searching the database
Your data matches 16 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: St000698
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000698: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000698: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1},{2},{3}}
=> [1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,2,3,4}}
=> [4]
=> [2,2]
=> [2]
=> 1
{{1,2,3},{4}}
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,2,4},{3}}
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,3,4},{2}}
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1},{2,3,4}}
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
{{1,2,3,4,5}}
=> [5]
=> [2,2,1]
=> [2,1]
=> 0
{{1,2,3,4},{5}}
=> [4,1]
=> [3,2]
=> [2]
=> 1
{{1,2,3,5},{4}}
=> [4,1]
=> [3,2]
=> [2]
=> 1
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
{{1,2,4,5},{3}}
=> [4,1]
=> [3,2]
=> [2]
=> 1
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
{{1,3,4,5},{2}}
=> [4,1]
=> [3,2]
=> [2]
=> 1
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
{{1},{2,3,4,5}}
=> [4,1]
=> [3,2]
=> [2]
=> 1
{{1},{2,3,4},{5}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
{{1},{2,3,5},{4}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
{{1,4,5},{2},{3}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
{{1},{2,4,5},{3}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
{{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
{{1},{2},{3,4,5}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
{{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
{{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
{{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
{{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
{{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 2
{{1,2,3,4,5,6}}
=> [6]
=> [2,2,2]
=> [2,2]
=> 2
{{1,2,3,4,5},{6}}
=> [5,1]
=> [2,2,1,1]
=> [2,1,1]
=> 2
{{1,2,3,4,6},{5}}
=> [5,1]
=> [2,2,1,1]
=> [2,1,1]
=> 2
{{1,2,3,4},{5,6}}
=> [4,2]
=> [4,2]
=> [2]
=> 1
{{1,2,3,4},{5},{6}}
=> [4,1,1]
=> [4,1,1]
=> [1,1]
=> 1
{{1,2,3,5,6},{4}}
=> [5,1]
=> [2,2,1,1]
=> [2,1,1]
=> 2
{{1,2,3,5},{4,6}}
=> [4,2]
=> [4,2]
=> [2]
=> 1
{{1,2,3,5},{4},{6}}
=> [4,1,1]
=> [4,1,1]
=> [1,1]
=> 1
{{1,2,3,6},{4,5}}
=> [4,2]
=> [4,2]
=> [2]
=> 1
{{1,2,3},{4,5,6}}
=> [3,3]
=> [3,2,1]
=> [2,1]
=> 0
{{1,2,3},{4,5},{6}}
=> [3,2,1]
=> [3,3]
=> [3]
=> 1
{{1,2,3,6},{4},{5}}
=> [4,1,1]
=> [4,1,1]
=> [1,1]
=> 1
{{1,2,3},{4,6},{5}}
=> [3,2,1]
=> [3,3]
=> [3]
=> 1
{{1,2,3},{4},{5,6}}
=> [3,2,1]
=> [3,3]
=> [3]
=> 1
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 2
{{1,2,4,5,6},{3}}
=> [5,1]
=> [2,2,1,1]
=> [2,1,1]
=> 2
Description
The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core.
For any positive integer $k$, one associates a $k$-core to a partition by repeatedly removing all rim hooks of size $k$.
This statistic counts the $2$-rim hooks that are removed in this process to obtain a $2$-core.
Matching statistic: St001195
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001195: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 17%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001195: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 17%
Values
{{1},{2},{3}}
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
{{1,2,3,4}}
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 1
{{1,2,3},{4}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
{{1,2,4},{3}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
{{1,3,4},{2}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
{{1},{2,3,4}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 1
{{1,2,3,4,5}}
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 0
{{1,2,3,4},{5}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1
{{1,2,3,5},{4}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
{{1,2,4,5},{3}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1
{{1,3,4,5},{2}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1
{{1},{2,3,4,5}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1
{{1},{2,3,4},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
{{1},{2,3,5},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1
{{1,4,5},{2},{3}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1
{{1},{2,4,5},{3}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
{{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1
{{1},{2},{3,4,5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
{{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1
{{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1
{{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1
{{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1
{{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2
{{1,2,3,4,5,6}}
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 2
{{1,2,3,4,5},{6}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 2
{{1,2,3,4,6},{5}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 2
{{1,2,3,4},{5,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 1
{{1,2,3,4},{5},{6}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 1
{{1,2,3,5,6},{4}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 2
{{1,2,3,5},{4,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 1
{{1,2,3,5},{4},{6}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 1
{{1,2,3,6},{4,5}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 1
{{1,2,3},{4,5,6}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> ? = 0
{{1,2,3},{4,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
{{1,2,3,6},{4},{5}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 1
{{1,2,3},{4,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
{{1,2,3},{4},{5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 2
{{1,2,4,5,6},{3}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 2
{{1,2,4,5},{3,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 1
{{1,2,4,5},{3},{6}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 1
{{1,2,4,6},{3,5}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 1
{{1,2,4},{3,5,6}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> ? = 0
{{1,2,4},{3,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
{{1,2,4,6},{3},{5}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 1
{{1,2,4},{3,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
{{1,2,4},{3},{5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
{{1,2,4},{3},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 2
{{1,2,5,6},{3,4}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 1
{{1,2,5},{3,4,6}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> ? = 0
{{1,2,5},{3,4},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
{{1,2,6},{3,4,5}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> ? = 0
{{1,2},{3,4,5,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 1
{{1,2},{3,4,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
{{1,2,6},{3,4},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
{{1,2},{3,4,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
{{1,2,5,6},{3},{4}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 1
{{1,2,5},{3,6},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
{{1,2,5},{3},{4,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
{{1,2,5},{3},{4},{6}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 2
{{1,2,6},{3,5},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
{{1,2},{3,5,6},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
{{1,2,6},{3},{4,5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
{{1,2},{3},{4,5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
{{1,2,6},{3},{4},{5}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 2
{{1,2},{3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 1
{{1,3,4,5,6},{2}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 2
{{1,3,4,5},{2,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 1
{{1,3,4,5},{2},{6}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 1
{{1,3,4,6},{2,5}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 1
{{1,3,4},{2,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
{{1,3,4},{2,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
{{1,3,4},{2},{5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
{{1,3,5},{2,4},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
{{1,3},{2,4,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
{{1,3,6},{2,4},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
{{1,3},{2,4,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
{{1,3,5},{2,6},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
{{1,3,5},{2},{4,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
{{1,3,6},{2,5},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
{{1,3},{2,5,6},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
{{1,3,6},{2},{4,5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
{{1,3},{2},{4,5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
{{1,4,5},{2,3},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
{{1,4},{2,3,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
{{1,4,6},{2,3},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
{{1,4},{2,3,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
{{1,5},{2,3,4},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
{{1,6},{2,3,4},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
Description
The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$.
