- FindStat

Your data matches 36 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001175
(load all 2 compositions to match this statistic)

Mapping

Mp00251: Graphs —clique sizes⟶ Integer partitions
St001175: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => [1]
 => 0
([],2)
 => [1,1]
 => 0
([(0,1)],2)
 => [2]
 => 0
([],3)
 => [1,1,1]
 => 0
([(1,2)],3)
 => [2,1]
 => 0
([(0,2),(1,2)],3)
 => [2,2]
 => 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => 0
([],4)
 => [1,1,1,1]
 => 0
([(2,3)],4)
 => [2,1,1]
 => 0
([(1,3),(2,3)],4)
 => [2,2,1]
 => 1
([(0,3),(1,3),(2,3)],4)
 => [2,2,2]
 => 2
([(0,3),(1,2)],4)
 => [2,2]
 => 1
([(0,3),(1,2),(2,3)],4)
 => [2,2,2]
 => 2
([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 0
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,2]
 => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,3]
 => 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 0
([],5)
 => [1,1,1,1,1]
 => 0
([(3,4)],5)
 => [2,1,1,1]
 => 0
([(2,4),(3,4)],5)
 => [2,2,1,1]
 => 1
([(1,4),(2,4),(3,4)],5)
 => [2,2,2,1]
 => 2
([(1,4),(2,3)],5)
 => [2,2,1]
 => 1
([(1,4),(2,3),(3,4)],5)
 => [2,2,2,1]
 => 2
([(0,1),(2,4),(3,4)],5)
 => [2,2,2]
 => 2
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 0
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2,1]
 => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2,2]
 => 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,3,1]
 => 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,2,2]
 => 2
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,2,2]
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,2]
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,3]
 => 2
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 0
([],6)
 => [1,1,1,1,1,1]
 => 0
([(4,5)],6)
 => [2,1,1,1,1]
 => 0
([(3,5),(4,5)],6)
 => [2,2,1,1,1]
 => 1
([(2,5),(3,4)],6)
 => [2,2,1,1]
 => 1
([(1,2),(3,5),(4,5)],6)
 => [2,2,2,1]
 => 2
([(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => 0
([(2,5),(3,4),(3,5),(4,5)],6)
 => [3,2,1,1]
 => 1
([(0,5),(1,4),(2,3)],6)
 => [2,2,2]
 => 2
([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,2,1]
 => 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,2,2]
 => 2
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [3,3,1]
 => 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => [3,2,2]
 => 2
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => 0
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2,1]
 => 1

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: St001033
(load all 3 compositions to match this statistic)

Mapping

Mp00251: Graphs —clique sizes⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001033: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => [1]
 => [1,0]
 => 0
([],2)
 => [1,1]
 => [1,1,0,0]
 => 0
([(0,1)],2)
 => [2]
 => [1,0,1,0]
 => 0
([],3)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
([(1,2)],3)
 => [2,1]
 => [1,0,1,1,0,0]
 => 0
([(0,2),(1,2)],3)
 => [2,2]
 => [1,1,1,0,0,0]
 => 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => [1,0,1,0,1,0]
 => 0
([],4)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 0
([(2,3)],4)
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 0
([(1,3),(2,3)],4)
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 1
([(0,3),(1,3),(2,3)],4)
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => 2
([(0,3),(1,2)],4)
 => [2,2]
 => [1,1,1,0,0,0]
 => 1
([(0,3),(1,2),(2,3)],4)
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => 2
([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 0
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 0
([],5)
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
([(3,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 0
([(2,4),(3,4)],5)
 => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1
([(1,4),(2,4),(3,4)],5)
 => [2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
([(1,4),(2,3)],5)
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 1
([(1,4),(2,3),(3,4)],5)
 => [2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
([(0,1),(2,4),(3,4)],5)
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => 2
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 0
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
([],6)
 => [1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 0
([(4,5)],6)
 => [2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 0
([(3,5),(4,5)],6)
 => [2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => 1
([(2,5),(3,4)],6)
 => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1
([(1,2),(3,5),(4,5)],6)
 => [2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
([(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 0
([(2,5),(3,4),(3,5),(4,5)],6)
 => [3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => 1
([(0,5),(1,4),(2,3)],6)
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => 2
([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => [3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 0
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 1

Description

The normalized area of the parallelogram polyomino associated with the Dyck path. The area of the smallest parallelogram polyomino equals the semilength of the Dyck path. This statistic is therefore the area of the parallelogram polyomino minus the semilength of the Dyck path. The area itself is equidistributed with [[St001034]] and with [[St000395]].

