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Your data matches 59 different statistics following compositions of up to 3 maps.
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Matching statistic: St000284
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Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000284: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000284: 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]
=> 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]
=> 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 Plancherel distribution on integer partitions.
This is defined as the distribution induced by the RSK shape of the uniform distribution on permutations. In other words, this is the size of the preimage of the map 'Robinson-Schensted tableau shape' from permutations to integer partitions.
Equivalently, this is given by the square of the number of standard Young tableaux of the given shape.
Matching statistic: St000782
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000782: Perfect matchings ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 20%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000782: Perfect matchings ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 20%
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)]
=> ? = 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]
=> [1,1,0,1,0,1,0,0]
=> [(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}
Matching statistic: St001491
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 20%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 20%
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 => ? = 1
[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 type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 20%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 20%
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] => ? = 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]
=> [1,1,0,1,0,1,0,0]
=> [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).
Matching statistic: St001771
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00163: Signed permutations —permutation⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001771: Signed permutations ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 20%
Mp00126: Permutations —cactus evacuation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001771: Signed permutations ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 20%
Values
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,2,3,4] => [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] => [1,2,3,4] => 0 = 1 - 1
[1,2,-3,4] => [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] => [1,2,3,4] => 0 = 1 - 1
[-1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,4,3] => [1,2,4,3] => [4,1,2,3] => [4,1,2,3] => 0 = 1 - 1
[1,2,-4,-3] => [1,2,4,3] => [4,1,2,3] => [4,1,2,3] => 0 = 1 - 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0 = 1 - 1
[1,-3,-2,4] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0 = 1 - 1
[1,4,3,2] => [1,4,3,2] => [4,3,1,2] => [4,3,1,2] => 0 = 1 - 1
[1,-4,3,-2] => [1,4,3,2] => [4,3,1,2] => [4,3,1,2] => 0 = 1 - 1
[2,1,3,4] => [2,1,3,4] => [2,3,4,1] => [2,3,4,1] => 0 = 1 - 1
[-2,-1,3,4] => [2,1,3,4] => [2,3,4,1] => [2,3,4,1] => 0 = 1 - 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0 = 1 - 1
[2,1,-4,-3] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0 = 1 - 1
[-2,-1,4,3] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0 = 1 - 1
[-2,-1,-4,-3] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0 = 1 - 1
[3,2,1,4] => [3,2,1,4] => [3,4,2,1] => [3,4,2,1] => 0 = 1 - 1
[-3,2,-1,4] => [3,2,1,4] => [3,4,2,1] => [3,4,2,1] => 0 = 1 - 1
[3,4,1,2] => [3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 0 = 1 - 1
[3,-4,1,-2] => [3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 0 = 1 - 1
[-3,4,-1,2] => [3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 0 = 1 - 1
[-3,-4,-1,-2] => [3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 0 = 1 - 1
[4,2,3,1] => [4,2,3,1] => [4,2,3,1] => [4,2,3,1] => 0 = 1 - 1
[-4,2,3,-1] => [4,2,3,1] => [4,2,3,1] => [4,2,3,1] => 0 = 1 - 1
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
[4,-3,-2,1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
[-4,3,2,-1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
