searching the database
Your data matches 14 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St001037
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
St001037: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001037: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> 0
[[],[]]
=> [1,0,1,0]
=> 0
[[[]]]
=> [1,1,0,0]
=> 0
[[],[],[]]
=> [1,0,1,0,1,0]
=> 0
[[],[[]]]
=> [1,0,1,1,0,0]
=> 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> 0
[[[],[]]]
=> [1,1,0,1,0,0]
=> 0
[[[[]]]]
=> [1,1,1,0,0,0]
=> 0
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> 0
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> 1
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> 1
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> 1
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> 0
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> 1
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> 0
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> 0
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> 0
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> 0
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> 0
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> 0
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> 0
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
Description
The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path.
Matching statistic: St000386
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000386: Dyck paths ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 100%
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000386: Dyck paths ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> [1,0]
=> 0
[[],[]]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[[[]]]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 0
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 0
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 0
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 0
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 0
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 0
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 0
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 0
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 0
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 0
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
[[],[],[],[],[],[[],[]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[[],[],[],[],[[]],[],[]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 1
[[],[],[],[],[[]],[[]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 2
[[],[],[],[],[[],[]],[]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 1
[[],[],[],[],[[[]]],[]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
=> ? = 1
[[],[],[],[],[[],[],[]]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 1
[[],[],[],[],[[],[[]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,1,0,0]
=> ? = 1
[[],[],[],[],[[[]],[]]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> ? = 2
[[],[],[],[],[[[[]]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> ? = 1
[[],[],[],[[]],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 2
[[],[],[],[[]],[[],[]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 2
[[],[],[],[[]],[[[]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,1,0,0]
=> ? = 2
[[],[],[],[[],[]],[],[]]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 1
[[],[],[],[[[]]],[],[]]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
=> ? = 1
[[],[],[],[[],[]],[[]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 2
[[],[],[],[[[]]],[[]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0,1,0]
=> ? = 2
[[],[],[],[[],[],[]],[]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 1
[[],[],[],[[],[[]]],[]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,1,0,0,0]
=> ? = 1
[[],[],[],[[[]],[]],[]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,1,0,0]
=> ? = 2
[[],[],[],[[[],[]]],[]]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0]
=> ? = 1
[[],[],[],[[[[]]]],[]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,1,0,0,0]
=> ? = 1
[[],[],[],[[],[],[],[]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[[],[],[],[[],[],[[]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,1,0,0]
=> ? = 1
[[],[],[],[[],[[]],[]]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,0,0,1,0]
=> ? = 2
[[],[],[],[[],[[[]]]]]
=> [1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,0,1,0,0]
=> ? = 1
[[],[],[],[[[]],[],[]]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0]
=> ? = 2
[[],[],[],[[[]],[[]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,1,0,0]
=> ? = 2
[[],[],[],[[[],[]],[]]]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> ? = 2
[[],[],[],[[[[]]],[]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,0]
=> ? = 2
[[],[],[],[[[],[[]]]]]
=> [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> ? = 2
[[],[],[],[[[[]],[]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,1,0,0,0]
=> ? = 1
[[],[],[],[[[[],[]]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> ? = 1
[[],[],[],[[[[[]]]]]]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,1,0,0,0]
=> ? = 1
[[],[],[[]],[],[],[],[]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[[],[],[[]],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 2
[[],[],[[]],[],[[],[]]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 2
[[],[],[[]],[],[[[]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,1,0,0]
=> ? = 2
[[],[],[[]],[[]],[],[]]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 2
[[],[],[[]],[[]],[[]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3
[[],[],[[]],[[],[]],[]]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 2
[[],[],[[]],[[[]]],[]]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,1,1,0,0,0]
=> ? = 2
[[],[],[[]],[[],[],[]]]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 2
[[],[],[[]],[[],[[]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 2
[[],[],[[]],[[[]],[]]]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,1,0,0,1,0]
=> ? = 3
[[],[],[[]],[[[],[]]]]
=> [1,0,1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,1,0,0,0]
=> ? = 2
[[],[],[[]],[[[[]]]]]
=> [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,1,0,1,0,0]
=> ? = 2
[[],[],[[],[]],[],[],[]]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1
[[],[],[[[]]],[],[],[]]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 1
[[],[],[[],[]],[],[[]]]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 2
[[],[],[[[]]],[],[[]]]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0,1,0]
=> ? = 2
Description
The number of factors DDU in a Dyck path.
Matching statistic: St000201
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
St000201: Binary trees ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 100%
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
St000201: Binary trees ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> [1,0]
=> [.,.]
=> 1 = 0 + 1
[[],[]]
=> [1,0,1,0]
=> [1,1,0,0]
=> [[.,.],.]
=> 1 = 0 + 1
[[[]]]
=> [1,1,0,0]
=> [1,0,1,0]
=> [.,[.,.]]
=> 1 = 0 + 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [[[.,.],.],.]
=> 1 = 0 + 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [[.,.],[.,.]]
=> 2 = 1 + 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> 1 = 0 + 1
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> 1 = 0 + 1
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [[.,[.,.]],.]
=> 1 = 0 + 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [[[[.,.],.],.],.]
=> 1 = 0 + 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [[[.,.],.],[.,.]]
=> 2 = 1 + 1
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [[.,.],[[.,.],.]]
=> 2 = 1 + 1
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [[.,.],[.,[.,.]]]
=> 2 = 1 + 1
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [[[.,.],[.,.]],.]
=> 2 = 1 + 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [.,[[[.,.],.],.]]
=> 1 = 0 + 1
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [.,[[.,.],[.,.]]]
=> 2 = 1 + 1
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> 1 = 0 + 1
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [[.,[[.,.],.]],.]
=> 1 = 0 + 1
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> 1 = 0 + 1
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [.,[[.,[.,.]],.]]
=> 1 = 0 + 1
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [[.,[.,.]],[.,.]]
=> 2 = 1 + 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [[[.,[.,.]],.],.]
=> 1 = 0 + 1
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [[.,[.,[.,.]]],.]
=> 1 = 0 + 1
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[[[[.,.],.],.],.],.]
=> 1 = 0 + 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[[[.,.],.],.],[.,.]]
=> 2 = 1 + 1
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> 2 = 1 + 1
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[[.,.],.],[.,[.,.]]]
=> 2 = 1 + 1
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[[[.,.],.],[.,.]],.]
=> 2 = 1 + 1
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[.,.],[[[.,.],.],.]]
=> 2 = 1 + 1
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[.,.],[[.,.],[.,.]]]
=> 3 = 2 + 1
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[.,.],[.,[[.,.],.]]]
=> 2 = 1 + 1
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[[.,.],[[.,.],.]],.]
=> 2 = 1 + 1
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[.,.],[.,[.,[.,.]]]]
=> 2 = 1 + 1
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[.,.],[[.,[.,.]],.]]
=> 2 = 1 + 1
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[[.,.],[.,.]],[.,.]]
=> 3 = 2 + 1
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[[[.,.],[.,.]],.],.]
=> 2 = 1 + 1
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[[.,.],[.,[.,.]]],.]
=> 2 = 1 + 1
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [.,[[[[.,.],.],.],.]]
=> 1 = 0 + 1
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [.,[[[.,.],.],[.,.]]]
=> 2 = 1 + 1
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [.,[[.,.],[[.,.],.]]]
=> 2 = 1 + 1
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [.,[[.,.],[.,[.,.]]]]
=> 2 = 1 + 1
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> 2 = 1 + 1
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> 1 = 0 + 1
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [[.,[[[.,.],.],.]],.]
=> 1 = 0 + 1
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[[.,.],[.,.]]]]
=> 2 = 1 + 1
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[.,[[.,.],.]],[.,.]]
=> 2 = 1 + 1
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> 1 = 0 + 1
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [.,[[.,[[.,.],.]],.]]
=> 1 = 0 + 1
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[.,[.,.]],[[.,.],.]]
=> 2 = 1 + 1
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [[[.,[[.,.],.]],.],.]
=> 1 = 0 + 1
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [[.,[.,[[.,.],.]]],.]
=> 1 = 0 + 1
[[],[],[],[],[],[[],[]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> ? = 1 + 1
[[],[],[],[],[],[[[]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [[[[[[[.,.],.],.],.],.],[.,.]],.]
=> ? = 1 + 1
[[],[],[],[],[[]],[],[]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [[[[[.,.],.],.],.],[[[.,.],.],.]]
=> ? = 1 + 1
[[],[],[],[],[[]],[[]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> [[[[[.,.],.],.],.],[[.,.],[.,.]]]
=> ? = 2 + 1
[[],[],[],[],[[],[]],[]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> [[[[[.,.],.],.],.],[.,[[.,.],.]]]
=> ? = 1 + 1
[[],[],[],[],[[],[],[]]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> ? = 1 + 1
[[],[],[],[],[[],[[]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,1,0,0]
=> [[[[[.,.],.],.],.],[[.,[.,.]],.]]
=> ? = 1 + 1
[[],[],[],[],[[[],[]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [[[[[[[.,.],.],.],.],[.,.]],.],.]