Matching statistic: St001596
(load all 4 compositions to match this statistic)

Mapping

Mp00251: Graphs —clique sizes⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001596: Skew partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => [1]
 => [[1],[]]
 => 0
([],2)
 => [1,1]
 => [[1,1],[]]
 => 0
([(0,1)],2)
 => [2]
 => [[2],[]]
 => 0
([],3)
 => [1,1,1]
 => [[1,1,1],[]]
 => 0
([(1,2)],3)
 => [2,1]
 => [[2,1],[]]
 => 0
([(0,2),(1,2)],3)
 => [2,2]
 => [[2,2],[]]
 => 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => [[3],[]]
 => 0
([],4)
 => [1,1,1,1]
 => [[1,1,1,1],[]]
 => 0
([(2,3)],4)
 => [2,1,1]
 => [[2,1,1],[]]
 => 0
([(1,3),(2,3)],4)
 => [2,2,1]
 => [[2,2,1],[]]
 => 1
([(0,3),(1,3),(2,3)],4)
 => [2,2,2]
 => [[2,2,2],[]]
 => 2
([(0,3),(1,2)],4)
 => [2,2]
 => [[2,2],[]]
 => 1
([(0,3),(1,2),(2,3)],4)
 => [2,2,2]
 => [[2,2,2],[]]
 => 2
([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [[3,1],[]]
 => 0
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,2]
 => [[3,2],[]]
 => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,3]
 => [[3,3],[]]
 => 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [[4],[]]
 => 0
([],5)
 => [1,1,1,1,1]
 => [[1,1,1,1,1],[]]
 => 0
([(3,4)],5)
 => [2,1,1,1]
 => [[2,1,1,1],[]]
 => 0
([(2,4),(3,4)],5)
 => [2,2,1,1]
 => [[2,2,1,1],[]]
 => 1
([(1,4),(2,4),(3,4)],5)
 => [2,2,2,1]
 => [[2,2,2,1],[]]
 => 2
([(1,4),(2,3)],5)
 => [2,2,1]
 => [[2,2,1],[]]
 => 1
([(1,4),(2,3),(3,4)],5)
 => [2,2,2,1]
 => [[2,2,2,1],[]]
 => 2
([(0,1),(2,4),(3,4)],5)
 => [2,2,2]
 => [[2,2,2],[]]
 => 2
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [[3,1,1],[]]
 => 0
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2,1]
 => [[3,2,1],[]]
 => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2,2]
 => [[3,2,2],[]]
 => 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,3,1]
 => [[3,3,1],[]]
 => 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,2,2]
 => [[3,2,2],[]]
 => 2
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [[3,2],[]]
 => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,2,2]
 => [[3,2,2],[]]
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [[3,3],[]]
 => 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [[4,1],[]]
 => 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,2]
 => [[4,2],[]]
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,3]
 => [[4,3],[]]
 => 2
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[5],[]]
 => 0
([],6)
 => [1,1,1,1,1,1]
 => [[1,1,1,1,1,1],[]]
 => 0
([(4,5)],6)
 => [2,1,1,1,1]
 => [[2,1,1,1,1],[]]
 => 0
([(3,5),(4,5)],6)
 => [2,2,1,1,1]
 => [[2,2,1,1,1],[]]
 => 1
([(2,5),(3,4)],6)
 => [2,2,1,1]
 => [[2,2,1,1],[]]
 => 1
([(1,2),(3,5),(4,5)],6)
 => [2,2,2,1]
 => [[2,2,2,1],[]]
 => 2
([(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [[3,1,1,1],[]]
 => 0
([(2,5),(3,4),(3,5),(4,5)],6)
 => [3,2,1,1]
 => [[3,2,1,1],[]]
 => 1
([(0,5),(1,4),(2,3)],6)
 => [2,2,2]
 => [[2,2,2],[]]
 => 2
([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [[3,2,1],[]]
 => 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,2,2]
 => [[3,2,2],[]]
 => 2
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [3,3,1]
 => [[3,3,1],[]]
 => 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => [3,2,2]
 => [[3,2,2],[]]
 => 2
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [[4,1,1],[]]
 => 0
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2,1]
 => [[4,2,1],[]]
 => 1

Description

The number of two-by-two squares inside a skew partition. This is, the number of cells

$(i,j)$ in a skew partition for which the box

$(i+1,j+1)$ is also a cell inside the skew partition.