[-4,-3,-2,-1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,3,4,-5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,3,-4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,3,-4,-5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,-3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,-3,4,-5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,-3,-4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,-2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,-2,3,4,-5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,-2,3,-4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,-2,-3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[-1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[-1,2,3,4,-5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[-1,2,3,-4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[-1,2,-3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[-1,-2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 1 - 1
[1,2,3,5,-4] => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 1 - 1
[1,2,3,-5,4] => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 1 - 1
[1,2,3,-5,-4] => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 1 - 1
[1,2,-3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 1 - 1
[1,2,-3,-5,-4] => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 1 - 1
[1,-2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 1 - 1
[1,-2,3,-5,-4] => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 1 - 1
[-1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 1 - 1
[-1,2,3,-5,-4] => [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] => [1,4,2,3,5] => [1,4,2,3,5] => 0 = 1 - 1
[1,2,4,3,-5] => [1,2,4,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => 0 = 1 - 1
[1,2,4,-3,5] => [1,2,4,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => 0 = 1 - 1
[1,2,-4,3,5] => [1,2,4,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => 0 = 1 - 1
[1,2,4,5,3] => [1,2,4,5,3] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 1 - 1
[1,2,4,-5,-3] => [1,2,4,5,3] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 1 - 1
[1,2,-4,5,-3] => [1,2,4,5,3] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 1 - 1
[1,2,-4,-5,3] => [1,2,4,5,3] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 1 - 1
[1,2,5,4,3] => [1,2,5,4,3] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 1 - 1
[1,2,5,4,-3] => [1,2,5,4,3] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 1 - 1
[1,2,5,-4,3] => [1,2,5,4,3] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 1 - 1
[1,2,-5,4,3] => [1,2,5,4,3] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 1 - 1
[1,2,-5,4,-3] => [1,2,5,4,3] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 1 - 1
[1,2,-5,-4,-3] => [1,2,5,4,3] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 1 - 1
[1,-2,5,4,3] => [1,2,5,4,3] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 1 - 1
[1,-2,-5,4,-3] => [1,2,5,4,3] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 1 - 1
[-1,2,5,4,3] => [1,2,5,4,3] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 1 - 1
[-1,2,-5,4,-3] => [1,2,5,4,3] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 1 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [3,1,5,2,4] => [3,1,5,2,4] => ? = 4 - 1
[1,3,2,-5,-4] => [1,3,2,5,4] => [3,1,5,2,4] => [3,1,5,2,4] => ? = 4 - 1
[1,-3,-2,5,4] => [1,3,2,5,4] => [3,1,5,2,4] => [3,1,5,2,4] => ? = 4 - 1
[1,-3,-2,-5,-4] => [1,3,2,5,4] => [3,1,5,2,4] => [3,1,5,2,4] => ? = 4 - 1
[-1,3,2,5,4] => [1,3,2,5,4] => [3,1,5,2,4] => [3,1,5,2,4] => ? = 1 - 1
[-1,3,2,-5,-4] => [1,3,2,5,4] => [3,1,5,2,4] => [3,1,5,2,4] => ? = 1 - 1
[-1,-3,-2,5,4] => [1,3,2,5,4] => [3,1,5,2,4] => [3,1,5,2,4] => ? = 1 - 1
[-1,-3,-2,-5,-4] => [1,3,2,5,4] => [3,1,5,2,4] => [3,1,5,2,4] => ? = 1 - 1
[1,3,5,4,2] => [1,3,5,4,2] => [5,3,1,2,4] => [5,3,1,2,4] => ? = 1 - 1
[1,3,-5,4,-2] => [1,3,5,4,2] => [5,3,1,2,4] => [5,3,1,2,4] => ? = 1 - 1
[1,-3,5,4,-2] => [1,3,5,4,2] => [5,3,1,2,4] => [5,3,1,2,4] => ? = 1 - 1
[1,-3,-5,4,2] => [1,3,5,4,2] => [5,3,1,2,4] => [5,3,1,2,4] => ? = 1 - 1
[1,4,3,5,2] => [1,4,3,5,2] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1 - 1
[1,4,3,-5,-2] => [1,4,3,5,2] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1 - 1
[1,-4,3,5,-2] => [1,4,3,5,2] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1 - 1
[1,-4,3,-5,2] => [1,4,3,5,2] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1 - 1
[1,4,5,2,3] => [1,4,5,2,3] => [4,5,1,2,3] => [4,5,1,2,3] => ? = 4 - 1
[1,4,-5,2,-3] => [1,4,5,2,3] => [4,5,1,2,3] => [4,5,1,2,3] => ? = 4 - 1
[1,-4,5,-2,3] => [1,4,5,2,3] => [4,5,1,2,3] => [4,5,1,2,3] => ? = 4 - 1
[1,-4,-5,-2,-3] => [1,4,5,2,3] => [4,5,1,2,3] => [4,5,1,2,3] => ? = 4 - 1
[-1,4,5,2,3] => [1,4,5,2,3] => [4,5,1,2,3] => [4,5,1,2,3] => ? = 1 - 1
[-1,4,-5,2,-3] => [1,4,5,2,3] => [4,5,1,2,3] => [4,5,1,2,3] => ? = 1 - 1
[-1,-4,5,-2,3] => [1,4,5,2,3] => [4,5,1,2,3] => [4,5,1,2,3] => ? = 1 - 1
[-1,-4,-5,-2,-3] => [1,4,5,2,3] => [4,5,1,2,3] => [4,5,1,2,3] => ? = 1 - 1
[1,5,2,4,3] => [1,5,2,4,3] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 1 - 1
[1,5,-2,4,-3] => [1,5,2,4,3] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 1 - 1
Description
The number of occurrences of the signed pattern 1-2 in a signed permutation.
