Processing math: 100%

Your data matches 113 different statistics following compositions of up to 3 maps.
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St001568: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[2]
=> 1
[1,1]
=> 2
[3]
=> 1
[2,1]
=> 1
[1,1,1]
=> 2
[4]
=> 1
[3,1]
=> 1
[2,2]
=> 1
[2,1,1]
=> 2
[1,1,1,1]
=> 2
[5]
=> 1
[4,1]
=> 1
[3,2]
=> 1
[3,1,1]
=> 2
[2,2,1]
=> 1
[2,1,1,1]
=> 2
[1,1,1,1,1]
=> 2
[6]
=> 1
[5,1]
=> 1
[4,2]
=> 1
[4,1,1]
=> 2
[3,3]
=> 1
[3,2,1]
=> 1
[3,1,1,1]
=> 2
[2,2,2]
=> 1
[2,2,1,1]
=> 3
[2,1,1,1,1]
=> 2
[1,1,1,1,1,1]
=> 2
[7]
=> 1
[6,1]
=> 1
[5,2]
=> 1
[5,1,1]
=> 2
[4,3]
=> 1
[4,2,1]
=> 1
[4,1,1,1]
=> 2
[3,3,1]
=> 1
[3,2,2]
=> 1
[3,2,1,1]
=> 2
[3,1,1,1,1]
=> 2
[2,2,2,1]
=> 1
[2,2,1,1,1]
=> 3
[2,1,1,1,1,1]
=> 2
[1,1,1,1,1,1,1]
=> 2
[8]
=> 1
[7,1]
=> 1
[6,2]
=> 1
[6,1,1]
=> 2
[5,3]
=> 1
[5,2,1]
=> 1
[5,1,1,1]
=> 2
Description
The smallest positive integer that does not appear twice in the partition.
Matching statistic: St000759
Mp00202: Integer partitions first row removalInteger partitions
Mp00044: Integer partitions conjugateInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
St000759: Integer partitions ⟶ ℤResult quality: 93% values known / values provided: 93%distinct values known / distinct values provided: 100%
Values
[2]
=> []
=> []
=> ?
=> ? = 2
[1,1]
=> [1]
=> [1]
=> []
=> 1
[3]
=> []
=> []
=> ?
=> ? = 2
[2,1]
=> [1]
=> [1]
=> []
=> 1
[1,1,1]
=> [1,1]
=> [2]
=> []
=> 1
[4]
=> []
=> []
=> ?
=> ? = 2
[3,1]
=> [1]
=> [1]
=> []
=> 1
[2,2]
=> [2]
=> [1,1]
=> [1]
=> 2
[2,1,1]
=> [1,1]
=> [2]
=> []
=> 1
[1,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> 1
[5]
=> []
=> []
=> ?
=> ? = 2
[4,1]
=> [1]
=> [1]
=> []
=> 1
[3,2]
=> [2]
=> [1,1]
=> [1]
=> 2
[3,1,1]
=> [1,1]
=> [2]
=> []
=> 1
[2,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 2
[2,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> []
=> 1
[6]
=> []
=> []
=> ?
=> ? = 3
[5,1]
=> [1]
=> [1]
=> []
=> 1
[4,2]
=> [2]
=> [1,1]
=> [1]
=> 2
[4,1,1]
=> [1,1]
=> [2]
=> []
=> 1
[3,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[3,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 2
[3,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> 1
[2,2,2]
=> [2,2]
=> [2,2]
=> [2]
=> 1
[2,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 2
[2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> []
=> 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> []
=> 1
[7]
=> []
=> []
=> ?
=> ? = 3
[6,1]
=> [1]
=> [1]
=> []
=> 1
[5,2]
=> [2]
=> [1,1]
=> [1]
=> 2
[5,1,1]
=> [1,1]
=> [2]
=> []
=> 1
[4,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[4,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 2
[4,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> 1
[3,3,1]
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 2
[3,2,2]
=> [2,2]
=> [2,2]
=> [2]
=> 1
[3,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 2
[3,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> []
=> 1
[2,2,2,1]
=> [2,2,1]
=> [3,2]
=> [2]
=> 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [4,1]
=> [1]
=> 2
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> []
=> 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [6]
=> []
=> 1
[8]
=> []
=> []
=> ?
