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Your data matches 110 different statistics following compositions of up to 3 maps.
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Matching statistic: St000010
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00232: Dyck paths —parallelogram poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> ([],1)
=> [1]
=> 1
[1,0,1,0]
=> ([(0,1)],2)
=> [2]
=> 1
[1,1,0,0]
=> ([(0,1)],2)
=> [2]
=> 1
[1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> [3]
=> 1
[1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> [3]
=> 1
[1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> [3]
=> 1
[1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> [3]
=> 1
[1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> 2
[1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 2
[1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 2
[1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> 2
[1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> 2
[1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [4,2]
=> 2
[1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [4,2]
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [5,1]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [5,1]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [5,1]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [5,1]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 2
Description
The length of the partition.
Matching statistic: St001484
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00232: Dyck paths —parallelogram poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St001484: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St001484: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> ([],1)
=> [1]
=> 1
[1,0,1,0]
=> ([(0,1)],2)
=> [2]
=> 1
[1,1,0,0]
=> ([(0,1)],2)
=> [2]
=> 1
[1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> [3]
=> 1
[1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> [3]
=> 1
[1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> [3]
=> 1
[1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> [3]
=> 1
[1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> 2
[1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 2
[1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 2
[1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> 2
[1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> 2
[1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [4,2]
=> 2
[1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [4,2]
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [5,1]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [5,1]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [5,1]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [5,1]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 2
Description
The number of singletons of an integer partition.
A singleton in an integer partition is a part that appear precisely once.
Matching statistic: St001924
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00232: Dyck paths —parallelogram poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St001924: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St001924: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> ([],1)
=> [1]
=> 1
[1,0,1,0]
=> ([(0,1)],2)
=> [2]
=> 1
[1,1,0,0]
=> ([(0,1)],2)
=> [2]
=> 1
[1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> [3]
=> 1
[1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> [3]
=> 1
[1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> [3]
=> 1
[1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> [3]
=> 1
[1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> 2
[1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 2
[1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 2
[1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> 2
[1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> 2
[1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [4,2]
=> 2
[1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [4,2]
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [5,1]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [5,1]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [5,1]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [5,1]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 2
Description
The number of cells in an integer partition whose arm and leg length coincide.
Matching statistic: St000147
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00232: Dyck paths —parallelogram poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> ([],1)
=> [1]
=> [1]
=> 1
[1,0,1,0]
=> ([(0,1)],2)
=> [2]
=> [1,1]
=> 1
[1,1,0,0]
=> ([(0,1)],2)
=> [2]
=> [1,1]
=> 1
[1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,1,1]
=> 1
[1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,1,1]
=> 1
[1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,1,1]
=> 1
[1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,1,1]
=> 1
[1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 2
[1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1]
=> 1
[1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1]
=> 1
[1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1]
=> 1
[1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1]
=> 1
[1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> [2,1,1,1]
=> 2
[1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1]
=> 1
[1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1]
=> 1
[1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1]
=> 1
[1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1]
=> 1
[1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> [2,1,1,1]
=> 2
[1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [2,1,1,1]
=> 2
[1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [2,1,1,1]
=> 2
[1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [4,2]
=> [2,2,1,1]
=> 2
[1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [4,2]
=> [2,2,1,1]
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> [2,2,1,1,1]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> [2,2,1,1,1]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> [2,2,1,1,1]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> [2,2,1,1,1]
=> 2
Description
The largest part of an integer partition.