Matching statistic: St000345
(load all 2 compositions to match this statistic)

Mapping

Mp00251: Graphs —clique sizes⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000345: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => [1]
 => []
 => 1 = 0 + 1
([],2)
 => [1,1]
 => [1]
 => 1 = 0 + 1
([(0,1)],2)
 => [2]
 => []
 => 1 = 0 + 1
([],3)
 => [1,1,1]
 => [1,1]
 => 1 = 0 + 1
([(1,2)],3)
 => [2,1]
 => [1]
 => 1 = 0 + 1
([(0,2),(1,2)],3)
 => [2,2]
 => [2]
 => 2 = 1 + 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => []
 => 1 = 0 + 1
([],4)
 => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
([(2,3)],4)
 => [2,1,1]
 => [1,1]
 => 1 = 0 + 1
([(1,3),(2,3)],4)
 => [2,2,1]
 => [2,1]
 => 2 = 1 + 1
([(0,3),(1,3),(2,3)],4)
 => [2,2,2]
 => [2,2]
 => 3 = 2 + 1
([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => 2 = 1 + 1
([(0,3),(1,2),(2,3)],4)
 => [2,2,2]
 => [2,2]
 => 3 = 2 + 1
([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 1 = 0 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,2]
 => [2]
 => 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,3]
 => [3]
 => 3 = 2 + 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => 1 = 0 + 1
([],5)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1 = 0 + 1
([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
([(2,4),(3,4)],5)
 => [2,2,1,1]
 => [2,1,1]
 => 2 = 1 + 1
([(1,4),(2,4),(3,4)],5)
 => [2,2,2,1]
 => [2,2,1]
 => 3 = 2 + 1
([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => 2 = 1 + 1
([(1,4),(2,3),(3,4)],5)
 => [2,2,2,1]
 => [2,2,1]
 => 3 = 2 + 1
([(0,1),(2,4),(3,4)],5)
 => [2,2,2]
 => [2,2]
 => 3 = 2 + 1
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 1 = 0 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2,1]
 => [2,1]
 => 2 = 1 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2,2]
 => [2,2]
 => 3 = 2 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,3,1]
 => [3,1]
 => 3 = 2 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,2,2]
 => [2,2]
 => 3 = 2 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 2 = 1 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,2,2]
 => [2,2]
 => 3 = 2 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [3]
 => 3 = 2 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 1 = 0 + 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,2]
 => [2]
 => 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,3]
 => [3]
 => 3 = 2 + 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => 1 = 0 + 1
([],6)
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1 = 0 + 1
([(4,5)],6)
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 1 = 0 + 1
([(3,5),(4,5)],6)
 => [2,2,1,1,1]
 => [2,1,1,1]
 => 2 = 1 + 1
([(2,5),(3,4)],6)
 => [2,2,1,1]
 => [2,1,1]
 => 2 = 1 + 1
([(1,2),(3,5),(4,5)],6)
 => [2,2,2,1]
 => [2,2,1]
 => 3 = 2 + 1
([(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
([(2,5),(3,4),(3,5),(4,5)],6)
 => [3,2,1,1]
 => [2,1,1]
 => 2 = 1 + 1
([(0,5),(1,4),(2,3)],6)
 => [2,2,2]
 => [2,2]
 => 3 = 2 + 1
([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [2,1]
 => 2 = 1 + 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,2,2]
 => [2,2]
 => 3 = 2 + 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [3,3,1]
 => [3,1]
 => 3 = 2 + 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => [3,2,2]
 => [2,2]
 => 3 = 2 + 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 1 = 0 + 1
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2,1]
 => [2,1]
 => 2 = 1 + 1

Description

The number of refinements of a partition. A partition

$\lambda$ refines a partition

$\mu$ if the parts of

$\mu$ can be subdivided to obtain the parts of

$\lambda$ .

Matching statistic: St000935
(load all 2 compositions to match this statistic)