=> ? = 1 + 1
[[],[],[],[],[[[[]]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> [[[[[[.,.],.],.],.],[.,[.,.]]],.]
=> ? = 1 + 1
[[],[],[],[[]],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> [[[[.,.],.],.],[[[.,.],.],[.,.]]]
=> ? = 2 + 1
[[],[],[],[[]],[[],[]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> [[[[.,.],.],.],[[.,.],[.,[.,.]]]]
=> ? = 2 + 1
[[],[],[],[[]],[[[]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,1,0,0]
=> [[[[.,.],.],.],[[[.,.],[.,.]],.]]
=> ? = 2 + 1
[[],[],[],[[],[]],[],[]]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> [[[[.,.],.],.],[.,[[[.,.],.],.]]]
=> ? = 1 + 1
[[],[],[],[[[]]],[],[]]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
=> [[[[[.,.],.],.],[[[.,.],.],.]],.]
=> ? = 1 + 1
[[],[],[],[[],[]],[[]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> [[[[.,.],.],.],[.,[[.,.],[.,.]]]]
=> ? = 2 + 1
[[],[],[],[[[]]],[[]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0,1,0]
=> [[[[[.,.],.],.],[[.,.],.]],[.,.]]
=> ? = 2 + 1
[[],[],[],[[],[],[]],[]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> [[[[.,.],.],.],[.,[.,[[.,.],.]]]]
=> ? = 1 + 1
[[],[],[],[[],[[]]],[]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,1,0,0,0]
=> [[[[.,.],.],.],[[.,[[.,.],.]],.]]
=> ? = 1 + 1
[[],[],[],[[[]],[]],[]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,1,0,0]
=> [[[[[.,.],.],.],[.,.]],[[.,.],.]]
=> ? = 2 + 1
[[],[],[],[[[[]]]],[]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,1,0,0,0]
=> [[[[[.,.],.],.],[.,[[.,.],.]]],.]
=> ? = 1 + 1
[[],[],[],[[],[],[],[]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> ? = 1 + 1
[[],[],[],[[],[],[[]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,1,0,0]
=> [[[[.,.],.],.],[.,[[.,[.,.]],.]]]
=> ? = 1 + 1
[[],[],[],[[],[[],[]]]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,1,0,0,0]
=> [[[[.,.],.],.],[[[.,[.,.]],.],.]]
=> ? = 1 + 1
[[],[],[],[[],[[[]]]]]
=> [1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,0,1,0,0]
=> [[[[.,.],.],.],[[.,[.,[.,.]]],.]]
=> ? = 1 + 1
[[],[],[],[[[]],[],[]]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0]
=> [[[[[.,.],.],.],[.,.]],[.,[.,.]]]
=> ? = 2 + 1
[[],[],[],[[[]],[[]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,1,0,0]
=> [[[[[.,.],.],.],[[.,.],[.,.]]],.]
=> ? = 2 + 1
[[],[],[],[[[[]]],[]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,0]
=> [[[[[.,.],.],.],[.,[.,.]]],[.,.]]
=> ? = 2 + 1
[[],[],[],[[[],[],[]]]]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [[[[[[[.,.],.],.],[.,.]],.],.],.]
=> ? = 1 + 1
[[],[],[],[[[[]],[]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,1,0,0,0]
=> [[[[[.,.],.],.],[[.,[.,.]],.]],.]
=> ? = 1 + 1
[[],[],[],[[[[],[]]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> [[[[[.,.],.],.],[.,[.,[.,.]]]],.]
=> ? = 1 + 1
[[],[],[],[[[[[]]]]]]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,1,0,0,0]
=> [[[[[[.,.],.],.],[.,[.,.]]],.],.]
=> ? = 1 + 1
[[],[],[[]],[],[],[],[]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [[[.,.],.],[[[[[.,.],.],.],.],.]]
=> ? = 1 + 1
[[],[],[[]],[],[],[[]]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [[[.,.],.],[[[[.,.],.],.],[.,.]]]
=> ? = 2 + 1
[[],[],[[]],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> [[[.,.],.],[[[.,.],.],[[.,.],.]]]
=> ? = 2 + 1
[[],[],[[]],[],[[],[]]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [[[.,.],.],[[[.,.],.],[.,[.,.]]]]
=> ? = 2 + 1
[[],[],[[]],[],[[[]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,1,0,0]
=> [[[.,.],.],[[[[.,.],.],[.,.]],.]]
=> ? = 2 + 1
[[],[],[[]],[[]],[],[]]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> [[[.,.],.],[[.,.],[[[.,.],.],.]]]
=> ? = 2 + 1
[[],[],[[]],[[]],[[]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> [[[.,.],.],[[.,.],[[.,.],[.,.]]]]
=> ? = 3 + 1
[[],[],[[]],[[],[]],[]]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> [[[.,.],.],[[.,.],[.,[[.,.],.]]]]
=> ? = 2 + 1
[[],[],[[]],[[[]]],[]]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,1,1,0,0,0]
=> [[[.,.],.],[[[.,.],[[.,.],.]],.]]
=> ? = 2 + 1
[[],[],[[]],[[],[],[]]]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> [[[.,.],.],[[.,.],[.,[.,[.,.]]]]]
=> ? = 2 + 1
[[],[],[[]],[[],[[]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,1,0,0]
=> [[[.,.],.],[[.,.],[[.,[.,.]],.]]]
=> ? = 2 + 1
[[],[],[[]],[[[]],[]]]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,1,0,0,1,0]
=> [[[.,.],.],[[[.,.],[.,.]],[.,.]]]
=> ? = 3 + 1
[[],[],[[]],[[[],[]]]]
=> [1,0,1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,1,0,0,0]
=> [[[.,.],.],[[[[.,.],[.,.]],.],.]]
=> ? = 2 + 1
[[],[],[[]],[[[[]]]]]
=> [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,1,0,1,0,0]
=> [[[.,.],.],[[[.,.],[.,[.,.]]],.]]
=> ? = 2 + 1
[[],[],[[],[]],[],[],[]]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> [[[.,.],.],[.,[[[[.,.],.],.],.]]]
=> ? = 1 + 1
[[],[],[[],[]],[],[[]]]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> [[[.,.],.],[.,[[[.,.],.],[.,.]]]]
=> ? = 2 + 1
[[],[],[[],[]],[[]],[]]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> [[[.,.],.],[.,[[.,.],[[.,.],.]]]]
=> ? = 2 + 1
[[],[],[[],[]],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> [[[.,.],.],[.,[[.,.],[.,[.,.]]]]]
=> ? = 2 + 1
[[],[],[[],[]],[[[]]]]
=> [1,0,1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,1,0,0]
=> [[[.,.],.],[.,[[[.,.],[.,.]],.]]]
=> ? = 2 + 1
Description
The number of leaf nodes in a binary tree.
Equivalently, the number of cherries [1] in the complete binary tree.
The number of binary trees of size $n$, at least $1$, with exactly one leaf node for is $2^{n-1}$, see [2].
The number of binary tree of size $n$, at least $3$, with exactly two leaf nodes is $n(n+1)2^{n-2}$, see [3].
Matching statistic: St000196
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
St000196: Binary trees ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 100%
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
St000196: Binary trees ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> [1,0]
=> [.,.]
=> 0
[[],[]]
=> [1,0,1,0]
=> [1,1,0,0]
=> [[.,.],.]
=> 0
[[[]]]
=> [1,1,0,0]
=> [1,0,1,0]
=> [.,[.,.]]
=> 0
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [[[.,.],.],.]
=> 0
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [[.,.],[.,.]]
=> 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> 0
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> 0
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [[.,[.,.]],.]
=> 0
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [[[[.,.],.],.],.]
=> 0
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [[[.,.],.],[.,.]]
=> 1
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [[.,.],[[.,.],.]]
=> 1
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [[.,.],[.,[.,.]]]
=> 1
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [[[.,.],[.,.]],.]
=> 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [.,[[[.,.],.],.]]
=> 0
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [.,[[.,.],[.,.]]]
=> 1
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> 0
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [[.,[[.,.],.]],.]
=> 0
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> 0
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [.,[[.,[.,.]],.]]
=> 0
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [[.,[.,.]],[.,.]]
=> 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [[[.,[.,.]],.],.]
=> 0
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [[.,[.,[.,.]]],.]
=> 0
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[[[[.,.],.],.],.],.]
=> 0
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[[[.,.],.],.],[.,.]]
=> 1
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> 1
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[[.,.],.],[.,[.,.]]]
=> 1
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[[[.,.],.],[.,.]],.]
=> 1
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[.,.],[[[.,.],.],.]]
=> 1
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[.,.],[[.,.],[.,.]]]
=> 2
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[.,.],[.,[[.,.],.]]]
=> 1
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[[.,.],[[.,.],.]],.]
=> 1
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[.,.],[.,[.,[.,.]]]]
=> 1
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[.,.],[[.,[.,.]],.]]
=> 1
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[[.,.],[.,.]],[.,.]]
=> 2
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[[[.,.],[.,.]],.],.]
=> 1
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[[.,.],[.,[.,.]]],.]