This is the number of pairs 1\leq i < j\leq n such that 0 < \pi(i) < -\pi(j).
Matching statistic: St001870
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00163: Signed permutations —permutation⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001870: Signed permutations ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 20%
Mp00126: Permutations —cactus evacuation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001870: Signed permutations ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 20%
Values
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,2,3,4] => [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] => [1,2,3,4] => 0 = 1 - 1
[1,2,-3,4] => [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] => [1,2,3,4] => 0 = 1 - 1
[-1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,4,3] => [1,2,4,3] => [4,1,2,3] => [4,1,2,3] => 0 = 1 - 1
[1,2,-4,-3] => [1,2,4,3] => [4,1,2,3] => [4,1,2,3] => 0 = 1 - 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0 = 1 - 1
[1,-3,-2,4] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0 = 1 - 1
[1,4,3,2] => [1,4,3,2] => [4,3,1,2] => [4,3,1,2] => 0 = 1 - 1
[1,-4,3,-2] => [1,4,3,2] => [4,3,1,2] => [4,3,1,2] => 0 = 1 - 1
[2,1,3,4] => [2,1,3,4] => [2,3,4,1] => [2,3,4,1] => 0 = 1 - 1
[-2,-1,3,4] => [2,1,3,4] => [2,3,4,1] => [2,3,4,1] => 0 = 1 - 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0 = 1 - 1
[2,1,-4,-3] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0 = 1 - 1
[-2,-1,4,3] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0 = 1 - 1
[-2,-1,-4,-3] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0 = 1 - 1
[3,2,1,4] => [3,2,1,4] => [3,4,2,1] => [3,4,2,1] => 0 = 1 - 1
[-3,2,-1,4] => [3,2,1,4] => [3,4,2,1] => [3,4,2,1] => 0 = 1 - 1
[3,4,1,2] => [3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 0 = 1 - 1
[3,-4,1,-2] => [3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 0 = 1 - 1
[-3,4,-1,2] => [3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 0 = 1 - 1
[-3,-4,-1,-2] => [3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 0 = 1 - 1
[4,2,3,1] => [4,2,3,1] => [4,2,3,1] => [4,2,3,1] => 0 = 1 - 1
[-4,2,3,-1] => [4,2,3,1] => [4,2,3,1] => [4,2,3,1] => 0 = 1 - 1
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
[4,-3,-2,1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
[-4,3,2,-1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
[-4,-3,-2,-1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,3,4,-5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,3,-4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,3,-4,-5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,-3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,-3,4,-5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,-3,-4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,-2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,-2,3,4,-5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,-2,3,-4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,-2,-3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[-1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[-1,2,3,4,-5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[-1,2,3,-4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[-1,2,-3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[-1,-2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 1 - 1
[1,2,3,5,-4] => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 1 - 1
[1,2,3,-5,4] => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 1 - 1
[1,2,3,-5,-4] => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 1 - 1
[1,2,-3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 1 - 1
[1,2,-3,-5,-4] => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 1 - 1
[1,-2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 1 - 1
[1,-2,3,-5,-4] => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 1 - 1