Mapping

Mp00251: Graphs —clique sizes⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000935: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => [1]
 => []
 => 1 = 0 + 1
([],2)
 => [1,1]
 => [1]
 => 1 = 0 + 1
([(0,1)],2)
 => [2]
 => []
 => 1 = 0 + 1
([],3)
 => [1,1,1]
 => [1,1]
 => 1 = 0 + 1
([(1,2)],3)
 => [2,1]
 => [1]
 => 1 = 0 + 1
([(0,2),(1,2)],3)
 => [2,2]
 => [2]
 => 2 = 1 + 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => []
 => 1 = 0 + 1
([],4)
 => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
([(2,3)],4)
 => [2,1,1]
 => [1,1]
 => 1 = 0 + 1
([(1,3),(2,3)],4)
 => [2,2,1]
 => [2,1]
 => 2 = 1 + 1
([(0,3),(1,3),(2,3)],4)
 => [2,2,2]
 => [2,2]
 => 3 = 2 + 1
([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => 2 = 1 + 1
([(0,3),(1,2),(2,3)],4)
 => [2,2,2]
 => [2,2]
 => 3 = 2 + 1
([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 1 = 0 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,2]
 => [2]
 => 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,3]
 => [3]
 => 3 = 2 + 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => 1 = 0 + 1
([],5)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1 = 0 + 1
([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
([(2,4),(3,4)],5)
 => [2,2,1,1]
 => [2,1,1]
 => 2 = 1 + 1
([(1,4),(2,4),(3,4)],5)
 => [2,2,2,1]
 => [2,2,1]
 => 3 = 2 + 1
([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => 2 = 1 + 1
([(1,4),(2,3),(3,4)],5)
 => [2,2,2,1]
 => [2,2,1]
 => 3 = 2 + 1
([(0,1),(2,4),(3,4)],5)
 => [2,2,2]
 => [2,2]
 => 3 = 2 + 1
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 1 = 0 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2,1]
 => [2,1]
 => 2 = 1 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2,2]
 => [2,2]
 => 3 = 2 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,3,1]
 => [3,1]
 => 3 = 2 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,2,2]
 => [2,2]
 => 3 = 2 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 2 = 1 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,2,2]
 => [2,2]
 => 3 = 2 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [3]
 => 3 = 2 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 1 = 0 + 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,2]
 => [2]
 => 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,3]
 => [3]
 => 3 = 2 + 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => 1 = 0 + 1
([],6)
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1 = 0 + 1
([(4,5)],6)
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 1 = 0 + 1
([(3,5),(4,5)],6)
 => [2,2,1,1,1]
 => [2,1,1,1]
 => 2 = 1 + 1
([(2,5),(3,4)],6)
 => [2,2,1,1]
 => [2,1,1]
 => 2 = 1 + 1
([(1,2),(3,5),(4,5)],6)
 => [2,2,2,1]
 => [2,2,1]
 => 3 = 2 + 1
([(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
([(2,5),(3,4),(3,5),(4,5)],6)
 => [3,2,1,1]
 => [2,1,1]
 => 2 = 1 + 1
([(0,5),(1,4),(2,3)],6)
 => [2,2,2]
 => [2,2]
 => 3 = 2 + 1
([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [2,1]
 => 2 = 1 + 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,2,2]
 => [2,2]
 => 3 = 2 + 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [3,3,1]
 => [3,1]
 => 3 = 2 + 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => [3,2,2]
 => [2,2]
 => 3 = 2 + 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 1 = 0 + 1
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2,1]
 => [2,1]
 => 2 = 1 + 1

Description

The number of ordered refinements of an integer partition. This is, for an integer partition

$\mu = (\mu_1,\ldots,\mu_n)$ the number of integer partition

$\lambda = (\lambda_1,\ldots,\lambda_m)$ such that there are indices

$1 = a_0 < \ldots < a_n = m$ with

$\mu_j = \lambda_{a_{j-1}} + \ldots + \lambda_{a_j-1}$ .

Matching statistic: St001389
(load all 2 compositions to match this statistic)

Mapping

Mp00251: Graphs —clique sizes⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001389: Integer partitions ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%

Values

([],1)
 => [1]
 => []
 => ? = 0 + 1
([],2)
 => [1,1]
 => [1]
 => 1 = 0 + 1
([(0,1)],2)
 => [2]
 => []
 => ? = 0 + 1
([],3)
 => [1,1,1]
 => [1,1]
 => 1 = 0 + 1
([(1,2)],3)
 => [2,1]
 => [1]
 => 1 = 0 + 1
([(0,2),(1,2)],3)
 => [2,2]
 => [2]
 => 2 = 1 + 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => []
 => ? = 0 + 1
([],4)
 => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
([(2,3)],4)
 => [2,1,1]
 => [1,1]
 => 1 = 0 + 1
([(1,3),(2,3)],4)
 => [2,2,1]
 => [2,1]
 => 2 = 1 + 1
([(0,3),(1,3),(2,3)],4)
 => [2,2,2]
 => [2,2]
 => 3 = 2 + 1
([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => 2 = 1 + 1
([(0,3),(1,2),(2,3)],4)
 => [2,2,2]
 => [2,2]
 => 3 = 2 + 1
([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 1 = 0 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,2]
 => [2]
 => 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,3]
 => [3]
 => 3 = 2 + 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? = 0 + 1
([],5)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1 = 0 + 1
([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
([(2,4),(3,4)],5)
 => [2,2,1,1]
 => [2,1,1]
 => 2 = 1 + 1
([(1,4),(2,4),(3,4)],5)
 => [2,2,2,1]
 => [2,2,1]
 => 3 = 2 + 1
([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => 2 = 1 + 1
([(1,4),(2,3),(3,4)],5)
 => [2,2,2,1]
 => [2,2,1]
 => 3 = 2 + 1
([(0,1),(2,4),(3,4)],5)
 => [2,2,2]
 => [2,2]
 => 3 = 2 + 1
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 1 = 0 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2,1]
 => [2,1]
 => 2 = 1 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2,2]
 => [2,2]
 => 3 = 2 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,3,1]
 => [3,1]
 => 3 = 2 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,2,2]
 => [2,2]
 => 3 = 2 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 2 = 1 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,2,2]
 => [2,2]
 => 3 = 2 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [3]
 => 3 = 2 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 1 = 0 + 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,2]
 => [2]
 => 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,3]
 => [3]
 => 3 = 2 + 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 0 + 1
([],6)
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1 = 0 + 1
([(4,5)],6)
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 1 = 0 + 1
([(3,5),(4,5)],6)
 => [2,2,1,1,1]
 => [2,1,1,1]
 => 2 = 1 + 1
([(2,5),(3,4)],6)
 => [2,2,1,1]
 => [2,1,1]
 => 2 = 1 + 1
([(1,2),(3,5),(4,5)],6)
 => [2,2,2,1]
 => [2,2,1]
 => 3 = 2 + 1
([(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
([(2,5),(3,4),(3,5),(4,5)],6)
 => [3,2,1,1]
 => [2,1,1]
 => 2 = 1 + 1
([(0,5),(1,4),(2,3)],6)
 => [2,2,2]
 => [2,2]
 => 3 = 2 + 1
([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [2,1]
 => 2 = 1 + 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,2,2]
 => [2,2]
 => 3 = 2 + 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [3,3,1]
 => [3,1]
 => 3 = 2 + 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => [3,2,2]
 => [2,2]
 => 3 = 2 + 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 1 = 0 + 1
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2,1]
 => [2,1]
 => 2 = 1 + 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => [3,3]
 => [3]
 => 3 = 2 + 1
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => 2 = 1 + 1
([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,3]
 => [3]
 => 3 = 2 + 1
([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 1 = 0 + 1
([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,2]
 => [2]
 => 2 = 1 + 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 0 + 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [7]
 => []
 => ? = 0 + 1

Description

The number of partitions of the same length below the given integer partition. For a partition

$\lambda_1 \geq \dots \lambda_k > 0$ , this number is

$\det\left( \binom{\lambda_{k+1-i}}{j-i+1} \right)_{1 \le i,j \le k}.$

Matching statistic: St001176

Mapping

Mp00251: Graphs —clique sizes⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%

