Your data matches 12 different statistics following compositions of up to 3 maps.
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Matching statistic: St000478
Mapping
St000478: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
 => 1
[1,1]
 => 0
[3]
 => 2
[2,1]
 => 0
[1,1,1]
 => 0
[4]
 => 2
[3,1]
 => 1
[2,2]
 => -1
[2,1,1]
 => 0
[1,1,1,1]
 => 0
[5]
 => 3
[4,1]
 => 1
[3,2]
 => 0
[3,1,1]
 => 0
[2,2,1]
 => 0
[2,1,1,1]
 => 0
[1,1,1,1,1]
 => 0
[6]
 => 3
[5,1]
 => 2
[4,2]
 => 0
[4,1,1]
 => 0
[3,3]
 => 0
[3,2,1]
 => 0
[3,1,1,1]
 => 0
[2,2,2]
 => 1
[2,2,1,1]
 => 0
[2,1,1,1,1]
 => 0
[1,1,1,1,1,1]
 => 0
[7]
 => 4
[6,1]
 => 2
[5,2]
 => 1
[5,1,1]
 => 0
[4,3]
 => 0
[4,2,1]
 => 0
[4,1,1,1]
 => 0
[3,3,1]
 => 0
[3,2,2]
 => 0
[3,2,1,1]
 => 0
[3,1,1,1,1]
 => 0
[2,2,2,1]
 => 0
[2,2,1,1,1]
 => 0
[2,1,1,1,1,1]
 => 0
[1,1,1,1,1,1,1]
 => 0
[8]
 => 4
[7,1]
 => 3
[6,2]
 => 1
[6,1,1]
 => 0
[5,3]
 => 2
[5,2,1]
 => 0
[5,1,1,1]
 => 0
Description
Another weight of a partition according to Alladi. According to Theorem 3.4 (Alladi 2012) in [1] $$ \sum_{\pi\in GG_1(r)} w_1(\pi) $$ equals the number of partitions of $r$ whose odd parts are all distinct. $GG_1(r)$ is the set of partitions of $r$ where consecutive entries differ by at least $2$, and consecutive even entries differ by at least $4$.
Matching statistic: St001175
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001175: Integer partitions ⟶ ℤResult quality: 30% ●values known / values provided: 75%●distinct values known / distinct values provided: 30%
Values
[2]
 => []
 => ?
 => ?
 => ? ∊ {0,1}
[1,1]
 => [1]
 => [1]
 => []
 => ? ∊ {0,1}
[3]
 => []
 => ?
 => ?
 => ? ∊ {0,2}
[2,1]
 => [1]
 => [1]
 => []
 => ? ∊ {0,2}
[1,1,1]
 => [1,1]
 => [1,1]
 => [1]
 => 0
[4]
 => []
 => ?
 => ?
 => ? ∊ {-1,1,2}
[3,1]
 => [1]
 => [1]
 => []
 => ? ∊ {-1,1,2}
[2,2]
 => [2]
 => [2]
 => []
 => ? ∊ {-1,1,2}
[2,1,1]
 => [1,1]
 => [1,1]
 => [1]
 => 0
[1,1,1,1]
 => [1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[5]
 => []
 => ?
 => ?
 => ? ∊ {0,0,1,3}
[4,1]
 => [1]
 => [1]
 => []
 => ? ∊ {0,0,1,3}
[3,2]
 => [2]
 => [2]
 => []
 => ? ∊ {0,0,1,3}
[3,1,1]
 => [1,1]
 => [1,1]
 => [1]
 => 0
[2,2,1]
 => [2,1]
 => [3]
 => []
 => ? ∊ {0,0,1,3}
[2,1,1,1]
 => [1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[6]
 => []
 => ?
 => ?
 => ? ∊ {0,0,1,2,3}
[5,1]
 => [1]
 => [1]
 => []
 => ? ∊ {0,0,1,2,3}
[4,2]
 => [2]
 => [2]
 => []
 => ? ∊ {0,0,1,2,3}
[4,1,1]
 => [1,1]
 => [1,1]
 => [1]
 => 0
[3,3]
 => [3]
 => [2,1]
 => [1]
 => 0
[3,2,1]
 => [2,1]
 => [3]
 => []
 => ? ∊ {0,0,1,2,3}
[3,1,1,1]
 => [1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[2,2,2]
 => [2,2]
 => [4]
 => []
 => ? ∊ {0,0,1,2,3}
[2,2,1,1]
 => [2,1,1]
 => [3,1]
 => [1]
 => 0
[2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[7]
 => []
 => ?
 => ?
 => ? ∊ {0,0,0,1,2,4}
[6,1]
 => [1]
 => [1]
 => []
 => ? ∊ {0,0,0,1,2,4}
[5,2]
 => [2]
 => [2]
 => []
 => ? ∊ {0,0,0,1,2,4}
[5,1,1]
 => [1,1]
 => [1,1]
 => [1]
 => 0
[4,3]
 => [3]
 => [2,1]
 => [1]
 => 0
[4,2,1]
 => [2,1]
 => [3]
 => []
 => ? ∊ {0,0,0,1,2,4}
[4,1,1,1]
 => [1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[3,3,1]
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 0
[3,2,2]
 => [2,2]
 => [4]
 => []
 => ? ∊ {0,0,0,1,2,4}
[3,2,1,1]
 => [2,1,1]
 => [3,1]
 => [1]
 => 0
[3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[2,2,2,1]
 => [2,2,1]
 => [5]
 => []
 => ? ∊ {0,0,0,1,2,4}
[2,2,1,1,1]
 => [2,1,1,1]
 => [3,1,1]
 => [1,1]
 => 0
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 0
[8]
 => []
 => ?
 => ?
 => ? ∊ {-2,-1,0,1,2,3,4}
[7,1]
 => [1]
 => [1]
 => []
 => ? ∊ {-2,-1,0,1,2,3,4}
[6,2]
 => [2]
 => [2]
 => []
 => ? ∊ {-2,-1,0,1,2,3,4}
[6,1,1]
 => [1,1]
 => [1,1]
 => [1]
 => 0
[5,3]
 => [3]
 => [2,1]
 => [1]
 => 0
[5,2,1]
 => [2,1]
 => [3]
 => []
 => ? ∊ {-2,-1,0,1,2,3,4}
[5,1,1,1]
 => [1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[4,4]
 => [4]
 => [2,2]
 => [2]
 => 0
[4,3,1]
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 0
[4,2,2]
 => [2,2]
 => [4]
 => []
 => ? ∊ {-2,-1,0,1,2,3,4}
[4,2,1,1]
 => [2,1,1]
 => [3,1]
 => [1]
 => 0
[4,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[3,3,2]
 => [3,2]
 => [4,1]
 => [1]
 => 0
[3,3,1,1]
 => [3,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
[3,2,2,1]
 => [2,2,1]
 => [5]
 => []
 => ? ∊ {-2,-1,0,1,2,3,4}
[3,2,1,1,1]
 => [2,1,1,1]
 => [3,1,1]
 => [1,1]
 => 0
[3,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[2,2,2,2]
 => [2,2,2]
 => [6]
 => []
 => ? ∊ {-2,-1,0,1,2,3,4}
[2,2,2,1,1]
 => [2,2,1,1]
 => [5,1]
 => [1]
 => 0
[2,2,1,1,1,1]
 => [2,1,1,1,1]
 => [3,1,1,1]
 => [1,1,1]
 => 0
[2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 0
[1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 0
[9]
 => []
 => ?
 => ?
 => ? ∊ {-1,-1,0,1,2,2,3,5}
[8,1]
 => [1]
 => [1]
 => []
 => ? ∊ {-1,-1,0,1,2,2,3,5}
[7,2]
 => [2]
 => [2]
 => []
 => ? ∊ {-1,-1,0,1,2,2,3,5}
[7,1,1]
 => [1,1]
 => [1,1]
 => [1]
 => 0
[6,3]
 => [3]
 => [2,1]
 => [1]
 => 0
[6,2,1]
 => [2,1]
 => [3]
 => []
 => ? ∊ {-1,-1,0,1,2,2,3,5}
[6,1,1,1]
 => [1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[5,4]
 => [4]
 => [2,2]
 => [2]
 => 0
[5,3,1]
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 0
[5,2,2]
 => [2,2]
 => [4]
 => []
 => ? ∊ {-1,-1,0,1,2,2,3,5}
[5,2,1,1]
 => [2,1,1]
 => [3,1]
 => [1]
 => 0
[5,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[4,4,1]
 => [4,1]
 => [3,2]
 => [2]
 => 0
[4,3,2]
 => [3,2]
 => [4,1]
 => [1]
 => 0
[4,3,1,1]
 => [3,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
[4,2,2,1]
 => [2,2,1]
 => [5]
 => []
 => ? ∊ {-1,-1,0,1,2,2,3,5}
[4,2,1,1,1]
 => [2,1,1,1]
 => [3,1,1]
 => [1,1]
 => 0
[4,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[3,3,3]
 => [3,3]
 => [3,2,1]
 => [2,1]
 => 0
[3,3,2,1]
 => [3,2,1]
 => [3,3]
 => [3]
 => 0
[3,2,2,2]
 => [2,2,2]
 => [6]
 => []
 => ? ∊ {-1,-1,0,1,2,2,3,5}
[2,2,2,2,1]
 => [2,2,2,1]
 => [7]
 => []
 => ? ∊ {-1,-1,0,1,2,2,3,5}
[10]
 => []
 => ?
 => ?
 => ? ∊ {-1,0,0,1,1,2,4,4,5}
[9,1]
 => [1]
 => [1]
 => []
 => ? ∊ {-1,0,0,1,1,2,4,4,5}
[8,2]
 => [2]
 => [2]
 => []
 => ? ∊ {-1,0,0,1,1,2,4,4,5}
[7,2,1]
 => [2,1]
 => [3]
 => []
 => ? ∊ {-1,0,0,1,1,2,4,4,5}
[6,2,2]
 => [2,2]
 => [4]
 => []
 => ? ∊ {-1,0,0,1,1,2,4,4,5}
[5,2,2,1]
 => [2,2,1]
 => [5]
 => []
 => ? ∊ {-1,0,0,1,1,2,4,4,5}
[4,2,2,2]
 => [2,2,2]
 => [6]
 => []
 => ? ∊ {-1,0,0,1,1,2,4,4,5}
[3,2,2,2,1]
 => [2,2,2,1]
 => [7]
 => []
 => ? ∊ {-1,0,0,1,1,2,4,4,5}
[2,2,2,2,2]
 => [2,2,2,2]
 => [8]
 => []
 => ? ∊ {-1,0,0,1,1,2,4,4,5}
[11]
 => []
 => ?