[-1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 1 - 1
[-1,2,3,-5,-4] => [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] => [1,4,2,3,5] => [1,4,2,3,5] => 0 = 1 - 1
[1,2,4,3,-5] => [1,2,4,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => 0 = 1 - 1
[1,2,4,-3,5] => [1,2,4,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => 0 = 1 - 1
[1,2,-4,3,5] => [1,2,4,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => 0 = 1 - 1
[1,2,4,5,3] => [1,2,4,5,3] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 1 - 1
[1,2,4,-5,-3] => [1,2,4,5,3] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 1 - 1
[1,2,-4,5,-3] => [1,2,4,5,3] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 1 - 1
[1,2,-4,-5,3] => [1,2,4,5,3] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 1 - 1
[1,2,5,4,3] => [1,2,5,4,3] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 1 - 1
[1,2,5,4,-3] => [1,2,5,4,3] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 1 - 1
[1,2,5,-4,3] => [1,2,5,4,3] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 1 - 1
[1,2,-5,4,3] => [1,2,5,4,3] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 1 - 1
[1,2,-5,4,-3] => [1,2,5,4,3] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 1 - 1
[1,2,-5,-4,-3] => [1,2,5,4,3] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 1 - 1
[1,-2,5,4,3] => [1,2,5,4,3] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 1 - 1
[1,-2,-5,4,-3] => [1,2,5,4,3] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 1 - 1
[-1,2,5,4,3] => [1,2,5,4,3] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 1 - 1
[-1,2,-5,4,-3] => [1,2,5,4,3] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 1 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [3,1,5,2,4] => [3,1,5,2,4] => ? = 4 - 1
[1,3,2,-5,-4] => [1,3,2,5,4] => [3,1,5,2,4] => [3,1,5,2,4] => ? = 4 - 1
[1,-3,-2,5,4] => [1,3,2,5,4] => [3,1,5,2,4] => [3,1,5,2,4] => ? = 4 - 1
[1,-3,-2,-5,-4] => [1,3,2,5,4] => [3,1,5,2,4] => [3,1,5,2,4] => ? = 4 - 1
[-1,3,2,5,4] => [1,3,2,5,4] => [3,1,5,2,4] => [3,1,5,2,4] => ? = 1 - 1
[-1,3,2,-5,-4] => [1,3,2,5,4] => [3,1,5,2,4] => [3,1,5,2,4] => ? = 1 - 1
[-1,-3,-2,5,4] => [1,3,2,5,4] => [3,1,5,2,4] => [3,1,5,2,4] => ? = 1 - 1
[-1,-3,-2,-5,-4] => [1,3,2,5,4] => [3,1,5,2,4] => [3,1,5,2,4] => ? = 1 - 1
[1,3,5,4,2] => [1,3,5,4,2] => [5,3,1,2,4] => [5,3,1,2,4] => ? = 1 - 1
[1,3,-5,4,-2] => [1,3,5,4,2] => [5,3,1,2,4] => [5,3,1,2,4] => ? = 1 - 1
[1,-3,5,4,-2] => [1,3,5,4,2] => [5,3,1,2,4] => [5,3,1,2,4] => ? = 1 - 1
[1,-3,-5,4,2] => [1,3,5,4,2] => [5,3,1,2,4] => [5,3,1,2,4] => ? = 1 - 1
[1,4,3,5,2] => [1,4,3,5,2] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1 - 1
[1,4,3,-5,-2] => [1,4,3,5,2] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1 - 1
[1,-4,3,5,-2] => [1,4,3,5,2] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1 - 1
[1,-4,3,-5,2] => [1,4,3,5,2] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1 - 1
[1,4,5,2,3] => [1,4,5,2,3] => [4,5,1,2,3] => [4,5,1,2,3] => ? = 4 - 1
[1,4,-5,2,-3] => [1,4,5,2,3] => [4,5,1,2,3] => [4,5,1,2,3] => ? = 4 - 1
[1,-4,5,-2,3] => [1,4,5,2,3] => [4,5,1,2,3] => [4,5,1,2,3] => ? = 4 - 1
[1,-4,-5,-2,-3] => [1,4,5,2,3] => [4,5,1,2,3] => [4,5,1,2,3] => ? = 4 - 1
[-1,4,5,2,3] => [1,4,5,2,3] => [4,5,1,2,3] => [4,5,1,2,3] => ? = 1 - 1
[-1,4,-5,2,-3] => [1,4,5,2,3] => [4,5,1,2,3] => [4,5,1,2,3] => ? = 1 - 1
[-1,-4,5,-2,3] => [1,4,5,2,3] => [4,5,1,2,3] => [4,5,1,2,3] => ? = 1 - 1
[-1,-4,-5,-2,-3] => [1,4,5,2,3] => [4,5,1,2,3] => [4,5,1,2,3] => ? = 1 - 1
[1,5,2,4,3] => [1,5,2,4,3] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 1 - 1
[1,5,-2,4,-3] => [1,5,2,4,3] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 1 - 1
Description
The number of positive entries followed by a negative entry in a signed permutation.
For a signed permutation \pi\in\mathfrak H_n, this is the number of positive entries followed by a negative entry in \pi(-n),\dots,\pi(-1),\pi(1),\dots,\pi(n).