=> ? = 3
[7,1]
=> [1]
=> [1]
=> []
=> 1
[6,2]
=> [2]
=> [1,1]
=> [1]
=> 2
[6,1,1]
=> [1,1]
=> [2]
=> []
=> 1
[5,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[5,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 2
[5,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> 1
[4,4]
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[4,3,1]
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 2
[4,2,2]
=> [2,2]
=> [2,2]
=> [2]
=> 1
[4,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 2
[4,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> []
=> 1
[3,3,2]
=> [3,2]
=> [2,2,1]
=> [2,1]
=> 3
[3,3,1,1]
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 2
[9]
=> []
=> []
=> ?
=> ? = 3
[10]
=> []
=> []
=> ?
=> ? = 3
Description
The smallest missing part in an integer partition. In [3], this is referred to as the mex, the minimal excluded part of the partition. For compositions, this is studied in [sec.3.2., 1].
Matching statistic: St001884
Mp00202: Integer partitions first row removalInteger partitions
Mp00095: Integer partitions to binary wordBinary words
Mp00136: Binary words rotate back-to-frontBinary words
St001884: Binary words ⟶ ℤResult quality: 93% values known / values provided: 93%distinct values known / distinct values provided: 100%
Values
[2]
=> []
=> => ? => ? = 2
[1,1]
=> [1]
=> 10 => 01 => 1
[3]
=> []
=> => ? => ? = 2
[2,1]
=> [1]
=> 10 => 01 => 1
[1,1,1]
=> [1,1]
=> 110 => 011 => 1
[4]
=> []
=> => ? => ? = 2
[3,1]
=> [1]
=> 10 => 01 => 1
[2,2]
=> [2]
=> 100 => 010 => 2
[2,1,1]
=> [1,1]
=> 110 => 011 => 1
[1,1,1,1]
=> [1,1,1]
=> 1110 => 0111 => 1
[5]
=> []
=> => ? => ? = 2
[4,1]
=> [1]
=> 10 => 01 => 1
[3,2]
=> [2]
=> 100 => 010 => 2
[3,1,1]
=> [1,1]
=> 110 => 011 => 1
[2,2,1]
=> [2,1]
=> 1010 => 0101 => 2
[2,1,1,1]
=> [1,1,1]
=> 1110 => 0111 => 1
[1,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 01111 => 1
[6]
=> []
=> => ? => ? = 3
[5,1]
=> [1]
=> 10 => 01 => 1
[4,2]
=> [2]
=> 100 => 010 => 2
[4,1,1]
=> [1,1]
=> 110 => 011 => 1
[3,3]
=> [3]
=> 1000 => 0100 => 2
[3,2,1]
=> [2,1]
=> 1010 => 0101 => 2
[3,1,1,1]
=> [1,1,1]
=> 1110 => 0111 => 1
[2,2,2]
=> [2,2]
=> 1100 => 0110 => 2
[2,2,1,1]
=> [2,1,1]
=> 10110 => 01011 => 1
[2,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 01111 => 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 011111 => 1
[7]
=> []
=> => ? => ? = 3
[6,1]
=> [1]
=> 10 => 01 => 1
[5,2]
=> [2]
=> 100 => 010 => 2
[5,1,1]
=> [1,1]
=> 110 => 011 => 1
[4,3]
=> [3]
=> 1000 => 0100 => 2
[4,2,1]
=> [2,1]
=> 1010 => 0101 => 2
[4,1,1,1]
=> [1,1,1]
=> 1110 => 0111 => 1
[3,3,1]
=> [3,1]
=> 10010 => 01001 => 2
[3,2,2]
=> [2,2]
=> 1100 => 0110 => 2
[3,2,1,1]
=> [2,1,1]
=> 10110 => 01011 => 1
[3,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 01111 => 1
[2,2,2,1]
=> [2,2,1]
=> 11010 => 01101 => 2
[2,2,1,1,1]
=> [2,1,1,1]
=> 101110 => 010111 => 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 011111 => 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 1111110 => 0111111 => 1
[8]
=> []
=> => ? => ? = 3
[7,1]
=> [1]
=> 10 => 01 => 1
[6,2]
=> [2]
=> 100 => 010 => 2
[6,1,1]
=> [1,1]
=> 110 => 011 => 1
[5,3]
=> [3]
=> 1000 => 0100 => 2
[5,2,1]
=> [2,1]
=> 1010 => 0101 => 2
[5,1,1,1]
=> [1,1,1]
=> 1110 => 0111 => 1
[4,4]
=> [4]
=> 10000 => 01000 => 2
[4,3,1]
=> [3,1]
=> 10010 => 01001 => 2
[4,2,2]
=> [2,2]
=> 1100 => 0110 => 2
[4,2,1,1]
=> [2,1,1]
=> 10110 => 01011 => 1
[4,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 01111 => 1
[3,3,2]
=> [3,2]
=> 10100 => 01010 => 3
[3,3,1,1]
=> [3,1,1]
=> 100110 => 010011 => 1
[9]
=> []
=> => ? => ? = 3
[10]
=> []
=> => ? => ? ∊ {3,3}
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> 1111111110 => 0111111111 => ? ∊ {3,3}
