? = 1 - 1
[1,2,4,-3,5] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 1 - 1
[1,2,-4,3,5] => [1,2,5,3,4] => [1,2,5,3,4] => ? = 1 - 1
[1,2,-4,-3,5] => [1,2,4,5,3] => [1,2,4,5,3] => ? = 1 - 1
[1,2,-4,-3,-5] => [1,2,4,5,3] => [1,2,4,5,3] => ? = 1 - 1
[1,-2,4,3,5] => [1,5,3,2,4] => [1,5,3,2,4] => ? = 1 - 1
[1,-2,-4,-3,5] => [1,5,3,4,2] => [1,5,3,4,2] => ? = 1 - 1
[-1,2,4,3,5] => [5,1,3,2,4] => [5,1,3,2,4] => ? = 1 - 1
[-1,2,-4,-3,5] => [5,1,3,4,2] => [5,1,3,4,2] => ? = 1 - 1
[1,2,4,5,3] => [1,2,4,5,3] => [1,2,4,5,3] => ? = 1 - 1
[1,2,4,-5,-3] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 1 - 1
[1,2,-4,5,-3] => [1,2,4,3,5] => [1,2,4,3,5] => ? = 1 - 1
[1,2,-4,-5,3] => [1,2,5,4,3] => [1,2,5,4,3] => ? = 1 - 1
[1,2,5,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => ? = 1 - 1
[1,2,5,-3,-4] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 1 - 1
[1,2,-5,3,-4] => [1,2,4,3,5] => [1,2,4,3,5] => ? = 1 - 1
[1,2,-5,-3,4] => [1,2,4,5,3] => [1,2,4,5,3] => ? = 1 - 1
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => ? = 1 - 1
[1,2,5,4,-3] => [1,2,4,3,5] => [1,2,4,3,5] => ? = 1 - 1
[1,2,5,-4,3] => [1,2,4,5,3] => [1,2,4,5,3] => ? = 1 - 1
[1,2,-5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => ? = 1 - 1
[1,2,-5,4,-3] => [1,2,4,3,5] => [1,2,4,3,5] => ? = 1 - 1
[1,2,-5,-4,-3] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 1 - 1
Description
The number of secondary dinversion pairs of the dyck path corresponding to a parking function.
Mp00245: Signed permutations standardizePermutations
Mp00305: Permutations parking functionParking functions
St000943: Parking functions ⟶ ℤResult quality: 3% values known / values provided: 3%distinct values known / distinct values provided: 25%
Values
[1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,3,-4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,-3,4] => [1,2,4,3] => [1,2,4,3] => 0 = 1 - 1
[1,-2,3,4] => [1,4,2,3] => [1,4,2,3] => 0 = 1 - 1
[-1,2,3,4] => [4,1,2,3] => [4,1,2,3] => 0 = 1 - 1
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0 = 1 - 1
[1,2,-4,-3] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0 = 1 - 1
[1,-3,-2,4] => [1,3,4,2] => [1,3,4,2] => 0 = 1 - 1
[1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 0 = 1 - 1
[1,-4,3,-2] => [1,3,2,4] => [1,3,2,4] => 0 = 1 - 1
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0 = 1 - 1
[-2,-1,3,4] => [3,4,1,2] => [3,4,1,2] => 0 = 1 - 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0 = 1 - 1
[2,1,-4,-3] => [2,1,3,4] => [2,1,3,4] => 0 = 1 - 1
[-2,-1,4,3] => [3,4,2,1] => [3,4,2,1] => 0 = 1 - 1
[-2,-1,-4,-3] => [3,4,1,2] => [3,4,1,2] => 0 = 1 - 1
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 0 = 1 - 1
[-3,2,-1,4] => [3,1,4,2] => [3,1,4,2] => 0 = 1 - 1
[3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 0 = 1 - 1
[3,-4,1,-2] => [2,3,1,4] => [2,3,1,4] => 0 = 1 - 1
[-3,4,-1,2] => [3,2,4,1] => [3,2,4,1] => 0 = 1 - 1
[-3,-4,-1,-2] => [2,1,4,3] => [2,1,4,3] => 0 = 1 - 1
[4,2,3,1] => [4,2,3,1] => [4,2,3,1] => 0 = 1 - 1
[-4,2,3,-1] => [3,1,2,4] => [3,1,2,4] => 0 = 1 - 1
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
[4,-3,-2,1] => [2,3,4,1] => [2,3,4,1] => 0 = 1 - 1
[-4,3,2,-1] => [3,2,1,4] => [3,2,1,4] => 0 = 1 - 1