=> 1
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [.,[[[[.,.],.],.],.]]
=> 0
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [.,[[[.,.],.],[.,.]]]
=> 1
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [.,[[.,.],[[.,.],.]]]
=> 1
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [.,[[.,.],[.,[.,.]]]]
=> 1
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> 1
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> 0
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [[.,[[[.,.],.],.]],.]
=> 0
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[[.,.],[.,.]]]]
=> 1
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[.,[[.,.],.]],[.,.]]
=> 1
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> 0
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [.,[[.,[[.,.],.]],.]]
=> 0
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[.,[.,.]],[[.,.],.]]
=> 1
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [[[.,[[.,.],.]],.],.]
=> 0
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [[.,[.,[[.,.],.]]],.]
=> 0
[[],[],[],[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[[[[[[[.,.],.],.],.],.],.],.],.]
=> ? = 0
[[],[],[],[],[],[],[[]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> ? = 1
[[],[],[],[],[],[[]],[]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [[[[[[.,.],.],.],.],.],[[.,.],.]]
=> ? = 1
[[],[],[],[],[],[[],[]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> ? = 1
[[],[],[],[],[],[[[]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [[[[[[[.,.],.],.],.],.],[.,.]],.]
=> ? = 1
[[],[],[],[],[[]],[],[]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [[[[[.,.],.],.],.],[[[.,.],.],.]]
=> ? = 1
[[],[],[],[],[[]],[[]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> [[[[[.,.],.],.],.],[[.,.],[.,.]]]
=> ? = 2
[[],[],[],[],[[],[]],[]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> [[[[[.,.],.],.],.],[.,[[.,.],.]]]
=> ? = 1
[[],[],[],[],[[[]]],[]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
=> [[[[[[.,.],.],.],.],[[.,.],.]],.]
=> ? = 1
[[],[],[],[],[[],[],[]]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> ? = 1
[[],[],[],[],[[],[[]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,1,0,0]
=> [[[[[.,.],.],.],.],[[.,[.,.]],.]]
=> ? = 1
[[],[],[],[],[[[]],[]]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> [[[[[[.,.],.],.],.],[.,.]],[.,.]]
=> ? = 2
[[],[],[],[],[[[],[]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [[[[[[[.,.],.],.],.],[.,.]],.],.]
=> ? = 1
[[],[],[],[],[[[[]]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> [[[[[[.,.],.],.],.],[.,[.,.]]],.]
=> ? = 1
[[],[],[],[[]],[],[],[]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [[[[.,.],.],.],[[[[.,.],.],.],.]]
=> ? = 1
[[],[],[],[[]],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> [[[[.,.],.],.],[[[.,.],.],[.,.]]]
=> ? = 2
[[],[],[],[[]],[[]],[]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [[[[.,.],.],.],[[.,.],[[.,.],.]]]
=> ? = 2
[[],[],[],[[]],[[],[]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> [[[[.,.],.],.],[[.,.],[.,[.,.]]]]
=> ? = 2
[[],[],[],[[]],[[[]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,1,0,0]
=> [[[[.,.],.],.],[[[.,.],[.,.]],.]]
=> ? = 2
[[],[],[],[[],[]],[],[]]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> [[[[.,.],.],.],[.,[[[.,.],.],.]]]
=> ? = 1
[[],[],[],[[[]]],[],[]]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
=> [[[[[.,.],.],.],[[[.,.],.],.]],.]
=> ? = 1
[[],[],[],[[],[]],[[]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> [[[[.,.],.],.],[.,[[.,.],[.,.]]]]
=> ? = 2
[[],[],[],[[[]]],[[]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0,1,0]
=> [[[[[.,.],.],.],[[.,.],.]],[.,.]]
=> ? = 2
[[],[],[],[[],[],[]],[]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> [[[[.,.],.],.],[.,[.,[[.,.],.]]]]
=> ? = 1
[[],[],[],[[],[[]]],[]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,1,0,0,0]
=> [[[[.,.],.],.],[[.,[[.,.],.]],.]]
=> ? = 1
[[],[],[],[[[]],[]],[]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,1,0,0]
=> [[[[[.,.],.],.],[.,.]],[[.,.],.]]
=> ? = 2
[[],[],[],[[[],[]]],[]]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0]
=> [[[[[[.,.],.],.],[[.,.],.]],.],.]
=> ? = 1
[[],[],[],[[[[]]]],[]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,1,0,0,0]
=> [[[[[.,.],.],.],[.,[[.,.],.]]],.]
=> ? = 1
[[],[],[],[[],[],[],[]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> ? = 1
[[],[],[],[[],[],[[]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,1,0,0]
=> [[[[.,.],.],.],[.,[[.,[.,.]],.]]]
=> ? = 1
[[],[],[],[[],[[]],[]]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,0,0,1,0]
=> [[[[.,.],.],.],[[.,[.,.]],[.,.]]]
=> ? = 2
[[],[],[],[[],[[],[]]]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,1,0,0,0]
=> [[[[.,.],.],.],[[[.,[.,.]],.],.]]
=> ? = 1
[[],[],[],[[],[[[]]]]]
=> [1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,0,1,0,0]
=> [[[[.,.],.],.],[[.,[.,[.,.]]],.]]
=> ? = 1
[[],[],[],[[[]],[],[]]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0]
=> [[[[[.,.],.],.],[.,.]],[.,[.,.]]]
=> ? = 2
[[],[],[],[[[]],[[]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,1,0,0]
=> [[[[[.,.],.],.],[[.,.],[.,.]]],.]
=> ? = 2
[[],[],[],[[[],[]],[]]]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [[[[[[.,.],.],.],[.,.]],.],[.,.]]
=> ? = 2
[[],[],[],[[[[]]],[]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,0]
=> [[[[[.,.],.],.],[.,[.,.]]],[.,.]]
=> ? = 2
[[],[],[],[[[],[],[]]]]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [[[[[[[.,.],.],.],[.,.]],.],.],.]
=> ? = 1
[[],[],[],[[[],[[]]]]]
=> [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> [[[[[[.,.],.],.],[.,.]],[.,.]],.]
=> ? = 2
[[],[],[],[[[[]],[]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,1,0,0,0]
=> [[[[[.,.],.],.],[[.,[.,.]],.]],.]
=> ? = 1
[[],[],[],[[[[],[]]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> [[[[[.,.],.],.],[.,[.,[.,.]]]],.]
=> ? = 1
[[],[],[],[[[[[]]]]]]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,1,0,0,0]
=> [[[[[[.,.],.],.],[.,[.,.]]],.],.]
=> ? = 1
[[],[],[[]],[],[],[],[]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [[[.,.],.],[[[[[.,.],.],.],.],.]]
=> ? = 1
[[],[],[[]],[],[],[[]]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [[[.,.],.],[[[[.,.],.],.],[.,.]]]
=> ? = 2
[[],[],[[]],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> [[[.,.],.],[[[.,.],.],[[.,.],.]]]
=> ? = 2
[[],[],[[]],[],[[],[]]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [[[.,.],.],[[[.,.],.],[.,[.,.]]]]
=> ? = 2
[[],[],[[]],[],[[[]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,1,0,0]
=> [[[.,.],.],[[[[.,.],.],[.,.]],.]]
=> ? = 2
[[],[],[[]],[[]],[],[]]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> [[[.,.],.],[[.,.],[[[.,.],.],.]]]
=> ? = 2
[[],[],[[]],[[]],[[]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> [[[.,.],.],[[.,.],[[.,.],[.,.]]]]
=> ? = 3
[[],[],[[]],[[],[]],[]]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> [[[.,.],.],[[.,.],[.,[[.,.],.]]]]
=> ? = 2
Description
The number of occurrences of the contiguous pattern {{{[[.,.],[.,.]]}}} in a binary tree.
Equivalently, this is the number of branches in the tree, i.e. the number of nodes with two children. Binary trees avoiding this pattern are counted by $2^{n-2}$.