Matching statistic: St001208
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001208: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 17%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001208: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 17%
Values
{{1},{2},{3}}
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1
{{1,2,3,4}}
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? = 1
{{1,2,3},{4}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 1
{{1,2,4},{3}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 1
{{1,3,4},{2}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 1
{{1},{2,3,4}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 1
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ? = 1
{{1,2,3,4,5}}
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? = 0
{{1,2,3,4},{5}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 1
{{1,2,3,5},{4}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 1
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
{{1,2,4,5},{3}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 1
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 1
{{1,3,4,5},{2}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 1
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 1
{{1},{2,3,4,5}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 1
{{1},{2,3,4},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
{{1},{2,3,5},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 1
{{1,4,5},{2},{3}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 1
{{1},{2,4,5},{3}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
{{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 1
{{1},{2},{3,4,5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
{{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 1
{{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 1
{{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 1
{{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 1
{{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 1
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 2
{{1,2,3,4,5,6}}
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => ? = 2
{{1,2,3,4,5},{6}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? = 2
{{1,2,3,4,6},{5}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? = 2
{{1,2,3,4},{5,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => ? = 1
{{1,2,3,4},{5},{6}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => ? = 1
{{1,2,3,5,6},{4}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? = 2
{{1,2,3,5},{4,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => ? = 1
{{1,2,3,5},{4},{6}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => ? = 1
{{1,2,3,6},{4,5}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => ? = 1
{{1,2,3},{4,5,6}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => ? = 0
{{1,2,3},{4,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
{{1,2,3,6},{4},{5}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => ? = 1
{{1,2,3},{4,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
{{1,2,3},{4},{5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ? = 2
{{1,2,4,5,6},{3}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? = 2
{{1,2,4,5},{3,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => ? = 1
{{1,2,4,5},{3},{6}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => ? = 1
{{1,2,4,6},{3,5}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => ? = 1
{{1,2,4},{3,5,6}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => ? = 0
{{1,2,4},{3,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
{{1,2,4,6},{3},{5}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => ? = 1
{{1,2,4},{3,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
{{1,2,4},{3},{5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
{{1,2,4},{3},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ? = 2
{{1,2,5,6},{3,4}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => ? = 1
{{1,2,5},{3,4,6}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => ? = 0
{{1,2,5},{3,4},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
{{1,2,6},{3,4,5}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => ? = 0
{{1,2},{3,4,5,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => ? = 1
{{1,2},{3,4,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
{{1,2,6},{3,4},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
{{1,2},{3,4,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
{{1,2,5,6},{3},{4}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => ? = 1
{{1,2,5},{3,6},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
{{1,2,5},{3},{4,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
{{1,2,5},{3},{4},{6}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ? = 2
{{1,2,6},{3,5},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
{{1,2},{3,5,6},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
{{1,2,6},{3},{4,5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
{{1,2},{3},{4,5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
{{1,2,6},{3},{4},{5}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ? = 2
{{1,2},{3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? = 1
{{1,3,4,5,6},{2}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? = 2
{{1,3,4,5},{2,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => ? = 1
{{1,3,4,5},{2},{6}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => ? = 1
{{1,3,4,6},{2,5}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => ? = 1
{{1,3,4},{2,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
{{1,3,4},{2,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
{{1,3,4},{2},{5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
{{1,3,5},{2,4},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
{{1,3},{2,4,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
{{1,3,6},{2,4},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
{{1,3},{2,4,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
{{1,3,5},{2,6},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
{{1,3,5},{2},{4,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
{{1,3,6},{2,5},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
{{1,3},{2,5,6},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
{{1,3,6},{2},{4,5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
{{1,3},{2},{4,5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
{{1,4,5},{2,3},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
{{1,4},{2,3,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
{{1,4,6},{2,3},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
{{1,4},{2,3,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
{{1,5},{2,3,4},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
{{1,6},{2,3,4},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
Description
The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$.
Matching statistic: St001001
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001001: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 17%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001001: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 17%
Values
{{1},{2},{3}}
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0 = 1 - 1
{{1,2,3,4}}
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 1 - 1
{{1,2,3},{4}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
{{1,2,4},{3}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
{{1,3,4},{2}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
{{1},{2,3,4}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
{{1,2,3,4,5}}
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 0 - 1
{{1,2,3,4},{5}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1 - 1
{{1,2,3,5},{4}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1 - 1
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
{{1,2,4,5},{3}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1 - 1
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1 - 1
{{1,3,4,5},{2}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1 - 1
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1 - 1
{{1},{2,3,4,5}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1 - 1
{{1},{2,3,4},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
{{1},{2,3,5},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1 - 1
{{1,4,5},{2},{3}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1 - 1
{{1},{2,4,5},{3}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
{{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1 - 1
{{1},{2},{3,4,5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
{{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1 - 1
{{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1 - 1
{{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1 - 1
{{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1 - 1
{{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1 - 1
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
{{1,2,3,4,5,6}}
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 2 - 1
{{1,2,3,4,5},{6}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 2 - 1
{{1,2,3,4,6},{5}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 2 - 1
{{1,2,3,4},{5,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 1 - 1
{{1,2,3,4},{5},{6}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 1 - 1
{{1,2,3,5,6},{4}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 2 - 1
{{1,2,3,5},{4,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 1 - 1
{{1,2,3,5},{4},{6}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 1 - 1
{{1,2,3,6},{4,5}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 1 - 1
{{1,2,3},{4,5,6}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> ? = 0 - 1
{{1,2,3},{4,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
{{1,2,3,6},{4},{5}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 1 - 1
{{1,2,3},{4,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
{{1,2,3},{4},{5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 2 - 1
{{1,2,4,5,6},{3}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 2 - 1
{{1,2,4,5},{3,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 1 - 1
{{1,2,4,5},{3},{6}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 1 - 1
{{1,2,4,6},{3,5}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 1 - 1
{{1,2,4},{3,5,6}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> ? = 0 - 1
{{1,2,4},{3,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
{{1,2,4,6},{3},{5}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 1 - 1
{{1,2,4},{3,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
{{1,2,4},{3},{5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