Values

([],1)
 => [1]
 => []
 => []
 => ? = 0
([],2)
 => [1,1]
 => [1]
 => [1]
 => 0
([(0,1)],2)
 => [2]
 => []
 => []
 => ? = 0
([],3)
 => [1,1,1]
 => [1,1]
 => [2]
 => 0
([(1,2)],3)
 => [2,1]
 => [1]
 => [1]
 => 0
([(0,2),(1,2)],3)
 => [2,2]
 => [2]
 => [1,1]
 => 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => []
 => []
 => ? = 0
([],4)
 => [1,1,1,1]
 => [1,1,1]
 => [3]
 => 0
([(2,3)],4)
 => [2,1,1]
 => [1,1]
 => [2]
 => 0
([(1,3),(2,3)],4)
 => [2,2,1]
 => [2,1]
 => [2,1]
 => 1
([(0,3),(1,3),(2,3)],4)
 => [2,2,2]
 => [2,2]
 => [2,2]
 => 2
([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => [1,1]
 => 1
([(0,3),(1,2),(2,3)],4)
 => [2,2,2]
 => [2,2]
 => [2,2]
 => 2
([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => [1]
 => 0
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,2]
 => [2]
 => [1,1]
 => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,3]
 => [3]
 => [1,1,1]
 => 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => []
 => ? = 0
([],5)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => 0
([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => [3]
 => 0
([(2,4),(3,4)],5)
 => [2,2,1,1]
 => [2,1,1]
 => [3,1]
 => 1
([(1,4),(2,4),(3,4)],5)
 => [2,2,2,1]
 => [2,2,1]
 => [3,2]
 => 2
([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => [2,1]
 => 1
([(1,4),(2,3),(3,4)],5)
 => [2,2,2,1]
 => [2,2,1]
 => [3,2]
 => 2
([(0,1),(2,4),(3,4)],5)
 => [2,2,2]
 => [2,2]
 => [2,2]
 => 2
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => [2]
 => 0
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2,1]
 => [2,1]
 => [2,1]
 => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2,2]
 => [2,2]
 => [2,2]
 => 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,3,1]
 => [3,1]
 => [2,1,1]
 => 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,2,2]
 => [2,2]
 => [2,2]
 => 2
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => [1,1]
 => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,2,2]
 => [2,2]
 => [2,2]
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [3]
 => [1,1,1]
 => 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => [1]
 => 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,2]
 => [2]
 => [1,1]
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,3]
 => [3]
 => [1,1,1]
 => 2
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => []
 => ? = 0
([],6)
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [5]
 => 0
([(4,5)],6)
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => 0
([(3,5),(4,5)],6)
 => [2,2,1,1,1]
 => [2,1,1,1]
 => [4,1]
 => 1
([(2,5),(3,4)],6)
 => [2,2,1,1]
 => [2,1,1]
 => [3,1]
 => 1
([(1,2),(3,5),(4,5)],6)
 => [2,2,2,1]
 => [2,2,1]
 => [3,2]
 => 2
([(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => [3]
 => 0
([(2,5),(3,4),(3,5),(4,5)],6)
 => [3,2,1,1]
 => [2,1,1]
 => [3,1]
 => 1
([(0,5),(1,4),(2,3)],6)
 => [2,2,2]
 => [2,2]
 => [2,2]
 => 2
([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [2,1]
 => [2,1]
 => 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,2,2]
 => [2,2]
 => [2,2]
 => 2
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [3,3,1]
 => [3,1]
 => [2,1,1]
 => 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => [3,2,2]
 => [2,2]
 => [2,2]
 => 2
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [2]
 => 0
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2,1]
 => [2,1]
 => [2,1]
 => 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => [3,3]
 => [3]
 => [1,1,1]
 => 2
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => [1,1]
 => 1
([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,3]
 => [3]
 => [1,1,1]
 => 2
([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => [1]
 => 0
([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,2]
 => [2]
 => [1,1]
 => 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => []
 => ? = 0
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [7]
 => []
 => []
 => ? = 0

Description

The size of a partition minus its first part. This is the number of boxes in its diagram that are not in the first row.

Matching statistic: St001229
(load all 2 compositions to match this statistic)

Mapping

Mp00251: Graphs —clique sizes⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001229: Dyck paths ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%

Values

([],1)
 => [1]
 => [1,0]
 => [1,0]
 => 0
([],2)
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0
([(0,1)],2)
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 0
([],3)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 0
([(1,2)],3)
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 0
([(0,2),(1,2)],3)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
([],4)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
([(2,3)],4)
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 0
([(1,3),(2,3)],4)
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 1
([(0,3),(1,3),(2,3)],4)
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2
([(0,3),(1,2)],4)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 1
([(0,3),(1,2),(2,3)],4)
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2
([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 0
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
([],5)
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
([(3,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 0
([(2,4),(3,4)],5)
 => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 1
([(1,4),(2,4),(3,4)],5)
 => [2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 2
([(1,4),(2,3)],5)
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 1
([(1,4),(2,3),(3,4)],5)
 => [2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 2
([(0,1),(2,4),(3,4)],5)
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 0
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 2
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 2
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
([],6)
 => [1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0
([(4,5)],6)
 => [2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 0
([(3,5),(4,5)],6)
 => [2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => 1
([(2,5),(3,4)],6)
 => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 1
([(1,2),(3,5),(4,5)],6)
 => [2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 2
([(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => 0
([(2,5),(3,4),(3,5),(4,5)],6)
 => [3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => 1
([(0,5),(1,4),(2,3)],6)
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2
([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 2
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => [3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 2
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => 0
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => 1
([],7)
 => [1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
([(5,6)],7)
 => [2,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
([(4,5),(4,6),(5,6)],7)
 => [3,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 0
([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => ? = 0
([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => ? = 0
([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 0
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0

Description

The vector space dimension of the first extension group between the Jacobson radical J and J^2. The vector space dimension of

$Ext_A^1(J,J^2)$ .