Matching statistic: St000159
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00182: Skew partitions —outer shape⟶ Integer partitions
St000159: Integer partitions ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 100%
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00182: Skew partitions —outer shape⟶ Integer partitions
St000159: Integer partitions ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> [[1],[]]
=> [1]
=> 1 = 0 + 1
[[],[]]
=> [1,0,1,0]
=> [[1,1],[]]
=> [1,1]
=> 1 = 0 + 1
[[[]]]
=> [1,1,0,0]
=> [[2],[]]
=> [2]
=> 1 = 0 + 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> [1,1,1]
=> 1 = 0 + 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> [2,1]
=> 2 = 1 + 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> [2,2]
=> 1 = 0 + 1
[[[],[]]]
=> [1,1,0,1,0,0]
=> [[3],[]]
=> [3]
=> 1 = 0 + 1
[[[[]]]]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> [2,2]
=> 1 = 0 + 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> [1,1,1,1]
=> 1 = 0 + 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> [2,1,1]
=> 2 = 1 + 1
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [2,2,1]
=> 2 = 1 + 1
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> [3,1]
=> 2 = 1 + 1
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> [2,2,1]
=> 2 = 1 + 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [2,2,2]
=> 1 = 0 + 1
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [3,2]
=> 2 = 1 + 1
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [3,3]
=> 1 = 0 + 1
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [2,2,2]
=> 1 = 0 + 1
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [[4],[]]
=> [4]
=> 1 = 0 + 1
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [3,3]
=> 1 = 0 + 1
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> [3,2]
=> 2 = 1 + 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> [2,2,2]
=> 1 = 0 + 1
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> [3,3]
=> 1 = 0 + 1
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> [1,1,1,1,1]
=> 1 = 0 + 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> [2,1,1,1]
=> 2 = 1 + 1
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [2,2,1,1]
=> 2 = 1 + 1
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> [3,1,1]
=> 2 = 1 + 1
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> [2,2,1,1]
=> 2 = 1 + 1
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [2,2,2,1]
=> 2 = 1 + 1
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [3,2,1]
=> 3 = 2 + 1
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [3,3,1]
=> 2 = 1 + 1
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [2,2,2,1]
=> 2 = 1 + 1
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> [4,1]
=> 2 = 1 + 1
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [3,3,1]
=> 2 = 1 + 1
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> [3,2,1]
=> 3 = 2 + 1
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> [2,2,2,1]
=> 2 = 1 + 1
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> [3,3,1]
=> 2 = 1 + 1
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [2,2,2,2]
=> 1 = 0 + 1
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [3,2,2]
=> 2 = 1 + 1
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [3,3,2]
=> 2 = 1 + 1
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [4,2]
=> 2 = 1 + 1
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [3,3,2]
=> 2 = 1 + 1
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [3,3,3]
=> 1 = 0 + 1
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [2,2,2,2]
=> 1 = 0 + 1
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [4,3]
=> 2 = 1 + 1
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [3,2,2]
=> 2 = 1 + 1
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [4,4]
=> 1 = 0 + 1
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [3,3,3]
=> 1 = 0 + 1
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [3,3,2]
=> 2 = 1 + 1
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> [2,2,2,2]
=> 1 = 0 + 1
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [3,3,3]
=> 1 = 0 + 1
[[],[],[],[[[]]],[]]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1,1],[1]]
=> ?
=> ? = 1 + 1
[[],[],[],[[],[[]]]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1,1,1],[1]]
=> ?
=> ? = 1 + 1
[[],[],[[]],[[[]]]]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1,1],[1,1]]
=> ?
=> ? = 2 + 1
[[],[],[[[]]],[],[]]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2,1,1],[1,1]]
=> ?
=> ? = 1 + 1
[[],[],[[[]]],[[]]]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2,1,1],[1]]
=> ?
=> ? = 2 + 1
[[],[],[[],[[]]],[]]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1,1],[2,1]]
=> ?
=> ? = 1 + 1
[[],[],[[[]],[]],[]]
=> [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2,1,1],[2]]
=> ?
=> ? = 2 + 1
[[],[],[[[],[]]],[]]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2,1,1],[1]]
=> ?
=> ? = 1 + 1
[[],[],[[[[]]]],[]]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3,1,1],[2]]
=> ?
=> ? = 1 + 1
[[],[],[[],[],[[]]]]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1,1],[2]]
=> ?
=> ? = 1 + 1
[[],[],[[],[[]],[]]]
=> [1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1,1],[1]]
=> ?
=> ? = 2 + 1
[[],[],[[],[[],[]]]]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3,1,1],[1,1]]
=> ?
=> ? = 1 + 1
[[],[],[[],[[[]]]]]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [[4,4,1,1],[1]]
=> ?
=> ? = 1 + 1
[[],[],[[[]],[[]]]]
=> [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2,1,1],[1]]
=> ?
=> ? = 2 + 1
[[],[],[[[[]],[]]]]
=> [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3,1,1],[1]]
=> ?
=> ? = 1 + 1
[[],[],[[[[[]]]]]]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1,1],[]]
=> ?
=> ? = 1 + 1
[[],[[]],[],[[[]]]]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [[3,3,2,2,1],[1,1,1]]
=> ?
=> ? = 2 + 1
[[],[[]],[[[]]],[]]
=> [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [[3,3,3,2,1],[2,1,1]]
=> ?
=> ? = 2 + 1
[[],[[]],[[],[[]]]]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [[4,4,2,1],[2,1]]
=> ?
=> ? = 2 + 1
[[],[[]],[[[]],[]]]
=> [1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [[4,3,2,1],[1,1]]
=> ?
=> ? = 3 + 1
[[],[[]],[[[],[]]]]
=> [1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [[3,3,3,2,1],[1,1,1]]
=> ?
=> ? = 2 + 1
[[],[[]],[[[[]]]]]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [[4,4,2,1],[1,1]]
=> ?
=> ? = 2 + 1
[[],[[],[]],[],[],[]]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [[3,3,3,3,1],[2,2,2]]
=> [3,3,3,3,1]
=> ? = 1 + 1
[[],[[[]]],[],[],[]]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,2,1],[1,1,1]]
=> ?
=> ? = 1 + 1
[[],[[[]]],[],[[]]]
=> [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [[3,2,2,2,1],[1,1]]
=> ?
=> ? = 2 + 1
[[],[[[]]],[[]],[]]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [[3,3,2,2,1],[2,1]]
=> ?
=> ? = 2 + 1
[[],[[],[]],[[[]]]]
=> [1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [[4,4,3,1],[2,2]]
=> ?
=> ? = 2 + 1
[[],[[[]]],[[],[]]]
=> [1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [[4,2,2,1],[1]]
=> ?
=> ? = 2 + 1
[[],[[[]]],[[[]]]]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [[3,3,2,2,1],[1,1]]
=> ?
=> ? = 2 + 1
[[],[[],[],[]],[],[]]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [[4,4,4,1],[3,3]]
=> [4,4,4,1]
=> ? = 1 + 1
[[],[[],[[]]],[],[]]
=> [1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [[3,3,3,3,1],[2,2,1]]
=> ?
=> ? = 1 + 1
[[],[[[]],[]],[],[]]
=> [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [[3,3,3,2,1],[2,2]]
=> ?
=> ? = 2 + 1
[[],[[[],[]]],[],[]]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [[2,2,2,2,2,1],[1,1]]
=> ?
=> ? = 1 + 1
[[],[[[[]]]],[],[]]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [[3,3,3,3,1],[2,2]]
=> ?
=> ? = 1 + 1
[[],[[],[[]]],[[]]]
=> [1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [[4,3,3,1],[2,1]]
=> ?
=> ? = 2 + 1
[[],[[[]],[]],[[]]]
=> [1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [[4,3,2,1],[2]]
=> ?
=> ? = 3 + 1
[[],[[[],[]]],[[]]]
=> [1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [[3,2,2,2,1],[1]]
=> ?
=> ? = 2 + 1
[[],[[[[]]]],[[]]]
=> [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [[4,3,3,1],[2]]
=> ?
=> ? = 2 + 1
[[],[[],[],[[]]],[]]
=> [1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [[4,4,4,1],[3,2]]
=> ?
=> ? = 1 + 1
[[],[[],[[]],[]],[]]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [[4,4,3,1],[3,1]]
=> ?
=> ? = 2 + 1
[[],[[],[[],[]]],[]]
=> [1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [[3,3,3,3,1],[2,1,1]]
=> ?
=> ? = 1 + 1
[[],[[],[[[]]]],[]]
=> [1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [[4,4,4,1],[3,1]]
=> ?
=> ? = 1 + 1
[[],[[[]],[],[]],[]]
=> [1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [[4,4,2,1],[3]]
=> ?
=> ? = 2 + 1
[[],[[[]],[[]]],[]]
=> [1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [[3,3,3,2,1],[2,1]]
=> ?
=> ? = 2 + 1
[[],[[[],[]],[]],[]]
=> [1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [[3,3,2,2,1],[2]]
=> ?
=> ? = 2 + 1
[[],[[[[]]],[]],[]]
=> [1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [[4,4,3,1],[3]]
=> ?
=> ? = 2 + 1
[[],[[[],[],[]]],[]]
=> [1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [[2,2,2,2,2,1],[1]]
=> ?
=> ? = 1 + 1
[[],[[[],[[]]]],[]]
=> [1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [[3,3,3,2,1],[2]]
=> ?
=> ? = 2 + 1
[[],[[[[]],[]]],[]]
=> [1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [[3,3,3,3,1],[2,1]]
=> ?
=> ? = 1 + 1
[[],[[[[],[]]]],[]]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [[4,4,4,1],[3]]
=> ?