[-4,-3,-2,-1] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 2 - 1
[1,2,3,4,-5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 1 - 1
[1,2,3,-4,5] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 1 - 1
[1,2,3,-4,-5] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 1 - 1
[1,2,-3,4,5] => [1,2,5,3,4] => [1,2,5,3,4] => ? = 1 - 1
[1,2,-3,4,-5] => [1,2,5,3,4] => [1,2,5,3,4] => ? = 1 - 1
[1,2,-3,-4,5] => [1,2,5,4,3] => [1,2,5,4,3] => ? = 1 - 1
[1,-2,3,4,5] => [1,5,2,3,4] => [1,5,2,3,4] => ? = 1 - 1
[1,-2,3,4,-5] => [1,5,2,3,4] => [1,5,2,3,4] => ? = 1 - 1
[1,-2,3,-4,5] => [1,5,2,4,3] => [1,5,2,4,3] => ? = 1 - 1
[1,-2,-3,4,5] => [1,5,4,2,3] => [1,5,4,2,3] => ? = 1 - 1
[-1,2,3,4,5] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 1 - 1
[-1,2,3,4,-5] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 1 - 1
[-1,2,3,-4,5] => [5,1,2,4,3] => [5,1,2,4,3] => ? = 1 - 1
[-1,2,-3,4,5] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 1 - 1
[-1,-2,3,4,5] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 1 - 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 1 - 1
[1,2,3,5,-4] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 1 - 1
[1,2,3,-5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 1 - 1
[1,2,3,-5,-4] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 1 - 1
[1,2,-3,5,4] => [1,2,5,4,3] => [1,2,5,4,3] => ? = 1 - 1
[1,2,-3,-5,-4] => [1,2,5,3,4] => [1,2,5,3,4] => ? = 1 - 1
[1,-2,3,5,4] => [1,5,2,4,3] => [1,5,2,4,3] => ? = 1 - 1
[1,-2,3,-5,-4] => [1,5,2,3,4] => [1,5,2,3,4] => ? = 1 - 1
[-1,2,3,5,4] => [5,1,2,4,3] => [5,1,2,4,3] => ? = 1 - 1
[-1,2,3,-5,-4] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 1 - 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ? = 1 - 1
[1,2,4,3,-5] => [1,2,4,3,5] => [1,2,4,3,5] => ? = 1 - 1
[1,2,4,-3,5] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 1 - 1
[1,2,-4,3,5] => [1,2,5,3,4] => [1,2,5,3,4] => ? = 1 - 1
[1,2,-4,-3,5] => [1,2,4,5,3] => [1,2,4,5,3] => ? = 1 - 1
[1,2,-4,-3,-5] => [1,2,4,5,3] => [1,2,4,5,3] => ? = 1 - 1
[1,-2,4,3,5] => [1,5,3,2,4] => [1,5,3,2,4] => ? = 1 - 1
[1,-2,-4,-3,5] => [1,5,3,4,2] => [1,5,3,4,2] => ? = 1 - 1
[-1,2,4,3,5] => [5,1,3,2,4] => [5,1,3,2,4] => ? = 1 - 1
[-1,2,-4,-3,5] => [5,1,3,4,2] => [5,1,3,4,2] => ? = 1 - 1
[1,2,4,5,3] => [1,2,4,5,3] => [1,2,4,5,3] => ? = 1 - 1
[1,2,4,-5,-3] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 1 - 1
[1,2,-4,5,-3] => [1,2,4,3,5] => [1,2,4,3,5] => ? = 1 - 1
[1,2,-4,-5,3] => [1,2,5,4,3] => [1,2,5,4,3] => ? = 1 - 1
[1,2,5,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => ? = 1 - 1
[1,2,5,-3,-4] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 1 - 1
[1,2,-5,3,-4] => [1,2,4,3,5] => [1,2,4,3,5] => ? = 1 - 1
[1,2,-5,-3,4] => [1,2,4,5,3] => [1,2,4,5,3] => ? = 1 - 1
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => ? = 1 - 1
[1,2,5,4,-3] => [1,2,4,3,5] => [1,2,4,3,5] => ? = 1 - 1
[1,2,5,-4,3] => [1,2,4,5,3] => [1,2,4,5,3] => ? = 1 - 1
[1,2,-5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => ? = 1 - 1
[1,2,-5,4,-3] => [1,2,4,3,5] => [1,2,4,3,5] => ? = 1 - 1
[1,2,-5,-4,-3] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 1 - 1
Description
The number of spots the most unlucky car had to go further in a parking function.