Description
The number of borders of a binary word. A border of a binary word w is a word which is both a prefix and a suffix of w.
Matching statistic: St000335
Mp00202: Integer partitions first row removalInteger partitions
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00142: Dyck paths promotionDyck paths
St000335: Dyck paths ⟶ ℤResult quality: 85% values known / values provided: 85%distinct values known / distinct values provided: 100%
Values
[2]
=> []
=> []
=> []
=> ? = 2
[1,1]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[3]
=> []
=> []
=> []
=> ? = 2
[2,1]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[1,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[4]
=> []
=> []
=> []
=> ? = 2
[3,1]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[2,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[2,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1
[5]
=> []
=> []
=> []
=> ? = 2
[4,1]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[3,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[3,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[6]
=> []
=> []
=> []
=> ? = 3
[5,1]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[4,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[4,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[3,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1
[2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,1,0,0]
=> 2
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[7]
=> []
=> []
=> []
=> ? = 3
[6,1]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[5,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[5,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[4,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1
[3,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[3,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,1,0,0]
=> 2
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[2,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 1
[8]
=> []
=> []
=> []
=> ? ∊ {1,3}
[7,1]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[6,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[6,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[5,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[5,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[5,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1
[4,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[4,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[4,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,1,0,0]
=> 2
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[3,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,3}
[9]
=> []
=> []
=> []
=> ? ∊ {1,1,2,3}
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,2,3}
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,2,3}
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,2,3}
[10]
=> []
=> []
=> []
=> ? ∊ {1,1,2,2,2,2,2,3}
[3,3,1,1,1,1]
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,2,2,2,2,2,3}
[3,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,2,2,2,2,2,3}
[3,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,2,2,2,2,2,3}
[2,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,2,2,2,2,2,3}
[2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,2,2,2,2,2,3}
[2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,2,2,2,2,2,3}
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,2,2,2,2,2,3}
Description
The difference of lower and upper interactions. An ''upper interaction'' in a Dyck path is the occurrence of a factor 0k1k with k1 (see [[St000331]]), and a ''lower interaction'' is the occurrence of a factor 1k0k with k1. In both cases, 1 denotes an up-step 0 denotes a a down-step.
Matching statistic: St001024
Mp00202: Integer partitions first row removalInteger partitions
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00143: Dyck paths inverse promotionDyck paths
St001024: Dyck paths ⟶ ℤResult quality: 83% values known / values provided: 83%distinct values known / distinct values provided: 100%
Values
[2]
=> []
=> []
=> []
=> ? = 2
[1,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[3]
=> []
=> []
=> []
=> ? = 2
[2,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,1,0,0,0]
=> 1
[4]
=> []
=> []
=> []
=> ? = 2
[3,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[2,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2
[2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,1,0,0,0]
=> 1
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[5]
=> []
=> []
=> []
=> ? = 2
[4,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[3,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2
[3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,1,0,0,0]
=> 1
[2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[6]
=> []
=> []
=> []
=> ? = 3
[5,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[4,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2
[4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,1,0,0,0]
=> 1
[3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
[7]
=> []
=> []
=> []
=> ? ∊ {1,1}
[6,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[5,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2
[5,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,1,0,0,0]
=> 1
[4,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1}
[8]
=> []
=> []
=> []
=> ? ∊ {1,1,1}
[7,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[6,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2
[6,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,1,0,0,0]
=> 1
[5,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[5,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[5,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[4,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[4,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[4,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[3,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 3
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[3,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1}
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1}
[9]
=> []
=> []
=> []
=> ? ∊ {1,1,1,1,1}
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1}
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1}
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1}
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1}
[10]
=> []
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1}
[4,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1}
[3,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1}
[3,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1}
[2,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1}
[2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1}
[2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1}
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1}
Description
Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path.
Mp00202: Integer partitions first row removalInteger partitions
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00118: Dyck paths swap returns and last descentDyck paths
St001503: Dyck paths ⟶ ℤResult quality: 83% values known / values provided: 83%distinct values known / distinct values provided: 100%
Values
[2]
=> []
=> []
=> []
=> ? = 2
[1,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[3]
=> []
=> []
=> []
=> ? = 2
[2,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[4]
=> []
=> []
=> []
=> ? = 2
[3,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[2,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[5]
=> []
=> []
=> []
=> ? = 2
[4,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[3,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2
[6]
=> []
=> []
=> []
=> ? = 2
[5,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[4,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 3
[7]
=> []
=> []
=> []
=> ? ∊ {1,2}
[6,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[5,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[5,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[4,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 3
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,2}
[8]
=> []
=> []
=> []
=> ? ∊ {2,2,2}
[7,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[6,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[6,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[5,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[5,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[5,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[4,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[4,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[4,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2
[3,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[3,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {2,2,2}
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {2,2,2}
[9]
=> []
=> []
=> []
=> ? ∊ {1,2,2,2,3}
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,2,2,2,3}
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {1,2,2,2,3}
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,2,2,2,3}
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,2,2,2,3}
[10]
=> []
=> []
=> []
=> ? ∊ {1,1,1,1,1,3,3,3}
[4,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,1,1,1,1,3,3,3}
[3,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {1,1,1,1,1,3,3,3}
[3,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,1,1,1,1,3,3,3}
[2,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> ? ∊ {1,1,1,1,1,3,3,3}
[2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {1,1,1,1,1,3,3,3}
[2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,1,1,1,1,3,3,3}
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,1,1,1,1,3,3,3}
Description
The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra.
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00132: Dyck paths switch returns and last double riseDyck paths
Mp00023: Dyck paths to non-crossing permutationPermutations
St001085: Permutations ⟶ ℤResult quality: 81% values known / values provided: 81%distinct values known / distinct values provided: 100%
Values