Matching statistic: St000378
Mp00232: Dyck paths —parallelogram poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000378: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 97%●distinct values known / distinct values provided: 75%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000378: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 97%●distinct values known / distinct values provided: 75%
Values
[1,0]
=> ([],1)
=> [1]
=> [1]
=> 1
[1,0,1,0]
=> ([(0,1)],2)
=> [2]
=> [1,1]
=> 1
[1,1,0,0]
=> ([(0,1)],2)
=> [2]
=> [1,1]
=> 1
[1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,1,1]
=> 1
[1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,1,1]
=> 1
[1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,1,1]
=> 1
[1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,1,1]
=> 1
[1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 2
[1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1]
=> 1
[1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1]
=> 1
[1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1]
=> 1
[1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1]
=> 1
[1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> [2,1,1,1]
=> 2
[1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1]
=> 1
[1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1]
=> 1
[1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1]
=> 1
[1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1]
=> 1
[1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> [2,1,1,1]
=> 2
[1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [2,1,1,1]
=> 2
[1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [2,1,1,1]
=> 2
[1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [4,2]
=> [2,2,1,1]
=> 2
[1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [4,2]
=> [2,2,1,1]
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> [2,2,1,1,1]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> [2,2,1,1,1]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> [2,2,1,1,1]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> [2,2,1,1,1]
=> 2
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> [7,4,2]
=> [3,3,2,2,1,1,1]
=> ? = 3
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> [7,4,2]
=> [3,3,2,2,1,1,1]
=> ? = 3
[1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> [7,4,2]
=> [3,3,2,2,1,1,1]
=> ? = 3
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> [7,4,2]
=> [3,3,2,2,1,1,1]
=> ? = 3
[1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> [7,5,3]
=> [3,3,3,2,2,1,1]
=> ? = 3
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> [7,5,3]
=> [3,3,3,2,2,1,1]
=> ? = 3
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)
=> [7,5,3,1]
=> [4,3,3,2,2,1,1]
=> ? = 4
[1,0,1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> ([(0,7),(1,10),(2,11),(3,8),(4,9),(5,2),(5,9),(6,3),(6,12),(7,4),(7,5),(8,10),(9,6),(9,11),(11,12),(12,1),(12,8)],13)
=> [8,5]
=> [2,2,2,2,2,1,1,1]
=> ? = 2
[1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> ([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
=> [8,5,2]
=> [3,3,2,2,2,1,1,1]
=> ? = 3
[1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> ([(0,7),(1,10),(2,11),(3,8),(4,9),(5,2),(5,9),(6,3),(6,12),(7,4),(7,5),(8,10),(9,6),(9,11),(11,12),(12,1),(12,8)],13)
=> [8,5]
=> [2,2,2,2,2,1,1,1]
=> ? = 2
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
=> [8,5,2]
=> [3,3,2,2,2,1,1,1]
=> ? = 3
[1,1,0,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> ([(0,7),(1,10),(2,11),(3,8),(4,9),(5,2),(5,9),(6,3),(6,12),(7,4),(7,5),(8,10),(9,6),(9,11),(11,12),(12,1),(12,8)],13)
=> [8,5]
=> [2,2,2,2,2,1,1,1]
=> ? = 2
[1,1,0,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> ([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
=> [8,5,2]
=> [3,3,2,2,2,1,1,1]
=> ? = 3
[1,1,0,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> ([(0,7),(1,10),(2,11),(3,8),(4,9),(5,2),(5,9),(6,3),(6,12),(7,4),(7,5),(8,10),(9,6),(9,11),(11,12),(12,1),(12,8)],13)
=> [8,5]
=> [2,2,2,2,2,1,1,1]
=> ? = 2
[1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
=> [8,5,2]
=> [3,3,2,2,2,1,1,1]
=> ? = 3
Description
The diagonal inversion number of an integer partition.
The dinv of a partition is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \in \{0,1\}$.
See also exercise 3.19 of [2].
This statistic is equidistributed with the length of the partition, see [3].