Matching statistic: St001895
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00163: Signed permutations —permutation⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001895: Signed permutations ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 20%
Mp00126: Permutations —cactus evacuation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001895: Signed permutations ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 20%
Values
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,2,3,4] => [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] => [1,2,3,4] => 0 = 1 - 1
[1,2,-3,4] => [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] => [1,2,3,4] => 0 = 1 - 1
[-1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,4,3] => [1,2,4,3] => [4,1,2,3] => [4,1,2,3] => 0 = 1 - 1
[1,2,-4,-3] => [1,2,4,3] => [4,1,2,3] => [4,1,2,3] => 0 = 1 - 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0 = 1 - 1
[1,-3,-2,4] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0 = 1 - 1
[1,4,3,2] => [1,4,3,2] => [4,3,1,2] => [4,3,1,2] => 0 = 1 - 1
[1,-4,3,-2] => [1,4,3,2] => [4,3,1,2] => [4,3,1,2] => 0 = 1 - 1
[2,1,3,4] => [2,1,3,4] => [2,3,4,1] => [2,3,4,1] => 0 = 1 - 1
[-2,-1,3,4] => [2,1,3,4] => [2,3,4,1] => [2,3,4,1] => 0 = 1 - 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0 = 1 - 1
[2,1,-4,-3] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0 = 1 - 1
[-2,-1,4,3] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0 = 1 - 1
[-2,-1,-4,-3] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0 = 1 - 1
[3,2,1,4] => [3,2,1,4] => [3,4,2,1] => [3,4,2,1] => 0 = 1 - 1
[-3,2,-1,4] => [3,2,1,4] => [3,4,2,1] => [3,4,2,1] => 0 = 1 - 1
[3,4,1,2] => [3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 0 = 1 - 1
[3,-4,1,-2] => [3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 0 = 1 - 1
[-3,4,-1,2] => [3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 0 = 1 - 1
[-3,-4,-1,-2] => [3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 0 = 1 - 1
[4,2,3,1] => [4,2,3,1] => [4,2,3,1] => [4,2,3,1] => 0 = 1 - 1
[-4,2,3,-1] => [4,2,3,1] => [4,2,3,1] => [4,2,3,1] => 0 = 1 - 1
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
[4,-3,-2,1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
[-4,3,2,-1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
[-4,-3,-2,-1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,3,4,-5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,3,-4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,3,-4,-5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,-3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,-3,4,-5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,-3,-4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,-2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,-2,3,4,-5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,-2,3,-4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,-2,-3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[-1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[-1,2,3,4,-5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[-1,2,3,-4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[-1,2,-3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[-1,-2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 1 - 1
[1,2,3,5,-4] => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 1 - 1
[1,2,3,-5,4] => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 1 - 1
[1,2,3,-5,-4] => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 1 - 1
[1,2,-3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 1 - 1
[1,2,-3,-5,-4] => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 1 - 1
[1,-2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 1 - 1
[1,-2,3,-5,-4] => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 1 - 1
[-1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 1 - 1
[-1,2,3,-5,-4] => [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] => [1,4,2,3,5] => [1,4,2,3,5] => 0 = 1 - 1
[1,2,4,3,-5] => [1,2,4,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => 0 = 1 - 1
[1,2,4,-3,5] => [1,2,4,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => 0 = 1 - 1
[1,2,-4,3,5] => [1,2,4,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => 0 = 1 - 1
[1,2,4,5,3] => [1,2,4,5,3] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 1 - 1
[1,2,4,-5,-3] => [1,2,4,5,3] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 1 - 1
[1,2,-4,5,-3] => [1,2,4,5,3] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 1 - 1
[1,2,-4,-5,3] => [1,2,4,5,3] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 1 - 1
[1,2,5,4,3] => [1,2,5,4,3] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 1 - 1
[1,2,5,4,-3] => [1,2,5,4,3] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 1 - 1
[1,2,5,-4,3] => [1,2,5,4,3] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 1 - 1
[1,2,-5,4,3] => [1,2,5,4,3] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 1 - 1
[1,2,-5,4,-3] => [1,2,5,4,3] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 1 - 1
[1,2,-5,-4,-3] => [1,2,5,4,3] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 1 - 1
[1,-2,5,4,3] => [1,2,5,4,3] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 1 - 1
[1,-2,-5,4,-3] => [1,2,5,4,3] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 1 - 1
[-1,2,5,4,3] => [1,2,5,4,3] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 1 - 1
[-1,2,-5,4,-3] => [1,2,5,4,3] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 1 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [3,1,5,2,4] => [3,1,5,2,4] => ? = 4 - 1
[1,3,2,-5,-4] => [1,3,2,5,4] => [3,1,5,2,4] => [3,1,5,2,4] => ? = 4 - 1
[1,-3,-2,5,4] => [1,3,2,5,4] => [3,1,5,2,4] => [3,1,5,2,4] => ? = 4 - 1
[1,-3,-2,-5,-4] => [1,3,2,5,4] => [3,1,5,2,4] => [3,1,5,2,4] => ? = 4 - 1
[-1,3,2,5,4] => [1,3,2,5,4] => [3,1,5,2,4] => [3,1,5,2,4] => ? = 1 - 1
[-1,3,2,-5,-4] => [1,3,2,5,4] => [3,1,5,2,4] => [3,1,5,2,4] => ? = 1 - 1
[-1,-3,-2,5,4] => [1,3,2,5,4] => [3,1,5,2,4] => [3,1,5,2,4] => ? = 1 - 1
[-1,-3,-2,-5,-4] => [1,3,2,5,4] => [3,1,5,2,4] => [3,1,5,2,4] => ? = 1 - 1
[1,3,5,4,2] => [1,3,5,4,2] => [5,3,1,2,4] => [5,3,1,2,4] => ? = 1 - 1
[1,3,-5,4,-2] => [1,3,5,4,2] => [5,3,1,2,4] => [5,3,1,2,4] => ? = 1 - 1
[1,-3,5,4,-2] => [1,3,5,4,2] => [5,3,1,2,4] => [5,3,1,2,4] => ? = 1 - 1
[1,-3,-5,4,2] => [1,3,5,4,2] => [5,3,1,2,4] => [5,3,1,2,4] => ? = 1 - 1
[1,4,3,5,2] => [1,4,3,5,2] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1 - 1
[1,4,3,-5,-2] => [1,4,3,5,2] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1 - 1
[1,-4,3,5,-2] => [1,4,3,5,2] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1 - 1
[1,-4,3,-5,2] => [1,4,3,5,2] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1 - 1
[1,4,5,2,3] => [1,4,5,2,3] => [4,5,1,2,3] => [4,5,1,2,3] => ? = 4 - 1
[1,4,-5,2,-3] => [1,4,5,2,3] => [4,5,1,2,3] => [4,5,1,2,3] => ? = 4 - 1
[1,-4,5,-2,3] => [1,4,5,2,3] => [4,5,1,2,3] => [4,5,1,2,3] => ? = 4 - 1
[1,-4,-5,-2,-3] => [1,4,5,2,3] => [4,5,1,2,3] => [4,5,1,2,3] => ? = 4 - 1
[-1,4,5,2,3] => [1,4,5,2,3] => [4,5,1,2,3] => [4,5,1,2,3] => ? = 1 - 1
[-1,4,-5,2,-3] => [1,4,5,2,3] => [4,5,1,2,3] => [4,5,1,2,3] => ? = 1 - 1
[-1,-4,5,-2,3] => [1,4,5,2,3] => [4,5,1,2,3] => [4,5,1,2,3] => ? = 1 - 1
[-1,-4,-5,-2,-3] => [1,4,5,2,3] => [4,5,1,2,3] => [4,5,1,2,3] => ? = 1 - 1
[1,5,2,4,3] => [1,5,2,4,3] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 1 - 1
[1,5,-2,4,-3] => [1,5,2,4,3] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 1 - 1
Description
The oddness of a signed permutation.
The direct sum of two signed permutations \sigma\in\mathfrak H_k and \tau\in\mathfrak H_m is the signed permutation in \mathfrak H_{k+m} obtained by concatenating \sigma with the result of increasing the absolute value of every entry in \tau by k.
This statistic records the number of blocks with an odd number of signs in the direct sum decomposition of a signed permutation.