Matching statistic: St001608

Mapping

Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001608: Integer partitions ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 67%

Values

([],1)
 => []
 => ?
 => ? = 0
([],2)
 => []
 => ?
 => ? = 0
([(0,1)],2)
 => [1]
 => []
 => ? = 0
([],3)
 => []
 => ?
 => ? = 0
([(1,2)],3)
 => [1]
 => []
 => ? = 0
([(0,2),(1,2)],3)
 => [1,1]
 => [1]
 => 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => []
 => ? = 0
([],4)
 => []
 => ?
 => ? = 0
([(2,3)],4)
 => [1]
 => []
 => ? = 0
([(1,3),(2,3)],4)
 => [1,1]
 => [1]
 => 1
([(0,3),(1,3),(2,3)],4)
 => [1,1,1]
 => [1,1]
 => 2
([(0,3),(1,2)],4)
 => [1,1]
 => [1]
 => 1
([(0,3),(1,2),(2,3)],4)
 => [1,1,1]
 => [1,1]
 => 2
([(1,2),(1,3),(2,3)],4)
 => [3]
 => []
 => ? = 0
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => []
 => ? = 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => []
 => ? = 0
([],5)
 => []
 => ?
 => ? = 0
([(3,4)],5)
 => [1]
 => []
 => ? = 0
([(2,4),(3,4)],5)
 => [1,1]
 => [1]
 => 1
([(1,4),(2,4),(3,4)],5)
 => [1,1,1]
 => [1,1]
 => 2
([(1,4),(2,3)],5)
 => [1,1]
 => [1]
 => 1
([(1,4),(2,3),(3,4)],5)
 => [1,1,1]
 => [1,1]
 => 2
([(0,1),(2,4),(3,4)],5)
 => [1,1,1]
 => [1,1]
 => 2
([(2,3),(2,4),(3,4)],5)
 => [3]
 => []
 => ? = 0
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1]
 => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 2
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1]
 => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [3]
 => 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => ? = 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6,1]
 => [1]
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => []
 => ? = 2
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [10]
 => []
 => ? = 0
([],6)
 => []
 => ?
 => ? = 0
([(4,5)],6)
 => [1]
 => []
 => ? = 0
([(3,5),(4,5)],6)
 => [1,1]
 => [1]
 => 1
([(2,5),(3,4)],6)
 => [1,1]
 => [1]
 => 1
([(1,2),(3,5),(4,5)],6)
 => [1,1,1]
 => [1,1]
 => 2
([(3,4),(3,5),(4,5)],6)
 => [3]
 => []
 => ? = 0
([(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1]
 => [1]
 => 1
([(0,5),(1,4),(2,3)],6)
 => [1,1,1]
 => [1,1]
 => 2
([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,1]
 => [1]
 => 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1]
 => [1,1]
 => 2
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [3,3]
 => [3]
 => 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => [3,1,1]
 => [1,1]
 => 2
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 0
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6,1]
 => [1]
 => 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => [3,3]
 => [3]
 => 2
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6,1]
 => [1]
 => 1
([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6,3]
 => [3]
 => 2
([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [10]
 => []
 => ? = 0
([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [10,1]
 => [1]
 => 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [15]
 => []
 => ? = 0
([],7)
 => []
 => ?
 => ? = 0
([(5,6)],7)
 => [1]
 => []
 => ? = 0
([(3,6),(4,5)],7)
 => [1,1]
 => [1]
 => 1
([(4,5),(4,6),(5,6)],7)
 => [3]
 => []
 => ? = 0
([(1,6),(2,5),(3,4)],7)
 => [1,1,1]
 => [1,1]
 => 2
([(2,3),(4,5),(4,6),(5,6)],7)
 => [3,1]
 => [1]
 => 1
([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [6]
 => []
 => ? = 0
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
 => [3,1,1]
 => [1,1]
 => 2
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
 => [3,3]
 => [3]
 => 2
([(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [6,1]
 => [1]
 => 1
([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [10]
 => []
 => ? = 0
([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [6,3]
 => [3]
 => 2
([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [10,1]
 => [1]
 => 1
([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [15]
 => []
 => ? = 0
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [21]
 => ?
 => ? = 0

Description

The number of coloured rooted trees such that the multiplicities of colours are given by a partition. In particular, the value on the partition

$(n)$ is the number of unlabelled rooted trees on

$n$ vertices, [[oeis:A000081]], whereas the value on the partition

$(1^n)$ is the number of labelled rooted trees [[oeis:A000169]].