Matching statistic: St001195
Mp00163: Signed permutations permutationPermutations
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
Mp00199: Dyck paths prime Dyck pathDyck paths
St001195: Dyck paths ⟶ ℤResult quality: 3% values known / values provided: 3%distinct values known / distinct values provided: 25%
Values
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,2,3,-4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,2,-3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,-2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[-1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[1,2,-4,-3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[1,-3,-2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,-4,3,-2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[-2,-1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[2,1,-4,-3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[-2,-1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[-2,-1,-4,-3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[-3,2,-1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[3,4,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[3,-4,1,-2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[-3,4,-1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[-3,-4,-1,-2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[-4,2,3,-1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[4,-3,-2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[-4,3,2,-1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[-4,-3,-2,-1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,2,3,4,-5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,2,3,-4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,2,3,-4,-5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,2,-3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,2,-3,4,-5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,2,-3,-4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,-2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,-2,3,4,-5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,-2,3,-4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,-2,-3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[-1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[-1,2,3,4,-5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[-1,2,3,-4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[-1,2,-3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[-1,-2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 1
[1,2,3,5,-4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 1
[1,2,3,-5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 1
[1,2,3,-5,-4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 1
[1,2,-3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 1
[1,2,-3,-5,-4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 1
[1,-2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 1
[1,-2,3,-5,-4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 1
[-1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 1
[-1,2,3,-5,-4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> ? = 1
[1,2,4,3,-5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> ? = 1
[1,2,4,-3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> ? = 1
[1,2,-4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> ? = 1
[1,2,-4,-3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> ? = 1
[1,2,-4,-3,-5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> ? = 1
[1,-2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> ? = 1
[1,-2,-4,-3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> ? = 1
[-1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> ? = 1
[-1,2,-4,-3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> ? = 1
[1,2,4,5,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 1
[1,2,4,-5,-3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 1
[1,2,-4,5,-3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 1
[1,2,-4,-5,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 1
[1,2,5,3,4] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 1
[1,2,5,-3,-4] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 1
[1,2,-5,3,-4] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 1
[1,2,-5,-3,4] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 1
[1,2,5,4,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 1
[1,2,5,4,-3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 1
[1,2,5,-4,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 1
[1,2,-5,4,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 1
[1,2,-5,4,-3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 1
[1,2,-5,-4,-3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 1
Description
The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$.
The following 37 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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 $K[x]/(x^n)$. St001490The number of connected components of a skew partition. St001768The number of reduced words of a signed permutation. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. 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. St001927Sparre Andersen's number of positives of a signed permutation. St000284The Plancherel distribution on integer partitions. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001128The exponens consonantiae of a 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. St001613The binary logarithm of the size of the center of a lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001616The number of neutral elements in a lattice. St001624The breadth of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001720The minimal length of a chain of small intervals in a lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. 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. St001568The smallest positive integer that does not appear twice in the partition. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition.