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Matching statistic: St000547
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00323: Integer partitions —Loehr-Warrington inverse⟶ Integer partitions
St000547: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00323: Integer partitions —Loehr-Warrington inverse⟶ Integer partitions
St000547: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-1] => [1]
=> [1]
=> 0
[1,-2] => [1]
=> [1]
=> 0
[-1,2] => [1]
=> [1]
=> 0
[-1,-2] => [1,1]
=> [2]
=> 1
[2,-1] => [2]
=> [1,1]
=> 1
[-2,1] => [2]
=> [1,1]
=> 1
[1,2,-3] => [1]
=> [1]
=> 0
[1,-2,3] => [1]
=> [1]
=> 0
[1,-2,-3] => [1,1]
=> [2]
=> 1
[-1,2,3] => [1]
=> [1]
=> 0
[-1,2,-3] => [1,1]
=> [2]
=> 1
[-1,-2,3] => [1,1]
=> [2]
=> 1
[-1,-2,-3] => [1,1,1]
=> [3]
=> 0
[1,3,-2] => [2]
=> [1,1]
=> 1
[1,-3,2] => [2]
=> [1,1]
=> 1
[-1,3,2] => [1]
=> [1]
=> 0
[-1,3,-2] => [2,1]
=> [1,1,1]
=> 1
[-1,-3,2] => [2,1]
=> [1,1,1]
=> 1
[-1,-3,-2] => [1]
=> [1]
=> 0
[2,1,-3] => [1]
=> [1]
=> 0
[2,-1,3] => [2]
=> [1,1]
=> 1
[2,-1,-3] => [2,1]
=> [1,1,1]
=> 1
[-2,1,3] => [2]
=> [1,1]
=> 1
[-2,1,-3] => [2,1]
=> [1,1,1]
=> 1
[-2,-1,-3] => [1]
=> [1]
=> 0
[2,3,-1] => [3]
=> [2,1]
=> 1
[2,-3,1] => [3]
=> [2,1]
=> 1
[-2,3,1] => [3]
=> [2,1]
=> 1
[-2,-3,-1] => [3]
=> [2,1]
=> 1
[3,1,-2] => [3]
=> [2,1]
=> 1
[3,-1,2] => [3]
=> [2,1]
=> 1
[-3,1,2] => [3]
=> [2,1]
=> 1
[-3,-1,-2] => [3]
=> [2,1]
=> 1
[3,2,-1] => [2]
=> [1,1]
=> 1
[3,-2,1] => [1]
=> [1]
=> 0
[3,-2,-1] => [2,1]
=> [1,1,1]
=> 1
[-3,2,1] => [2]
=> [1,1]
=> 1
[-3,-2,1] => [2,1]
=> [1,1,1]
=> 1
[-3,-2,-1] => [1]
=> [1]
=> 0
[1,2,3,-4] => [1]
=> [1]
=> 0
[1,2,-3,4] => [1]
=> [1]
=> 0
[1,2,-3,-4] => [1,1]
=> [2]
=> 1
[1,-2,3,4] => [1]
=> [1]
=> 0
[1,-2,3,-4] => [1,1]
=> [2]
=> 1
[1,-2,-3,4] => [1,1]
=> [2]
=> 1
[1,-2,-3,-4] => [1,1,1]
=> [3]
=> 0
[-1,2,3,4] => [1]
=> [1]
=> 0
[-1,2,3,-4] => [1,1]
=> [2]
=> 1
[-1,2,-3,4] => [1,1]
=> [2]
=> 1
[-1,2,-3,-4] => [1,1,1]
=> [3]
=> 0
Description
The number of even non-empty partial sums of an integer partition.
Matching statistic: St000782
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000782: Perfect matchings ⟶ ℤResult quality: 20% ●values known / values provided: 21%●distinct values known / distinct values provided: 20%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000782: Perfect matchings ⟶ ℤResult quality: 20% ●values known / values provided: 21%●distinct values known / distinct values provided: 20%
Values
[-1] => [1]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> ? = 0
[1,-2] => [1]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> ? = 0
[-1,2] => [1]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> ? = 0
[-1,-2] => [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[2,-1] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[-2,1] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[1,2,-3] => [1]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> ? = 0
[1,-2,3] => [1]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> ? = 0
[1,-2,-3] => [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[-1,2,3] => [1]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> ? = 0
[-1,2,-3] => [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[-1,-2,3] => [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[-1,-2,-3] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> ? = 0
[1,3,-2] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[1,-3,2] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[-1,3,2] => [1]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> ? = 0
[-1,3,-2] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1
[-1,-3,2] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1
[-1,-3,-2] => [1]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> ? = 0
[2,1,-3] => [1]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> ? = 0
[2,-1,3] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[2,-1,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1
[-2,1,3] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[-2,1,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1
[-2,-1,-3] => [1]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> ? = 0
[2,3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 1
[2,-3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 1
[-2,3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 1
[-2,-3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 1
[3,1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 1
[3,-1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 1
[-3,1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 1
[-3,-1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 1
[3,2,-1] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[3,-2,1] => [1]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> ? = 0
[3,-2,-1] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1
[-3,2,1] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[-3,-2,1] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1
[-3,-2,-1] => [1]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> ? = 0
[1,2,3,-4] => [1]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> ? = 0
[1,2,-3,4] => [1]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> ? = 0
[1,2,-3,-4] => [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[1,-2,3,4] => [1]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> ? = 0
[1,-2,3,-4] => [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[1,-2,-3,4] => [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[1,-2,-3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> ? = 0
[-1,2,3,4] => [1]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> ? = 0
[-1,2,3,-4] => [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[-1,2,-3,4] => [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[-1,2,-3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> ? = 0
[-1,-2,3,4] => [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[-1,-2,3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> ? = 0
[-1,-2,-3,4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> ? = 0
[-1,-2,-3,-4] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> ? = 1
[1,2,4,-3] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[1,2,-4,3] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[1,-2,4,3] => [1]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> ? = 0
[1,-2,4,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1
[1,-2,-4,3] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1
[1,-2,-4,-3] => [1]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> ? = 0
[-1,2,4,3] => [1]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> ? = 0
[-1,2,4,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1
[-1,2,-4,3] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1
[-1,2,-4,-3] => [1]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> ? = 0
[-1,-2,4,3] => [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(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,-2,-4,-3] => [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[1,3,2,-4] => [1]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> ? = 0
[1,3,-2,4] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[1,3,-2,-4] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1
[1,-3,2,4] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[1,-3,2,-4] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1
[1,-3,-2,-4] => [1]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> ? = 0
[-1,3,2,4] => [1]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> ? = 0
[-1,3,2,-4] => [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[-1,3,-2,4] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 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]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 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] => [1]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> ? = 0
[-1,-3,-2,-4] => [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[1,3,4,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 1
[1,3,-4,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 1
[1,-3,4,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 1
[1,-3,-4,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 1
[-1,3,4,2] => [1]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> ? = 0
[-1,3,4,-2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 2
[-1,3,-4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 2
[-1,3,-4,-2] => [1]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> ? = 0
[1,4,3,-2] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[1,4,-3,-2] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1
[1,-4,3,2] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[1,-4,-3,2] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1
[-1,4,3,-2] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1
[-1,4,-3,2] => [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[-1,-4,3,2] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1
[-1,-4,-3,-2] => [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[2,1,-3,-4] => [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[2,-1,3,4] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 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
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 20%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 20%
Values
[-1] => [1]
=> [1,0,1,0]
=> 1010 => 0
[1,-2] => [1]
=> [1,0,1,0]
=> 1010 => 0
[-1,2] => [1]
=> [1,0,1,0]
=> 1010 => 0
[-1,-2] => [1,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 1
[2,-1] => [2]
=> [1,1,0,0,1,0]
=> 110010 => ? = 1
[-2,1] => [2]
=> [1,1,0,0,1,0]
=> 110010 => ? = 1
[1,2,-3] => [1]
=> [1,0,1,0]
=> 1010 => 0
[1,-2,3] => [1]
=> [1,0,1,0]
=> 1010 => 0
[1,-2,-3] => [1,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 1
[-1,2,3] => [1]
=> [1,0,1,0]
=> 1010 => 0
[-1,2,-3] => [1,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 1
[-1,-2,3] => [1,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 1
[-1,-2,-3] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 0
[1,3,-2] => [2]
=> [1,1,0,0,1,0]
=> 110010 => ? = 1
[1,-3,2] => [2]
=> [1,1,0,0,1,0]
=> 110010 => ? = 1
[-1,3,2] => [1]
=> [1,0,1,0]
=> 1010 => 0
[-1,3,-2] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => ? = 1
[-1,-3,2] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => ? = 1
[-1,-3,-2] => [1]
=> [1,0,1,0]
=> 1010 => 0
[2,1,-3] => [1]
=> [1,0,1,0]
=> 1010 => 0
[2,-1,3] => [2]
=> [1,1,0,0,1,0]
=> 110010 => ? = 1
[2,-1,-3] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => ? = 1
[-2,1,3] => [2]
=> [1,1,0,0,1,0]
=> 110010 => ? = 1
[-2,1,-3] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => ? = 1
[-2,-1,-3] => [1]
=> [1,0,1,0]
=> 1010 => 0
[2,3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 1
[2,-3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 1
[-2,3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 1
[-2,-3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 1
[3,1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 1
[3,-1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 1
[-3,1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 1
[-3,-1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 1
[3,2,-1] => [2]
=> [1,1,0,0,1,0]
=> 110010 => ? = 1
[3,-2,1] => [1]
=> [1,0,1,0]
=> 1010 => 0
[3,-2,-1] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => ? = 1
[-3,2,1] => [2]
=> [1,1,0,0,1,0]
=> 110010 => ? = 1
[-3,-2,1] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => ? = 1
[-3,-2,-1] => [1]
=> [1,0,1,0]
=> 1010 => 0
[1,2,3,-4] => [1]
=> [1,0,1,0]
=> 1010 => 0
[1,2,-3,4] => [1]
=> [1,0,1,0]
=> 1010 => 0
[1,2,-3,-4] => [1,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 1
[1,-2,3,4] => [1]
=> [1,0,1,0]
=> 1010 => 0
[1,-2,3,-4] => [1,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 1
[1,-2,-3,4] => [1,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 1
[1,-2,-3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 0
[-1,2,3,4] => [1]
=> [1,0,1,0]
=> 1010 => 0
[-1,2,3,-4] => [1,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 1
[-1,2,-3,4] => [1,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 1
[-1,2,-3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 0
[-1,-2,3,4] => [1,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 1
[-1,-2,3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 0
[-1,-2,-3,4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 0
[-1,-2,-3,-4] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? = 1
[1,2,4,-3] => [2]
=> [1,1,0,0,1,0]
=> 110010 => ? = 1
[1,2,-4,3] => [2]
=> [1,1,0,0,1,0]
=> 110010 => ? = 1
[1,-2,4,3] => [1]
=> [1,0,1,0]
=> 1010 => 0
[1,-2,4,-3] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => ? = 1
[1,-2,-4,3] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => ? = 1
[1,-2,-4,-3] => [1]
=> [1,0,1,0]
=> 1010 => 0
[-1,2,4,3] => [1]
=> [1,0,1,0]
=> 1010 => 0
[-1,2,4,-3] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => ? = 1
[-1,2,-4,3] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => ? = 1
[-1,2,-4,-3] => [1]
=> [1,0,1,0]
=> 1010 => 0
[-1,-2,4,3] => [1,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 1
[-1,-2,4,-3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 1
[-1,-2,-4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 1
[-1,-2,-4,-3] => [1,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 1
[1,3,2,-4] => [1]
=> [1,0,1,0]
=> 1010 => 0
[1,3,-2,4] => [2]
=> [1,1,0,0,1,0]
=> 110010 => ? = 1
[1,3,-2,-4] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => ? = 1
[1,-3,-2,-4] => [1]
=> [1,0,1,0]
=> 1010 => 0
[-1,3,2,4] => [1]
=> [1,0,1,0]
=> 1010 => 0
[-1,-3,-2,4] => [1]
=> [1,0,1,0]
=> 1010 => 0
[-1,3,4,2] => [1]
=> [1,0,1,0]
=> 1010 => 0
[-1,3,-4,-2] => [1]
=> [1,0,1,0]
=> 1010 => 0
[-1,-3,4,-2] => [1]
=> [1,0,1,0]
=> 1010 => 0
[-1,-3,-4,2] => [1]
=> [1,0,1,0]
=> 1010 => 0
[-1,4,2,3] => [1]
=> [1,0,1,0]
=> 1010 => 0
[-1,4,-2,-3] => [1]
=> [1,0,1,0]
=> 1010 => 0
[-1,-4,2,-3] => [1]
=> [1,0,1,0]
=> 1010 => 0
[-1,-4,-2,3] => [1]
=> [1,0,1,0]
=> 1010 => 0
[1,4,-3,2] => [1]
=> [1,0,1,0]
=> 1010 => 0
[1,-4,-3,-2] => [1]
=> [1,0,1,0]
=> 1010 => 0
[-1,4,3,2] => [1]
=> [1,0,1,0]
=> 1010 => 0
[-1,-4,3,-2] => [1]
=> [1,0,1,0]
=> 1010 => 0
[2,1,3,-4] => [1]
=> [1,0,1,0]
=> 1010 => 0
[2,1,-3,4] => [1]
=> [1,0,1,0]
=> 1010 => 0
[-2,-1,3,-4] => [1]
=> [1,0,1,0]
=> 1010 => 0
[-2,-1,-3,4] => [1]
=> [1,0,1,0]
=> 1010 => 0
[2,3,1,-4] => [1]
=> [1,0,1,0]
=> 1010 => 0
[2,-3,-1,-4] => [1]
=> [1,0,1,0]
=> 1010 => 0
[-2,3,-1,-4] => [1]
=> [1,0,1,0]
=> 1010 => 0
[-2,-3,1,-4] => [1]
=> [1,0,1,0]
=> 1010 => 0
[2,4,-3,1] => [1]
=> [1,0,1,0]
=> 1010 => 0
[2,-4,-3,-1] => [1]
=> [1,0,1,0]
=> 1010 => 0
[-2,4,-3,-1] => [1]
=> [1,0,1,0]
=> 1010 => 0
[-2,-4,-3,1] => [1]
=> [1,0,1,0]
=> 1010 => 0
[3,1,2,-4] => [1]
=> [1,0,1,0]
=> 1010 => 0
[3,-1,-2,-4] => [1]
=> [1,0,1,0]
=> 1010 => 0
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.
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