{{1,2,4},{3},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 2 - 1
{{1,2,5,6},{3,4}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 1 - 1
{{1,2,5},{3,4,6}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> ? = 0 - 1
{{1,2,5},{3,4},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
{{1,2,6},{3,4,5}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> ? = 0 - 1
{{1,2},{3,4,5,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 1 - 1
{{1,2},{3,4,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
{{1,2,6},{3,4},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
{{1,2},{3,4,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
{{1,2,5,6},{3},{4}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 1 - 1
{{1,2,5},{3,6},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
{{1,2,5},{3},{4,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
{{1,2,5},{3},{4},{6}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 2 - 1
{{1,2,6},{3,5},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
{{1,2},{3,5,6},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
{{1,2,6},{3},{4,5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
{{1,2},{3},{4,5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
{{1,2,6},{3},{4},{5}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 2 - 1
{{1,2},{3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 1 - 1
{{1,3,4,5,6},{2}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 2 - 1
{{1,3,4,5},{2,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 1 - 1
{{1,3,4,5},{2},{6}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 1 - 1
{{1,3,4,6},{2,5}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 1 - 1
{{1,3,4},{2,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
{{1,3,4},{2,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
{{1,3,4},{2},{5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
{{1,3,5},{2,4},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
{{1,3},{2,4,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
{{1,3,6},{2,4},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
{{1,3},{2,4,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
{{1,3,5},{2,6},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
{{1,3,5},{2},{4,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
{{1,3,6},{2,5},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
{{1,3},{2,5,6},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
{{1,3,6},{2},{4,5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
{{1,3},{2},{4,5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
{{1,4,5},{2,3},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
{{1,4},{2,3,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
{{1,4,6},{2,3},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
{{1,4},{2,3,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
{{1,5},{2,3,4},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
{{1,6},{2,3,4},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
Description
The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001371
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001371: Binary words ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 17%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001371: Binary words ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 17%
Values
{{1},{2},{3}}
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0 = 1 - 1
{{1,2,3,4}}
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? = 1 - 1
{{1,2,3},{4}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 0 = 1 - 1
{{1,2,4},{3}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 0 = 1 - 1
{{1,3,4},{2}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 0 = 1 - 1
{{1},{2,3,4}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 0 = 1 - 1
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? = 1 - 1
{{1,2,3,4,5}}
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 111110000010 => ? = 0 - 1
{{1,2,3,4},{5}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 1 - 1
{{1,2,3,5},{4}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 1 - 1
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0 = 1 - 1
{{1,2,4,5},{3}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 1 - 1
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0 = 1 - 1
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0 = 1 - 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 1 - 1
{{1,3,4,5},{2}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 1 - 1
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0 = 1 - 1
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0 = 1 - 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 1 - 1
{{1},{2,3,4,5}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 1 - 1
{{1},{2,3,4},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0 = 1 - 1
{{1},{2,3,5},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0 = 1 - 1
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 1 - 1
{{1,4,5},{2},{3}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0 = 1 - 1
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 1 - 1
{{1},{2,4,5},{3}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0 = 1 - 1
{{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 1 - 1
{{1},{2},{3,4,5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0 = 1 - 1
{{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 1 - 1
{{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 1 - 1
{{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 1 - 1
{{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 1 - 1
{{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 1 - 1
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 101111100000 => ? = 2 - 1
{{1,2,3,4,5,6}}
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 11111100000010 => ? = 2 - 1
{{1,2,3,4,5},{6}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 111101000010 => ? = 2 - 1
{{1,2,3,4,6},{5}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 111101000010 => ? = 2 - 1
{{1,2,3,4},{5,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? = 1 - 1
{{1,2,3,4},{5},{6}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 1 - 1
{{1,2,3,5,6},{4}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 111101000010 => ? = 2 - 1
{{1,2,3,5},{4,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? = 1 - 1
{{1,2,3,5},{4},{6}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 1 - 1
{{1,2,3,6},{4,5}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? = 1 - 1
{{1,2,3},{4,5,6}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => ? = 0 - 1
{{1,2,3},{4,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
{{1,2,3,6},{4},{5}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 1 - 1
{{1,2,3},{4,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
{{1,2,3},{4},{5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 2 - 1
{{1,2,4,5,6},{3}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 111101000010 => ? = 2 - 1
{{1,2,4,5},{3,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? = 1 - 1
{{1,2,4,5},{3},{6}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 1 - 1
{{1,2,4,6},{3,5}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? = 1 - 1
{{1,2,4},{3,5,6}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => ? = 0 - 1
{{1,2,4},{3,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
{{1,2,4,6},{3},{5}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 1 - 1
{{1,2,4},{3,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
{{1,2,4},{3},{5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
{{1,2,4},{3},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 2 - 1
{{1,2,5,6},{3,4}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? = 1 - 1
{{1,2,5},{3,4,6}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => ? = 0 - 1
{{1,2,5},{3,4},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
{{1,2,6},{3,4,5}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => ? = 0 - 1
{{1,2},{3,4,5,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? = 1 - 1
{{1,2},{3,4,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
{{1,2,6},{3,4},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
{{1,2},{3,4,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
{{1,2,5,6},{3},{4}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 1 - 1
{{1,2,5},{3,6},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
{{1,2,5},{3},{4,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
{{1,2,5},{3},{4},{6}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 2 - 1
{{1,2,6},{3,5},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
{{1,2},{3,5,6},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
{{1,2,6},{3},{4,5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
{{1,2},{3},{4,5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
{{1,2,6},{3},{4},{5}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 2 - 1
{{1,2},{3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 101111010000 => ? = 1 - 1
{{1,3,4,5,6},{2}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 111101000010 => ? = 2 - 1
{{1,3,4,5},{2,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? = 1 - 1
{{1,3,4,5},{2},{6}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 1 - 1
{{1,3,4,6},{2,5}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? = 1 - 1
{{1,3,4},{2,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
{{1,3,4},{2,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
{{1,3,4},{2},{5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
{{1,3,5},{2,4},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
{{1,3},{2,4,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
{{1,3,6},{2,4},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
{{1,3},{2,4,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
{{1,3,5},{2,6},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
{{1,3,5},{2},{4,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
{{1,3,6},{2,5},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
{{1,3},{2,5,6},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
{{1,3,6},{2},{4,5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
{{1,3},{2},{4,5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
{{1,4,5},{2,3},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
{{1,4},{2,3,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
{{1,4,6},{2,3},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
{{1,4},{2,3,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
{{1,5},{2,3,4},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
{{1,6},{2,3,4},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
Description
The length of the longest Yamanouchi prefix of a binary word.