 => ?
 => ? ∊ {-2,0,0,1,2,2,3,4,4,6}
[10,1]
 => [1]
 => [1]
 => []
 => ? ∊ {-2,0,0,1,2,2,3,4,4,6}
[9,2]
 => [2]
 => [2]
 => []
 => ? ∊ {-2,0,0,1,2,2,3,4,4,6}
[8,2,1]
 => [2,1]
 => [3]
 => []
 => ? ∊ {-2,0,0,1,2,2,3,4,4,6}
Description
The size of a partition minus the hook length of the base cell. This is, the number of boxes in the diagram of a partition that are neither in the first row nor in the first column.
Matching statistic: St000205
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000205: Integer partitions ⟶ ℤResult quality: 30% ●values known / values provided: 74%●distinct values known / distinct values provided: 30%
Values
[2]
 => []
 => ?
 => ?
 => ? ∊ {0,1}
[1,1]
 => [1]
 => [1]
 => []
 => ? ∊ {0,1}
[3]
 => []
 => ?
 => ?
 => ? ∊ {0,2}
[2,1]
 => [1]
 => [1]
 => []
 => ? ∊ {0,2}
[1,1,1]
 => [1,1]
 => [1,1]
 => [1]
 => 0
[4]
 => []
 => ?
 => ?
 => ? ∊ {-1,1,2}
[3,1]
 => [1]
 => [1]
 => []
 => ? ∊ {-1,1,2}
[2,2]
 => [2]
 => [2]
 => []
 => ? ∊ {-1,1,2}
[2,1,1]
 => [1,1]
 => [1,1]
 => [1]
 => 0
[1,1,1,1]
 => [1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[5]
 => []
 => ?
 => ?
 => ? ∊ {0,0,1,3}
[4,1]
 => [1]
 => [1]
 => []
 => ? ∊ {0,0,1,3}
[3,2]
 => [2]
 => [2]
 => []
 => ? ∊ {0,0,1,3}
[3,1,1]
 => [1,1]
 => [1,1]
 => [1]
 => 0
[2,2,1]
 => [2,1]
 => [3]
 => []
 => ? ∊ {0,0,1,3}
[2,1,1,1]
 => [1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[6]
 => []
 => ?
 => ?
 => ? ∊ {0,0,1,2,3}
[5,1]
 => [1]
 => [1]
 => []
 => ? ∊ {0,0,1,2,3}
[4,2]
 => [2]
 => [2]
 => []
 => ? ∊ {0,0,1,2,3}
[4,1,1]
 => [1,1]
 => [1,1]
 => [1]
 => 0
[3,3]
 => [3]
 => [2,1]
 => [1]
 => 0
[3,2,1]
 => [2,1]
 => [3]
 => []
 => ? ∊ {0,0,1,2,3}
[3,1,1,1]
 => [1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[2,2,2]
 => [2,2]
 => [4]
 => []
 => ? ∊ {0,0,1,2,3}
[2,2,1,1]
 => [2,1,1]
 => [3,1]
 => [1]
 => 0
[2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[7]
 => []
 => ?
 => ?
 => ? ∊ {0,0,0,1,2,4}
[6,1]
 => [1]
 => [1]
 => []
 => ? ∊ {0,0,0,1,2,4}
[5,2]
 => [2]
 => [2]
 => []
 => ? ∊ {0,0,0,1,2,4}
[5,1,1]
 => [1,1]
 => [1,1]
 => [1]
 => 0
[4,3]
 => [3]
 => [2,1]
 => [1]
 => 0
[4,2,1]
 => [2,1]
 => [3]
 => []
 => ? ∊ {0,0,0,1,2,4}
[4,1,1,1]
 => [1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[3,3,1]
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 0
[3,2,2]
 => [2,2]
 => [4]
 => []
 => ? ∊ {0,0,0,1,2,4}
[3,2,1,1]
 => [2,1,1]
 => [3,1]
 => [1]
 => 0
[3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[2,2,2,1]
 => [2,2,1]
 => [5]
 => []
 => ? ∊ {0,0,0,1,2,4}
[2,2,1,1,1]
 => [2,1,1,1]
 => [3,1,1]
 => [1,1]
 => 0
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 0
[8]
 => []
 => ?
 => ?
 => ? ∊ {-2,-1,0,1,2,3,4}
[7,1]
 => [1]
 => [1]
 => []
 => ? ∊ {-2,-1,0,1,2,3,4}
[6,2]
 => [2]
 => [2]
 => []
 => ? ∊ {-2,-1,0,1,2,3,4}
[6,1,1]
 => [1,1]
 => [1,1]
 => [1]
 => 0
[5,3]
 => [3]
 => [2,1]
 => [1]
 => 0
[5,2,1]
 => [2,1]
 => [3]
 => []
 => ? ∊ {-2,-1,0,1,2,3,4}
[5,1,1,1]
 => [1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[4,4]
 => [4]
 => [2,2]
 => [2]
 => 0
[4,3,1]
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 0
[4,2,2]
 => [2,2]
 => [4]
 => []
 => ? ∊ {-2,-1,0,1,2,3,4}
[4,2,1,1]
 => [2,1,1]
 => [3,1]
 => [1]
 => 0
[4,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[3,3,2]
 => [3,2]
 => [4,1]
 => [1]
 => 0
[3,3,1,1]
 => [3,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
[3,2,2,1]
 => [2,2,1]
 => [5]
 => []
 => ? ∊ {-2,-1,0,1,2,3,4}
[3,2,1,1,1]
 => [2,1,1,1]
 => [3,1,1]
 => [1,1]
 => 0
[3,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[2,2,2,2]
 => [2,2,2]
 => [6]
 => []
 => ? ∊ {-2,-1,0,1,2,3,4}
[2,2,2,1,1]
 => [2,2,1,1]
 => [5,1]
 => [1]
 => 0
[2,2,1,1,1,1]
 => [2,1,1,1,1]
 => [3,1,1,1]
 => [1,1,1]
 => 0
[2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 0
[1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 0
[9]
 => []
 => ?
 => ?
 => ? ∊ {-1,-1,0,1,2,2,3,5}
[8,1]
 => [1]
 => [1]
 => []
 => ? ∊ {-1,-1,0,1,2,2,3,5}
[7,2]
 => [2]
 => [2]
 => []
 => ? ∊ {-1,-1,0,1,2,2,3,5}
[7,1,1]
 => [1,1]
 => [1,1]
 => [1]
 => 0
[6,3]
 => [3]
 => [2,1]
 => [1]
 => 0
[6,2,1]
 => [2,1]
 => [3]
 => []
 => ? ∊ {-1,-1,0,1,2,2,3,5}
[6,1,1,1]
 => [1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[5,4]
 => [4]
 => [2,2]
 => [2]
 => 0
[5,3,1]
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 0
[5,2,2]
 => [2,2]
 => [4]
 => []
 => ? ∊ {-1,-1,0,1,2,2,3,5}
[5,2,1,1]
 => [2,1,1]
 => [3,1]
 => [1]
 => 0
[5,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[4,4,1]
 => [4,1]
 => [3,2]
 => [2]
 => 0
[4,3,2]
 => [3,2]
 => [4,1]
 => [1]
 => 0
[4,3,1,1]
 => [3,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
[4,2,2,1]
 => [2,2,1]
 => [5]
 => []
 => ? ∊ {-1,-1,0,1,2,2,3,5}
[4,2,1,1,1]
 => [2,1,1,1]
 => [3,1,1]
 => [1,1]
 => 0
[4,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[3,3,3]
 => [3,3]
 => [3,2,1]
 => [2,1]
 => 0
[3,3,2,1]
 => [3,2,1]
 => [3,3]
 => [3]
 => 0
[3,2,2,2]
 => [2,2,2]
 => [6]
 => []
 => ? ∊ {-1,-1,0,1,2,2,3,5}
[2,2,2,2,1]
 => [2,2,2,1]
 => [7]
 => []
 => ? ∊ {-1,-1,0,1,2,2,3,5}
[10]
 => []
 => ?
 => ?
 => ? ∊ {-1,0,0,1,1,2,4,4,5}
[9,1]
 => [1]
 => [1]
 => []
 => ? ∊ {-1,0,0,1,1,2,4,4,5}
[8,2]
 => [2]
 => [2]
 => []
 => ? ∊ {-1,0,0,1,1,2,4,4,5}
[7,2,1]
 => [2,1]
 => [3]
 => []
 => ? ∊ {-1,0,0,1,1,2,4,4,5}
[6,2,2]
 => [2,2]
 => [4]
 => []
 => ? ∊ {-1,0,0,1,1,2,4,4,5}
[5,2,2,1]
 => [2,2,1]
 => [5]
 => []
 => ? ∊ {-1,0,0,1,1,2,4,4,5}
[4,2,2,2]
 => [2,2,2]
 => [6]
 => []
 => ? ∊ {-1,0,0,1,1,2,4,4,5}
[3,2,2,2,1]
 => [2,2,2,1]
 => [7]
 => []
 => ? ∊ {-1,0,0,1,1,2,4,4,5}
[2,2,2,2,2]
 => [2,2,2,2]
 => [8]
 => []
 => ? ∊ {-1,0,0,1,1,2,4,4,5}
[11]
 => []
 => ?
 => ?
 => ? ∊ {-2,0,0,0,1,2,2,3,4,4,6}
[10,1]
 => [1]
 => [1]
 => []
 => ? ∊ {-2,0,0,0,1,2,2,3,4,4,6}
[9,2]
 => [2]
 => [2]
 => []
 => ? ∊ {-2,0,0,0,1,2,2,3,4,4,6}
[8,2,1]
 => [2,1]
 => [3]
 => []
 => ? ∊ {-2,0,0,0,1,2,2,3,4,4,6}
Description
Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. Given $\lambda$ count how many ''integer partitions'' $w$ (weight) there are, such that $P_{\lambda,w}$ is non-integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has at least one non-integral vertex.
Matching statistic: St000206
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000206: Integer partitions ⟶ ℤResult quality: 30% ●values known / values provided: 74%●distinct values known / distinct values provided: 30%
Values
[2]
 => []
 => ?
 => ?
 => ? ∊ {0,1}
[1,1]
 => [1]
 => [1]
 => []
 => ? ∊ {0,1}
[3]
 => []
 => ?
 => ?
 => ? ∊ {0,2}
[2,1]
 => [1]
 => [1]
 => []
 => ? ∊ {0,2}
[1,1,1]
 => [1,1]
 => [1,1]
 => [1]
 => 0
[4]
 => []
 => ?
 => ?
 => ? ∊ {-1,1,2}
[3,1]
 => [1]
 => [1]
 => []
 => ? ∊ {-1,1,2}
[2,2]
 => [2]
 => [2]
 => []
 => ? ∊ {-1,1,2}
[2,1,1]
 => [1,1]
 => [1,1]
 => [1]
 => 0
[1,1,1,1]
 => [1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[5]
 => []
 => ?
 => ?
 => ? ∊ {0,0,1,3}
[4,1]
 => [1]
 => [1]
 => []
 => ? ∊ {0,0,1,3}
[3,2]
 => [2]
 => [2]
 => []
 => ? ∊ {0,0,1,3}
[3,1,1]
 => [1,1]
 => [1,1]
 => [1]
 => 0
[2,2,1]
 => [2,1]
 => [3]
 => []
 => ? ∊ {0,0,1,3}
[2,1,1,1]
 => [1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[6]
 => []
 => ?
 => ?
 => ? ∊ {0,0,1,2,3}
[5,1]
 => [1]
 => [1]
 => []
 => ? ∊ {0,0,1,2,3}
[4,2]
 => [2]
 => [2]
 => []
 => ? ∊ {0,0,1,2,3}
[4,1,1]
 => [1,1]
 => [1,1]
 => [1]
 => 0
[3,3]
 => [3]
 => [2,1]
 => [1]
 => 0
[3,2,1]
 => [2,1]
 => [3]
 => []
 => ? ∊ {0,0,1,2,3}
[3,1,1,1]
 => [1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[2,2,2]
 => [2,2]
 => [4]
 => []
 => ? ∊ {0,0,1,2,3}
[2,2,1,1]
 => [2,1,1]
 => [3,1]
 => [1]
 => 0
[2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[7]
 => []
 => ?
 => ?
 => ? ∊ {0,0,0,1,2,4}
[6,1]
 => [1]
 => [1]
 => []
 => ? ∊ {0,0,0,1,2,4}
[5,2]
 => [2]
 => [2]
 => []
 => ? ∊ {0,0,0,1,2,4}
[5,1,1]
 => [1,1]
 => [1,1]
 => [1]
 => 0
[4,3]
 => [3]
 => [2,1]
 => [1]
 => 0
[4,2,1]
 => [2,1]
 => [3]
 => []
 => ? ∊ {0,0,0,1,2,4}
[4,1,1,1]
 => [1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[3,3,1]
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 0
[3,2,2]
 => [2,2]
 => [4]
 => []
 => ? ∊ {0,0,0,1,2,4}
[3,2,1,1]
 => [2,1,1]
 => [3,1]
 => [1]
 => 0
[3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[2,2,2,1]
 => [2,2,1]
 => [5]
 => []
 => ? ∊ {0,0,0,1,2,4}
[2,2,1,1,1]
 => [2,1,1,1]
 => [3,1,1]
 => [1,1]
 => 0
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 0
[8]
 => []
 => ?
 => ?
 => ? ∊ {-2,-1,0,1,2,3,4}
[7,1]
 => [1]
 => [1]
 => []
 => ? ∊ {-2,-1,0,1,2,3,4}
[6,2]
 => [2]
 => [2]
 => []
 => ? ∊ {-2,-1,0,1,2,3,4}
[6,1,1]
 => [1,1]
 => [1,1]
 => [1]
 => 0
[5,3]
 => [3]
 => [2,1]
 => [1]
 => 0
[5,2,1]
 => [2,1]
 => [3]
 => []
 => ? ∊ {-2,-1,0,1,2,3,4}
[5,1,1,1]
 => [1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[4,4]
 => [4]
 => [2,2]
 => [2]
 => 0
[4,3,1]
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 0
[4,2,2]
 => [2,2]
 => [4]
 => []
 => ? ∊ {-2,-1,0,1,2,3,4}
[4,2,1,1]
 => [2,1,1]
 => [3,1]
 => [1]
 => 0
[4,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[3,3,2]
 => [3,2]
 => [4,1]
 => [1]
 => 0
[3,3,1,1]
 => [3,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
[3,2,2,1]
 => [2,2,1]
 => [5]
 => []
 => ? ∊ {-2,-1,0,1,2,3,4}
[3,2,1,1,1]
 => [2,1,1,1]
 => [3,1,1]
 => [1,1]
 => 0
[3,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[2,2,2,2]
 => [2,2,2]
 => [6]
 => []
 => ? ∊ {-2,-1,0,1,2,3,4}
[2,2,2,1,1]
 => [2,2,1,1]
 => [5,1]
 => [1]
 => 0
[2,2,1,1,1,1]
 => [2,1,1,1,1]
 => [3,1,1,1]
 => [1,1,1]
 => 0
[2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 0
[1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 0
[9]
 => []
 => ?
 => ?
 => ? ∊ {-1,-1,0,1,2,2,3,5}
[8,1]
 => [1]
 => [1]
 => []
 => ? ∊ {-1,-1,0,1,2,2,3,5}
[7,2]
 => [2]
 => [2]
 => []
 => ? ∊ {-1,-1,0,1,2,2,3,5}
[7,1,1]
 => [1,1]
 => [1,1]
 => [1]
 => 0
[6,3]
 => [3]
 => [2,1]
 => [1]
 => 0
[6,2,1]
 => [2,1]
 => [3]
 => []
 => ? ∊ {-1,-1,0,1,2,2,3,5}
[6,1,1,1]
 => [1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[5,4]
 => [4]
 => [2,2]
 => [2]
 => 0
[5,3,1]
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 0
[5,2,2]
 => [2,2]
 => [4]
 => []
 => ? ∊ {-1,-1,0,1,2,2,3,5}
[5,2,1,1]
 => [2,1,1]
 => [3,1]
 => [1]
 => 0
[5,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[4,4,1]
 => [4,1]
 => [3,2]
 => [2]
 => 0
[4,3,2]
 => [3,2]
 => [4,1]
 => [1]
 => 0
[4,3,1,1]
 => [3,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
[4,2,2,1]
 => [2,2,1]
 => [5]
 => []
 => ? ∊ {-1,-1,0,1,2,2,3,5}
[4,2,1,1,1]
 => [2,1,1,1]
 => [3,1,1]
 => [1,1]
 => 0
[4,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[3,3,3]
 => [3,3]
 => [3,2,1]
 => [2,1]
 => 0
[3,3,2,1]
 => [3,2,1]
 => [3,3]
 => [3]
 => 0
[3,2,2,2]
 => [2,2,2]
 => [6]
 => []
 => ? ∊ {-1,-1,0,1,2,2,3,5}
[2,2,2,2,1]
 => [2,2,2,1]
 => [7]
 => []
 => ? ∊ {-1,-1,0,1,2,2,3,5}
[10]
 => []
 => ?
 => ?
 => ? ∊ {-1,0,0,1,1,2,4,4,5}
[9,1]
 => [1]
 => [1]
 => []
 => ? ∊ {-1,0,0,1,1,2,4,4,5}
[8,2]
 => [2]
 => [2]
 => []
 => ? ∊ {-1,0,0,1,1,2,4,4,5}
[7,2,1]
 => [2,1]
 => [3]
 => []
 => ? ∊ {-1,0,0,1,1,2,4,4,5}
[6,2,2]
 => [2,2]
 => [4]
 => []
 => ? ∊ {-1,0,0,1,1,2,4,4,5}
[5,2,2,1]
 => [2,2,1]
 => [5]
 => []
 => ? ∊ {-1,0,0,1,1,2,4,4,5}
[4,2,2,2]
 => [2,2,2]
 => [6]
 => []
 => ? ∊ {-1,0,0,1,1,2,4,4,5}
[3,2,2,2,1]
 => [2,2,2,1]
 => [7]
 => []
 => ? ∊ {-1,0,0,1,1,2,4,4,5}
[2,2,2,2,2]
 => [2,2,2,2]
 => [8]
 => []
 => ? ∊ {-1,0,0,1,1,2,4,4,5}
[11]
 => []
 => ?
 => ?
 => ? ∊ {-2,0,0,0,1,2,2,3,4,4,6}
[10,1]
 => [1]
 => [1]
 => []
 => ? ∊ {-2,0,0,0,1,2,2,3,4,4,6}
[9,2]
 => [2]
 => [2]
 => []
 => ? ∊ {-2,0,0,0,1,2,2,3,4,4,6}
[8,2,1]
 => [2,1]
 => [3]
 => []
 => ? ∊ {-2,0,0,0,1,2,2,3,4,4,6}
Description
Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. Given $\lambda$ count how many ''integer compositions'' $w$ (weight) there are, such that $P_{\lambda,w}$ is non-integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has at least one non-integral vertex. See also [[St000205]]. Each value in this statistic is greater than or equal to corresponding value in [[St000205]].
Matching statistic: St001140
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001140: Dyck paths ⟶ ℤResult quality: 30% ●values known / values provided: 73%●distinct values known / distinct values provided: 30%
Values
[2]
 => []
 => ?
 => ?
 => ? ∊ {0,1}
[1,1]
 => [1]
 => []
 => []
 => ? ∊ {0,1}
[3]
 => []
 => ?
 => ?
 => ? ∊ {0,2}
[2,1]
 => [1]
 => []
 => []
 => ? ∊ {0,2}
[1,1,1]
 => [1,1]
 => [1]
 => [1,0,1,0]
 => 0
[4]
 => []
 => ?
 => ?
 => ? ∊ {-1,1,2}
[3,1]
 => [1]
 => []
 => []
 => ? ∊ {-1,1,2}
[2,2]
 => [2]
 => []
 => []
 => ? ∊ {-1,1,2}
[2,1,1]
 => [1,1]
 => [1]
 => [1,0,1,0]
 => 0
[1,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[5]
 => []
 => ?
 => ?
 => ? ∊ {0,1,3}
[4,1]
 => [1]
 => []
 => []
 => ? ∊ {0,1,3}
[3,2]
 => [2]
 => []
 => []
 => ? ∊ {0,1,3}
[3,1,1]
 => [1,1]
 => [1]
 => [1,0,1,0]
 => 0
[2,2,1]
 => [2,1]
 => [1]
 => [1,0,1,0]
 => 0
[2,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 0
[6]
 => []
 => ?
 => ?
 => ? ∊ {0,1,2,3}
[5,1]
 => [1]
 => []
 => []
 => ? ∊ {0,1,2,3}
[4,2]
 => [2]
 => []
 => []
 => ? ∊ {0,1,2,3}
[4,1,1]
 => [1,1]
 => [1]
 => [1,0,1,0]
 => 0
[3,3]
 => [3]
 => []
 => []
 => ? ∊ {0,1,2,3}
[3,2,1]
 => [2,1]
 => [1]
 => [1,0,1,0]
 => 0
[3,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[2,2,2]
 => [2,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0
[2,2,1,1]
 => [2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 0
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0
[7]
 => []
 => ?
 => ?
 => ? ∊ {0,1,2,4}
[6,1]
 => [1]
 => []
 => []
 => ? ∊ {0,1,2,4}
[5,2]
 => [2]
 => []
 => []
 => ? ∊ {0,1,2,4}
[5,1,1]
 => [1,1]
 => [1]
 => [1,0,1,0]
 => 0
[4,3]
 => [3]
 => []
 => []
 => ? ∊ {0,1,2,4}
[4,2,1]
 => [2,1]
 => [1]
 => [1,0,1,0]
 => 0
[4,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[3,3,1]
 => [3,1]
 => [1]
 => [1,0,1,0]
 => 0
[3,2,2]
 => [2,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0
[3,2,1,1]
 => [2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 0
[2,2,2,1]
 => [2,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => 0
[2,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 0
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 0
[8]
 => []
 => ?
 => ?
 => ? ∊ {-2,-1,1,2,3,4}
[7,1]
 => [1]
 => []
 => []
 => ? ∊ {-2,-1,1,2,3,4}
[6,2]
 => [2]
 => []
 => []
 => ? ∊ {-2,-1,1,2,3,4}
[6,1,1]
 => [1,1]
 => [1]
 => [1,0,1,0]
 => 0
[5,3]
 => [3]
 => []
 => []
 => ? ∊ {-2,-1,1,2,3,4}
[5,2,1]
 => [2,1]
 => [1]
 => [1,0,1,0]
 => 0
[5,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[4,4]
 => [4]
 => []
 => []
 => ? ∊ {-2,-1,1,2,3,4}
[4,3,1]
 => [3,1]
 => [1]
 => [1,0,1,0]
 => 0
[4,2,2]
 => [2,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0
[4,2,1,1]
 => [2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[4,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 0
[3,3,2]
 => [3,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0
[3,3,1,1]
 => [3,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[3,2,2,1]
 => [2,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => 0
[3,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 0
[3,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0
[2,2,2,2]
 => [2,2,2]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 0
[2,2,2,1,1]
 => [2,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 0
[2,2,1,1,1,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0
[2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 0
[1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {-2,-1,1,2,3,4}
[9]
 => []
 => ?
 => ?
 => ? ∊ {-1,-1,1,2,2,3,5}
[8,1]
 => [1]
 => []
 => []
 => ? ∊ {-1,-1,1,2,2,3,5}
[7,2]
 => [2]
 => []
 => []
 => ? ∊ {-1,-1,1,2,2,3,5}
[7,1,1]
 => [1,1]
 => [1]
 => [1,0,1,0]
 => 0
[6,3]
 => [3]
 => []
 => []
 => ? ∊ {-1,-1,1,2,2,3,5}
[6,2,1]
 => [2,1]
 => [1]
 => [1,0,1,0]
 => 0
[6,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[5,4]
 => [4]
 => []
 => []
 => ? ∊ {-1,-1,1,2,2,3,5}
[5,3,1]
 => [3,1]
 => [1]
 => [1,0,1,0]
 => 0
[5,2,2]
 => [2,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0
[5,2,1,1]
 => [2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[5,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 0
[4,4,1]
 => [4,1]
 => [1]
 => [1,0,1,0]
 => 0
[4,3,2]
 => [3,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0
[2,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {-1,-1,1,2,2,3,5}
[1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? ∊ {-1,-1,1,2,2,3,5}
[10]
 => []
 => ?
 => ?
 => ? ∊ {-1,0,0,0,1,1,2,4,4,5}
[9,1]
 => [1]
 => []
 => []
 => ? ∊ {-1,0,0,0,1,1,2,4,4,5}
[8,2]
 => [2]
 => []
 => []
 => ? ∊ {-1,0,0,0,1,1,2,4,4,5}
[7,3]
 => [3]
 => []
 => []
 => ? ∊ {-1,0,0,0,1,1,2,4,4,5}
[6,4]
 => [4]
 => []
 => []
 => ? ∊ {-1,0,0,0,1,1,2,4,4,5}
[5,5]
 => [5]
 => []
 => []
 => ? ∊ {-1,0,0,0,1,1,2,4,4,5}
[3,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {-1,0,0,0,1,1,2,4,4,5}
[2,2,1,1,1,1,1,1]
 => [2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {-1,0,0,0,1,1,2,4,4,5}
[2,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? ∊ {-1,0,0,0,1,1,2,4,4,5}
[1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {-1,0,0,0,1,1,2,4,4,5}
[11]
 => []
 => ?
 => ?
 => ? ∊ {-2,0,0,0,0,0,0,2,2,3,4,4,6}
[10,1]
 => [1]
 => []
 => []
 => ? ∊ {-2,0,0,0,0,0,0,2,2,3,4,4,6}
[9,2]
 => [2]
 => []
 => []
 => ? ∊ {-2,0,0,0,0,0,0,2,2,3,4,4,6}
[8,3]
 => [3]
 => []
 => []
 => ? ∊ {-2,0,0,0,0,0,0,2,2,3,4,4,6}
[7,4]
 => [4]
 => []
 => []
 => ? ∊ {-2,0,0,0,0,0,0,2,2,3,4,4,6}
[6,5]
 => [5]
 => []
 => []
 => ? ∊ {-2,0,0,0,0,0,0,2,2,3,4,4,6}
[4,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {-2,0,0,0,0,0,0,2,2,3,4,4,6}
[3,2,1,1,1,1,1,1]
 => [2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {-2,0,0,0,0,0,0,2,2,3,4,4,6}
[3,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? ∊ {-2,0,0,0,0,0,0,2,2,3,4,4,6}
Description
Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra.
Matching statistic: St001292
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001292: Dyck paths ⟶ ℤResult quality: 20% ●values known / values provided: 73%●distinct values known / distinct values provided: 20%
Values
[2]
 => []
 => ?
 => ?
 => ? ∊ {0,1}
[1,1]
 => [1]
 => []
 => []
 => ? ∊ {0,1}
[3]
 => []
 => ?
 => ?
 => ? ∊ {0,2}
[2,1]
 => [1]
 => []
 => []
 => ? ∊ {0,2}
[1,1,1]
 => [1,1]
 => [1]
 => [1,0,1,0]
 => 0
[4]
 => []
 => ?
 => ?
 => ? ∊ {-1,1,2}
[3,1]
 => [1]
 => []
 => []
 => ? ∊ {-1,1,2}
[2,2]
 => [2]
 => []
 => []
 => ? ∊ {-1,1,2}
[2,1,1]
 => [1,1]
 => [1]
 => [1,0,1,0]
 => 0
[1,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[5]
 => []
 => ?
 => ?
 => ? ∊ {0,1,3}
[4,1]
 => [1]
 => []
 => []
 => ? ∊ {0,1,3}
[3,2]
 => [2]
 => []
 => []
 => ? ∊ {0,1,3}
[3,1,1]
 => [1,1]
 => [1]
 => [1,0,1,0]
 => 0
[2,2,1]
 => [2,1]
 => [1]
 => [1,0,1,0]
 => 0
[2,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 0
[6]
 => []
 => ?
 => ?
 => ? ∊ {0,1,2,3}
[5,1]
 => [1]
 => []
 => []
 => ? ∊ {0,1,2,3}
[4,2]
 => [2]
 => []
 => []
 => ? ∊ {0,1,2,3}
[4,1,1]
 => [1,1]
 => [1]
 => [1,0,1,0]
 => 0
[3,3]
 => [3]
 => []
 => []
 => ? ∊ {0,1,2,3}
[3,2,1]
 => [2,1]
 => [1]
 => [1,0,1,0]
 => 0
[3,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[2,2,2]
 => [2,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0
[2,2,1,1]
 => [2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 0
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0
[7]
 => []
 => ?
 => ?
 => ? ∊ {0,1,2,4}
[6,1]
 => [1]
 => []
 => []
 => ? ∊ {0,1,2,4}
[5,2]
 => [2]
 => []
 => []
 => ? ∊ {0,1,2,4}
[5,1,1]
 => [1,1]
 => [1]
 => [1,0,1,0]
 => 0
[4,3]
 => [3]
 => []
 => []
 => ? ∊ {0,1,2,4}
[4,2,1]
 => [2,1]
 => [1]
 => [1,0,1,0]
 => 0
[4,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[3,3,1]
 => [3,1]
 => [1]
 => [1,0,1,0]
 => 0
[3,2,2]
 => [2,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0
[3,2,1,1]
 => [2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 0
[2,2,2,1]
 => [2,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => 0
[2,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 0
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 0
[8]
 => []
 => ?