=> ? = 1 + 1
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: St000318
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00182: Skew partitions —outer shape⟶ Integer partitions
St000318: Integer partitions ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 100%
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00182: Skew partitions —outer shape⟶ Integer partitions
St000318: Integer partitions ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> [[1],[]]
=> [1]
=> 2 = 0 + 2
[[],[]]
=> [1,0,1,0]
=> [[1,1],[]]
=> [1,1]
=> 2 = 0 + 2
[[[]]]
=> [1,1,0,0]
=> [[2],[]]
=> [2]
=> 2 = 0 + 2
[[],[],[]]
=> [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> [1,1,1]
=> 2 = 0 + 2
[[],[[]]]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> [2,1]
=> 3 = 1 + 2
[[[]],[]]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> [2,2]
=> 2 = 0 + 2
[[[],[]]]
=> [1,1,0,1,0,0]
=> [[3],[]]
=> [3]
=> 2 = 0 + 2
[[[[]]]]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> [2,2]
=> 2 = 0 + 2
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> [1,1,1,1]
=> 2 = 0 + 2
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> [2,1,1]
=> 3 = 1 + 2
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [2,2,1]
=> 3 = 1 + 2
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> [3,1]
=> 3 = 1 + 2
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> [2,2,1]
=> 3 = 1 + 2
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [2,2,2]
=> 2 = 0 + 2
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [3,2]
=> 3 = 1 + 2
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [3,3]
=> 2 = 0 + 2
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [2,2,2]
=> 2 = 0 + 2
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [[4],[]]
=> [4]
=> 2 = 0 + 2
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [3,3]
=> 2 = 0 + 2
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> [3,2]
=> 3 = 1 + 2
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> [2,2,2]
=> 2 = 0 + 2
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> [3,3]
=> 2 = 0 + 2
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> [1,1,1,1,1]
=> 2 = 0 + 2
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> [2,1,1,1]
=> 3 = 1 + 2
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [2,2,1,1]
=> 3 = 1 + 2
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> [3,1,1]
=> 3 = 1 + 2
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> [2,2,1,1]
=> 3 = 1 + 2
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [2,2,2,1]
=> 3 = 1 + 2
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [3,2,1]
=> 4 = 2 + 2
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [3,3,1]
=> 3 = 1 + 2
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [2,2,2,1]
=> 3 = 1 + 2
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> [4,1]
=> 3 = 1 + 2
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [3,3,1]
=> 3 = 1 + 2
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> [3,2,1]
=> 4 = 2 + 2
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> [2,2,2,1]
=> 3 = 1 + 2
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> [3,3,1]
=> 3 = 1 + 2
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [2,2,2,2]
=> 2 = 0 + 2
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [3,2,2]
=> 3 = 1 + 2
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [3,3,2]
=> 3 = 1 + 2
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [4,2]
=> 3 = 1 + 2
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [3,3,2]
=> 3 = 1 + 2
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [3,3,3]
=> 2 = 0 + 2
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [2,2,2,2]
=> 2 = 0 + 2
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [4,3]
=> 3 = 1 + 2
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [3,2,2]
=> 3 = 1 + 2
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [4,4]
=> 2 = 0 + 2
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [3,3,3]
=> 2 = 0 + 2
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [3,3,2]
=> 3 = 1 + 2
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> [2,2,2,2]
=> 2 = 0 + 2
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [3,3,3]
=> 2 = 0 + 2
[[[],[]],[],[],[]]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [[3,3,3,3],[2,2,2]]
=> [3,3,3,3]
=> ? = 0 + 2
[[[],[],[]],[],[]]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [[4,4,4],[3,3]]
=> [4,4,4]
=> ? = 0 + 2
[[[],[[]]],[],[]]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [[3,3,3,3],[2,2,1]]
=> [3,3,3,3]
=> ? = 0 + 2
[[[[[]]]],[],[]]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [[3,3,3,3],[2,2]]
=> [3,3,3,3]
=> ? = 0 + 2
[[[],[],[[]]],[]]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [[4,4,4],[3,2]]
=> [4,4,4]
=> ? = 0 + 2
[[[],[[],[]]],[]]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [[3,3,3,3],[2,1,1]]
=> [3,3,3,3]
=> ? = 0 + 2
[[[],[[[]]]],[]]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [[4,4,4],[3,1]]
=> [4,4,4]
=> ? = 0 + 2
[[[[[]],[]]],[]]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [[3,3,3,3],[2,1]]
=> [3,3,3,3]
=> ? = 0 + 2
[[[[[],[]]]],[]]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[4,4,4],[3]]
=> [4,4,4]
=> ? = 0 + 2
[[[[[[]]]]],[]]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[3,3,3,3],[2]]
=> [3,3,3,3]
=> ? = 0 + 2
[[[],[],[[],[]]]]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [[4,4,4],[2,2]]
=> [4,4,4]
=> ? = 0 + 2
[[[],[[],[],[]]]]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [[3,3,3,3],[1,1,1]]
=> [3,3,3,3]
=> ? = 0 + 2
[[[],[[[]],[]]]]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [[4,4,4],[2,1]]
=> [4,4,4]
=> ? = 0 + 2
[[[],[[[[]]]]]]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [[4,4,4],[1,1]]
=> [4,4,4]
=> ? = 0 + 2
[[[[[]],[],[]]]]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [[3,3,3,3],[1,1]]
=> [3,3,3,3]
=> ? = 0 + 2
[[[[[],[]],[]]]]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [[4,4,4],[2]]
=> [4,4,4]
=> ? = 0 + 2
[[[[[[]]],[]]]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[3,3,3,3],[1]]
=> [3,3,3,3]
=> ? = 0 + 2
[[[[[],[[]]]]]]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [[4,4,4],[1]]
=> [4,4,4]
=> ? = 0 + 2
[[[[[[],[]]]]]]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [[3,3,3,3],[]]
=> [3,3,3,3]
=> ? = 0 + 2
[[[[[[[]]]]]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [[4,4,4],[]]
=> [4,4,4]
=> ? = 0 + 2
[[],[],[],[[[]]],[]]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1,1],[1]]
=> ?
=> ? = 1 + 2
[[],[],[],[[],[[]]]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1,1,1],[1]]
=> ?
=> ? = 1 + 2
[[],[],[[]],[[[]]]]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1,1],[1,1]]
=> ?
=> ? = 2 + 2
[[],[],[[[]]],[],[]]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2,1,1],[1,1]]
=> ?
=> ? = 1 + 2
[[],[],[[[]]],[[]]]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2,1,1],[1]]
=> ?
=> ? = 2 + 2
[[],[],[[],[[]]],[]]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1,1],[2,1]]
=> ?
=> ? = 1 + 2
[[],[],[[[]],[]],[]]
=> [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2,1,1],[2]]
=> ?
=> ? = 2 + 2
[[],[],[[[],[]]],[]]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2,1,1],[1]]
=> ?
=> ? = 1 + 2
[[],[],[[[[]]]],[]]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3,1,1],[2]]
=> ?
=> ? = 1 + 2
[[],[],[[],[],[[]]]]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1,1],[2]]
=> ?
=> ? = 1 + 2
[[],[],[[],[[]],[]]]
=> [1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1,1],[1]]
=> ?
=> ? = 2 + 2
[[],[],[[],[[],[]]]]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3,1,1],[1,1]]
=> ?
=> ? = 1 + 2
[[],[],[[],[[[]]]]]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [[4,4,1,1],[1]]
=> ?
=> ? = 1 + 2
[[],[],[[[]],[[]]]]
=> [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2,1,1],[1]]
=> ?
=> ? = 2 + 2
[[],[],[[[[]],[]]]]
=> [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3,1,1],[1]]
=> ?
=> ? = 1 + 2
[[],[],[[[[[]]]]]]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1,1],[]]
=> ?
=> ? = 1 + 2
[[],[[]],[],[],[],[]]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [[2,2,2,2,2,1],[1,1,1,1]]
=> [2,2,2,2,2,1]
=> ? = 1 + 2
[[],[[]],[],[[[]]]]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [[3,3,2,2,1],[1,1,1]]
=> ?
=> ? = 2 + 2
[[],[[]],[[[]]],[]]
=> [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [[3,3,3,2,1],[2,1,1]]
=> ?
=> ? = 2 + 2
[[],[[]],[[],[[]]]]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [[4,4,2,1],[2,1]]
=> ?
=> ? = 2 + 2
[[],[[]],[[[]],[]]]
=> [1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [[4,3,2,1],[1,1]]
=> ?
=> ? = 3 + 2
[[],[[]],[[[],[]]]]
=> [1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [[3,3,3,2,1],[1,1,1]]
=> ?
=> ? = 2 + 2
[[],[[]],[[[[]]]]]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [[4,4,2,1],[1,1]]
=> ?
=> ? = 2 + 2
[[],[[],[]],[],[],[]]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [[3,3,3,3,1],[2,2,2]]
=> [3,3,3,3,1]
=> ? = 1 + 2
[[],[[[]]],[],[],[]]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,2,1],[1,1,1]]
=> ?
=> ? = 1 + 2
[[],[[[]]],[],[[]]]
=> [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [[3,2,2,2,1],[1,1]]
=> ?
=> ? = 2 + 2
[[],[[[]]],[[]],[]]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [[3,3,2,2,1],[2,1]]
=> ?
=> ? = 2 + 2
[[],[[],[]],[[[]]]]
=> [1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [[4,4,3,1],[2,2]]
=> ?
=> ? = 2 + 2
[[],[[[]]],[[],[]]]
=> [1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [[4,2,2,1],[1]]
=> ?