This is the largest index $i$ such that in each of the prefixes $w_1$, $w_1w_2$, $w_1w_2\dots w_i$ the number of zeros is greater than or equal to the number of ones.
Matching statistic: St001730
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001730: Binary words ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 17%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001730: Binary words ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 17%
Values
{{1},{2},{3}}
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0 = 1 - 1
{{1,2,3,4}}
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? = 1 - 1
{{1,2,3},{4}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 0 = 1 - 1
{{1,2,4},{3}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 0 = 1 - 1
{{1,3,4},{2}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 0 = 1 - 1
{{1},{2,3,4}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 0 = 1 - 1
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? = 1 - 1
{{1,2,3,4,5}}
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 111110000010 => ? = 0 - 1
{{1,2,3,4},{5}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 1 - 1
{{1,2,3,5},{4}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 1 - 1
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0 = 1 - 1
{{1,2,4,5},{3}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 1 - 1
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0 = 1 - 1
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0 = 1 - 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 1 - 1
{{1,3,4,5},{2}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 1 - 1
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0 = 1 - 1
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0 = 1 - 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 1 - 1
{{1},{2,3,4,5}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 1 - 1
{{1},{2,3,4},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0 = 1 - 1
{{1},{2,3,5},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0 = 1 - 1
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 1 - 1
{{1,4,5},{2},{3}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0 = 1 - 1
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 1 - 1
{{1},{2,4,5},{3}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0 = 1 - 1
{{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 1 - 1
{{1},{2},{3,4,5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0 = 1 - 1
{{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 1 - 1
{{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 1 - 1
{{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 1 - 1
{{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 1 - 1
{{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 1 - 1
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 101111100000 => ? = 2 - 1
{{1,2,3,4,5,6}}
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 11111100000010 => ? = 2 - 1
{{1,2,3,4,5},{6}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 111101000010 => ? = 2 - 1
{{1,2,3,4,6},{5}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 111101000010 => ? = 2 - 1
{{1,2,3,4},{5,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? = 1 - 1
{{1,2,3,4},{5},{6}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 1 - 1
{{1,2,3,5,6},{4}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 111101000010 => ? = 2 - 1
{{1,2,3,5},{4,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? = 1 - 1
{{1,2,3,5},{4},{6}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 1 - 1
{{1,2,3,6},{4,5}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? = 1 - 1
{{1,2,3},{4,5,6}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => ? = 0 - 1
{{1,2,3},{4,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
{{1,2,3,6},{4},{5}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 1 - 1
{{1,2,3},{4,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
{{1,2,3},{4},{5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 2 - 1
{{1,2,4,5,6},{3}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 111101000010 => ? = 2 - 1
{{1,2,4,5},{3,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? = 1 - 1
{{1,2,4,5},{3},{6}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 1 - 1
{{1,2,4,6},{3,5}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? = 1 - 1
{{1,2,4},{3,5,6}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => ? = 0 - 1
{{1,2,4},{3,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
{{1,2,4,6},{3},{5}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 1 - 1
{{1,2,4},{3,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
{{1,2,4},{3},{5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
{{1,2,4},{3},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 2 - 1
{{1,2,5,6},{3,4}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? = 1 - 1
{{1,2,5},{3,4,6}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => ? = 0 - 1
{{1,2,5},{3,4},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
{{1,2,6},{3,4,5}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => ? = 0 - 1
{{1,2},{3,4,5,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? = 1 - 1
{{1,2},{3,4,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
{{1,2,6},{3,4},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
{{1,2},{3,4,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
{{1,2,5,6},{3},{4}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 1 - 1
{{1,2,5},{3,6},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
{{1,2,5},{3},{4,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
{{1,2,5},{3},{4},{6}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 2 - 1
{{1,2,6},{3,5},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
{{1,2},{3,5,6},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
{{1,2,6},{3},{4,5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
{{1,2},{3},{4,5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
{{1,2,6},{3},{4},{5}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 2 - 1
{{1,2},{3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 101111010000 => ? = 1 - 1
{{1,3,4,5,6},{2}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 111101000010 => ? = 2 - 1
{{1,3,4,5},{2,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? = 1 - 1
{{1,3,4,5},{2},{6}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 1 - 1
{{1,3,4,6},{2,5}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? = 1 - 1
{{1,3,4},{2,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
{{1,3,4},{2,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
{{1,3,4},{2},{5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
{{1,3,5},{2,4},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
{{1,3},{2,4,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
{{1,3,6},{2,4},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
{{1,3},{2,4,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
{{1,3,5},{2,6},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
{{1,3,5},{2},{4,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
{{1,3,6},{2,5},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
{{1,3},{2,5,6},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
{{1,3,6},{2},{4,5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
{{1,3},{2},{4,5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
{{1,4,5},{2,3},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
{{1,4},{2,3,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
{{1,4,6},{2,3},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
{{1,4},{2,3,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
{{1,5},{2,3,4},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
{{1,6},{2,3,4},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
Description
The number of times the path corresponding to a binary word crosses the base line.
Interpret each $0$ as a step $(1,-1)$ and $1$ as a step $(1,1)$. Then this statistic counts the number of times the path crosses the $x$-axis.