Matching statistic: St000068
Mp00163: Signed permutations —permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000068: Posets ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 20%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000068: Posets ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 20%
Values
[1,2,3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,3,-4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,-3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,-2,3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[-1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[1,2,-4,-3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1
[1,-3,-2,4] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1
[1,4,3,2] => [1,4,3,2] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1
[1,-4,3,-2] => [1,4,3,2] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[-2,-1,3,4] => [2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[2,1,-4,-3] => [2,1,4,3] => [2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[-2,-1,4,3] => [2,1,4,3] => [2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[-2,-1,-4,-3] => [2,1,4,3] => [2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[3,2,1,4] => [3,2,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1
[-3,2,-1,4] => [3,2,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1
[3,4,1,2] => [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[3,-4,1,-2] => [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[-3,4,-1,2] => [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[-3,-4,-1,-2] => [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[4,2,3,1] => [4,2,3,1] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[-4,2,3,-1] => [4,2,3,1] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[4,3,2,1] => [4,3,2,1] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1
[4,-3,-2,1] => [4,3,2,1] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1
[-4,3,2,-1] => [4,3,2,1] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1
[-4,-3,-2,-1] => [4,3,2,1] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,3,4,-5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,3,-4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,3,-4,-5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,-3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,-3,4,-5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,-3,-4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,-2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,-2,3,4,-5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,-2,3,-4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,-2,-3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[-1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[-1,2,3,4,-5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[-1,2,3,-4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[-1,2,-3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[-1,-2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1
[1,2,3,5,-4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1
[1,2,3,-5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1
[1,2,3,-5,-4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1
[1,2,-3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1
[1,2,-3,-5,-4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1
[1,-2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1
[1,-2,3,-5,-4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[1,2,4,3,-5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[1,2,4,-3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[1,2,-4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[1,2,-4,-3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[1,2,-4,-3,-5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[1,-2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[1,-2,-4,-3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[-1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[-1,2,-4,-3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[1,2,4,5,3] => [1,2,4,5,3] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[1,2,4,-5,-3] => [1,2,4,5,3] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[1,2,-4,5,-3] => [1,2,4,5,3] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[1,2,-4,-5,3] => [1,2,4,5,3] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[1,2,5,4,-3] => [1,2,5,4,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[1,2,5,-4,3] => [1,2,5,4,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[1,2,-5,4,3] => [1,2,5,4,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[1,2,-5,4,-3] => [1,2,5,4,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[1,2,-5,-4,-3] => [1,2,5,4,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[1,-2,5,4,3] => [1,2,5,4,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[1,-2,-5,4,-3] => [1,2,5,4,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[-1,2,5,4,3] => [1,2,5,4,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[-1,2,-5,4,-3] => [1,2,5,4,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[1,3,2,4,-5] => [1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[1,3,2,-4,5] => [1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[1,3,-2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[1,-3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[1,-3,-2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[1,-3,-2,4,-5] => [1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[1,-3,-2,-4,5] => [1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[-1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[-1,-3,-2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 4
[1,3,2,-5,-4] => [1,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 4
[1,-3,-2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 4
[1,-3,-2,-5,-4] => [1,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 4
[-1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 1
[-1,3,2,-5,-4] => [1,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 1
[-1,-3,-2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 1
[-1,-3,-2,-5,-4] => [1,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 1
[1,3,4,2,5] => [1,3,4,2,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 1
[1,3,-4,-2,5] => [1,3,4,2,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 1
[1,-3,4,-2,5] => [1,3,4,2,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 1
[1,-3,-4,2,5] => [1,3,4,2,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 1
Description
The number of minimal elements in a poset.
Matching statistic: St001625
Mp00163: Signed permutations —permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001625: Lattices ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 20%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001625: Lattices ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 20%
Values
[1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,2,3,-4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,2,-3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,-2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[-1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,2,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 0 = 1 - 1
[1,2,-4,-3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 0 = 1 - 1
[1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 0 = 1 - 1
[1,-3,-2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 0 = 1 - 1
[1,4,3,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
=> ? = 1 - 1
[1,-4,3,-2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
=> ? = 1 - 1
[2,1,3,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 0 = 1 - 1
[-2,-1,3,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 0 = 1 - 1
[2,1,4,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 0 = 1 - 1
[2,1,-4,-3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 0 = 1 - 1
[-2,-1,4,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 0 = 1 - 1
[-2,-1,-4,-3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 0 = 1 - 1