 => ?
 => ? ∊ {-2,-1,1,2,3,4}
[7,1]
 => [1]
 => []
 => []
 => ? ∊ {-2,-1,1,2,3,4}
[6,2]
 => [2]
 => []
 => []
 => ? ∊ {-2,-1,1,2,3,4}
[6,1,1]
 => [1,1]
 => [1]
 => [1,0,1,0]
 => 0
[5,3]
 => [3]
 => []
 => []
 => ? ∊ {-2,-1,1,2,3,4}
[5,2,1]
 => [2,1]
 => [1]
 => [1,0,1,0]
 => 0
[5,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[4,4]
 => [4]
 => []
 => []
 => ? ∊ {-2,-1,1,2,3,4}
[4,3,1]
 => [3,1]
 => [1]
 => [1,0,1,0]
 => 0
[4,2,2]
 => [2,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0
[4,2,1,1]
 => [2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[4,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 0
[3,3,2]
 => [3,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0
[3,3,1,1]
 => [3,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[3,2,2,1]
 => [2,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => 0
[3,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 0
[3,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0
[2,2,2,2]
 => [2,2,2]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 0
[2,2,2,1,1]
 => [2,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 0
[2,2,1,1,1,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0
[2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 0
[1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {-2,-1,1,2,3,4}
[9]
 => []
 => ?
 => ?
 => ? ∊ {-1,-1,1,2,2,3,5}
[8,1]
 => [1]
 => []
 => []
 => ? ∊ {-1,-1,1,2,2,3,5}
[7,2]
 => [2]
 => []
 => []
 => ? ∊ {-1,-1,1,2,2,3,5}
[7,1,1]
 => [1,1]
 => [1]
 => [1,0,1,0]
 => 0
[6,3]
 => [3]
 => []
 => []
 => ? ∊ {-1,-1,1,2,2,3,5}
[6,2,1]
 => [2,1]
 => [1]
 => [1,0,1,0]
 => 0
[6,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[5,4]
 => [4]
 => []
 => []
 => ? ∊ {-1,-1,1,2,2,3,5}
[5,3,1]
 => [3,1]
 => [1]
 => [1,0,1,0]
 => 0
[5,2,2]
 => [2,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0
[5,2,1,1]
 => [2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[5,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 0
[4,4,1]
 => [4,1]
 => [1]
 => [1,0,1,0]
 => 0
[4,3,2]
 => [3,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0
[2,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {-1,-1,1,2,2,3,5}
[1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? ∊ {-1,-1,1,2,2,3,5}
[10]
 => []
 => ?
 => ?
 => ? ∊ {-1,0,0,0,1,1,2,4,4,5}
[9,1]
 => [1]
 => []
 => []
 => ? ∊ {-1,0,0,0,1,1,2,4,4,5}
[8,2]
 => [2]
 => []
 => []
 => ? ∊ {-1,0,0,0,1,1,2,4,4,5}
[7,3]
 => [3]
 => []
 => []
 => ? ∊ {-1,0,0,0,1,1,2,4,4,5}
[6,4]
 => [4]
 => []
 => []
 => ? ∊ {-1,0,0,0,1,1,2,4,4,5}
[5,5]
 => [5]
 => []
 => []
 => ? ∊ {-1,0,0,0,1,1,2,4,4,5}
[3,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {-1,0,0,0,1,1,2,4,4,5}
[2,2,1,1,1,1,1,1]
 => [2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {-1,0,0,0,1,1,2,4,4,5}
[2,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? ∊ {-1,0,0,0,1,1,2,4,4,5}
[1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {-1,0,0,0,1,1,2,4,4,5}
[11]
 => []
 => ?
 => ?
 => ? ∊ {-2,0,0,0,0,0,0,2,2,3,4,4,6}
[10,1]
 => [1]
 => []
 => []
 => ? ∊ {-2,0,0,0,0,0,0,2,2,3,4,4,6}
[9,2]
 => [2]
 => []
 => []
 => ? ∊ {-2,0,0,0,0,0,0,2,2,3,4,4,6}
[8,3]
 => [3]
 => []
 => []
 => ? ∊ {-2,0,0,0,0,0,0,2,2,3,4,4,6}
[7,4]
 => [4]
 => []
 => []
 => ? ∊ {-2,0,0,0,0,0,0,2,2,3,4,4,6}
[6,5]
 => [5]
 => []
 => []
 => ? ∊ {-2,0,0,0,0,0,0,2,2,3,4,4,6}
[4,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {-2,0,0,0,0,0,0,2,2,3,4,4,6}
[3,2,1,1,1,1,1,1]
 => [2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {-2,0,0,0,0,0,0,2,2,3,4,4,6}
[3,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? ∊ {-2,0,0,0,0,0,0,2,2,3,4,4,6}
Description
The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. Here $A$ is the Nakayama algebra associated to a Dyck path as given in [[DyckPaths/NakayamaAlgebras]].
Matching statistic: St000980
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000980: Dyck paths ⟶ ℤResult quality: 30% ●values known / values provided: 69%●distinct values known / distinct values provided: 30%
Values
[2]
 => []
 => ?
 => ?
 => ? ∊ {0,1}
[1,1]
 => [1]
 => []
 => []
 => ? ∊ {0,1}
[3]
 => []
 => ?
 => ?
 => ? ∊ {0,0,2}
[2,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,2}
[1,1,1]
 => [1,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,2}
[4]
 => []
 => ?
 => ?
 => ? ∊ {-1,0,1,2}
[3,1]
 => [1]
 => []
 => []
 => ? ∊ {-1,0,1,2}
[2,2]
 => [2]
 => []
 => []
 => ? ∊ {-1,0,1,2}
[2,1,1]
 => [1,1]
 => [1]
 => [1,0]
 => ? ∊ {-1,0,1,2}
[1,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[5]
 => []
 => ?
 => ?
 => ? ∊ {0,0,0,1,3}
[4,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,0,1,3}
[3,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,0,1,3}
[3,1,1]
 => [1,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,0,1,3}
[2,2,1]
 => [2,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,0,1,3}
[2,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
[6]
 => []
 => ?
 => ?
 => ? ∊ {0,0,0,1,2,3}
[5,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,0,1,2,3}
[4,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,0,1,2,3}
[4,1,1]
 => [1,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,0,1,2,3}
[3,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,1,2,3}
[3,2,1]
 => [2,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,0,1,2,3}
[3,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[2,2,2]
 => [2,2]
 => [2]
 => [1,0,1,0]
 => 0
[2,2,1,1]
 => [2,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 0
[7]
 => []
 => ?
 => ?
 => ? ∊ {0,0,0,0,1,2,4}
[6,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,0,0,1,2,4}
[5,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,0,0,1,2,4}
[5,1,1]
 => [1,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,0,0,1,2,4}
[4,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,0,1,2,4}
[4,2,1]
 => [2,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,0,0,1,2,4}
[4,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[3,3,1]
 => [3,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,0,0,1,2,4}
[3,2,2]
 => [2,2]
 => [2]
 => [1,0,1,0]
 => 0
[3,2,1,1]
 => [2,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
[2,2,2,1]
 => [2,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 0
[2,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 0
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
[8]
 => []
 => ?
 => ?
 => ? ∊ {-2,-1,0,0,1,2,3,4}
[7,1]
 => [1]
 => []
 => []
 => ? ∊ {-2,-1,0,0,1,2,3,4}
[6,2]
 => [2]
 => []
 => []
 => ? ∊ {-2,-1,0,0,1,2,3,4}
[6,1,1]
 => [1,1]
 => [1]
 => [1,0]
 => ? ∊ {-2,-1,0,0,1,2,3,4}
[5,3]
 => [3]
 => []
 => []
 => ? ∊ {-2,-1,0,0,1,2,3,4}
[5,2,1]
 => [2,1]
 => [1]
 => [1,0]
 => ? ∊ {-2,-1,0,0,1,2,3,4}
[5,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[4,4]
 => [4]
 => []
 => []
 => ? ∊ {-2,-1,0,0,1,2,3,4}
[4,3,1]
 => [3,1]
 => [1]
 => [1,0]
 => ? ∊ {-2,-1,0,0,1,2,3,4}
[4,2,2]
 => [2,2]
 => [2]
 => [1,0,1,0]
 => 0
[4,2,1,1]
 => [2,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[4,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
[3,3,2]
 => [3,2]
 => [2]
 => [1,0,1,0]
 => 0
[3,3,1,1]
 => [3,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[3,2,2,1]
 => [2,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 0
[3,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
[3,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 0
[2,2,2,2]
 => [2,2,2]
 => [2,2]
 => [1,1,1,0,0,0]
 => 0
[2,2,2,1,1]
 => [2,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 0
[2,2,1,1,1,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 0
[2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
[1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 0
[9]
 => []
 => ?
 => ?
 => ? ∊ {-1,-1,0,0,1,2,2,3,5}
[8,1]
 => [1]
 => []
 => []
 => ? ∊ {-1,-1,0,0,1,2,2,3,5}
[7,2]
 => [2]
 => []
 => []
 => ? ∊ {-1,-1,0,0,1,2,2,3,5}
[7,1,1]
 => [1,1]
 => [1]
 => [1,0]
 => ? ∊ {-1,-1,0,0,1,2,2,3,5}
[6,3]
 => [3]
 => []
 => []
 => ? ∊ {-1,-1,0,0,1,2,2,3,5}
[6,2,1]
 => [2,1]
 => [1]
 => [1,0]
 => ? ∊ {-1,-1,0,0,1,2,2,3,5}
[6,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[5,4]
 => [4]
 => []
 => []
 => ? ∊ {-1,-1,0,0,1,2,2,3,5}
[5,3,1]
 => [3,1]
 => [1]
 => [1,0]
 => ? ∊ {-1,-1,0,0,1,2,2,3,5}
[5,2,2]
 => [2,2]
 => [2]
 => [1,0,1,0]
 => 0
[5,2,1,1]
 => [2,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[5,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
[4,4,1]
 => [4,1]
 => [1]
 => [1,0]
 => ? ∊ {-1,-1,0,0,1,2,2,3,5}
[4,3,2]
 => [3,2]
 => [2]
 => [1,0,1,0]
 => 0
[4,3,1,1]
 => [3,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[4,2,2,1]
 => [2,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 0
[4,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
[4,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 0
[3,3,3]
 => [3,3]
 => [3]
 => [1,0,1,0,1,0]
 => 0
[3,3,2,1]
 => [3,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 0
[3,3,1,1,1]
 => [3,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
[3,2,2,2]
 => [2,2,2]
 => [2,2]
 => [1,1,1,0,0,0]
 => 0
[3,2,2,1,1]
 => [2,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 0
[3,2,1,1,1,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 0
[3,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
[2,2,2,2,1]
 => [2,2,2,1]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 0
[2,2,2,1,1,1]
 => [2,2,1,1,1]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 0
[2,2,1,1,1,1,1]
 => [2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
[2,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 0
[10]
 => []
 => ?