=> ? = 2 + 2
[[],[[[]]],[[[]]]]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [[3,3,2,2,1],[1,1]]
=> ?
=> ? = 2 + 2
Description
The number of addable cells of the Ferrers diagram of an integer partition.
Matching statistic: St001124
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00182: Skew partitions —outer shape⟶ Integer partitions
St001124: Integer partitions ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 100%
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00182: Skew partitions —outer shape⟶ Integer partitions
St001124: Integer partitions ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> [[1],[]]
=> [1]
=> ? = 0
[[],[]]
=> [1,0,1,0]
=> [[1,1],[]]
=> [1,1]
=> 0
[[[]]]
=> [1,1,0,0]
=> [[2],[]]
=> [2]
=> 0
[[],[],[]]
=> [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> [1,1,1]
=> 0
[[],[[]]]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> [2,1]
=> 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> [2,2]
=> 0
[[[],[]]]
=> [1,1,0,1,0,0]
=> [[3],[]]
=> [3]
=> 0
[[[[]]]]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> [2,2]
=> 0
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> [1,1,1,1]
=> 0
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> [2,1,1]
=> 1
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [2,2,1]
=> 1
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> [3,1]
=> 1
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> [2,2,1]
=> 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [2,2,2]
=> 0
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [3,2]
=> 1
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [3,3]
=> 0
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [2,2,2]
=> 0
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [[4],[]]
=> [4]
=> 0
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [3,3]
=> 0
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> [3,2]
=> 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> [2,2,2]
=> 0
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> [3,3]
=> 0
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> [1,1,1,1,1]
=> 0
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> [2,1,1,1]
=> 1
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [2,2,1,1]
=> 1
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> [3,1,1]
=> 1
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> [2,2,1,1]
=> 1
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [2,2,2,1]
=> 1
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [3,2,1]
=> 2
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [3,3,1]
=> 1
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [2,2,2,1]
=> 1
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> [4,1]
=> 1
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [3,3,1]
=> 1
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> [3,2,1]
=> 2
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> [2,2,2,1]
=> 1
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> [3,3,1]
=> 1
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [2,2,2,2]
=> 0
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [3,2,2]
=> 1
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [3,3,2]
=> 1
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [4,2]
=> 1
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [3,3,2]
=> 1
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [3,3,3]
=> 0
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [2,2,2,2]
=> 0
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [4,3]
=> 1
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [3,2,2]
=> 1
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [4,4]
=> 0
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [3,3,3]
=> 0
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [3,3,2]
=> 1
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> [2,2,2,2]
=> 0
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [3,3,3]
=> 0
[[[],[],[],[]]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> [5]
=> 0
[[[],[]],[],[],[]]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [[3,3,3,3],[2,2,2]]
=> [3,3,3,3]
=> ? = 0
[[[],[],[]],[],[]]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [[4,4,4],[3,3]]
=> [4,4,4]
=> ? = 0
[[[],[[]]],[],[]]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [[3,3,3,3],[2,2,1]]
=> [3,3,3,3]
=> ? = 0
[[[[[]]]],[],[]]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [[3,3,3,3],[2,2]]
=> [3,3,3,3]
=> ? = 0
[[[],[],[[]]],[]]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [[4,4,4],[3,2]]
=> [4,4,4]
=> ? = 0
[[[],[[],[]]],[]]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [[3,3,3,3],[2,1,1]]
=> [3,3,3,3]
=> ? = 0
[[[],[[[]]]],[]]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [[4,4,4],[3,1]]
=> [4,4,4]
=> ? = 0
[[[[[]],[]]],[]]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [[3,3,3,3],[2,1]]
=> [3,3,3,3]
=> ? = 0
[[[[[],[]]]],[]]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[4,4,4],[3]]
=> [4,4,4]
=> ? = 0
[[[[[[]]]]],[]]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[3,3,3,3],[2]]
=> [3,3,3,3]
=> ? = 0
[[[],[],[[],[]]]]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [[4,4,4],[2,2]]
=> [4,4,4]
=> ? = 0
[[[],[[],[],[]]]]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [[3,3,3,3],[1,1,1]]
=> [3,3,3,3]
=> ? = 0
[[[],[[[]],[]]]]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [[4,4,4],[2,1]]
=> [4,4,4]
=> ? = 0
[[[],[[[[]]]]]]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [[4,4,4],[1,1]]
=> [4,4,4]
=> ? = 0
[[[[[]],[],[]]]]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [[3,3,3,3],[1,1]]
=> [3,3,3,3]
=> ? = 0
[[[[[],[]],[]]]]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [[4,4,4],[2]]
=> [4,4,4]
=> ? = 0
[[[[[[]]],[]]]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[3,3,3,3],[1]]
=> [3,3,3,3]
=> ? = 0
[[[[[],[[]]]]]]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [[4,4,4],[1]]
=> [4,4,4]
=> ? = 0
[[[[[[],[]]]]]]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [[3,3,3,3],[]]
=> [3,3,3,3]
=> ? = 0
[[[[[[[]]]]]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [[4,4,4],[]]
=> [4,4,4]
=> ? = 0
[[],[],[],[[[]]],[]]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1,1],[1]]
=> ?
=> ? = 1
[[],[],[],[[],[[]]]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1,1,1],[1]]
=> ?
=> ? = 1
[[],[],[[]],[[[]]]]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1,1],[1,1]]
=> ?
=> ? = 2
[[],[],[[[]]],[],[]]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2,1,1],[1,1]]
=> ?
=> ? = 1
[[],[],[[[]]],[[]]]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2,1,1],[1]]
=> ?
=> ? = 2
[[],[],[[],[[]]],[]]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1,1],[2,1]]
=> ?
=> ? = 1
[[],[],[[[]],[]],[]]
=> [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2,1,1],[2]]
=> ?
=> ? = 2
[[],[],[[[],[]]],[]]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2,1,1],[1]]
=> ?
=> ? = 1
[[],[],[[[[]]]],[]]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3,1,1],[2]]
=> ?
=> ? = 1
[[],[],[[],[],[[]]]]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1,1],[2]]
=> ?
=> ? = 1
[[],[],[[],[[]],[]]]
=> [1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1,1],[1]]
=> ?
=> ? = 2
[[],[],[[],[[],[]]]]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3,1,1],[1,1]]
=> ?
=> ? = 1
[[],[],[[],[[[]]]]]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [[4,4,1,1],[1]]
=> ?
=> ? = 1
[[],[],[[[]],[[]]]]
=> [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2,1,1],[1]]
=> ?
=> ? = 2
[[],[],[[[[]],[]]]]
=> [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3,1,1],[1]]
=> ?
=> ? = 1
[[],[],[[[[[]]]]]]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1,1],[]]
=> ?
=> ? = 1
[[],[[]],[],[],[],[]]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [[2,2,2,2,2,1],[1,1,1,1]]
=> [2,2,2,2,2,1]
=> ? = 1
[[],[[]],[],[[[]]]]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [[3,3,2,2,1],[1,1,1]]
=> ?
=> ? = 2
[[],[[]],[[[]]],[]]
=> [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [[3,3,3,2,1],[2,1,1]]
=> ?
=> ? = 2
[[],[[]],[[],[[]]]]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [[4,4,2,1],[2,1]]
=> ?
=> ? = 2
[[],[[]],[[[]],[]]]
=> [1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [[4,3,2,1],[1,1]]
=> ?
=> ? = 3
[[],[[]],[[[],[]]]]
=> [1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [[3,3,3,2,1],[1,1,1]]
=> ?
=> ? = 2
[[],[[]],[[[[]]]]]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [[4,4,2,1],[1,1]]
=> ?
=> ? = 2
[[],[[],[]],[],[],[]]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [[3,3,3,3,1],[2,2,2]]
=> [3,3,3,3,1]
=> ? = 1
[[],[[[]]],[],[],[]]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,2,1],[1,1,1]]
=> ?
=> ? = 1
[[],[[[]]],[],[[]]]
=> [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [[3,2,2,2,1],[1,1]]
=> ?
=> ? = 2
[[],[[[]]],[[]],[]]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [[3,3,2,2,1],[2,1]]
=> ?
=> ? = 2
[[],[[],[]],[[[]]]]
=> [1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [[4,4,3,1],[2,2]]
=> ?
=> ? = 2
[[],[[[]]],[[],[]]]
=> [1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [[4,2,2,1],[1]]
=> ?
=> ? = 2
Description
The multiplicity of the standard representation in the Kronecker square corresponding to a partition.
The Kronecker coefficient is the multiplicity $g_{\mu,\nu}^\lambda$ of the Specht module $S^\lambda$ in $S^\mu\otimes S^\nu$:
$$ S^\mu\otimes S^\nu = \bigoplus_\lambda g_{\mu,\nu}^\lambda S^\lambda $$
This statistic records the Kronecker coefficient $g_{\lambda,\lambda}^{(n-1)1}$, for $\lambda\vdash n > 1$. For $n\leq1$ the statistic is undefined.
It follows from [3, Prop.4.1] (or, slightly easier from [3, Thm.4.2]) that this is one less than [[St000159]], the number of distinct parts of the partition.