Matching statistic: St000533
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00232: Dyck paths —parallelogram poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000533: Integer partitions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000533: Integer partitions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> ([],1)
=> [1]
=> 1
[1,0,1,0]
=> ([(0,1)],2)
=> [2]
=> 1
[1,1,0,0]
=> ([(0,1)],2)
=> [2]
=> 1
[1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> [3]
=> 1
[1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> [3]
=> 1
[1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> [3]
=> 1
[1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> [3]
=> 1
[1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> 2
[1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 2
[1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 2
[1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> 2
[1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> 2
[1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [4,2]
=> 2
[1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [4,2]
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [5,1]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [5,1]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [5,1]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [5,1]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 2
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> [7,4,2]
=> ? = 3
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> [7,4,2]
=> ? = 3
[1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> [7,4,2]
=> ? = 3
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> [7,4,2]
=> ? = 3
[1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> [7,5,3]
=> ? = 3
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> [7,5,3]
=> ? = 3
[1,0,1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> ([(0,7),(1,10),(2,11),(3,8),(4,9),(5,2),(5,9),(6,3),(6,12),(7,4),(7,5),(8,10),(9,6),(9,11),(11,12),(12,1),(12,8)],13)
=> [8,5]
=> ? = 2
[1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [8,5,1]
=> ? = 3
[1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> ([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
=> [8,5,2]
=> ? = 3
[1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> ([(0,7),(1,10),(2,11),(3,8),(4,9),(5,2),(5,9),(6,3),(6,12),(7,4),(7,5),(8,10),(9,6),(9,11),(11,12),(12,1),(12,8)],13)
=> [8,5]
=> ? = 2
[1,0,1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [8,5,1]
=> ? = 3
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
=> [8,5,2]
=> ? = 3
[1,1,0,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> ([(0,7),(1,10),(2,11),(3,8),(4,9),(5,2),(5,9),(6,3),(6,12),(7,4),(7,5),(8,10),(9,6),(9,11),(11,12),(12,1),(12,8)],13)
=> [8,5]
=> ? = 2
[1,1,0,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [8,5,1]
=> ? = 3
[1,1,0,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> ([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
=> [8,5,2]
=> ? = 3
[1,1,0,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> ([(0,7),(1,10),(2,11),(3,8),(4,9),(5,2),(5,9),(6,3),(6,12),(7,4),(7,5),(8,10),(9,6),(9,11),(11,12),(12,1),(12,8)],13)
=> [8,5]
=> ? = 2
[1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [8,5,1]
=> ? = 3
[1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
=> [8,5,2]
=> ? = 3
Description
The minimum of the number of parts and the size of the first part of an integer partition.
This is also an upper bound on the maximal number of non-attacking rooks that can be placed on the Ferrers board.
Matching statistic: St000159
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00232: Dyck paths —parallelogram poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000159: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 96%●distinct values known / distinct values provided: 75%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000159: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 96%●distinct values known / distinct values provided: 75%
Values
[1,0]
=> ([],1)
=> [1]
=> 1
[1,0,1,0]
=> ([(0,1)],2)
=> [2]
=> 1
[1,1,0,0]
=> ([(0,1)],2)
=> [2]
=> 1
[1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> [3]
=> 1
[1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> [3]
=> 1
[1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> [3]
=> 1
[1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> [3]
=> 1
[1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> 2
[1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 2
[1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 2
[1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> 2
[1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> 2
[1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [4,2]
=> 2
[1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [4,2]
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [5,1]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [5,1]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [5,1]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [5,1]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 2
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> [7,4,2]
=> ? = 3
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> [7,4,2]
=> ? = 3
[1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> [7,4,2]
=> ? = 3
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> [7,4,2]
=> ? = 3
[1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> [7,5,3]
=> ? = 3
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> [7,5,3]
=> ? = 3
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)
=> [7,5,3,1]
=> ? = 4
[1,0,1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> ([(0,7),(1,10),(2,11),(3,8),(4,9),(5,2),(5,9),(6,3),(6,12),(7,4),(7,5),(8,10),(9,6),(9,11),(11,12),(12,1),(12,8)],13)
=> [8,5]
=> ? = 2
[1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [8,5,1]
=> ? = 3
[1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> ([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
=> [8,5,2]
=> ? = 3
[1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> ([(0,7),(1,10),(2,11),(3,8),(4,9),(5,2),(5,9),(6,3),(6,12),(7,4),(7,5),(8,10),(9,6),(9,11),(11,12),(12,1),(12,8)],13)
=> [8,5]
=> ? = 2
[1,0,1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [8,5,1]
=> ? = 3
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
=> [8,5,2]
=> ? = 3
[1,1,0,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> ([(0,7),(1,10),(2,11),(3,8),(4,9),(5,2),(5,9),(6,3),(6,12),(7,4),(7,5),(8,10),(9,6),(9,11),(11,12),(12,1),(12,8)],13)
=> [8,5]
=> ? = 2
[1,1,0,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [8,5,1]
=> ? = 3
[1,1,0,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> ([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
=> [8,5,2]
=> ? = 3
[1,1,0,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> ([(0,7),(1,10),(2,11),(3,8),(4,9),(5,2),(5,9),(6,3),(6,12),(7,4),(7,5),(8,10),(9,6),(9,11),(11,12),(12,1),(12,8)],13)
=> [8,5]
=> ? = 2
[1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [8,5,1]
=> ? = 3
[1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
=> [8,5,2]
=> ? = 3
Description
The number of distinct parts of the integer partition.