[2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => 1 = 2 - 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => 0 = 1 - 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1 = 2 - 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0 = 1 - 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 0 = 1 - 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 1 = 2 - 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 0 = 1 - 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1 = 2 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0 = 1 - 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => 0 = 1 - 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,4,3,2,1,6] => 1 = 2 - 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 0 = 1 - 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1 = 2 - 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 1 = 2 - 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 0 = 1 - 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => 0 = 1 - 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [6,5,3,4,2,1] => 0 = 1 - 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => 1 = 2 - 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,5,4,3,2,6] => 0 = 1 - 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 1 = 2 - 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => 1 = 2 - 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => 1 = 2 - 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0 = 1 - 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 0 = 1 - 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => 0 = 1 - 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => 0 = 1 - 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,6,4,5,3,2] => 0 = 1 - 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [7,6,4,5,3,2,1] => ? = 3 - 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7,6,5,4,3,2,1,8] => 1 = 2 - 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,6,5,4,3,2,7] => ? ∊ {1,3} - 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,5,4,3,6] => 1 = 2 - 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,2,4,3,1,6] => 1 = 2 - 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1 = 2 - 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 0 = 1 - 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => 1 = 2 - 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => 0 = 1 - 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 1 = 2 - 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0 = 1 - 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,6,4,3,5,2] => 0 = 1 - 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [5,2,3,4,1] => 0 = 1 - 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,6,4,5,3,1] => 0 = 1 - 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,7,6,4,5,3,2] => 0 = 1 - 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [8,7,6,4,5,3,2,1] => ? ∊ {1,3} - 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8,7,6,5,4,3,2,1,9] => 1 = 2 - 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,7,6,5,4,3,2,8] => ? ∊ {1,2,2,3} - 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,6,5,4,3,7] => ? ∊ {1,2,2,3} - 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [6,3,5,4,2,1,7] => 1 = 2 - 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [3,2,1,5,4,6] => 1 = 2 - 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,5,4,3,6] => 0 = 1 - 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,5,4,3,1,6] => 1 = 2 - 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,4,3,2,6,1] => 1 = 2 - 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 0 = 1 - 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => 2 = 3 - 1
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 0 = 1 - 1
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,5,4,3,6,2] => 0 = 1 - 1
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [2,7,6,4,5,3,1] => ? ∊ {1,2,2,3} - 1
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [9,8,7,5,6,4,3,2,1] => ? ∊ {1,2,2,3} - 1
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,8,7,6,5,4,3,2,9] => ? ∊ {1,2,2,2,3,3} - 1
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,1,7,6,5,4,3,8] => ? ∊ {1,2,2,2,3,3} - 1
[6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [3,2,1,6,5,4,7] => ? ∊ {1,2,2,2,3,3} - 1
[2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [7,2,6,4,5,3,1] => ? ∊ {1,2,2,2,3,3} - 1
[2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [2,8,7,5,6,4,3,1] => ? ∊ {1,2,2,2,3,3} - 1
[1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [10,9,8,7,5,6,4,3,2,1] => ? ∊ {1,2,2,2,3,3} - 1
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,9,8,7,6,5,4,3,2,10] => ? ∊ {1,1,1,1,2,2,2,2,3,3,3,3,3} - 1
[8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,1,8,7,6,5,4,3,9] => ? ∊ {1,1,1,1,2,2,2,2,3,3,3,3,3} - 1
[7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [3,2,1,7,6,5,4,8] => ? ∊ {1,1,1,1,2,2,2,2,3,3,3,3,3} - 1
[7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,2,7,6,5,4,3,8] => ? ∊ {1,1,1,1,2,2,2,2,3,3,3,3,3} - 1
[6,4]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? ∊ {1,1,1,1,2,2,2,2,3,3,3,3,3} - 1
[5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [6,5,4,3,2,7,1] => ? ∊ {1,1,1,1,2,2,2,2,3,3,3,3,3} - 1
[3,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [2,7,5,4,6,3,1] => ? ∊ {1,1,1,1,2,2,2,2,3,3,3,3,3} - 1
[3,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,2,8,7,5,6,4,3] => ? ∊ {1,1,1,1,2,2,2,2,3,3,3,3,3} - 1
[2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [7,6,4,3,5,2,1] => ? ∊ {1,1,1,1,2,2,2,2,3,3,3,3,3} - 1
[2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [7,3,6,4,5,2,1] => ? ∊ {1,1,1,1,2,2,2,2,3,3,3,3,3} - 1
[2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0]
=> [8,2,7,5,6,4,3,1] => ? ∊ {1,1,1,1,2,2,2,2,3,3,3,3,3} - 1
[2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [2,9,8,7,5,6,4,3,1] => ? ∊ {1,1,1,1,2,2,2,2,3,3,3,3,3} - 1
[1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
=> [11,10,9,8,6,7,5,4,3,2,1] => ? ∊ {1,1,1,1,2,2,2,2,3,3,3,3,3} - 1
Description
The number of occurrences of the vincular pattern |21-3 in a permutation. This is the number of occurrences of the pattern 213, where the first matched entry is the first entry of the permutation and the other two matched entries are consecutive. In other words, this is the number of ascents whose bottom value is strictly smaller and the top value is strictly larger than the first entry of the permutation.
Matching statistic: St001220
Mp00202: Integer partitions first row removalInteger partitions
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00129: Dyck paths to 321-avoiding permutation (Billey-Jockusch-Stanley)Permutations
St001220: Permutations ⟶ ℤResult quality: 79% values known / values provided: 79%distinct values known / distinct values provided: 100%
Values
[2]
=> []
=> []
=> [] => ? ∊ {1,2}
[1,1]
=> [1]
=> [1,0]
=> [1] => ? ∊ {1,2}
[3]
=> []
=> []
=> [] => ? ∊ {1,2}
[2,1]
=> [1]
=> [1,0]
=> [1] => ? ∊ {1,2}
[1,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 1
[4]
=> []
=> []
=> [] => ? ∊ {2,2}
[3,1]
=> [1]
=> [1,0]
=> [1] => ? ∊ {2,2}
[2,2]
=> [2]
=> [1,0,1,0]
=> [2,1] => 1
[2,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 1
[1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [3,1,2] => 1
[5]
=> []
=> []
=> [] => ? ∊ {2,2}
[4,1]
=> [1]
=> [1,0]
=> [1] => ? ∊ {2,2}
[3,2]
=> [2]
=> [1,0,1,0]
=> [2,1] => 1
[3,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 1
[2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 1
[2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [3,1,2] => 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 2
[6]
=> []
=> []
=> [] => ? ∊ {2,3}
[5,1]
=> [1]
=> [1,0]
=> [1] => ? ∊ {2,3}
[4,2]
=> [2]
=> [1,0,1,0]
=> [2,1] => 1
[4,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 1
[3,3]
=> [3]
=> [1,0,1,0,1,0]
=> [2,3,1] => 1
[3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 1
[3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [3,1,2] => 1
[2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 1
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => 2
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 2
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => 2
[7]
=> []
=> []
=> [] => ? ∊ {2,2}
[6,1]
=> [1]
=> [1,0]
=> [1] => ? ∊ {2,2}
[5,2]
=> [2]
=> [1,0,1,0]
=> [2,1] => 1
[5,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 1
[4,3]
=> [3]
=> [1,0,1,0,1,0]
=> [2,3,1] => 1
[4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 1
[4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [3,1,2] => 1
[3,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 1
[3,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 1
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => 2
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 2
[2,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => 2
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => 2
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,5,6,1,2] => 3
[8]
=> []
=> []
=> [] => ? ∊ {2,2,2}
[7,1]
=> [1]
=> [1,0]
=> [1] => ? ∊ {2,2,2}
[6,2]
=> [2]
=> [1,0,1,0]
=> [2,1] => 1
[6,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 1
[5,3]
=> [3]
=> [1,0,1,0,1,0]
=> [2,3,1] => 1
[5,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 1
[5,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [3,1,2] => 1
[4,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 1
[4,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 1
[4,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 1
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => 2
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 2
[3,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 1
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => 2
[3,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => 2
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => 2
[2,2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 1
[2,2,2,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => 2
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,4,5,6,1,3] => 3
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,5,6,1,2] => 3
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,5,6,7,1,2] => ? ∊ {2,2,2}
[9]
=> []
=> []
=> [] => ? ∊ {1,2,2,2,2}
[8,1]
=> [1]
=> [1,0]
=> [1] => ? ∊ {1,2,2,2,2}
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,4,5,6,7,1,3] => ? ∊ {1,2,2,2,2}
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,5,6,7,1,2] => ? ∊ {1,2,2,2,2}
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,5,6,7,8,1,2] => ? ∊ {1,2,2,2,2}
[10]
=> []
=> []
=> [] => ? ∊ {1,1,2,2,2,2,2,2,3}
[9,1]
=> [1]
=> [1,0]
=> [1] => ? ∊ {1,1,2,2,2,2,2,2,3}
[3,3,1,1,1,1]
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,3,5,6,7,1,4] => ? ∊ {1,1,2,2,2,2,2,2,3}
[3,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,4,5,6,7,1,3] => ? ∊ {1,1,2,2,2,2,2,2,3}
[3,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,5,6,7,1,2] => ? ∊ {1,1,2,2,2,2,2,2,3}
[2,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,4,5,6,7,2,3] => ? ∊ {1,1,2,2,2,2,2,2,3}
[2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,4,5,6,7,8,1,3] => ? ∊ {1,1,2,2,2,2,2,2,3}
[2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,5,6,7,8,1,2] => ? ∊ {1,1,2,2,2,2,2,2,3}
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,5,6,7,8,9,1,2] => ? ∊ {1,1,2,2,2,2,2,2,3}
Description
The width of a permutation. Let w be a permutation. The interval [e,w] in the weak order is ranked, and we define ri=ri(w) to be the number of elements at rank i in [e,w], where i{0,,(w)}. The ''width'' of w is the maximum of {r0,r1,,r(w)}. See [1].