 => ?
 => ? ∊ {-1,0,0,0,0,1,1,2,4,4,5}
[9,1]
 => [1]
 => []
 => []
 => ? ∊ {-1,0,0,0,0,1,1,2,4,4,5}
[8,2]
 => [2]
 => []
 => []
 => ? ∊ {-1,0,0,0,0,1,1,2,4,4,5}
[8,1,1]
 => [1,1]
 => [1]
 => [1,0]
 => ? ∊ {-1,0,0,0,0,1,1,2,4,4,5}
[7,3]
 => [3]
 => []
 => []
 => ? ∊ {-1,0,0,0,0,1,1,2,4,4,5}
[7,2,1]
 => [2,1]
 => [1]
 => [1,0]
 => ? ∊ {-1,0,0,0,0,1,1,2,4,4,5}
Description
The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. For example, the path $111011010000$ has three peaks in positions $03, 15, 26$. The boxes below $03$ are $01,02,\textbf{12}$, the boxes below $15$ are $\textbf{12},13,14,\textbf{23},\textbf{24},\textbf{34}$, and the boxes below $26$ are $\textbf{23},\textbf{24},25,\textbf{34},35,45$. We thus obtain the four boxes in positions $12,23,24,34$ that are below at least two peaks.
Matching statistic: St001001
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001001: Dyck paths ⟶ ℤResult quality: 30% ●values known / values provided: 66%●distinct values known / distinct values provided: 30%
Values
[2]
 => []
 => ?
 => ?
 => ? ∊ {0,1}
[1,1]
 => [1]
 => []
 => []
 => ? ∊ {0,1}
[3]
 => []
 => ?
 => ?
 => ? ∊ {0,2}
[2,1]
 => [1]
 => []
 => []
 => ? ∊ {0,2}
[1,1,1]
 => [1,1]
 => [1]
 => [1,0,1,0]
 => 0
[4]
 => []
 => ?
 => ?
 => ? ∊ {-1,1,2}
[3,1]
 => [1]
 => []
 => []
 => ? ∊ {-1,1,2}
[2,2]
 => [2]
 => []
 => []
 => ? ∊ {-1,1,2}
[2,1,1]
 => [1,1]
 => [1]
 => [1,0,1,0]
 => 0
[1,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[5]
 => []
 => ?
 => ?
 => ? ∊ {0,1,3}
[4,1]
 => [1]
 => []
 => []
 => ? ∊ {0,1,3}
[3,2]
 => [2]
 => []
 => []
 => ? ∊ {0,1,3}
[3,1,1]
 => [1,1]
 => [1]
 => [1,0,1,0]
 => 0
[2,2,1]
 => [2,1]
 => [1]
 => [1,0,1,0]
 => 0
[2,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 0
[6]
 => []
 => ?
 => ?
 => ? ∊ {0,1,2,3}
[5,1]
 => [1]
 => []
 => []
 => ? ∊ {0,1,2,3}
[4,2]
 => [2]
 => []
 => []
 => ? ∊ {0,1,2,3}
[4,1,1]
 => [1,1]
 => [1]
 => [1,0,1,0]
 => 0
[3,3]
 => [3]
 => []
 => []
 => ? ∊ {0,1,2,3}
[3,2,1]
 => [2,1]
 => [1]
 => [1,0,1,0]
 => 0
[3,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[2,2,2]
 => [2,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0
[2,2,1,1]
 => [2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 0
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0
[7]
 => []
 => ?
 => ?
 => ? ∊ {0,0,1,2,4}
[6,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,1,2,4}
[5,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,1,2,4}
[5,1,1]
 => [1,1]
 => [1]
 => [1,0,1,0]
 => 0
[4,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,1,2,4}
[4,2,1]
 => [2,1]
 => [1]
 => [1,0,1,0]
 => 0
[4,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[3,3,1]
 => [3,1]
 => [1]
 => [1,0,1,0]
 => 0
[3,2,2]
 => [2,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0
[3,2,1,1]
 => [2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 0
[2,2,2,1]
 => [2,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => 0
[2,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 0
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,1,2,4}
[8]
 => []
 => ?
 => ?
 => ? ∊ {-2,-1,0,1,2,3,4}
[7,1]
 => [1]
 => []
 => []
 => ? ∊ {-2,-1,0,1,2,3,4}
[6,2]
 => [2]
 => []
 => []
 => ? ∊ {-2,-1,0,1,2,3,4}
[6,1,1]
 => [1,1]
 => [1]
 => [1,0,1,0]
 => 0
[5,3]
 => [3]
 => []
 => []
 => ? ∊ {-2,-1,0,1,2,3,4}
[5,2,1]
 => [2,1]
 => [1]
 => [1,0,1,0]
 => 0
[5,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[4,4]
 => [4]
 => []
 => []
 => ? ∊ {-2,-1,0,1,2,3,4}
[4,3,1]
 => [3,1]
 => [1]
 => [1,0,1,0]
 => 0
[4,2,2]
 => [2,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0
[4,2,1,1]
 => [2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[4,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 0
[3,3,2]
 => [3,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0
[3,3,1,1]
 => [3,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[3,2,2,1]
 => [2,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => 0
[3,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 0
[3,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0
[2,2,2,2]
 => [2,2,2]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 0
[2,2,2,1,1]
 => [2,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 0
[2,2,1,1,1,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0
[2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {-2,-1,0,1,2,3,4}
[1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {-2,-1,0,1,2,3,4}
[9]
 => []
 => ?
 => ?
 => ? ∊ {-1,-1,0,0,1,2,2,3,5}
[8,1]
 => [1]
 => []
 => []
 => ? ∊ {-1,-1,0,0,1,2,2,3,5}
[7,2]
 => [2]
 => []
 => []
 => ? ∊ {-1,-1,0,0,1,2,2,3,5}
[7,1,1]
 => [1,1]
 => [1]
 => [1,0,1,0]
 => 0
[6,3]
 => [3]
 => []
 => []
 => ? ∊ {-1,-1,0,0,1,2,2,3,5}
[6,2,1]
 => [2,1]
 => [1]
 => [1,0,1,0]
 => 0
[6,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[5,4]
 => [4]
 => []
 => []
 => ? ∊ {-1,-1,0,0,1,2,2,3,5}
[5,3,1]
 => [3,1]
 => [1]
 => [1,0,1,0]
 => 0
[5,2,2]
 => [2,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0
[5,2,1,1]
 => [2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[5,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 0
[4,4,1]
 => [4,1]
 => [1]
 => [1,0,1,0]
 => 0
[4,3,2]
 => [3,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0
[4,3,1,1]
 => [3,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[4,2,2,1]
 => [2,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => 0
[3,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {-1,-1,0,0,1,2,2,3,5}
[2,2,1,1,1,1,1]
 => [2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {-1,-1,0,0,1,2,2,3,5}
[2,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {-1,-1,0,0,1,2,2,3,5}
[1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? ∊ {-1,-1,0,0,1,2,2,3,5}
[10]
 => []
 => ?
 => ?
 => ? ∊ {-1,0,0,0,0,0,0,1,1,2,4,4,5}
[9,1]
 => [1]
 => []
 => []
 => ? ∊ {-1,0,0,0,0,0,0,1,1,2,4,4,5}
[8,2]
 => [2]
 => []
 => []
 => ? ∊ {-1,0,0,0,0,0,0,1,1,2,4,4,5}
[7,3]
 => [3]
 => []
 => []
 => ? ∊ {-1,0,0,0,0,0,0,1,1,2,4,4,5}
[6,4]
 => [4]
 => []
 => []
 => ? ∊ {-1,0,0,0,0,0,0,1,1,2,4,4,5}
[5,5]
 => [5]
 => []
 => []
 => ? ∊ {-1,0,0,0,0,0,0,1,1,2,4,4,5}
[4,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {-1,0,0,0,0,0,0,1,1,2,4,4,5}
[3,2,1,1,1,1,1]
 => [2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {-1,0,0,0,0,0,0,1,1,2,4,4,5}
[3,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {-1,0,0,0,0,0,0,1,1,2,4,4,5}
[2,2,2,1,1,1,1]
 => [2,2,1,1,1,1]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => ? ∊ {-1,0,0,0,0,0,0,1,1,2,4,4,5}
[2,2,1,1,1,1,1,1]
 => [2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {-1,0,0,0,0,0,0,1,1,2,4,4,5}
[2,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? ∊ {-1,0,0,0,0,0,0,1,1,2,4,4,5}
[1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {-1,0,0,0,0,0,0,1,1,2,4,4,5}
[11]
 => []
 => ?
 => ?
 => ? ∊ {-2,0,0,0,0,0,0,0,0,0,0,0,2,2,3,4,4,6}
[10,1]
 => [1]
 => []
 => []
 => ? ∊ {-2,0,0,0,0,0,0,0,0,0,0,0,2,2,3,4,4,6}
Description
The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001435
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001435: Skew partitions ⟶ ℤResult quality: 10% ●values known / values provided: 65%●distinct values known / distinct values provided: 10%
Values
[2]
 => []
 => ?
 => ?
 => ? ∊ {0,1}
[1,1]
 => [1]
 => []
 => [[],[]]
 => ? ∊ {0,1}
[3]
 => []
 => ?