Matching statistic: St001803
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St001803: Standard tableaux ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 17%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St001803: Standard tableaux ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 17%
Values
{{1},{2},{3}}
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 0 = 1 - 1
{{1,2,3,4}}
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 1 - 1
{{1,2,3},{4}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 0 = 1 - 1
{{1,2,4},{3}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 0 = 1 - 1
{{1,3,4},{2}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 0 = 1 - 1
{{1},{2,3,4}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 0 = 1 - 1
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? = 1 - 1
{{1,2,3,4,5}}
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,11],[6,7,8,9,10,12]]
=> ? = 0 - 1
{{1,2,3,4},{5}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 1 - 1
{{1,2,3,5},{4}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 1 - 1
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0 = 1 - 1
{{1,2,4,5},{3}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 1 - 1
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0 = 1 - 1
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0 = 1 - 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 1 - 1
{{1,3,4,5},{2}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 1 - 1
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0 = 1 - 1
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0 = 1 - 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 1 - 1
{{1},{2,3,4,5}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 1 - 1
{{1},{2,3,4},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0 = 1 - 1
{{1},{2,3,5},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0 = 1 - 1
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 1 - 1
{{1,4,5},{2},{3}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0 = 1 - 1
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 1 - 1
{{1},{2,4,5},{3}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0 = 1 - 1
{{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 1 - 1
{{1},{2},{3,4,5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0 = 1 - 1
{{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 1 - 1
{{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 1 - 1
{{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 1 - 1
{{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 1 - 1
{{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 1 - 1
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,3,4,5,6,7],[2,8,9,10,11,12]]
=> ? = 2 - 1
{{1,2,3,4,5,6}}
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,6,13],[7,8,9,10,11,12,14]]
=> ? = 2 - 1
{{1,2,3,4,5},{6}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[1,2,3,4,6,11],[5,7,8,9,10,12]]
=> ? = 2 - 1
{{1,2,3,4,6},{5}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[1,2,3,4,6,11],[5,7,8,9,10,12]]
=> ? = 2 - 1
{{1,2,3,4},{5,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? = 1 - 1
{{1,2,3,4},{5},{6}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 1 - 1
{{1,2,3,5,6},{4}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[1,2,3,4,6,11],[5,7,8,9,10,12]]
=> ? = 2 - 1
{{1,2,3,5},{4,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? = 1 - 1
{{1,2,3,5},{4},{6}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 1 - 1
{{1,2,3,6},{4,5}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? = 1 - 1
{{1,2,3},{4,5,6}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[1,2,3,7,8],[4,5,6,9,10]]
=> ? = 0 - 1
{{1,2,3},{4,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0 = 1 - 1
{{1,2,3,6},{4},{5}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 1 - 1
{{1,2,3},{4,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0 = 1 - 1
{{1,2,3},{4},{5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0 = 1 - 1
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? = 2 - 1
{{1,2,4,5,6},{3}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[1,2,3,4,6,11],[5,7,8,9,10,12]]
=> ? = 2 - 1
{{1,2,4,5},{3,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? = 1 - 1
{{1,2,4,5},{3},{6}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 1 - 1
{{1,2,4,6},{3,5}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? = 1 - 1
{{1,2,4},{3,5,6}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[1,2,3,7,8],[4,5,6,9,10]]
=> ? = 0 - 1
{{1,2,4},{3,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0 = 1 - 1
{{1,2,4,6},{3},{5}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 1 - 1
{{1,2,4},{3,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0 = 1 - 1
{{1,2,4},{3},{5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0 = 1 - 1
{{1,2,4},{3},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? = 2 - 1
{{1,2,5,6},{3,4}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? = 1 - 1
{{1,2,5},{3,4,6}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[1,2,3,7,8],[4,5,6,9,10]]
=> ? = 0 - 1
{{1,2,5},{3,4},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0 = 1 - 1
{{1,2,6},{3,4,5}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[1,2,3,7,8],[4,5,6,9,10]]
=> ? = 0 - 1
{{1,2},{3,4,5,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? = 1 - 1
{{1,2},{3,4,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0 = 1 - 1
{{1,2,6},{3,4},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0 = 1 - 1
{{1,2},{3,4,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0 = 1 - 1
{{1,2,5,6},{3},{4}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 1 - 1
{{1,2,5},{3,6},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0 = 1 - 1
{{1,2,5},{3},{4,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0 = 1 - 1
{{1,2,5},{3},{4},{6}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? = 2 - 1
{{1,2,6},{3,5},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0 = 1 - 1
{{1,2},{3,5,6},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0 = 1 - 1
{{1,2,6},{3},{4,5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0 = 1 - 1
{{1,2},{3},{4,5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0 = 1 - 1
{{1,2,6},{3},{4},{5}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? = 2 - 1
{{1,2},{3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [[1,3,4,5,6,8],[2,7,9,10,11,12]]
=> ? = 1 - 1
{{1,3,4,5,6},{2}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[1,2,3,4,6,11],[5,7,8,9,10,12]]
=> ? = 2 - 1
{{1,3,4,5},{2,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? = 1 - 1
{{1,3,4,5},{2},{6}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 1 - 1
{{1,3,4,6},{2,5}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? = 1 - 1
{{1,3,4},{2,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0 = 1 - 1
{{1,3,4},{2,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0 = 1 - 1
{{1,3,4},{2},{5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0 = 1 - 1
{{1,3,5},{2,4},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0 = 1 - 1
{{1,3},{2,4,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0 = 1 - 1
{{1,3,6},{2,4},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0 = 1 - 1
{{1,3},{2,4,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0 = 1 - 1
{{1,3,5},{2,6},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0 = 1 - 1
{{1,3,5},{2},{4,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0 = 1 - 1
{{1,3,6},{2,5},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0 = 1 - 1
{{1,3},{2,5,6},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0 = 1 - 1
{{1,3,6},{2},{4,5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0 = 1 - 1
{{1,3},{2},{4,5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0 = 1 - 1
{{1,4,5},{2,3},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0 = 1 - 1
{{1,4},{2,3,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0 = 1 - 1
{{1,4,6},{2,3},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0 = 1 - 1
{{1,4},{2,3,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0 = 1 - 1
{{1,5},{2,3,4},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0 = 1 - 1
{{1,6},{2,3,4},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0 = 1 - 1
Description
The maximal overlap of the cylindrical tableau associated with a tableau.
A cylindrical tableau associated with a standard Young tableau $T$ is the skew row-strict tableau obtained by gluing two copies of $T$ such that the inner shape is a rectangle.
The overlap, recorded in this statistic, equals $\max_C\big(2\ell(T) - \ell(C)\big)$, where $\ell$ denotes the number of rows of a tableau and the maximum is taken over all cylindrical tableaux.
In particular, the statistic equals $0$, if and only if the last entry of the first row is larger than or equal to the first entry of the last row. Moreover, the statistic attains its maximal value, the number of rows of the tableau minus 1, if and only if the tableau consists of a single column.