Matching statistic: St001914

Mapping

Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001914: Integer partitions ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 67%

Values

([],1)
 => []
 => ?
 => ? = 0
([],2)
 => []
 => ?
 => ? = 0
([(0,1)],2)
 => [1]
 => []
 => ? = 0
([],3)
 => []
 => ?
 => ? = 0
([(1,2)],3)
 => [1]
 => []
 => ? = 0
([(0,2),(1,2)],3)
 => [1,1]
 => [1]
 => 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => []
 => ? = 0
([],4)
 => []
 => ?
 => ? = 0
([(2,3)],4)
 => [1]
 => []
 => ? = 0
([(1,3),(2,3)],4)
 => [1,1]
 => [1]
 => 1
([(0,3),(1,3),(2,3)],4)
 => [1,1,1]
 => [1,1]
 => 2
([(0,3),(1,2)],4)
 => [1,1]
 => [1]
 => 1
([(0,3),(1,2),(2,3)],4)
 => [1,1,1]
 => [1,1]
 => 2
([(1,2),(1,3),(2,3)],4)
 => [3]
 => []
 => ? = 0
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => []
 => ? = 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => []
 => ? = 0
([],5)
 => []
 => ?
 => ? = 0
([(3,4)],5)
 => [1]
 => []
 => ? = 0
([(2,4),(3,4)],5)
 => [1,1]
 => [1]
 => 1
([(1,4),(2,4),(3,4)],5)
 => [1,1,1]
 => [1,1]
 => 2
([(1,4),(2,3)],5)
 => [1,1]
 => [1]
 => 1
([(1,4),(2,3),(3,4)],5)
 => [1,1,1]
 => [1,1]
 => 2
([(0,1),(2,4),(3,4)],5)
 => [1,1,1]
 => [1,1]
 => 2
([(2,3),(2,4),(3,4)],5)
 => [3]
 => []
 => ? = 0
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1]
 => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 2
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1]
 => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [3]
 => 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => ? = 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6,1]
 => [1]
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => []
 => ? = 2
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [10]
 => []
 => ? = 0
([],6)
 => []
 => ?
 => ? = 0
([(4,5)],6)
 => [1]
 => []
 => ? = 0
([(3,5),(4,5)],6)
 => [1,1]
 => [1]
 => 1
([(2,5),(3,4)],6)
 => [1,1]
 => [1]
 => 1
([(1,2),(3,5),(4,5)],6)
 => [1,1,1]
 => [1,1]
 => 2
([(3,4),(3,5),(4,5)],6)
 => [3]
 => []
 => ? = 0
([(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1]
 => [1]
 => 1
([(0,5),(1,4),(2,3)],6)
 => [1,1,1]
 => [1,1]
 => 2
([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,1]
 => [1]
 => 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1]
 => [1,1]
 => 2
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [3,3]
 => [3]
 => 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => [3,1,1]
 => [1,1]
 => 2
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 0
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6,1]
 => [1]
 => 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => [3,3]
 => [3]
 => 2
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6,1]
 => [1]
 => 1
([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6,3]
 => [3]
 => 2
([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [10]
 => []
 => ? = 0
([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [10,1]
 => [1]
 => 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [15]
 => []
 => ? = 0
([],7)
 => []
 => ?
 => ? = 0
([(5,6)],7)
 => [1]
 => []
 => ? = 0
([(3,6),(4,5)],7)
 => [1,1]
 => [1]
 => 1
([(4,5),(4,6),(5,6)],7)
 => [3]
 => []
 => ? = 0
([(1,6),(2,5),(3,4)],7)
 => [1,1,1]
 => [1,1]
 => 2
([(2,3),(4,5),(4,6),(5,6)],7)
 => [3,1]
 => [1]
 => 1
([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [6]
 => []
 => ? = 0
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
 => [3,1,1]
 => [1,1]
 => 2
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
 => [3,3]
 => [3]
 => 2
([(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [6,1]
 => [1]
 => 1
([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [10]
 => []
 => ? = 0
([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [6,3]
 => [3]
 => 2
([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [10,1]
 => [1]
 => 1
([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [15]
 => []
 => ? = 0
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [21]
 => ?
 => ? = 0

Description

The size of the orbit of an integer partition in Bulgarian solitaire. Bulgarian solitaire is the dynamical system where a move consists of removing the first column of the Ferrers diagram and inserting it as a row. This statistic returns the number of partitions that can be obtained from the given partition.

The following 26 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001498The normalised height of a Nakayama algebra with magnitude 1. St001881The number of factors of a lattice as a Cartesian product of lattices. St001200The number of simple modules in

$eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000939The number of characters of the symmetric group whose value on the partition is positive. St000284The Plancherel distribution on integer partitions. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001128The exponens consonantiae of a partition. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000640The rank of the largest boolean interval in a poset. St001877Number of indecomposable injective modules with projective dimension 2. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001875The number of simple modules with projective dimension at most 1. St001876The number of 2-regular simple modules in the incidence algebra of the lattice.