 => ?
 => ? ∊ {0,2}
[2,1]
 => [1]
 => []
 => [[],[]]
 => ? ∊ {0,2}
[1,1,1]
 => [1,1]
 => [1]
 => [[1],[]]
 => 0
[4]
 => []
 => ?
 => ?
 => ? ∊ {-1,1,2}
[3,1]
 => [1]
 => []
 => [[],[]]
 => ? ∊ {-1,1,2}
[2,2]
 => [2]
 => []
 => [[],[]]
 => ? ∊ {-1,1,2}
[2,1,1]
 => [1,1]
 => [1]
 => [[1],[]]
 => 0
[1,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0
[5]
 => []
 => ?
 => ?
 => ? ∊ {0,1,3}
[4,1]
 => [1]
 => []
 => [[],[]]
 => ? ∊ {0,1,3}
[3,2]
 => [2]
 => []
 => [[],[]]
 => ? ∊ {0,1,3}
[3,1,1]
 => [1,1]
 => [1]
 => [[1],[]]
 => 0
[2,2,1]
 => [2,1]
 => [1]
 => [[1],[]]
 => 0
[2,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0
[1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [[1,1,1],[]]
 => 0
[6]
 => []
 => ?
 => ?
 => ? ∊ {0,1,2,3}
[5,1]
 => [1]
 => []
 => [[],[]]
 => ? ∊ {0,1,2,3}
[4,2]
 => [2]
 => []
 => [[],[]]
 => ? ∊ {0,1,2,3}
[4,1,1]
 => [1,1]
 => [1]
 => [[1],[]]
 => 0
[3,3]
 => [3]
 => []
 => [[],[]]
 => ? ∊ {0,1,2,3}
[3,2,1]
 => [2,1]
 => [1]
 => [[1],[]]
 => 0
[3,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0
[2,2,2]
 => [2,2]
 => [2]
 => [[2],[]]
 => 0
[2,2,1,1]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0
[2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [[1,1,1],[]]
 => 0
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [[1,1,1,1],[]]
 => 0
[7]
 => []
 => ?
 => ?
 => ? ∊ {0,1,2,4}
[6,1]
 => [1]
 => []
 => [[],[]]
 => ? ∊ {0,1,2,4}
[5,2]
 => [2]
 => []
 => [[],[]]
 => ? ∊ {0,1,2,4}
[5,1,1]
 => [1,1]
 => [1]
 => [[1],[]]
 => 0
[4,3]
 => [3]
 => []
 => [[],[]]
 => ? ∊ {0,1,2,4}
[4,2,1]
 => [2,1]
 => [1]
 => [[1],[]]
 => 0
[4,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0
[3,3,1]
 => [3,1]
 => [1]
 => [[1],[]]
 => 0
[3,2,2]
 => [2,2]
 => [2]
 => [[2],[]]
 => 0
[3,2,1,1]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0
[3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [[1,1,1],[]]
 => 0
[2,2,2,1]
 => [2,2,1]
 => [2,1]
 => [[2,1],[]]
 => 0
[2,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => [[1,1,1],[]]
 => 0
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [[1,1,1,1],[]]
 => 0
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [[1,1,1,1,1],[]]
 => 0
[8]
 => []
 => ?
 => ?
 => ? ∊ {-2,-1,1,2,3,4}
[7,1]
 => [1]
 => []
 => [[],[]]
 => ? ∊ {-2,-1,1,2,3,4}
[6,2]
 => [2]
 => []
 => [[],[]]
 => ? ∊ {-2,-1,1,2,3,4}
[6,1,1]
 => [1,1]
 => [1]
 => [[1],[]]
 => 0
[5,3]
 => [3]
 => []
 => [[],[]]
 => ? ∊ {-2,-1,1,2,3,4}
[5,2,1]
 => [2,1]
 => [1]
 => [[1],[]]
 => 0
[5,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0
[4,4]
 => [4]
 => []
 => [[],[]]
 => ? ∊ {-2,-1,1,2,3,4}
[4,3,1]
 => [3,1]
 => [1]
 => [[1],[]]
 => 0
[4,2,2]
 => [2,2]
 => [2]
 => [[2],[]]
 => 0
[4,2,1,1]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0
[4,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [[1,1,1],[]]
 => 0
[3,3,2]
 => [3,2]
 => [2]
 => [[2],[]]
 => 0
[3,3,1,1]
 => [3,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0
[3,2,2,1]
 => [2,2,1]
 => [2,1]
 => [[2,1],[]]
 => 0
[3,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => [[1,1,1],[]]
 => 0
[3,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [[1,1,1,1],[]]
 => 0
[2,2,2,2]
 => [2,2,2]
 => [2,2]
 => [[2,2],[]]
 => 0
[2,2,2,1,1]
 => [2,2,1,1]
 => [2,1,1]
 => [[2,1,1],[]]
 => 0
[2,2,1,1,1,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [[1,1,1,1],[]]
 => 0
[2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [[1,1,1,1,1],[]]
 => 0
[1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [[1,1,1,1,1,1],[]]
 => ? ∊ {-2,-1,1,2,3,4}
[9]
 => []
 => ?
 => ?
 => ? ∊ {-1,-1,1,2,2,3,5}
[8,1]
 => [1]
 => []
 => [[],[]]
 => ? ∊ {-1,-1,1,2,2,3,5}
[7,2]
 => [2]
 => []
 => [[],[]]
 => ? ∊ {-1,-1,1,2,2,3,5}
[7,1,1]
 => [1,1]
 => [1]
 => [[1],[]]
 => 0
[6,3]
 => [3]
 => []
 => [[],[]]
 => ? ∊ {-1,-1,1,2,2,3,5}
[6,2,1]
 => [2,1]
 => [1]
 => [[1],[]]
 => 0
[6,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0
[5,4]
 => [4]
 => []
 => [[],[]]
 => ? ∊ {-1,-1,1,2,2,3,5}
[5,3,1]
 => [3,1]
 => [1]
 => [[1],[]]
 => 0
[5,2,2]
 => [2,2]
 => [2]
 => [[2],[]]
 => 0
[5,2,1,1]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0
[5,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [[1,1,1],[]]
 => 0
[4,4,1]
 => [4,1]
 => [1]
 => [[1],[]]
 => 0
[4,3,2]
 => [3,2]
 => [2]
 => [[2],[]]
 => 0
[2,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [[1,1,1,1,1,1],[]]
 => ? ∊ {-1,-1,1,2,2,3,5}
[1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [[1,1,1,1,1,1,1],[]]
 => ? ∊ {-1,-1,1,2,2,3,5}
[10]
 => []
 => ?
 => ?
 => ? ∊ {-1,0,0,0,0,0,0,1,1,2,4,4,5}
[9,1]
 => [1]
 => []
 => [[],[]]
 => ? ∊ {-1,0,0,0,0,0,0,1,1,2,4,4,5}
[8,2]
 => [2]
 => []
 => [[],[]]
 => ? ∊ {-1,0,0,0,0,0,0,1,1,2,4,4,5}
[7,3]
 => [3]
 => []
 => [[],[]]
 => ? ∊ {-1,0,0,0,0,0,0,1,1,2,4,4,5}
[6,4]
 => [4]
 => []
 => [[],[]]
 => ? ∊ {-1,0,0,0,0,0,0,1,1,2,4,4,5}
[5,5]
 => [5]
 => []
 => [[],[]]
 => ? ∊ {-1,0,0,0,0,0,0,1,1,2,4,4,5}
[3,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [[1,1,1,1,1,1],[]]
 => ? ∊ {-1,0,0,0,0,0,0,1,1,2,4,4,5}
[2,2,2,2,2]
 => [2,2,2,2]
 => [2,2,2]
 => [[2,2,2],[]]
 => ? ∊ {-1,0,0,0,0,0,0,1,1,2,4,4,5}
[2,2,2,2,1,1]
 => [2,2,2,1,1]
 => [2,2,1,1]
 => [[2,2,1,1],[]]
 => ? ∊ {-1,0,0,0,0,0,0,1,1,2,4,4,5}
[2,2,2,1,1,1,1]
 => [2,2,1,1,1,1]
 => [2,1,1,1,1]
 => [[2,1,1,1,1],[]]
 => ? ∊ {-1,0,0,0,0,0,0,1,1,2,4,4,5}
[2,2,1,1,1,1,1,1]
 => [2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [[1,1,1,1,1,1],[]]
 => ? ∊ {-1,0,0,0,0,0,0,1,1,2,4,4,5}
[2,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [[1,1,1,1,1,1,1],[]]
 => ? ∊ {-1,0,0,0,0,0,0,1,1,2,4,4,5}
[1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [[1,1,1,1,1,1,1,1],[]]
 => ? ∊ {-1,0,0,0,0,0,0,1,1,2,4,4,5}
[11]
 => []
 => ?
 => ?
 => ? ∊ {-2,0,0,0,0,0,0,0,0,0,0,1,2,2,3,4,4,6}
[10,1]
 => [1]
 => []
 => [[],[]]
 => ? ∊ {-2,0,0,0,0,0,0,0,0,0,0,1,2,2,3,4,4,6}
[9,2]
 => [2]
 => []
 => [[],[]]
 => ? ∊ {-2,0,0,0,0,0,0,0,0,0,0,1,2,2,3,4,4,6}
[8,3]
 => [3]
 => []
 => [[],[]]
 => ? ∊ {-2,0,0,0,0,0,0,0,0,0,0,1,2,2,3,4,4,6}
[7,4]
 => [4]
 => []
 => [[],[]]
 => ? ∊ {-2,0,0,0,0,0,0,0,0,0,0,1,2,2,3,4,4,6}
[6,5]
 => [5]
 => []
 => [[],[]]
 => ? ∊ {-2,0,0,0,0,0,0,0,0,0,0,1,2,2,3,4,4,6}
Description
The number of missing boxes in the first row.
Matching statistic: St001438
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001438: Skew partitions ⟶ ℤResult quality: 10% ●values known / values provided: 65%●distinct values known / distinct values provided: 10%
Values
[2]
 => []
 => ?
 => ?
