Your data matches 6 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001587
Mapping
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001587: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],3)
 => [3,3]
 => [3]
 => 0
([(2,3)],4)
 => [4,4,4]
 => [4,4]
 => 2
([(0,1),(0,2),(0,3)],4)
 => [3,3]
 => [3]
 => 0
([(0,3),(1,3),(2,3)],4)
 => [3,3]
 => [3]
 => 0
([(0,3),(1,2)],4)
 => [4,2]
 => [2]
 => 1
([(0,3),(1,2),(1,3)],4)
 => [3,2]
 => [2]
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [2]
 => 1
([(1,2),(1,3),(1,4)],5)
 => [15,15]
 => [15]
 => 0
([(0,2),(0,3),(0,4),(4,1)],5)
 => [4,4,4]
 => [4,4]
 => 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [3]
 => 0
([(1,2),(1,3),(2,4),(3,4)],5)
 => [5,5]
 => [5]
 => 0
([(0,3),(0,4),(3,2),(4,1)],5)
 => [4,2]
 => [2]
 => 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [3,2]
 => [2]
 => 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2]
 => [2]
 => 1
([(2,3),(3,4)],5)
 => [5,5,5,5]
 => [5,5,5]
 => 0
([(1,4),(4,2),(4,3)],5)
 => [5,5]
 => [5]
 => 0
([(0,4),(4,1),(4,2),(4,3)],5)
 => [3,3]
 => [3]
 => 0
([(1,4),(2,4),(4,3)],5)
 => [5,5]
 => [5]
 => 0
([(0,4),(1,4),(4,2),(4,3)],5)
 => [2,2]
 => [2]
 => 1
([(1,4),(2,4),(3,4)],5)
 => [15,15]
 => [15]
 => 0
([(0,4),(1,4),(2,4),(4,3)],5)
 => [3,3]
 => [3]
 => 0
([(0,4),(1,4),(2,3)],5)
 => [10,10]
 => [10]
 => 5
([(0,4),(1,3),(2,3),(2,4)],5)
 => [12,4]
 => [4]
 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6,6]
 => [6]
 => 3
([(0,4),(1,4),(2,3),(2,4)],5)
 => [10,4,4]
 => [4,4]
 => 2
([(0,4),(1,4),(2,3),(3,4)],5)
 => [4,4,4]
 => [4,4]
 => 2
([(1,4),(2,3),(2,4)],5)
 => [15,5,5]
 => [5,5]
 => 0
([(0,4),(1,2),(1,4),(2,3)],5)
 => [5,4]
 => [4]
 => 2
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [5,5,5,5]
 => [5,5,5]
 => 0
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [2,2]
 => [2]
 => 1
([(0,4),(1,2),(1,3)],5)
 => [10,10]
 => [10]
 => 5
([(0,4),(1,2),(1,3),(1,4)],5)
 => [10,4,4]
 => [4,4]
 => 2
([(0,4),(1,2),(1,3),(3,4)],5)
 => [4,4,3]
 => [4,3]
 => 2
([(0,3),(0,4),(1,2),(1,4)],5)
 => [12,4]
 => [4]
 => 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [6,6]
 => [6]
 => 3
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => [5,3]
 => [3]
 => 0
([(0,3),(1,2),(1,4),(3,4)],5)
 => [5,4]
 => [4]
 => 2
([(0,3),(1,4),(4,2)],5)
 => [5,5]
 => [5]
 => 0
([(0,3),(1,2),(2,4),(3,4)],5)
 => [4,2]
 => [2]
 => 1
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
 => [15,15]
 => [15]
 => 0
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => [3,3]
 => [3]
 => 0
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
 => [10,10]
 => [10]
 => 5
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5)],6)
 => [12,4]
 => [4]
 => 2
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6,6]
 => [6]
 => 3
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
 => [10,4,4]
 => [4,4]
 => 2
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
 => [5,5]
 => [5]
 => 0
([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
 => [15,5,5]
 => [5,5]
 => 0
([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [5,5,5,5]
 => [5,5,5]
 => 0
([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
 => [15,15]
 => [15]
 => 0
Description
Half of the largest even part of an integer partition. The largest even part is recorded by [[St000995]].
Matching statistic: St001487
(load all 2 compositions to match this statistic)
Mapping
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001487: Skew partitions ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 17%
Values
([],3)
 => [3,3]
 => [[3,3],[]]
 => ? = 0
([(2,3)],4)
 => [4,4,4]
 => [[4,4,4],[]]
 => ? = 2
([(0,1),(0,2),(0,3)],4)
 => [3,3]
 => [[3,3],[]]
 => ? = 0
([(0,3),(1,3),(2,3)],4)
 => [3,3]
 => [[3,3],[]]
 => ? = 0
([(0,3),(1,2)],4)
 => [4,2]
 => [[4,2],[]]
 => ? = 1
([(0,3),(1,2),(1,3)],4)
 => [3,2]
 => [[3,2],[]]
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [[2,2],[]]
 => 1
([(1,2),(1,3),(1,4)],5)
 => [15,15]
 => [[15,15],[]]
 => ? = 0
([(0,2),(0,3),(0,4),(4,1)],5)
 => [4,4,4]
 => [[4,4,4],[]]
 => ? = 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [[3,3],[]]
 => ? = 0
([(1,2),(1,3),(2,4),(3,4)],5)
 => [5,5]
 => [[5,5],[]]
 => ? = 0
([(0,3),(0,4),(3,2),(4,1)],5)
 => [4,2]
 => [[4,2],[]]
 => ? = 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [3,2]
 => [[3,2],[]]
 => 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2]
 => [[2,2],[]]
 => 1
([(2,3),(3,4)],5)
 => [5,5,5,5]
 => [[5,5,5,5],[]]
 => ? = 0
([(1,4),(4,2),(4,3)],5)
 => [5,5]
 => [[5,5],[]]
 => ? = 0
([(0,4),(4,1),(4,2),(4,3)],5)
 => [3,3]
 => [[3,3],[]]
 => ? = 0
([(1,4),(2,4),(4,3)],5)
 => [5,5]
 => [[5,5],[]]
 => ? = 0
([(0,4),(1,4),(4,2),(4,3)],5)
 => [2,2]
 => [[2,2],[]]
 => 1
([(1,4),(2,4),(3,4)],5)
 => [15,15]
 => [[15,15],[]]
 => ? = 0
([(0,4),(1,4),(2,4),(4,3)],5)
 => [3,3]
 => [[3,3],[]]
 => ? = 0
([(0,4),(1,4),(2,3)],5)
 => [10,10]
 => [[10,10],[]]
 => ? = 5
([(0,4),(1,3),(2,3),(2,4)],5)
 => [12,4]
 => [[12,4],[]]
 => ? = 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6,6]
 => [[6,6],[]]
 => ? = 3
([(0,4),(1,4),(2,3),(2,4)],5)
 => [10,4,4]
 => [[10,4,4],[]]
 => ? = 2
([(0,4),(1,4),(2,3),(3,4)],5)
 => [4,4,4]
 => [[4,4,4],[]]
 => ? = 2
([(1,4),(2,3),(2,4)],5)
 => [15,5,5]
 => [[15,5,5],[]]
 => ? = 0
([(0,4),(1,2),(1,4),(2,3)],5)
 => [5,4]
 => [[5,4],[]]
 => ? = 2
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => [3,2]
 => [[3,2],[]]
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [5,5,5,5]
 => [[5,5,5,5],[]]
 => ? = 0
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [2,2]
 => [[2,2],[]]
 => 1
([(0,4),(1,2),(1,3)],5)
 => [10,10]
 => [[10,10],[]]
 => ? = 5
([(0,4),(1,2),(1,3),(1,4)],5)
 => [10,4,4]
 => [[10,4,4],[]]
 => ? = 2
([(0,4),(1,2),(1,3),(3,4)],5)
 => [4,4,3]
 => [[4,4,3],[]]
 => ? = 2
([(0,3),(0,4),(1,2),(1,4)],5)
 => [12,4]
 => [[12,4],[]]
 => ? = 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [6,6]
 => [[6,6],[]]
 => ? = 3
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => [5,3]
 => [[5,3],[]]
 => ? = 0
([(0,3),(1,2),(1,4),(3,4)],5)
 => [5,4]
 => [[5,4],[]]
 => ? = 2
([(0,3),(1,4),(4,2)],5)
 => [5,5]
 => [[5,5],[]]
 => ? = 0
([(0,3),(1,2),(2,4),(3,4)],5)
 => [4,2]
 => [[4,2],[]]
 => ? = 1
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
 => [15,15]
 => [[15,15],[]]
 => ? = 0
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => [3,3]
 => [[3,3],[]]
 => ? = 0
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
 => [10,10]
 => [[10,10],[]]
 => ? = 5
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5)],6)
 => [12,4]
 => [[12,4],[]]
 => ? = 2
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6,6]
 => [[6,6],[]]
 => ? = 3
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
 => [10,4,4]
 => [[10,4,4],[]]
 => ? = 2
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
 => [5,5]
 => [[5,5],[]]
 => ? = 0
([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
 => [15,5,5]
 => [[15,5,5],[]]
 => ? = 0
([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [5,5,5,5]
 => [[5,5,5,5],[]]
 => ? = 0
([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
 => [15,15]
 => [[15,15],[]]
 => ? = 0
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => [2,2]
 => [[2,2],[]]
 => 1
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => [2,2]
 => [[2,2],[]]
 => 1
([(0,4),(0,5),(4,3),(5,1),(5,2)],6)
 => [10,10]
 => [[10,10],[]]
 => ? = 5
([(0,3),(0,4),(3,5),(4,1),(4,2),(4,5)],6)
 => [10,4,4]
 => [[10,4,4],[]]
 => ? = 2
([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
 => [12,4]
 => [[12,4],[]]
 => ? = 2
([(0,1),(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => [6,6]
 => [[6,6],[]]
 => ? = 3
([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
 => [3,3]
 => [[3,3],[]]
 => ? = 0
([(0,5),(1,5),(5,2),(5,3),(5,4)],6)
 => [6,6]
 => [[6,6],[]]
 => ? = 3
([(0,5),(1,5),(2,5),(5,3),(5,4)],6)
 => [6,6]
 => [[6,6],[]]
 => ? = 3
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
 => [2,2]
 => [[2,2],[]]
 => 1
([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
 => [2,2]
 => [[2,2],[]]
 => 1
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
 => [3,2]
 => [[3,2],[]]
 => 1
([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
 => [2,2]
 => [[2,2],[]]
 => 1
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => [3,2]
 => [[3,2],[]]
 => 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2)],6)
 => [2,2]
 => [[2,2],[]]
 => 1
([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
 => [3,2]
 => [[3,2],[]]
 => 1
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6),(6,1)],7)
 => [2,2]
 => [[2,2],[]]
 => 1
([(0,6),(1,6),(2,5),(3,5),(4,2),(4,3),(6,4)],7)
 => [2,2]
 => [[2,2],[]]
 => 1
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => [2,2]
 => [[2,2],[]]
 => 1
([(0,6),(1,6),(4,5),(5,2),(5,3),(6,4)],7)
 => [2,2]
 => [[2,2],[]]
 => 1
([(0,6),(1,6),(2,5),(3,5),(5,4),(6,2),(6,3)],7)
 => [2,2]
 => [[2,2],[]]
 => 1
([(0,5),(0,6),(1,5),(1,6),(2,3),(4,2),(5,4),(6,4)],7)
 => [2,2]
 => [[2,2],[]]
 => 1
([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => [3,2]
 => [[3,2],[]]
 => 1
([(0,3),(0,4),(3,6),(4,6),(5,1),(5,2),(6,5)],7)
 => [2,2]
 => [[2,2],[]]
 => 1
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => [3,2]
 => [[3,2],[]]
 => 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2),(4,6),(5,6)],7)
 => [2,2]
 => [[2,2],[]]
 => 1
([(0,5),(1,6),(2,6),(5,1),(5,2),(6,3),(6,4)],7)
 => [2,2]
 => [[2,2],[]]
 => 1
([(0,6),(1,3),(1,6),(3,5),(4,2),(5,4),(6,5)],7)
 => [3,2]
 => [[3,2],[]]
 => 1
([(0,3),(1,5),(1,6),(2,5),(2,6),(3,4),(4,1),(4,2)],7)
 => [2,2]
 => [[2,2],[]]
 => 1
([(0,4),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3)],7)
 => [3,2]
 => [[3,2],[]]
 => 1
([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => [3,2]
 => [[3,2],[]]
 => 1
Description
The number of inner corners of a skew partition.
Matching statistic: St001490
(load all 3 compositions to match this statistic)
Mapping
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001490: Skew partitions ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 17%
Values
([],3)
 => [3,3]
 => [[3,3],[]]
 => ? = 0
([(2,3)],4)
 => [4,4,4]
 => [[4,4,4],[]]
 => ? = 2
([(0,1),(0,2),(0,3)],4)
 => [3,3]
 => [[3,3],[]]
 => ? = 0
([(0,3),(1,3),(2,3)],4)
 => [3,3]
 => [[3,3],[]]
 => ? = 0
([(0,3),(1,2)],4)
 => [4,2]
 => [[4,2],[]]
 => ? = 1
([(0,3),(1,2),(1,3)],4)
 => [3,2]
 => [[3,2],[]]
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [[2,2],[]]
 => 1
([(1,2),(1,3),(1,4)],5)
 => [15,15]
 => [[15,15],[]]
 => ? = 0
([(0,2),(0,3),(0,4),(4,1)],5)
 => [4,4,4]
 => [[4,4,4],[]]
 => ? = 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [[3,3],[]]
 => ? = 0
([(1,2),(1,3),(2,4),(3,4)],5)
 => [5,5]
 => [[5,5],[]]
 => ? = 0
([(0,3),(0,4),(3,2),(4,1)],5)
 => [4,2]
 => [[4,2],[]]
 => ? = 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [3,2]
 => [[3,2],[]]
 => 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2]
 => [[2,2],[]]
 => 1
([(2,3),(3,4)],5)
 => [5,5,5,5]
 => [[5,5,5,5],[]]
 => ? = 0
([(1,4),(4,2),(4,3)],5)
 => [5,5]
 => [[5,5],[]]
 => ? = 0
([(0,4),(4,1),(4,2),(4,3)],5)
 => [3,3]
 => [[3,3],[]]
 => ? = 0
([(1,4),(2,4),(4,3)],5)
 => [5,5]
 => [[5,5],[]]
 => ? = 0
([(0,4),(1,4),(4,2),(4,3)],5)
 => [2,2]
 => [[2,2],[]]
 => 1
([(1,4),(2,4),(3,4)],5)
 => [15,15]
 => [[15,15],[]]
 => ? = 0
([(0,4),(1,4),(2,4),(4,3)],5)
 => [3,3]
 => [[3,3],[]]
 => ? = 0
([(0,4),(1,4),(2,3)],5)
 => [10,10]
 => [[10,10],[]]
 => ? = 5
([(0,4),(1,3),(2,3),(2,4)],5)
 => [12,4]
 => [[12,4],[]]
 => ? = 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6,6]
 => [[6,6],[]]
 => ? = 3
([(0,4),(1,4),(2,3),(2,4)],5)
 => [10,4,4]
 => [[10,4,4],[]]
 => ? = 2
([(0,4),(1,4),(2,3),(3,4)],5)
 => [4,4,4]
 => [[4,4,4],[]]
 => ? = 2
([(1,4),(2,3),(2,4)],5)
 => [15,5,5]
 => [[15,5,5],[]]
 => ? = 0
([(0,4),(1,2),(1,4),(2,3)],5)
 => [5,4]
 => [[5,4],[]]
 => ? = 2
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => [3,2]
 => [[3,2],[]]
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [5,5,5,5]
 => [[5,5,5,5],[]]
 => ? = 0
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [2,2]
 => [[2,2],[]]
 => 1
([(0,4),(1,2),(1,3)],5)
 => [10,10]
 => [[10,10],[]]
 => ? = 5
([(0,4),(1,2),(1,3),(1,4)],5)
 => [10,4,4]
 => [[10,4,4],[]]
 => ? = 2
([(0,4),(1,2),(1,3),(3,4)],5)
 => [4,4,3]
 => [[4,4,3],[]]
 => ? = 2
([(0,3),(0,4),(1,2),(1,4)],5)
 => [12,4]
 => [[12,4],[]]
 => ? = 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [6,6]
 => [[6,6],[]]
 => ? = 3
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => [5,3]
 => [[5,3],[]]
 => ? = 0
([(0,3),(1,2),(1,4),(3,4)],5)
 => [5,4]
 => [[5,4],[]]
 => ? = 2
([(0,3),(1,4),(4,2)],5)
 => [5,5]
 => [[5,5],[]]
 => ? = 0
([(0,3),(1,2),(2,4),(3,4)],5)
 => [4,2]
 => [[4,2],[]]
 => ? = 1
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
 => [15,15]
 => [[15,15],[]]
 => ? = 0
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => [3,3]
 => [[3,3],[]]
 => ? = 0
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
 => [10,10]
 => [[10,10],[]]
 => ? = 5
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5)],6)
 => [12,4]
 => [[12,4],[]]
 => ? = 2
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6,6]
 => [[6,6],[]]
 => ? = 3
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
 => [10,4,4]
 => [[10,4,4],[]]
 => ? = 2
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
 => [5,5]
 => [[5,5],[]]
 => ? = 0
([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
 => [15,5,5]
 => [[15,5,5],[]]
 => ? = 0
([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [5,5,5,5]
 => [[5,5,5,5],[]]
 => ? = 0
([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
 => [15,15]
 => [[15,15],[]]
 => ? = 0
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => [2,2]
 => [[2,2],[]]
 => 1
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => [2,2]
 => [[2,2],[]]
 => 1
([(0,4),(0,5),(4,3),(5,1),(5,2)],6)
 => [10,10]
 => [[10,10],[]]
 => ? = 5
([(0,3),(0,4),(3,5),(4,1),(4,2),(4,5)],6)
 => [10,4,4]
 => [[10,4,4],[]]
 => ? = 2
([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
 => [12,4]
 => [[12,4],[]]
 => ? = 2
([(0,1),(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => [6,6]
 => [[6,6],[]]
 => ? = 3
([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
 => [3,3]
 => [[3,3],[]]
 => ? = 0
([(0,5),(1,5),(5,2),(5,3),(5,4)],6)
 => [6,6]
 => [[6,6],[]]
 => ? = 3
([(0,5),(1,5),(2,5),(5,3),(5,4)],6)
 => [6,6]
 => [[6,6],[]]
 => ? = 3
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
 => [2,2]
 => [[2,2],[]]
 => 1
([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
 => [2,2]
 => [[2,2],[]]
 => 1
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
 => [3,2]
 => [[3,2],[]]
 => 1
([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
 => [2,2]
 => [[2,2],[]]
 => 1
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => [3,2]
 => [[3,2],[]]
 => 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2)],6)
 => [2,2]
 => [[2,2],[]]
 => 1
([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
 => [3,2]
 => [[3,2],[]]
 => 1
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6),(6,1)],7)
 => [2,2]
 => [[2,2],[]]
 => 1
([(0,6),(1,6),(2,5),(3,5),(4,2),(4,3),(6,4)],7)
 => [2,2]
 => [[2,2],[]]
 => 1
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => [2,2]
 => [[2,2],[]]
 => 1
([(0,6),(1,6),(4,5),(5,2),(5,3),(6,4)],7)
 => [2,2]
 => [[2,2],[]]
 => 1
([(0,6),(1,6),(2,5),(3,5),(5,4),(6,2),(6,3)],7)
 => [2,2]
 => [[2,2],[]]
 => 1
([(0,5),(0,6),(1,5),(1,6),(2,3),(4,2),(5,4),(6,4)],7)
 => [2,2]
 => [[2,2],[]]
 => 1
([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => [3,2]
 => [[3,2],[]]
 => 1
([(0,3),(0,4),(3,6),(4,6),(5,1),(5,2),(6,5)],7)
 => [2,2]
 => [[2,2],[]]
 => 1
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => [3,2]
 => [[3,2],[]]
 => 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2),(4,6),(5,6)],7)
 => [2,2]
 => [[2,2],[]]
 => 1
([(0,5),(1,6),(2,6),(5,1),(5,2),(6,3),(6,4)],7)
 => [2,2]
 => [[2,2],[]]
 => 1
([(0,6),(1,3),(1,6),(3,5),(4,2),(5,4),(6,5)],7)
 => [3,2]
 => [[3,2],[]]
 => 1
([(0,3),(1,5),(1,6),(2,5),(2,6),(3,4),(4,1),(4,2)],7)
 => [2,2]
 => [[2,2],[]]
 => 1
([(0,4),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3)],7)
 => [3,2]
 => [[3,2],[]]
 => 1
([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => [3,2]
 => [[3,2],[]]
 => 1
Description
The number of connected components of a skew partition.
Matching statistic: St001435
(load all 2 compositions to match this statistic)
Mapping
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001435: Skew partitions ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 17%
Values
([],3)
 => [3,3]
 => [[3,3],[]]
 => ? = 0 - 1
([(2,3)],4)
 => [4,4,4]
 => [[4,4,4],[]]
 => ? = 2 - 1
([(0,1),(0,2),(0,3)],4)
 => [3,3]
 => [[3,3],[]]
 => ? = 0 - 1
([(0,3),(1,3),(2,3)],4)
 => [3,3]
 => [[3,3],[]]
 => ? = 0 - 1
([(0,3),(1,2)],4)
 => [4,2]
 => [[4,2],[]]
 => ? = 1 - 1
([(0,3),(1,2),(1,3)],4)
 => [3,2]
 => [[3,2],[]]
 => 0 = 1 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [[2,2],[]]
 => 0 = 1 - 1
([(1,2),(1,3),(1,4)],5)
 => [15,15]
 => [[15,15],[]]
 => ? = 0 - 1
([(0,2),(0,3),(0,4),(4,1)],5)
 => [4,4,4]
 => [[4,4,4],[]]
 => ? = 2 - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [[3,3],[]]
 => ? = 0 - 1
([(1,2),(1,3),(2,4),(3,4)],5)
 => [5,5]
 => [[5,5],[]]
 => ? = 0 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
 => [4,2]
 => [[4,2],[]]
 => ? = 1 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [3,2]
 => [[3,2],[]]
 => 0 = 1 - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2]
 => [[2,2],[]]
 => 0 = 1 - 1
([(2,3),(3,4)],5)
 => [5,5,5,5]
 => [[5,5,5,5],[]]
 => ? = 0 - 1
([(1,4),(4,2),(4,3)],5)
 => [5,5]
 => [[5,5],[]]
 => ? = 0 - 1
([(0,4),(4,1),(4,2),(4,3)],5)
 => [3,3]
 => [[3,3],[]]
 => ? = 0 - 1
([(1,4),(2,4),(4,3)],5)
 => [5,5]
 => [[5,5],[]]
 => ? = 0 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
 => [2,2]
 => [[2,2],[]]
 => 0 = 1 - 1
([(1,4),(2,4),(3,4)],5)
 => [15,15]
 => [[15,15],[]]
 => ? = 0 - 1
([(0,4),(1,4),(2,4),(4,3)],5)
 => [3,3]
 => [[3,3],[]]
 => ? = 0 - 1
([(0,4),(1,4),(2,3)],5)
 => [10,10]
 => [[10,10],[]]
 => ? = 5 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [12,4]
 => [[12,4],[]]
 => ? = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6,6]
 => [[6,6],[]]
 => ? = 3 - 1
([(0,4),(1,4),(2,3),(2,4)],5)
 => [10,4,4]
 => [[10,4,4],[]]
 => ? = 2 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [4,4,4]
 => [[4,4,4],[]]
 => ? = 2 - 1
([(1,4),(2,3),(2,4)],5)
 => [15,5,5]
 => [[15,5,5],[]]
 => ? = 0 - 1
([(0,4),(1,2),(1,4),(2,3)],5)
 => [5,4]
 => [[5,4],[]]
 => ? = 2 - 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => [3,2]
 => [[3,2],[]]
 => 0 = 1 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [5,5,5,5]
 => [[5,5,5,5],[]]
 => ? = 0 - 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [2,2]
 => [[2,2],[]]
 => 0 = 1 - 1
([(0,4),(1,2),(1,3)],5)
 => [10,10]
 => [[10,10],[]]
 => ? = 5 - 1
([(0,4),(1,2),(1,3),(1,4)],5)
 => [10,4,4]
 => [[10,4,4],[]]
 => ? = 2 - 1
([(0,4),(1,2),(1,3),(3,4)],5)
 => [4,4,3]
 => [[4,4,3],[]]
 => ? = 2 - 1
([(0,3),(0,4),(1,2),(1,4)],5)
 => [12,4]
 => [[12,4],[]]
 => ? = 2 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [6,6]
 => [[6,6],[]]
 => ? = 3 - 1
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => [5,3]
 => [[5,3],[]]
 => ? = 0 - 1
([(0,3),(1,2),(1,4),(3,4)],5)
 => [5,4]
 => [[5,4],[]]
 => ? = 2 - 1
([(0,3),(1,4),(4,2)],5)
 => [5,5]
 => [[5,5],[]]
 => ? = 0 - 1
([(0,3),(1,2),(2,4),(3,4)],5)
 => [4,2]
 => [[4,2],[]]
 => ? = 1 - 1
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
 => [15,15]
 => [[15,15],[]]
 => ? = 0 - 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => [3,3]
 => [[3,3],[]]
 => ? = 0 - 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
 => [10,10]
 => [[10,10],[]]
 => ? = 5 - 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5)],6)
 => [12,4]
 => [[12,4],[]]
 => ? = 2 - 1
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6,6]
 => [[6,6],[]]
 => ? = 3 - 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
 => [10,4,4]
 => [[10,4,4],[]]
 => ? = 2 - 1
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
 => [5,5]
 => [[5,5],[]]
 => ? = 0 - 1
([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
 => [15,5,5]
 => [[15,5,5],[]]
 => ? = 0 - 1
([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [5,5,5,5]
 => [[5,5,5,5],[]]
 => ? = 0 - 1
([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
 => [15,15]
 => [[15,15],[]]
 => ? = 0 - 1
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => [2,2]
 => [[2,2],[]]
 => 0 = 1 - 1
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => [2,2]
 => [[2,2],[]]
 => 0 = 1 - 1
([(0,4),(0,5),(4,3),(5,1),(5,2)],6)
 => [10,10]
 => [[10,10],[]]
 => ? = 5 - 1
([(0,3),(0,4),(3,5),(4,1),(4,2),(4,5)],6)
 => [10,4,4]
 => [[10,4,4],[]]
 => ? = 2 - 1
([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
 => [12,4]
 => [[12,4],[]]
 => ? = 2 - 1
([(0,1),(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => [6,6]
 => [[6,6],[]]
 => ? = 3 - 1
([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
 => [3,3]
 => [[3,3],[]]
 => ? = 0 - 1
([(0,5),(1,5),(5,2),(5,3),(5,4)],6)
 => [6,6]
 => [[6,6],[]]
 => ? = 3 - 1
([(0,5),(1,5),(2,5),(5,3),(5,4)],6)
 => [6,6]
 => [[6,6],[]]
 => ? = 3 - 1
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
 => [2,2]
 => [[2,2],[]]
 => 0 = 1 - 1
([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
 => [2,2]
 => [[2,2],[]]
 => 0 = 1 - 1
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
 => [3,2]
 => [[3,2],[]]
 => 0 = 1 - 1
([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
 => [2,2]
 => [[2,2],[]]
 => 0 = 1 - 1
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => [3,2]
 => [[3,2],[]]
 => 0 = 1 - 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2)],6)
 => [2,2]
 => [[2,2],[]]
 => 0 = 1 - 1
([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
 => [3,2]
 => [[3,2],[]]
 => 0 = 1 - 1
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6),(6,1)],7)
 => [2,2]
 => [[2,2],[]]
 => 0 = 1 - 1
([(0,6),(1,6),(2,5),(3,5),(4,2),(4,3),(6,4)],7)
 => [2,2]
 => [[2,2],[]]
 => 0 = 1 - 1
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => [2,2]
 => [[2,2],[]]
 => 0 = 1 - 1
([(0,6),(1,6),(4,5),(5,2),(5,3),(6,4)],7)
 => [2,2]
 => [[2,2],[]]
 => 0 = 1 - 1
([(0,6),(1,6),(2,5),(3,5),(5,4),(6,2),(6,3)],7)
 => [2,2]
 => [[2,2],[]]
 => 0 = 1 - 1
([(0,5),(0,6),(1,5),(1,6),(2,3),(4,2),(5,4),(6,4)],7)
 => [2,2]
 => [[2,2],[]]
 => 0 = 1 - 1
([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => [3,2]
 => [[3,2],[]]
 => 0 = 1 - 1
([(0,3),(0,4),(3,6),(4,6),(5,1),(5,2),(6,5)],7)
 => [2,2]
 => [[2,2],[]]
 => 0 = 1 - 1
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => [3,2]
 => [[3,2],[]]
 => 0 = 1 - 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2),(4,6),(5,6)],7)
 => [2,2]
 => [[2,2],[]]
 => 0 = 1 - 1
([(0,5),(1,6),(2,6),(5,1),(5,2),(6,3),(6,4)],7)
 => [2,2]
 => [[2,2],[]]
 => 0 = 1 - 1
([(0,6),(1,3),(1,6),(3,5),(4,2),(5,4),(6,5)],7)
 => [3,2]
 => [[3,2],[]]
 => 0 = 1 - 1
([(0,3),(1,5),(1,6),(2,5),(2,6),(3,4),(4,1),(4,2)],7)
 => [2,2]
 => [[2,2],[]]
 => 0 = 1 - 1
([(0,4),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3)],7)
 => [3,2]
 => [[3,2],[]]
 => 0 = 1 - 1
([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => [3,2]
 => [[3,2],[]]
 => 0 = 1 - 1
Description
The number of missing boxes in the first row.
Matching statistic: St001438
(load all 2 compositions to match this statistic)
Mapping
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001438: Skew partitions ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 17%
Values
([],3)
 => [3,3]
 => [[3,3],[]]
 => ? = 0 - 1
([(2,3)],4)
 => [4,4,4]
 => [[4,4,4],[]]
 => ? = 2 - 1
([(0,1),(0,2),(0,3)],4)
 => [3,3]
 => [[3,3],[]]
 => ? = 0 - 1
([(0,3),(1,3),(2,3)],4)
 => [3,3]
 => [[3,3],[]]
 => ? = 0 - 1
([(0,3),(1,2)],4)
 => [4,2]
 => [[4,2],[]]
 => ? = 1 - 1
([(0,3),(1,2),(1,3)],4)
 => [3,2]
 => [[3,2],[]]
 => 0 = 1 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [[2,2],[]]
 => 0 = 1 - 1
([(1,2),(1,3),(1,4)],5)
 => [15,15]
 => [[15,15],[]]
 => ? = 0 - 1
([(0,2),(0,3),(0,4),(4,1)],5)
 => [4,4,4]
 => [[4,4,4],[]]
 => ? = 2 - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [[3,3],[]]
 => ? = 0 - 1
([(1,2),(1,3),(2,4),(3,4)],5)
 => [5,5]
 => [[5,5],[]]
 => ? = 0 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
 => [4,2]
 => [[4,2],[]]
 => ? = 1 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [3,2]
 => [[3,2],[]]
 => 0 = 1 - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2]
 => [[2,2],[]]
 => 0 = 1 - 1
([(2,3),(3,4)],5)
 => [5,5,5,5]
 => [[5,5,5,5],[]]
 => ? = 0 - 1
([(1,4),(4,2),(4,3)],5)
 => [5,5]
 => [[5,5],[]]
 => ? = 0 - 1
([(0,4),(4,1),(4,2),(4,3)],5)
 => [3,3]
 => [[3,3],[]]
 => ? = 0 - 1
([(1,4),(2,4),(4,3)],5)
 => [5,5]
 => [[5,5],[]]
 => ? = 0 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
 => [2,2]
 => [[2,2],[]]
 => 0 = 1 - 1
([(1,4),(2,4),(3,4)],5)
 => [15,15]
 => [[15,15],[]]
 => ? = 0 - 1
([(0,4),(1,4),(2,4),(4,3)],5)
 => [3,3]
 => [[3,3],[]]
 => ? = 0 - 1
([(0,4),(1,4),(2,3)],5)
 => [10,10]
 => [[10,10],[]]
 => ? = 5 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [12,4]
 => [[12,4],[]]
 => ? = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6,6]
 => [[6,6],[]]
 => ? = 3 - 1
([(0,4),(1,4),(2,3),(2,4)],5)
 => [10,4,4]
 => [[10,4,4],[]]
 => ? = 2 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [4,4,4]
 => [[4,4,4],[]]
 => ? = 2 - 1
([(1,4),(2,3),(2,4)],5)
 => [15,5,5]
 => [[15,5,5],[]]
 => ? = 0 - 1
([(0,4),(1,2),(1,4),(2,3)],5)
 => [5,4]
 => [[5,4],[]]
 => ? = 2 - 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => [3,2]
 => [[3,2],[]]
 => 0 = 1 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [5,5,5,5]
 => [[5,5,5,5],[]]
 => ? = 0 - 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [2,2]
 => [[2,2],[]]
 => 0 = 1 - 1
([(0,4),(1,2),(1,3)],5)
 => [10,10]
 => [[10,10],[]]
 => ? = 5 - 1
([(0,4),(1,2),(1,3),(1,4)],5)
 => [10,4,4]
 => [[10,4,4],[]]
 => ? = 2 - 1
([(0,4),(1,2),(1,3),(3,4)],5)
 => [4,4,3]
 => [[4,4,3],[]]
 => ? = 2 - 1
([(0,3),(0,4),(1,2),(1,4)],5)
 => [12,4]
 => [[12,4],[]]
 => ? = 2 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [6,6]
 => [[6,6],[]]
 => ? = 3 - 1
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => [5,3]
 => [[5,3],[]]
 => ? = 0 - 1
([(0,3),(1,2),(1,4),(3,4)],5)
 => [5,4]
 => [[5,4],[]]
 => ? = 2 - 1
([(0,3),(1,4),(4,2)],5)
 => [5,5]
 => [[5,5],[]]
 => ? = 0 - 1
([(0,3),(1,2),(2,4),(3,4)],5)
 => [4,2]
 => [[4,2],[]]
 => ? = 1 - 1
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
 => [15,15]
 => [[15,15],[]]
 => ? = 0 - 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => [3,3]
 => [[3,3],[]]
 => ? = 0 - 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
 => [10,10]
 => [[10,10],[]]
 => ? = 5 - 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5)],6)
 => [12,4]
 => [[12,4],[]]
 => ? = 2 - 1
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6,6]
 => [[6,6],[]]
 => ? = 3 - 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
 => [10,4,4]
 => [[10,4,4],[]]
 => ? = 2 - 1
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
 => [5,5]
 => [[5,5],[]]
 => ? = 0 - 1
([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
 => [15,5,5]
 => [[15,5,5],[]]
 => ? = 0 - 1
([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [5,5,5,5]
 => [[5,5,5,5],[]]
 => ? = 0 - 1
([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
 => [15,15]
 => [[15,15],[]]
 => ? = 0 - 1
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => [2,2]
 => [[2,2],[]]
 => 0 = 1 - 1
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => [2,2]
 => [[2,2],[]]
 => 0 = 1 - 1
([(0,4),(0,5),(4,3),(5,1),(5,2)],6)
 => [10,10]
 => [[10,10],[]]
 => ? = 5 - 1
([(0,3),(0,4),(3,5),(4,1),(4,2),(4,5)],6)
 => [10,4,4]
 => [[10,4,4],[]]
 => ? = 2 - 1
([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
 => [12,4]
 => [[12,4],[]]
 => ? = 2 - 1
([(0,1),(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => [6,6]
 => [[6,6],[]]
 => ? = 3 - 1
([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
 => [3,3]
 => [[3,3],[]]
 => ? = 0 - 1
([(0,5),(1,5),(5,2),(5,3),(5,4)],6)
 => [6,6]
 => [[6,6],[]]
 => ? = 3 - 1
([(0,5),(1,5),(2,5),(5,3),(5,4)],6)
 => [6,6]
 => [[6,6],[]]
 => ? = 3 - 1
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
 => [2,2]
 => [[2,2],[]]
 => 0 = 1 - 1
([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
 => [2,2]
 => [[2,2],[]]
 => 0 = 1 - 1
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
 => [3,2]
 => [[3,2],[]]
 => 0 = 1 - 1
([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
 => [2,2]
 => [[2,2],[]]
 => 0 = 1 - 1
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => [3,2]
 => [[3,2],[]]
 => 0 = 1 - 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2)],6)
 => [2,2]
 => [[2,2],[]]
 => 0 = 1 - 1
([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
 => [3,2]
 => [[3,2],[]]
 => 0 = 1 - 1
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6),(6,1)],7)
 => [2,2]
 => [[2,2],[]]
 => 0 = 1 - 1
([(0,6),(1,6),(2,5),(3,5),(4,2),(4,3),(6,4)],7)
 => [2,2]
 => [[2,2],[]]
 => 0 = 1 - 1
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => [2,2]
 => [[2,2],[]]
 => 0 = 1 - 1
([(0,6),(1,6),(4,5),(5,2),(5,3),(6,4)],7)
 => [2,2]
 => [[2,2],[]]
 => 0 = 1 - 1
([(0,6),(1,6),(2,5),(3,5),(5,4),(6,2),(6,3)],7)
 => [2,2]
 => [[2,2],[]]
 => 0 = 1 - 1
([(0,5),(0,6),(1,5),(1,6),(2,3),(4,2),(5,4),(6,4)],7)
 => [2,2]
 => [[2,2],[]]
 => 0 = 1 - 1
([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => [3,2]
 => [[3,2],[]]
 => 0 = 1 - 1
([(0,3),(0,4),(3,6),(4,6),(5,1),(5,2),(6,5)],7)
 => [2,2]
 => [[2,2],[]]
 => 0 = 1 - 1
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => [3,2]
 => [[3,2],[]]
 => 0 = 1 - 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2),(4,6),(5,6)],7)
 => [2,2]
 => [[2,2],[]]
 => 0 = 1 - 1
([(0,5),(1,6),(2,6),(5,1),(5,2),(6,3),(6,4)],7)
 => [2,2]
 => [[2,2],[]]
 => 0 = 1 - 1
([(0,6),(1,3),(1,6),(3,5),(4,2),(5,4),(6,5)],7)
 => [3,2]
 => [[3,2],[]]
 => 0 = 1 - 1
([(0,3),(1,5),(1,6),(2,5),(2,6),(3,4),(4,1),(4,2)],7)
 => [2,2]
 => [[2,2],[]]
 => 0 = 1 - 1
([(0,4),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3)],7)
 => [3,2]
 => [[3,2],[]]
 => 0 = 1 - 1
([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => [3,2]
 => [[3,2],[]]
 => 0 = 1 - 1
Description
The number of missing boxes of a skew partition.
Matching statistic: St001491
(load all 6 compositions to match this statistic)
Mapping
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 17%
Values
([],3)
 => [3,3]
 => 11000 => ? = 0
([(2,3)],4)
 => [4,4,4]
 => 1110000 => ? = 2
([(0,1),(0,2),(0,3)],4)
 => [3,3]
 => 11000 => ? = 0
([(0,3),(1,3),(2,3)],4)
 => [3,3]
 => 11000 => ? = 0
([(0,3),(1,2)],4)
 => [4,2]
 => 100100 => ? = 1
([(0,3),(1,2),(1,3)],4)
 => [3,2]
 => 10100 => ? = 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => 1100 => 1
([(1,2),(1,3),(1,4)],5)
 => [15,15]
 => 11000000000000000 => ? = 0
([(0,2),(0,3),(0,4),(4,1)],5)
 => [4,4,4]
 => 1110000 => ? = 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => 11000 => ? = 0
([(1,2),(1,3),(2,4),(3,4)],5)
 => [5,5]
 => 1100000 => ? = 0
([(0,3),(0,4),(3,2),(4,1)],5)
 => [4,2]
 => 100100 => ? = 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [3,2]
 => 10100 => ? = 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2]
 => 1100 => 1
([(2,3),(3,4)],5)
 => [5,5,5,5]
 => 111100000 => ? = 0
([(1,4),(4,2),(4,3)],5)
 => [5,5]
 => 1100000 => ? = 0
([(0,4),(4,1),(4,2),(4,3)],5)
 => [3,3]
 => 11000 => ? = 0
([(1,4),(2,4),(4,3)],5)
 => [5,5]
 => 1100000 => ? = 0
([(0,4),(1,4),(4,2),(4,3)],5)
 => [2,2]
 => 1100 => 1
([(1,4),(2,4),(3,4)],5)
 => [15,15]
 => 11000000000000000 => ? = 0
([(0,4),(1,4),(2,4),(4,3)],5)
 => [3,3]
 => 11000 => ? = 0
([(0,4),(1,4),(2,3)],5)
 => [10,10]
 => 110000000000 => ? = 5
([(0,4),(1,3),(2,3),(2,4)],5)
 => [12,4]
 => 10000000010000 => ? = 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6,6]
 => 11000000 => ? = 3
([(0,4),(1,4),(2,3),(2,4)],5)
 => [10,4,4]
 => 1000000110000 => ? = 2
([(0,4),(1,4),(2,3),(3,4)],5)
 => [4,4,4]
 => 1110000 => ? = 2
([(1,4),(2,3),(2,4)],5)
 => [15,5,5]
 => 100000000001100000 => ? = 0
([(0,4),(1,2),(1,4),(2,3)],5)
 => [5,4]
 => 1010000 => ? = 2
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => [3,2]
 => 10100 => ? = 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [5,5,5,5]
 => 111100000 => ? = 0
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [2,2]
 => 1100 => 1
([(0,4),(1,2),(1,3)],5)
 => [10,10]
 => 110000000000 => ? = 5
([(0,4),(1,2),(1,3),(1,4)],5)
 => [10,4,4]
 => 1000000110000 => ? = 2
([(0,4),(1,2),(1,3),(3,4)],5)
 => [4,4,3]
 => 1101000 => ? = 2
([(0,3),(0,4),(1,2),(1,4)],5)
 => [12,4]
 => 10000000010000 => ? = 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [6,6]
 => 11000000 => ? = 3
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => [5,3]
 => 1001000 => ? = 0
([(0,3),(1,2),(1,4),(3,4)],5)
 => [5,4]
 => 1010000 => ? = 2
([(0,3),(1,4),(4,2)],5)
 => [5,5]
 => 1100000 => ? = 0
([(0,3),(1,2),(2,4),(3,4)],5)
 => [4,2]
 => 100100 => ? = 1
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
 => [15,15]
 => 11000000000000000 => ? = 0
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => [3,3]
 => 11000 => ? = 0
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
 => [10,10]
 => 110000000000 => ? = 5
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5)],6)
 => [12,4]
 => 10000000010000 => ? = 2
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6,6]
 => 11000000 => ? = 3
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
 => [10,4,4]
 => 1000000110000 => ? = 2
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
 => [5,5]
 => 1100000 => ? = 0
([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
 => [15,5,5]
 => 100000000001100000 => ? = 0
([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [5,5,5,5]
 => 111100000 => ? = 0
([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
 => [15,15]
 => 11000000000000000 => ? = 0
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => [2,2]
 => 1100 => 1
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => [2,2]
 => 1100 => 1
([(0,4),(0,5),(4,3),(5,1),(5,2)],6)
 => [10,10]
 => 110000000000 => ? = 5
([(0,3),(0,4),(3,5),(4,1),(4,2),(4,5)],6)
 => [10,4,4]
 => 1000000110000 => ? = 2
([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
 => [12,4]
 => 10000000010000 => ? = 2
([(0,1),(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => [6,6]
 => 11000000 => ? = 3
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
 => [2,2]
 => 1100 => 1
([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
 => [2,2]
 => 1100 => 1
([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
 => [2,2]
 => 1100 => 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2)],6)
 => [2,2]
 => 1100 => 1
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6),(6,1)],7)
 => [2,2]
 => 1100 => 1
([(0,6),(1,6),(2,5),(3,5),(4,2),(4,3),(6,4)],7)
 => [2,2]
 => 1100 => 1
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => [2,2]
 => 1100 => 1
([(0,6),(1,6),(4,5),(5,2),(5,3),(6,4)],7)
 => [2,2]
 => 1100 => 1
([(0,6),(1,6),(2,5),(3,5),(5,4),(6,2),(6,3)],7)
 => [2,2]
 => 1100 => 1
([(0,5),(0,6),(1,5),(1,6),(2,3),(4,2),(5,4),(6,4)],7)
 => [2,2]
 => 1100 => 1
([(0,3),(0,4),(3,6),(4,6),(5,1),(5,2),(6,5)],7)
 => [2,2]
 => 1100 => 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2),(4,6),(5,6)],7)
 => [2,2]
 => 1100 => 1
([(0,5),(1,6),(2,6),(5,1),(5,2),(6,3),(6,4)],7)
 => [2,2]
 => 1100 => 1
([(0,3),(1,5),(1,6),(2,5),(2,6),(3,4),(4,1),(4,2)],7)
 => [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.