[3,2,1,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> ? = 1 - 1
[-3,2,-1,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> ? = 1 - 1
[3,4,1,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 - 1
[3,-4,1,-2] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 - 1
[-3,4,-1,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 - 1
[-3,-4,-1,-2] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 - 1
[4,2,3,1] => [4,2,3,1] => ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? = 1 - 1
[-4,2,3,-1] => [4,2,3,1] => ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? = 1 - 1
[4,3,2,1] => [4,3,2,1] => ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 1 - 1
[4,-3,-2,1] => [4,3,2,1] => ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 1 - 1
[-4,3,2,-1] => [4,3,2,1] => ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 1 - 1
[-4,-3,-2,-1] => [4,3,2,1] => ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 1 - 1
[1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,2,3,4,-5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,2,3,-4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,2,3,-4,-5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,2,-3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,2,-3,4,-5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,2,-3,-4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,-2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,-2,3,4,-5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,-2,3,-4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,-2,-3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[-1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[-1,2,3,4,-5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[-1,2,3,-4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[-1,2,-3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[-1,-2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> 0 = 1 - 1
[1,2,3,5,-4] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> 0 = 1 - 1
[1,2,3,-5,4] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> 0 = 1 - 1
[1,2,3,-5,-4] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> 0 = 1 - 1
[1,2,-3,5,4] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> 0 = 1 - 1
[1,2,-3,-5,-4] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> 0 = 1 - 1
[1,-2,3,5,4] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> 0 = 1 - 1
[1,-2,3,-5,-4] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> 0 = 1 - 1
[-1,2,3,5,4] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> 0 = 1 - 1
[-1,2,3,-5,-4] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> 0 = 1 - 1
[1,2,4,3,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 0 = 1 - 1
[1,2,4,3,-5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 0 = 1 - 1
[1,2,4,-3,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 0 = 1 - 1
[1,2,-4,3,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 0 = 1 - 1
[1,2,-4,-3,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 0 = 1 - 1
[1,2,-4,-3,-5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 0 = 1 - 1
[1,-2,4,3,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 0 = 1 - 1
[1,-2,-4,-3,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 0 = 1 - 1
[1,2,4,5,3] => [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 1 - 1
[1,2,4,-5,-3] => [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 1 - 1
[1,2,-4,5,-3] => [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 1 - 1
[1,2,-4,-5,3] => [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 1 - 1
[1,2,5,3,4] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 1 - 1
[1,2,5,-3,-4] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 1 - 1
[1,2,-5,3,-4] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 1 - 1
[1,2,-5,-3,4] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 1 - 1
[1,2,5,4,3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
=> ? = 1 - 1
[1,2,5,4,-3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
=> ? = 1 - 1
[1,2,5,-4,3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
=> ? = 1 - 1
[1,2,-5,4,3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
=> ? = 1 - 1
[1,2,-5,4,-3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
=> ? = 1 - 1
[1,2,-5,-4,-3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
=> ? = 1 - 1
[1,-2,5,4,3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
=> ? = 1 - 1
[1,-2,-5,4,-3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
=> ? = 1 - 1
[-1,2,5,4,3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
=> ? = 1 - 1
[-1,2,-5,4,-3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
=> ? = 1 - 1
[1,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> ? = 4 - 1
[1,3,2,-5,-4] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> ? = 4 - 1
[1,-3,-2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> ? = 4 - 1
[1,-3,-2,-5,-4] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> ? = 4 - 1
[-1,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> ? = 1 - 1
[-1,3,2,-5,-4] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> ? = 1 - 1
[-1,-3,-2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> ? = 1 - 1
[-1,-3,-2,-5,-4] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> ? = 1 - 1
[1,3,4,2,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 1 - 1
[1,3,-4,-2,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 1 - 1
[1,-3,4,-2,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 1 - 1
[1,-3,-4,2,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 1 - 1
[1,3,5,4,2] => [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> ? = 1 - 1
[1,3,-5,4,-2] => [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> ? = 1 - 1
[1,-3,5,4,-2] => [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> ? = 1 - 1
[1,-3,-5,4,2] => [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> ? = 1 - 1
[1,4,2,3,5] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 1 - 1
[1,4,-2,-3,5] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 1 - 1
Description
The Möbius invariant of a lattice.
The '''Möbius invariant''' of a lattice L is the value of the Möbius function applied to least and greatest element, that is \mu(L)=\mu_L(\hat{0},\hat{1}), where \hat{0} is the least element of L and \hat{1} is the greatest element of L.
For the definition of the Möbius function, see [[St000914]].
Matching statistic: St001713
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00082: Standard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
St001713: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 20%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00082: Standard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
St001713: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 20%
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]]
=> ? = 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]
=> [[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.
The following 49 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001301The first Betti number of the order complex associated with the 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. St001195The global dimension of the algebra A/AfA of the corresponding Nakayama algebra A with minimal left faithful projective-injective module Af. St001208The number of connected components of the quiver of A/T when T is the 1-tilting module corresponding to the permutation in the Auslander algebra A of K[x]/(x^n). 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. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. 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. St001964The interval resolution global dimension of a poset. St000181The number of connected components of the Hasse diagram for the poset. St000635The number of strictly order preserving maps of a poset into itself. St001890The maximum magnitude of the Möbius function of a poset. 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.
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