This statistic is also the number of removeable cells of the partition, and the number of valleys of the Dyck path tracing the shape of the partition.
Matching statistic: St001432
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00232: Dyck paths —parallelogram poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St001432: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 96%●distinct values known / distinct values provided: 75%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St001432: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 96%●distinct values known / distinct values provided: 75%
Values
[1,0]
=> ([],1)
=> [1]
=> 1
[1,0,1,0]
=> ([(0,1)],2)
=> [2]
=> 1
[1,1,0,0]
=> ([(0,1)],2)
=> [2]
=> 1
[1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> [3]
=> 1
[1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> [3]
=> 1
[1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> [3]
=> 1
[1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> [3]
=> 1
[1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> 2
[1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 2
[1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 2
[1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> 2
[1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> 2
[1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [4,2]
=> 2
[1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [4,2]
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [5,1]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [5,1]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [5,1]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [5,1]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 2
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> [7,4,2]
=> ? = 3
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> [7,4,2]
=> ? = 3
[1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> [7,4,2]
=> ? = 3
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> [7,4,2]
=> ? = 3
[1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> [7,5,3]
=> ? = 3
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> [7,5,3]
=> ? = 3
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)
=> [7,5,3,1]
=> ? = 4
[1,0,1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> ([(0,7),(1,10),(2,11),(3,8),(4,9),(5,2),(5,9),(6,3),(6,12),(7,4),(7,5),(8,10),(9,6),(9,11),(11,12),(12,1),(12,8)],13)
=> [8,5]
=> ? = 2
[1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [8,5,1]
=> ? = 3
[1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> ([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
=> [8,5,2]
=> ? = 3
[1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> ([(0,7),(1,10),(2,11),(3,8),(4,9),(5,2),(5,9),(6,3),(6,12),(7,4),(7,5),(8,10),(9,6),(9,11),(11,12),(12,1),(12,8)],13)
=> [8,5]
=> ? = 2
[1,0,1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [8,5,1]
=> ? = 3
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
=> [8,5,2]
=> ? = 3
[1,1,0,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> ([(0,7),(1,10),(2,11),(3,8),(4,9),(5,2),(5,9),(6,3),(6,12),(7,4),(7,5),(8,10),(9,6),(9,11),(11,12),(12,1),(12,8)],13)
=> [8,5]
=> ? = 2
[1,1,0,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [8,5,1]
=> ? = 3
[1,1,0,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> ([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
=> [8,5,2]
=> ? = 3
[1,1,0,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> ([(0,7),(1,10),(2,11),(3,8),(4,9),(5,2),(5,9),(6,3),(6,12),(7,4),(7,5),(8,10),(9,6),(9,11),(11,12),(12,1),(12,8)],13)
=> [8,5]
=> ? = 2
[1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [8,5,1]
=> ? = 3
[1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
=> [8,5,2]
=> ? = 3
Description
The order dimension of the partition.
Given a partition $\lambda$, let $I(\lambda)$ be the principal order ideal in the Young lattice generated by $\lambda$. The order dimension of a partition is defined as the order dimension of the poset $I(\lambda)$.