Matching statistic: St001804
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St001804: Standard tableaux ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 17%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St001804: Standard tableaux ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 17%
Values
{{1},{2},{3}}
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 2 = 1 + 1
{{1,2,3,4}}
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 1 + 1
{{1,2,3},{4}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 2 = 1 + 1
{{1,2,4},{3}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 2 = 1 + 1
{{1,3,4},{2}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 2 = 1 + 1
{{1},{2,3,4}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 2 = 1 + 1
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? = 1 + 1
{{1,2,3,4,5}}
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,11],[6,7,8,9,10,12]]
=> ? = 0 + 1
{{1,2,3,4},{5}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 1 + 1
{{1,2,3,5},{4}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 1 + 1
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 1 + 1
{{1,2,4,5},{3}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 1 + 1
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 1 + 1
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 1 + 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 1 + 1
{{1,3,4,5},{2}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 1 + 1
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 1 + 1
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 1 + 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 1 + 1
{{1},{2,3,4,5}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 1 + 1
{{1},{2,3,4},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 1 + 1
{{1},{2,3,5},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 1 + 1
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 1 + 1
{{1,4,5},{2},{3}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 1 + 1
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 1 + 1
{{1},{2,4,5},{3}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 1 + 1
{{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 1 + 1
{{1},{2},{3,4,5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 1 + 1
{{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 1 + 1
{{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 1 + 1
{{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 1 + 1
{{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 1 + 1
{{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 1 + 1
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,3,4,5,6,7],[2,8,9,10,11,12]]
=> ? = 2 + 1
{{1,2,3,4,5,6}}
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,6,13],[7,8,9,10,11,12,14]]
=> ? = 2 + 1
{{1,2,3,4,5},{6}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[1,2,3,4,6,11],[5,7,8,9,10,12]]
=> ? = 2 + 1
{{1,2,3,4,6},{5}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[1,2,3,4,6,11],[5,7,8,9,10,12]]
=> ? = 2 + 1
{{1,2,3,4},{5,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? = 1 + 1
{{1,2,3,4},{5},{6}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 1 + 1
{{1,2,3,5,6},{4}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[1,2,3,4,6,11],[5,7,8,9,10,12]]
=> ? = 2 + 1
{{1,2,3,5},{4,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? = 1 + 1
{{1,2,3,5},{4},{6}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 1 + 1
{{1,2,3,6},{4,5}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? = 1 + 1
{{1,2,3},{4,5,6}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[1,2,3,7,8],[4,5,6,9,10]]
=> ? = 0 + 1
{{1,2,3},{4,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 1 + 1
{{1,2,3,6},{4},{5}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 1 + 1
{{1,2,3},{4,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 1 + 1
{{1,2,3},{4},{5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 1 + 1
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? = 2 + 1
{{1,2,4,5,6},{3}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[1,2,3,4,6,11],[5,7,8,9,10,12]]
=> ? = 2 + 1
{{1,2,4,5},{3,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? = 1 + 1
{{1,2,4,5},{3},{6}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 1 + 1
{{1,2,4,6},{3,5}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? = 1 + 1
{{1,2,4},{3,5,6}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[1,2,3,7,8],[4,5,6,9,10]]
=> ? = 0 + 1
{{1,2,4},{3,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 1 + 1
{{1,2,4,6},{3},{5}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 1 + 1
{{1,2,4},{3,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 1 + 1
{{1,2,4},{3},{5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 1 + 1
{{1,2,4},{3},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? = 2 + 1
{{1,2,5,6},{3,4}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? = 1 + 1
{{1,2,5},{3,4,6}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[1,2,3,7,8],[4,5,6,9,10]]
=> ? = 0 + 1
{{1,2,5},{3,4},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 1 + 1
{{1,2,6},{3,4,5}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[1,2,3,7,8],[4,5,6,9,10]]
=> ? = 0 + 1
{{1,2},{3,4,5,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? = 1 + 1
{{1,2},{3,4,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 1 + 1
{{1,2,6},{3,4},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 1 + 1
{{1,2},{3,4,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 1 + 1
{{1,2,5,6},{3},{4}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 1 + 1
{{1,2,5},{3,6},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 1 + 1
{{1,2,5},{3},{4,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 1 + 1
{{1,2,5},{3},{4},{6}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? = 2 + 1
{{1,2,6},{3,5},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 1 + 1
{{1,2},{3,5,6},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 1 + 1
{{1,2,6},{3},{4,5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 1 + 1
{{1,2},{3},{4,5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 1 + 1
{{1,2,6},{3},{4},{5}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? = 2 + 1
{{1,2},{3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [[1,3,4,5,6,8],[2,7,9,10,11,12]]
=> ? = 1 + 1
{{1,3,4,5,6},{2}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[1,2,3,4,6,11],[5,7,8,9,10,12]]
=> ? = 2 + 1
{{1,3,4,5},{2,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? = 1 + 1
{{1,3,4,5},{2},{6}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 1 + 1
{{1,3,4,6},{2,5}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? = 1 + 1
{{1,3,4},{2,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 1 + 1
{{1,3,4},{2,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 1 + 1
{{1,3,4},{2},{5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 1 + 1
{{1,3,5},{2,4},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 1 + 1
{{1,3},{2,4,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 1 + 1
{{1,3,6},{2,4},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 1 + 1
{{1,3},{2,4,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 1 + 1
{{1,3,5},{2,6},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 1 + 1
{{1,3,5},{2},{4,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 1 + 1
{{1,3,6},{2,5},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 1 + 1
{{1,3},{2,5,6},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 1 + 1
{{1,3,6},{2},{4,5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 1 + 1
{{1,3},{2},{4,5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 1 + 1
{{1,4,5},{2,3},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 1 + 1
{{1,4},{2,3,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 1 + 1
{{1,4,6},{2,3},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 1 + 1
{{1,4},{2,3,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 1 + 1
{{1,5},{2,3,4},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 1 + 1
{{1,6},{2,3,4},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 1 + 1
Description
The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau.
A cylindrical tableau associated with a standard Young tableau $T$ is the skew row-strict tableau obtained by gluing two copies of $T$ such that the inner shape is a rectangle.