Matching statistic: St000069
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St000069: Posets ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 100%
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St000069: Posets ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> [[1],[]]
=> ([],1)
=> 1 = 0 + 1
[[],[]]
=> [1,0,1,0]
=> [[1,1],[]]
=> ([(0,1)],2)
=> 1 = 0 + 1
[[[]]]
=> [1,1,0,0]
=> [[2],[]]
=> ([(0,1)],2)
=> 1 = 0 + 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> 2 = 1 + 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[[[],[]]]
=> [1,1,0,1,0,0]
=> [[3],[]]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[[[]]]]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 2 = 1 + 1
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 2 = 1 + 1
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 2 = 1 + 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 2 = 1 + 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2 = 1 + 1
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> 2 = 1 + 1
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 2 = 1 + 1
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> 2 = 1 + 1
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> 2 = 1 + 1
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> ([(0,3),(0,4),(1,2),(1,4)],5)
=> 3 = 2 + 1
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> 2 = 1 + 1
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> ([(0,4),(1,2),(1,4),(2,3),(2,5),(4,5)],6)
=> 2 = 1 + 1
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2 = 1 + 1
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> ([(0,2),(0,4),(1,3),(1,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
=> 3 = 2 + 1
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
=> 2 = 1 + 1
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> 2 = 1 + 1
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 1 = 0 + 1
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> 2 = 1 + 1
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> 2 = 1 + 1
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> ([(0,4),(1,2),(1,3),(2,5),(3,4),(3,5)],6)
=> 2 = 1 + 1
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 1 = 0 + 1
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> ([(0,3),(1,2),(1,4),(2,5),(3,4),(4,5)],6)
=> 1 = 0 + 1
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> 2 = 1 + 1
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> ([(0,2),(0,4),(1,3),(1,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 1 = 0 + 1
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> ([(0,4),(1,2),(1,3),(2,5),(3,4),(3,5)],6)
=> 2 = 1 + 1
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> ([(0,6),(1,3),(1,6),(2,4),(3,2),(3,5),(5,4),(6,5)],7)
=> 1 = 0 + 1
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> ([(0,2),(0,3),(1,5),(2,6),(3,5),(3,6),(5,4),(6,4)],7)
=> 1 = 0 + 1
[[],[],[],[[[]]],[]]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1,1],[1]]
=> ?
=> ? = 1 + 1
[[],[],[],[[],[[]]]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1,1,1],[1]]
=> ?
=> ? = 1 + 1
[[],[],[[]],[[[]]]]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1,1],[1,1]]
=> ?
=> ? = 2 + 1
[[],[],[[[]]],[],[]]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2,1,1],[1,1]]
=> ?
=> ? = 1 + 1
[[],[],[[[]]],[[]]]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2,1,1],[1]]
=> ?
=> ? = 2 + 1
[[],[],[[],[[]]],[]]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1,1],[2,1]]
=> ?
=> ? = 1 + 1
[[],[],[[[]],[]],[]]
=> [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2,1,1],[2]]
=> ?
=> ? = 2 + 1
[[],[],[[[],[]]],[]]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2,1,1],[1]]
=> ?
=> ? = 1 + 1
[[],[],[[[[]]]],[]]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3,1,1],[2]]
=> ?
=> ? = 1 + 1
[[],[],[[],[],[[]]]]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1,1],[2]]
=> ?
=> ? = 1 + 1
[[],[],[[],[[]],[]]]
=> [1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1,1],[1]]
=> ?
=> ? = 2 + 1
[[],[],[[],[[],[]]]]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3,1,1],[1,1]]
=> ?
=> ? = 1 + 1
[[],[],[[],[[[]]]]]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [[4,4,1,1],[1]]
=> ?
=> ? = 1 + 1
[[],[],[[[]],[[]]]]
=> [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2,1,1],[1]]
=> ?
=> ? = 2 + 1
[[],[],[[[[]],[]]]]
=> [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3,1,1],[1]]
=> ?
=> ? = 1 + 1
[[],[],[[[[[]]]]]]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1,1],[]]
=> ([(0,5),(0,6),(2,9),(3,1),(4,3),(4,8),(5,4),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10)],11)
=> ? = 1 + 1
[[],[[]],[],[[[]]]]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [[3,3,2,2,1],[1,1,1]]
=> ?
=> ? = 2 + 1
[[],[[]],[[[]]],[]]
=> [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [[3,3,3,2,1],[2,1,1]]
=> ?
=> ? = 2 + 1
[[],[[]],[[],[[]]]]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [[4,4,2,1],[2,1]]
=> ?
=> ? = 2 + 1
[[],[[]],[[[]],[]]]
=> [1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [[4,3,2,1],[1,1]]
=> ?
=> ? = 3 + 1
[[],[[]],[[[],[]]]]
=> [1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [[3,3,3,2,1],[1,1,1]]
=> ?
=> ? = 2 + 1
[[],[[]],[[[[]]]]]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [[4,4,2,1],[1,1]]
=> ?
=> ? = 2 + 1
[[],[[[]]],[],[],[]]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,2,1],[1,1,1]]
=> ?
=> ? = 1 + 1
[[],[[[]]],[],[[]]]
=> [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [[3,2,2,2,1],[1,1]]
=> ?
=> ? = 2 + 1
[[],[[[]]],[[]],[]]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [[3,3,2,2,1],[2,1]]
=> ?
=> ? = 2 + 1
[[],[[],[]],[[[]]]]
=> [1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [[4,4,3,1],[2,2]]
=> ?
=> ? = 2 + 1
[[],[[[]]],[[],[]]]
=> [1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [[4,2,2,1],[1]]
=> ?
=> ? = 2 + 1
[[],[[[]]],[[[]]]]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [[3,3,2,2,1],[1,1]]
=> ?
=> ? = 2 + 1
[[],[[],[[]]],[],[]]
=> [1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [[3,3,3,3,1],[2,2,1]]
=> ?
=> ? = 1 + 1
[[],[[[]],[]],[],[]]
=> [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [[3,3,3,2,1],[2,2]]
=> ?
=> ? = 2 + 1
[[],[[[],[]]],[],[]]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [[2,2,2,2,2,1],[1,1]]
=> ?
=> ? = 1 + 1
[[],[[[[]]]],[],[]]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [[3,3,3,3,1],[2,2]]
=> ?
=> ? = 1 + 1
[[],[[],[[]]],[[]]]
=> [1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [[4,3,3,1],[2,1]]
=> ?
=> ? = 2 + 1
[[],[[[]],[]],[[]]]
=> [1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [[4,3,2,1],[2]]
=> ?
=> ? = 3 + 1
[[],[[[],[]]],[[]]]
=> [1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [[3,2,2,2,1],[1]]
=> ?
=> ? = 2 + 1
[[],[[[[]]]],[[]]]
=> [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [[4,3,3,1],[2]]
=> ?
=> ? = 2 + 1
[[],[[],[],[[]]],[]]
=> [1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [[4,4,4,1],[3,2]]
=> ?
=> ? = 1 + 1
[[],[[],[[]],[]],[]]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [[4,4,3,1],[3,1]]
=> ?
=> ? = 2 + 1
[[],[[],[[],[]]],[]]
=> [1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [[3,3,3,3,1],[2,1,1]]
=> ?
=> ? = 1 + 1
[[],[[],[[[]]]],[]]
=> [1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [[4,4,4,1],[3,1]]
=> ?
=> ? = 1 + 1
[[],[[[]],[],[]],[]]
=> [1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [[4,4,2,1],[3]]
=> ?
=> ? = 2 + 1
[[],[[[]],[[]]],[]]
=> [1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [[3,3,3,2,1],[2,1]]
=> ?
=> ? = 2 + 1
[[],[[[],[]],[]],[]]
=> [1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [[3,3,2,2,1],[2]]
=> ?
=> ? = 2 + 1
[[],[[[[]]],[]],[]]
=> [1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [[4,4,3,1],[3]]
=> ?
=> ? = 2 + 1
[[],[[[],[],[]]],[]]
=> [1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [[2,2,2,2,2,1],[1]]
=> ?
=> ? = 1 + 1
[[],[[[],[[]]]],[]]
=> [1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [[3,3,3,2,1],[2]]
=> ?
=> ? = 2 + 1
[[],[[[[]],[]]],[]]
=> [1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [[3,3,3,3,1],[2,1]]
=> ?
=> ? = 1 + 1
[[],[[[[],[]]]],[]]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [[4,4,4,1],[3]]
=> ?
=> ? = 1 + 1
[[],[[[[[]]]]],[]]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [[3,3,3,3,1],[2]]
=> ?
=> ? = 1 + 1
[[],[[],[],[],[[]]]]
=> [1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [[5,5,1],[3]]
=> ?
=> ? = 1 + 1
Description
The number of maximal elements of a poset.