Matching statistic: St000481
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00232: Dyck paths —parallelogram poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000481: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 96%●distinct values known / distinct values provided: 75%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000481: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 96%●distinct values known / distinct values provided: 75%
Values
[1,0]
=> ([],1)
=> [1]
=> 0 = 1 - 1
[1,0,1,0]
=> ([(0,1)],2)
=> [2]
=> 0 = 1 - 1
[1,1,0,0]
=> ([(0,1)],2)
=> [2]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> [3]
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> [3]
=> 0 = 1 - 1
[1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> [3]
=> 0 = 1 - 1
[1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> [3]
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [4,2]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [4,2]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [5,1]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [5,1]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [5,1]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [5,1]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 1 = 2 - 1
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> [7,4,2]
=> ? = 3 - 1
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> [7,4,2]
=> ? = 3 - 1
[1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> [7,4,2]
=> ? = 3 - 1
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> [7,4,2]
=> ? = 3 - 1
[1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> [7,5,3]
=> ? = 3 - 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> [7,5,3]
=> ? = 3 - 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)
=> [7,5,3,1]
=> ? = 4 - 1
[1,0,1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> ([(0,7),(1,10),(2,11),(3,8),(4,9),(5,2),(5,9),(6,3),(6,12),(7,4),(7,5),(8,10),(9,6),(9,11),(11,12),(12,1),(12,8)],13)
=> [8,5]
=> ? = 2 - 1
[1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [8,5,1]
=> ? = 3 - 1
[1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> ([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
=> [8,5,2]
=> ? = 3 - 1
[1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> ([(0,7),(1,10),(2,11),(3,8),(4,9),(5,2),(5,9),(6,3),(6,12),(7,4),(7,5),(8,10),(9,6),(9,11),(11,12),(12,1),(12,8)],13)
=> [8,5]
=> ? = 2 - 1
[1,0,1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [8,5,1]
=> ? = 3 - 1
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
=> [8,5,2]
=> ? = 3 - 1
[1,1,0,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> ([(0,7),(1,10),(2,11),(3,8),(4,9),(5,2),(5,9),(6,3),(6,12),(7,4),(7,5),(8,10),(9,6),(9,11),(11,12),(12,1),(12,8)],13)
=> [8,5]
=> ? = 2 - 1
[1,1,0,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [8,5,1]
=> ? = 3 - 1
[1,1,0,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> ([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
=> [8,5,2]
=> ? = 3 - 1
[1,1,0,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> ([(0,7),(1,10),(2,11),(3,8),(4,9),(5,2),(5,9),(6,3),(6,12),(7,4),(7,5),(8,10),(9,6),(9,11),(11,12),(12,1),(12,8)],13)
=> [8,5]
=> ? = 2 - 1
[1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [8,5,1]
=> ? = 3 - 1
[1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
=> [8,5,2]
=> ? = 3 - 1
Description
The number of upper covers of a partition in dominance order.
Matching statistic: St000480
Mp00232: Dyck paths —parallelogram poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000480: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 96%●distinct values known / distinct values provided: 75%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000480: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 96%●distinct values known / distinct values provided: 75%
Values
[1,0]
=> ([],1)
=> [1]
=> [1]
=> 0 = 1 - 1
[1,0,1,0]
=> ([(0,1)],2)
=> [2]
=> [1,1]
=> 0 = 1 - 1
[1,1,0,0]
=> ([(0,1)],2)
=> [2]
=> [1,1]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,1,1]
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,1,1]
=> 0 = 1 - 1
[1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,1,1]
=> 0 = 1 - 1
[1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,1,1]
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> [2,1,1,1]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> [2,1,1,1]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [2,1,1,1]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [2,1,1,1]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [4,2]
=> [2,2,1,1]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [4,2]
=> [2,2,1,1]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> [2,2,1,1,1]
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> [2,2,1,1,1]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> [2,2,1,1,1]
=> 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> [2,2,1,1,1]
=> 1 = 2 - 1
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> [7,4,2]
=> [3,3,2,2,1,1,1]
=> ? = 3 - 1
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> [7,4,2]
=> [3,3,2,2,1,1,1]
=> ? = 3 - 1
[1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> [7,4,2]
=> [3,3,2,2,1,1,1]
=> ? = 3 - 1
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> [7,4,2]
=> [3,3,2,2,1,1,1]
=> ? = 3 - 1
[1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> [7,5,3]
=> [3,3,3,2,2,1,1]
=> ? = 3 - 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> [7,5,3]
=> [3,3,3,2,2,1,1]
=> ? = 3 - 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)
=> [7,5,3,1]
=> [4,3,3,2,2,1,1]
=> ? = 4 - 1
[1,0,1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> ([(0,7),(1,10),(2,11),(3,8),(4,9),(5,2),(5,9),(6,3),(6,12),(7,4),(7,5),(8,10),(9,6),(9,11),(11,12),(12,1),(12,8)],13)
=> [8,5]
=> [2,2,2,2,2,1,1,1]
=> ? = 2 - 1
[1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [8,5,1]
=> [3,2,2,2,2,1,1,1]
=> ? = 3 - 1
[1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> ([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