This statistic equals $\max_C\big(\ell(C) - \ell(T)\big)$, where $\ell$ denotes the number of rows of a tableau and the maximum is taken over all cylindrical tableaux.
Matching statistic: St001207
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 17%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 17%
Values
{{1},{2},{3}}
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 1 + 2
{{1,2,3,4}}
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ? = 1 + 2
{{1,2,3},{4}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 3 = 1 + 2
{{1,2,4},{3}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 3 = 1 + 2
{{1,3,4},{2}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 3 = 1 + 2
{{1},{2,3,4}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 3 = 1 + 2
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ? = 1 + 2
{{1,2,3,4,5}}
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => ? = 0 + 2
{{1,2,3,4},{5}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => ? = 1 + 2
{{1,2,3,5},{4}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => ? = 1 + 2
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 3 = 1 + 2
{{1,2,4,5},{3}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => ? = 1 + 2
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 3 = 1 + 2
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 3 = 1 + 2
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => ? = 1 + 2
{{1,3,4,5},{2}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => ? = 1 + 2
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 3 = 1 + 2
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 3 = 1 + 2
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => ? = 1 + 2
{{1},{2,3,4,5}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => ? = 1 + 2
{{1},{2,3,4},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 3 = 1 + 2
{{1},{2,3,5},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 3 = 1 + 2
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => ? = 1 + 2
{{1,4,5},{2},{3}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 3 = 1 + 2
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => ? = 1 + 2
{{1},{2,4,5},{3}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 3 = 1 + 2
{{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => ? = 1 + 2
{{1},{2},{3,4,5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 3 = 1 + 2
{{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => ? = 1 + 2
{{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => ? = 1 + 2
{{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => ? = 1 + 2
{{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => ? = 1 + 2
{{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => ? = 1 + 2
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ? = 2 + 2
{{1,2,3,4,5,6}}
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => ? = 2 + 2
{{1,2,3,4,5},{6}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => ? = 2 + 2
{{1,2,3,4,6},{5}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => ? = 2 + 2
{{1,2,3,4},{5,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 1 + 2
{{1,2,3,4},{5},{6}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => ? = 1 + 2
{{1,2,3,5,6},{4}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => ? = 2 + 2
{{1,2,3,5},{4,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 1 + 2
{{1,2,3,5},{4},{6}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => ? = 1 + 2
{{1,2,3,6},{4,5}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 1 + 2
{{1,2,3},{4,5,6}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => ? = 0 + 2
{{1,2,3},{4,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 3 = 1 + 2
{{1,2,3,6},{4},{5}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => ? = 1 + 2
{{1,2,3},{4,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 3 = 1 + 2
{{1,2,3},{4},{5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 3 = 1 + 2
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => ? = 2 + 2
{{1,2,4,5,6},{3}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => ? = 2 + 2
{{1,2,4,5},{3,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 1 + 2
{{1,2,4,5},{3},{6}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => ? = 1 + 2
{{1,2,4,6},{3,5}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 1 + 2
{{1,2,4},{3,5,6}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => ? = 0 + 2
{{1,2,4},{3,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 3 = 1 + 2
{{1,2,4,6},{3},{5}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => ? = 1 + 2
{{1,2,4},{3,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 3 = 1 + 2
{{1,2,4},{3},{5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 3 = 1 + 2
{{1,2,4},{3},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => ? = 2 + 2
{{1,2,5,6},{3,4}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 1 + 2
{{1,2,5},{3,4,6}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => ? = 0 + 2
{{1,2,5},{3,4},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 3 = 1 + 2
{{1,2,6},{3,4,5}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => ? = 0 + 2
{{1,2},{3,4,5,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 1 + 2
{{1,2},{3,4,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 3 = 1 + 2
{{1,2,6},{3,4},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 3 = 1 + 2
{{1,2},{3,4,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 3 = 1 + 2
{{1,2,5,6},{3},{4}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => ? = 1 + 2
{{1,2,5},{3,6},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 3 = 1 + 2
{{1,2,5},{3},{4,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 3 = 1 + 2
{{1,2,5},{3},{4},{6}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => ? = 2 + 2
{{1,2,6},{3,5},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 3 = 1 + 2
{{1,2},{3,5,6},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 3 = 1 + 2
{{1,2,6},{3},{4,5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 3 = 1 + 2
{{1,2},{3},{4,5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 3 = 1 + 2
{{1,2,6},{3},{4},{5}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => ? = 2 + 2
{{1,2},{3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => ? = 1 + 2
{{1,3,4,5,6},{2}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => ? = 2 + 2
{{1,3,4,5},{2,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 1 + 2
{{1,3,4,5},{2},{6}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => ? = 1 + 2
{{1,3,4,6},{2,5}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 1 + 2
{{1,3,4},{2,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 3 = 1 + 2
{{1,3,4},{2,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 3 = 1 + 2
{{1,3,4},{2},{5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 3 = 1 + 2
{{1,3,5},{2,4},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 3 = 1 + 2
{{1,3},{2,4,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 3 = 1 + 2
{{1,3,6},{2,4},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 3 = 1 + 2
{{1,3},{2,4,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 3 = 1 + 2
{{1,3,5},{2,6},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 3 = 1 + 2
{{1,3,5},{2},{4,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 3 = 1 + 2
{{1,3,6},{2,5},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 3 = 1 + 2
{{1,3},{2,5,6},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 3 = 1 + 2
{{1,3,6},{2},{4,5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 3 = 1 + 2
{{1,3},{2},{4,5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 3 = 1 + 2
{{1,4,5},{2,3},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 3 = 1 + 2
{{1,4},{2,3,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 3 = 1 + 2
{{1,4,6},{2,3},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 3 = 1 + 2
{{1,4},{2,3,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 3 = 1 + 2
{{1,5},{2,3,4},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 3 = 1 + 2
{{1,6},{2,3,4},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 3 = 1 + 2
Description
The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
Matching statistic: St000744
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St000744: Standard tableaux ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 17%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St000744: Standard tableaux ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 17%
Values
{{1},{2},{3}}