Matching statistic: St000353
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000353: Permutations ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 75%
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000353: Permutations ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 75%
Values
[[]]
=> [1,0]
=> [1,0]
=> [1] => ? = 0
[[],[]]
=> [1,0,1,0]
=> [1,1,0,0]
=> [2,1] => 0
[[[]]]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,2] => 0
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => 0
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 0
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 1
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 0
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 0
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 0
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 0
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 0
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 1
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => 1
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 1
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 2
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 1
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => 1
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 1
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 1
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => 2
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,3,5,2,1] => 1
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => 1
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 0
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 1
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 1
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 1
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 1
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 0
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => 0
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 1
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => 1
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 0
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 1
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,5,4,2,1] => 0
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => 0
[[[],[],[],[]]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
[[],[],[],[],[],[],[]]
=> [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]
=> [7,6,5,4,3,2,1] => ? = 0
[[],[],[],[],[],[[]]]
=> [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]
=> [6,5,4,3,2,1,7] => ? = 1
[[],[],[],[],[[]],[]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,4,3,2,1,7,6] => ? = 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]
=> [5,4,3,2,1,6,7] => ? = 1
[[],[],[],[],[[[]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [6,5,4,3,2,7,1] => ? = 1
[[],[],[],[[]],[],[]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,3,2,1,7,6,5] => ? = 1
[[],[],[],[[]],[[]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 2
[[],[],[],[[],[]],[]]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,3,2,1,5,7,6] => ? = 1
[[],[],[],[[[]]],[]]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [5,4,3,2,7,6,1] => ? = 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]
=> [4,3,2,1,5,6,7] => ? = 1
[[],[],[],[[],[[]]]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,0,0]
=> [4,3,2,1,6,7,5] => ? = 1
[[],[],[],[[[]],[]]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [5,4,3,2,6,1,7] => ? = 2
[[],[],[],[[[],[]]]]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [6,5,4,3,7,2,1] => ? = 1
[[],[],[],[[[[]]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [5,4,3,2,6,7,1] => ? = 1
[[],[],[[]],[],[],[]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,2,1,7,6,5,4] => ? = 1
[[],[],[[]],[],[[]]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [3,2,1,6,5,4,7] => ? = 2
[[],[],[[]],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [3,2,1,5,4,7,6] => ? = 2
[[],[],[[]],[[],[]]]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [3,2,1,5,4,6,7] => ? = 2
[[],[],[[]],[[[]]]]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [3,2,1,6,5,7,4] => ? = 2
[[],[],[[],[]],[],[]]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [3,2,1,4,7,6,5] => ? = 1
[[],[],[[[]]],[],[]]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [4,3,2,7,6,5,1] => ? = 1
[[],[],[[],[]],[[]]]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [3,2,1,4,6,5,7] => ? = 2
[[],[],[[[]]],[[]]]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> [4,3,2,6,5,1,7] => ? = 2
[[],[],[[],[],[]],[]]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [3,2,1,4,5,7,6] => ? = 1
[[],[],[[],[[]]],[]]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [3,2,1,5,7,6,4] => ? = 1
[[],[],[[[]],[]],[]]
=> [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0,1,1,0,0]
=> [4,3,2,5,1,7,6] => ? = 2
[[],[],[[[],[]]],[]]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [5,4,3,7,6,2,1] => ? = 1
[[],[],[[[[]]]],[]]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> [4,3,2,5,7,6,1] => ? = 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]
=> [3,2,1,4,5,6,7] => ? = 1
[[],[],[[],[],[[]]]]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> [3,2,1,4,6,7,5] => ? = 1
[[],[],[[],[[]],[]]]
=> [1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,0]
=> [3,2,1,5,6,4,7] => ? = 2
[[],[],[[],[[],[]]]]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> [3,2,1,6,7,5,4] => ? = 1
[[],[],[[],[[[]]]]]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,0]
=> [3,2,1,5,6,7,4] => ? = 1
[[],[],[[[]],[],[]]]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [4,3,2,5,1,6,7] => ? = 2
[[],[],[[[]],[[]]]]
=> [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> [4,3,2,6,5,7,1] => ? = 2
[[],[],[[[],[]],[]]]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [5,4,3,6,2,1,7] => ? = 2
[[],[],[[[[]]],[]]]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [4,3,2,5,6,1,7] => ? = 2
[[],[],[[[],[],[]]]]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [6,5,4,7,3,2,1] => ? = 1
[[],[],[[[],[[]]]]]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [5,4,3,6,2,7,1] => ? = 2
[[],[],[[[[]],[]]]]
=> [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> [4,3,2,6,7,5,1] => ? = 1
[[],[],[[[[],[]]]]]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [4,3,2,5,6,7,1] => ? = 1
[[],[],[[[[[]]]]]]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [5,4,3,6,7,2,1] => ? = 1
[[],[[]],[],[],[],[]]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,7,6,5,4,3] => ? = 1
[[],[[]],[],[],[[]]]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,6,5,4,3,7] => ? = 2
[[],[[]],[],[[]],[]]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [2,1,5,4,3,7,6] => ? = 2
[[],[[]],[],[[],[]]]
=> [1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,5,4,3,6,7] => ? = 2
[[],[[]],[],[[[]]]]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> [2,1,6,5,4,7,3] => ? = 2
[[],[[]],[[]],[],[]]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,7,6,5] => ? = 2
[[],[[]],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5,7] => ? = 3
Description
The number of inner valleys of a permutation.
The number of valleys including the boundary is [[St000099]].
Matching statistic: St000256
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000256: Integer partitions ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 75%
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000256: Integer partitions ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 75%
Values
[[]]
=> [1,0]
=> [1,0]
=> []
=> 0
[[],[]]
=> [1,0,1,0]
=> [1,1,0,0]
=> []
=> 0
[[[]]]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1]
=> 0
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> 0
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2]
=> 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1]
=> 0
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> 0
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [1]
=> 0
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> 0
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 1
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 1
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> 1
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 0
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 1
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 0
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 0
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 0
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 0
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 0
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 0
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 1
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 1
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 1
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 1
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 1
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 2
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 1
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 1
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 1
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> 1
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> 2
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 1
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 1
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 1
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 1
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 1
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 1
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 0
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 0
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 1
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> 1
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 0
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 0
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> 1
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> 0
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 0
[[],[],[[]],[[]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> ? = 2
[[],[],[[],[]],[]]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> ? = 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]
=> [5,4,3]
=> ? = 1
[[],[[]],[],[[]]]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> ? = 2
[[],[[]],[[]],[]]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> ? = 2
[[],[[]],[[],[]]]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> ? = 2
[[],[[],[]],[],[]]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> ? = 1
[[],[[],[]],[[]]]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> ? = 2
[[],[[],[],[]],[]]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> ? = 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]
=> [5,4,3,2]
=> ? = 1
[[],[[],[],[[]]]]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2]
=> ? = 1
[[],[[],[[]],[]]]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2]
=> ? = 2
[[],[[],[[[]]]]]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2]
=> ? = 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]
=> [5,4,2]
=> ? = 2
[[[]],[],[[]],[]]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> ? = 1
[[[]],[],[[],[]]]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> ? = 1
[[[]],[[]],[],[]]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> ? = 1
[[[]],[[]],[[]]]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> ? = 2
[[[]],[[],[]],[]]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> ? = 1
[[[]],[[],[],[]]]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> ? = 1
[[[]],[[],[[]]]]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1,1]
=> ? = 1
[[[]],[[[]],[]]]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> ? = 2
[[[],[]],[],[[]]]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> ? = 1
[[[],[]],[[]],[]]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> ? = 1
[[[],[]],[[],[]]]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> ? = 1
[[[],[]],[[[]]]]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2,1]
=> ? = 1
[[[[]]],[[],[]]]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1]
=> ? = 1
[[[],[],[]],[],[]]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> ? = 0
[[[],[],[]],[[]]]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> ? = 1
[[[],[[]]],[[]]]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> ? = 1
[[[[]],[]],[[]]]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1]
=> ? = 2
[[[],[],[],[]],[]]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> ? = 0
[[[],[],[[]]],[]]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> ? = 0
[[[],[[]],[]],[]]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> ? = 1
[[[[]],[],[]],[]]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1]
=> ? = 1
[[[[[]]],[]],[]]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1]
=> ? = 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]
=> [5,4,3,2,1]
=> ? = 0
[[[],[],[],[[]]]]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> ? = 0
[[[],[],[[]],[]]]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> ? = 1
[[[],[],[[[]]]]]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> ? = 0
[[[],[[]],[],[]]]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> ? = 1
[[[],[[[]]],[]]]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> ? = 1
[[[],[[[],[]]]]]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> ? = 0
[[[[]],[],[],[]]]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1]
=> ? = 1
[[[[]],[],[[]]]]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1]
=> ? = 1
[[[[[]]],[],[]]]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1]
=> ? = 1
[[[[[],[]]],[]]]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1]
=> ? = 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]
=> [6,5]
=> ? = 1
[[],[],[],[[]],[],[]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,4,4]
=> ? = 1
[[],[],[],[[]],[[]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [6,4,4]
=> ? = 2
Description
The number of parts from which one can substract 2 and still get an integer partition.
The following 4 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000092The number of outer peaks of a permutation. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001960The number of descents of a permutation minus one if its first entry is not one. St001487The number of inner corners of a skew partition.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!