=> [8,5,2]
=> [3,3,2,2,2,1,1,1]
=> ? = 3 - 1
[1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> ([(0,7),(1,10),(2,11),(3,8),(4,9),(5,2),(5,9),(6,3),(6,12),(7,4),(7,5),(8,10),(9,6),(9,11),(11,12),(12,1),(12,8)],13)
=> [8,5]
=> [2,2,2,2,2,1,1,1]
=> ? = 2 - 1
[1,0,1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [8,5,1]
=> [3,2,2,2,2,1,1,1]
=> ? = 3 - 1
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
=> [8,5,2]
=> [3,3,2,2,2,1,1,1]
=> ? = 3 - 1
[1,1,0,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> ([(0,7),(1,10),(2,11),(3,8),(4,9),(5,2),(5,9),(6,3),(6,12),(7,4),(7,5),(8,10),(9,6),(9,11),(11,12),(12,1),(12,8)],13)
=> [8,5]
=> [2,2,2,2,2,1,1,1]
=> ? = 2 - 1
[1,1,0,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [8,5,1]
=> [3,2,2,2,2,1,1,1]
=> ? = 3 - 1
[1,1,0,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> ([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
=> [8,5,2]
=> [3,3,2,2,2,1,1,1]
=> ? = 3 - 1
[1,1,0,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> ([(0,7),(1,10),(2,11),(3,8),(4,9),(5,2),(5,9),(6,3),(6,12),(7,4),(7,5),(8,10),(9,6),(9,11),(11,12),(12,1),(12,8)],13)
=> [8,5]
=> [2,2,2,2,2,1,1,1]
=> ? = 2 - 1
[1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [8,5,1]
=> [3,2,2,2,2,1,1,1]
=> ? = 3 - 1
[1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
=> [8,5,2]
=> [3,3,2,2,2,1,1,1]
=> ? = 3 - 1
Description
The number of lower covers of a partition in dominance order.
According to [1], Corollary 2.4, the maximum number of elements one element (apparently for $n\neq 2$) can cover is
$$
\frac{1}{2}(\sqrt{1+8n}-3)
$$
and an element which covers this number of elements is given by $(c+i,c,c-1,\dots,3,2,1)$, where $1\leq i\leq c+2$.
The following 100 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000759The smallest missing part in an integer partition. St000225Difference between largest and smallest parts in a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000668The least common multiple of the parts of the partition. St001128The exponens consonantiae of a partition. St001372The length of a longest cyclic run of ones of a binary word. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St000097The order of the largest clique of the graph. St000527The width of the poset. St000783The side length of the largest staircase partition fitting into a partition. St000318The number of addable cells of the Ferrers diagram of an integer partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000093The cardinality of a maximal independent set of vertices of a graph. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000098The chromatic number of a graph. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001029The size of the core of a graph. St001716The 1-improper chromatic number of a graph. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000254The nesting number of a set partition. St000062The length of the longest increasing subsequence of the permutation. St000308The height of the tree associated to a permutation. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001335The cardinality of a minimal cycle-isolating set of a graph. St000544The cop number of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001883The mutual visibility number of a graph. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St000272The treewidth of a graph. St000535The rank-width of a graph. St000536The pathwidth of a graph. St000537The cutwidth of a graph. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001331The size of the minimal feedback vertex set. St001358The largest degree of a regular subgraph of a graph. St001638The book thickness of a graph. St001644The dimension of a graph. St001743The discrepancy of a graph. St001792The arboricity of a graph. St001826The maximal number of leaves on a vertex of a graph. St001962The proper pathwidth of a graph. St001057The Grundy value of the game of creating an independent set in a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St000454The largest eigenvalue of a graph if it is integral. St000455The second largest eigenvalue of a graph if it is integral. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St001330The hat guessing number of a graph. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000640The rank of the largest boolean interval in a poset. St000822The Hadwiger number of the graph. St001734The lettericity of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001352The number of internal nodes in the modular decomposition of a graph. St001597The Frobenius rank of a skew partition. St001642The Prague dimension of a graph. St001323The independence gap of a graph. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000243The number of cyclic valleys and cyclic peaks of a permutation. St001877Number of indecomposable injective modules with projective dimension 2. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001960The number of descents of a permutation minus one if its first entry is not one. St000264The girth of a graph, which is not a tree. St000908The length of the shortest maximal antichain in a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000914The sum of the values of the Möbius function of a poset. St001964The interval resolution global dimension of a poset. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001095The number of non-isomorphic posets with precisely one further covering relation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000711The number of big exceedences of a permutation. St000805The number of peaks of the associated bargraph. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001728The number of invisible descents of a permutation. St000252The number of nodes of degree 3 of a binary tree. St001823The Stasinski-Voll length of a signed permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St001946The number of descents in a parking function. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order.
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