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 4 = 1 + 3
{{1,2,3,4}}
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 1 + 3
{{1,2,3},{4}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 4 = 1 + 3
{{1,2,4},{3}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 4 = 1 + 3
{{1,3,4},{2}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 4 = 1 + 3
{{1},{2,3,4}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 4 = 1 + 3
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? = 1 + 3
{{1,2,3,4,5}}
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,11],[6,7,8,9,10,12]]
=> ? = 0 + 3
{{1,2,3,4},{5}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 1 + 3
{{1,2,3,5},{4}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 1 + 3
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 4 = 1 + 3
{{1,2,4,5},{3}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 1 + 3
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 4 = 1 + 3
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 4 = 1 + 3
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 1 + 3
{{1,3,4,5},{2}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 1 + 3
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 4 = 1 + 3
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 4 = 1 + 3
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 1 + 3
{{1},{2,3,4,5}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 1 + 3
{{1},{2,3,4},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 4 = 1 + 3
{{1},{2,3,5},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 4 = 1 + 3
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 1 + 3
{{1,4,5},{2},{3}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 4 = 1 + 3
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 1 + 3
{{1},{2,4,5},{3}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 4 = 1 + 3
{{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 1 + 3
{{1},{2},{3,4,5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 4 = 1 + 3
{{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 1 + 3
{{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 1 + 3
{{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 1 + 3
{{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 1 + 3
{{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 1 + 3
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,3,4,5,6,7],[2,8,9,10,11,12]]
=> ? = 2 + 3
{{1,2,3,4,5,6}}
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,6,13],[7,8,9,10,11,12,14]]
=> ? = 2 + 3
{{1,2,3,4,5},{6}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[1,2,3,4,6,11],[5,7,8,9,10,12]]
=> ? = 2 + 3
{{1,2,3,4,6},{5}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[1,2,3,4,6,11],[5,7,8,9,10,12]]
=> ? = 2 + 3
{{1,2,3,4},{5,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? = 1 + 3
{{1,2,3,4},{5},{6}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 1 + 3
{{1,2,3,5,6},{4}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[1,2,3,4,6,11],[5,7,8,9,10,12]]
=> ? = 2 + 3
{{1,2,3,5},{4,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? = 1 + 3
{{1,2,3,5},{4},{6}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 1 + 3
{{1,2,3,6},{4,5}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? = 1 + 3
{{1,2,3},{4,5,6}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[1,2,3,7,8],[4,5,6,9,10]]
=> ? = 0 + 3
{{1,2,3},{4,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 4 = 1 + 3
{{1,2,3,6},{4},{5}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 1 + 3
{{1,2,3},{4,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 4 = 1 + 3
{{1,2,3},{4},{5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 4 = 1 + 3
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? = 2 + 3
{{1,2,4,5,6},{3}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[1,2,3,4,6,11],[5,7,8,9,10,12]]
=> ? = 2 + 3
{{1,2,4,5},{3,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? = 1 + 3
{{1,2,4,5},{3},{6}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 1 + 3
{{1,2,4,6},{3,5}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? = 1 + 3
{{1,2,4},{3,5,6}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[1,2,3,7,8],[4,5,6,9,10]]
=> ? = 0 + 3
{{1,2,4},{3,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 4 = 1 + 3
{{1,2,4,6},{3},{5}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 1 + 3
{{1,2,4},{3,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 4 = 1 + 3
{{1,2,4},{3},{5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 4 = 1 + 3
{{1,2,4},{3},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? = 2 + 3
{{1,2,5,6},{3,4}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? = 1 + 3
{{1,2,5},{3,4,6}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[1,2,3,7,8],[4,5,6,9,10]]
=> ? = 0 + 3
{{1,2,5},{3,4},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 4 = 1 + 3
{{1,2,6},{3,4,5}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[1,2,3,7,8],[4,5,6,9,10]]
=> ? = 0 + 3
{{1,2},{3,4,5,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? = 1 + 3
{{1,2},{3,4,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 4 = 1 + 3
{{1,2,6},{3,4},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 4 = 1 + 3
{{1,2},{3,4,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 4 = 1 + 3
{{1,2,5,6},{3},{4}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 1 + 3
{{1,2,5},{3,6},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 4 = 1 + 3
{{1,2,5},{3},{4,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 4 = 1 + 3
{{1,2,5},{3},{4},{6}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? = 2 + 3
{{1,2,6},{3,5},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 4 = 1 + 3
{{1,2},{3,5,6},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 4 = 1 + 3
{{1,2,6},{3},{4,5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 4 = 1 + 3
{{1,2},{3},{4,5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 4 = 1 + 3
{{1,2,6},{3},{4},{5}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? = 2 + 3
{{1,2},{3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [[1,3,4,5,6,8],[2,7,9,10,11,12]]
=> ? = 1 + 3
{{1,3,4,5,6},{2}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[1,2,3,4,6,11],[5,7,8,9,10,12]]
=> ? = 2 + 3
{{1,3,4,5},{2,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? = 1 + 3
{{1,3,4,5},{2},{6}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 1 + 3
{{1,3,4,6},{2,5}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? = 1 + 3
{{1,3,4},{2,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 4 = 1 + 3
{{1,3,4},{2,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 4 = 1 + 3
{{1,3,4},{2},{5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 4 = 1 + 3
{{1,3,5},{2,4},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 4 = 1 + 3
{{1,3},{2,4,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 4 = 1 + 3
{{1,3,6},{2,4},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 4 = 1 + 3
{{1,3},{2,4,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 4 = 1 + 3
{{1,3,5},{2,6},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 4 = 1 + 3
{{1,3,5},{2},{4,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 4 = 1 + 3
{{1,3,6},{2,5},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 4 = 1 + 3
{{1,3},{2,5,6},{4}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 4 = 1 + 3
{{1,3,6},{2},{4,5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 4 = 1 + 3
{{1,3},{2},{4,5,6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 4 = 1 + 3
{{1,4,5},{2,3},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 4 = 1 + 3
{{1,4},{2,3,5},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 4 = 1 + 3
{{1,4,6},{2,3},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 4 = 1 + 3
{{1,4},{2,3,6},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 4 = 1 + 3
{{1,5},{2,3,4},{6}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 4 = 1 + 3
{{1,6},{2,3,4},{5}}
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 4 = 1 + 3
Description
The length of the path to the largest entry in a standard Young tableau.
The following 6 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St000044The number of vertices of the unicellular map given by a perfect matching. St000017The number of inversions of a standard tableau. St001721The degree of a binary word. St000016The number of attacking pairs of a standard tableau. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset.
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!