 => ? ∊ {0,1}
[1,1]
 => [1]
 => []
 => [[],[]]
 => ? ∊ {0,1}
[3]
 => []
 => ?
 => ?
 => ? ∊ {0,2}
[2,1]
 => [1]
 => []
 => [[],[]]
 => ? ∊ {0,2}
[1,1,1]
 => [1,1]
 => [1]
 => [[1],[]]
 => 0
[4]
 => []
 => ?
 => ?
 => ? ∊ {-1,1,2}
[3,1]
 => [1]
 => []
 => [[],[]]
 => ? ∊ {-1,1,2}
[2,2]
 => [2]
 => []
 => [[],[]]
 => ? ∊ {-1,1,2}
[2,1,1]
 => [1,1]
 => [1]
 => [[1],[]]
 => 0
[1,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0
[5]
 => []
 => ?
 => ?
 => ? ∊ {0,1,3}
[4,1]
 => [1]
 => []
 => [[],[]]
 => ? ∊ {0,1,3}
[3,2]
 => [2]
 => []
 => [[],[]]
 => ? ∊ {0,1,3}
[3,1,1]
 => [1,1]
 => [1]
 => [[1],[]]
 => 0
[2,2,1]
 => [2,1]
 => [1]
 => [[1],[]]
 => 0
[2,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0
[1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [[1,1,1],[]]
 => 0
[6]
 => []
 => ?
 => ?
 => ? ∊ {0,1,2,3}
[5,1]
 => [1]
 => []
 => [[],[]]
 => ? ∊ {0,1,2,3}
[4,2]
 => [2]
 => []
 => [[],[]]
 => ? ∊ {0,1,2,3}
[4,1,1]
 => [1,1]
 => [1]
 => [[1],[]]
 => 0
[3,3]
 => [3]
 => []
 => [[],[]]
 => ? ∊ {0,1,2,3}
[3,2,1]
 => [2,1]
 => [1]
 => [[1],[]]
 => 0
[3,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0
[2,2,2]
 => [2,2]
 => [2]
 => [[2],[]]
 => 0
[2,2,1,1]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0
[2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [[1,1,1],[]]
 => 0
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [[1,1,1,1],[]]
 => 0
[7]
 => []
 => ?
 => ?
 => ? ∊ {0,1,2,4}
[6,1]
 => [1]
 => []
 => [[],[]]
 => ? ∊ {0,1,2,4}
[5,2]
 => [2]
 => []
 => [[],[]]
 => ? ∊ {0,1,2,4}
[5,1,1]
 => [1,1]
 => [1]
 => [[1],[]]
 => 0
[4,3]
 => [3]
 => []
 => [[],[]]
 => ? ∊ {0,1,2,4}
[4,2,1]
 => [2,1]
 => [1]
 => [[1],[]]
 => 0
[4,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0
[3,3,1]
 => [3,1]
 => [1]
 => [[1],[]]
 => 0
[3,2,2]
 => [2,2]
 => [2]
 => [[2],[]]
 => 0
[3,2,1,1]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0
[3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [[1,1,1],[]]
 => 0
[2,2,2,1]
 => [2,2,1]
 => [2,1]
 => [[2,1],[]]
 => 0
[2,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => [[1,1,1],[]]
 => 0
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [[1,1,1,1],[]]
 => 0
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [[1,1,1,1,1],[]]
 => 0
[8]
 => []
 => ?
 => ?
 => ? ∊ {-2,-1,1,2,3,4}
[7,1]
 => [1]
 => []
 => [[],[]]
 => ? ∊ {-2,-1,1,2,3,4}
[6,2]
 => [2]
 => []
 => [[],[]]
 => ? ∊ {-2,-1,1,2,3,4}
[6,1,1]
 => [1,1]
 => [1]
 => [[1],[]]
 => 0
[5,3]
 => [3]
 => []
 => [[],[]]
 => ? ∊ {-2,-1,1,2,3,4}
[5,2,1]
 => [2,1]
 => [1]
 => [[1],[]]
 => 0
[5,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0
[4,4]
 => [4]
 => []
 => [[],[]]
 => ? ∊ {-2,-1,1,2,3,4}
[4,3,1]
 => [3,1]
 => [1]
 => [[1],[]]
 => 0
[4,2,2]
 => [2,2]
 => [2]
 => [[2],[]]
 => 0
[4,2,1,1]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0
[4,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [[1,1,1],[]]
 => 0
[3,3,2]
 => [3,2]
 => [2]
 => [[2],[]]
 => 0
[3,3,1,1]
 => [3,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0
[3,2,2,1]
 => [2,2,1]
 => [2,1]
 => [[2,1],[]]
 => 0
[3,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => [[1,1,1],[]]
 => 0
[3,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [[1,1,1,1],[]]
 => 0
[2,2,2,2]
 => [2,2,2]
 => [2,2]
 => [[2,2],[]]
 => 0
[2,2,2,1,1]
 => [2,2,1,1]
 => [2,1,1]
 => [[2,1,1],[]]
 => 0
[2,2,1,1,1,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [[1,1,1,1],[]]
 => 0
[2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [[1,1,1,1,1],[]]
 => 0
[1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [[1,1,1,1,1,1],[]]
 => ? ∊ {-2,-1,1,2,3,4}
[9]
 => []
 => ?
 => ?
 => ? ∊ {-1,-1,1,2,2,3,5}
[8,1]
 => [1]
 => []
 => [[],[]]
 => ? ∊ {-1,-1,1,2,2,3,5}
[7,2]
 => [2]
 => []
 => [[],[]]
 => ? ∊ {-1,-1,1,2,2,3,5}
[7,1,1]
 => [1,1]
 => [1]
 => [[1],[]]
 => 0
[6,3]
 => [3]
 => []
 => [[],[]]
 => ? ∊ {-1,-1,1,2,2,3,5}
[6,2,1]
 => [2,1]
 => [1]
 => [[1],[]]
 => 0
[6,1,1,1]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0
[5,4]
 => [4]
 => []
 => [[],[]]
 => ? ∊ {-1,-1,1,2,2,3,5}
[5,3,1]
 => [3,1]
 => [1]
 => [[1],[]]
 => 0
[5,2,2]
 => [2,2]
 => [2]
 => [[2],[]]
 => 0
[5,2,1,1]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0
[5,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [[1,1,1],[]]
 => 0
[4,4,1]
 => [4,1]
 => [1]
 => [[1],[]]
 => 0
[4,3,2]
 => [3,2]
 => [2]
 => [[2],[]]
 => 0
[2,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [[1,1,1,1,1,1],[]]
 => ? ∊ {-1,-1,1,2,2,3,5}
[1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [[1,1,1,1,1,1,1],[]]
 => ? ∊ {-1,-1,1,2,2,3,5}
[10]
 => []
 => ?
 => ?
 => ? ∊ {-1,0,0,0,0,0,0,1,1,2,4,4,5}
[9,1]
 => [1]
 => []
 => [[],[]]
 => ? ∊ {-1,0,0,0,0,0,0,1,1,2,4,4,5}
[8,2]
 => [2]
 => []
 => [[],[]]
 => ? ∊ {-1,0,0,0,0,0,0,1,1,2,4,4,5}
[7,3]
 => [3]
 => []
 => [[],[]]
 => ? ∊ {-1,0,0,0,0,0,0,1,1,2,4,4,5}
[6,4]
 => [4]
 => []
 => [[],[]]
 => ? ∊ {-1,0,0,0,0,0,0,1,1,2,4,4,5}
[5,5]
 => [5]
 => []
 => [[],[]]
 => ? ∊ {-1,0,0,0,0,0,0,1,1,2,4,4,5}
[3,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [[1,1,1,1,1,1],[]]
 => ? ∊ {-1,0,0,0,0,0,0,1,1,2,4,4,5}
[2,2,2,2,2]
 => [2,2,2,2]
 => [2,2,2]
 => [[2,2,2],[]]
 => ? ∊ {-1,0,0,0,0,0,0,1,1,2,4,4,5}
[2,2,2,2,1,1]
 => [2,2,2,1,1]
 => [2,2,1,1]
 => [[2,2,1,1],[]]
 => ? ∊ {-1,0,0,0,0,0,0,1,1,2,4,4,5}
[2,2,2,1,1,1,1]
 => [2,2,1,1,1,1]
 => [2,1,1,1,1]
 => [[2,1,1,1,1],[]]
 => ? ∊ {-1,0,0,0,0,0,0,1,1,2,4,4,5}
[2,2,1,1,1,1,1,1]
 => [2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [[1,1,1,1,1,1],[]]
 => ? ∊ {-1,0,0,0,0,0,0,1,1,2,4,4,5}
[2,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [[1,1,1,1,1,1,1],[]]
 => ? ∊ {-1,0,0,0,0,0,0,1,1,2,4,4,5}
[1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [[1,1,1,1,1,1,1,1],[]]
 => ? ∊ {-1,0,0,0,0,0,0,1,1,2,4,4,5}
[11]
 => []
 => ?
 => ?
 => ? ∊ {-2,0,0,0,0,0,0,0,0,0,0,1,2,2,3,4,4,6}
[10,1]
 => [1]
 => []
 => [[],[]]
 => ? ∊ {-2,0,0,0,0,0,0,0,0,0,0,1,2,2,3,4,4,6}
[9,2]
 => [2]
 => []
 => [[],[]]
 => ? ∊ {-2,0,0,0,0,0,0,0,0,0,0,1,2,2,3,4,4,6}
[8,3]
 => [3]
 => []
 => [[],[]]
 => ? ∊ {-2,0,0,0,0,0,0,0,0,0,0,1,2,2,3,4,4,6}
[7,4]
 => [4]
 => []
 => [[],[]]
 => ? ∊ {-2,0,0,0,0,0,0,0,0,0,0,1,2,2,3,4,4,6}
[6,5]
 => [5]
 => []
 => [[],[]]
 => ? ∊ {-2,0,0,0,0,0,0,0,0,0,0,1,2,2,3,4,4,6}
Description
The number of missing boxes of a skew partition.
The following 2 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001498The normalised height of a Nakayama algebra with magnitude 1. St001816Eigenvalues of the top-to-random operator acting on a simple module.