Matching statistic: St001465
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00101: Dyck paths decomposition reverseDyck paths
Mp00024: Dyck paths to 321-avoiding permutationPermutations
St001465: Permutations ⟶ ℤResult quality: 77% values known / values provided: 77%distinct values known / distinct values provided: 100%
Values
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,2] => 0 = 1 - 1
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [2,1] => 1 = 2 - 1
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0 = 1 - 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,3,2] => 1 = 2 - 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => 0 = 1 - 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0 = 1 - 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => 1 = 2 - 1
[2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [2,1,3] => 1 = 2 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,3,4,2] => 0 = 1 - 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 0 = 1 - 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0 = 1 - 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,2,3,5,4] => 1 = 2 - 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => 1 = 2 - 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,2,4,5,3] => 0 = 1 - 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => 1 = 2 - 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => 0 = 1 - 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0 = 1 - 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => 0 = 1 - 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,2,3,4,6,5] => 1 = 2 - 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,2,4,3,5] => 1 = 2 - 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,2,3,5,6,4] => 0 = 1 - 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => 0 = 1 - 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => 1 = 2 - 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,2,4,5,6,3] => 0 = 1 - 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => 2 = 3 - 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => 1 = 2 - 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,3,4,5,6,2] => 0 = 1 - 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 0 = 1 - 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7] => 0 = 1 - 1
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,2,3,4,5,7,6] => 1 = 2 - 1
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,2,3,5,4,6] => 1 = 2 - 1
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,2,3,4,6,7,5] => 0 = 1 - 1
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => 0 = 1 - 1
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,2,4,3,5,6] => 1 = 2 - 1
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,2,3,5,6,7,4] => 0 = 1 - 1
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => 0 = 1 - 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => 2 = 3 - 1
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,3,2,4,5,6] => 1 = 2 - 1
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,2,4,5,6,7,3] => 0 = 1 - 1
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => 1 = 2 - 1
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,3,4,5,6] => 1 = 2 - 1
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,3,4,5,6,7,2] => 0 = 1 - 1
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 0 = 1 - 1
[8]
=> [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]
=> [1,2,3,4,5,6,7,8] => 0 = 1 - 1
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,8,7] => 1 = 2 - 1
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,2,3,4,6,5,7] => 1 = 2 - 1
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,2,3,4,5,7,8,6] => ? ∊ {1,1,2,3} - 1
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,2,4,6,3,5] => 0 = 1 - 1
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [1,2,3,5,4,6,7] => 1 = 2 - 1
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,2,3,4,6,7,8,5] => ? ∊ {1,1,2,3} - 1
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => 0 = 1 - 1
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,3,6,2,4,5] => 0 = 1 - 1
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,2,4,5,6,7,8,3] => ? ∊ {1,1,2,3} - 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,3,4,5,6,7,8,2] => ? ∊ {1,1,2,3} - 1
[7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,8,9,7] => ? ∊ {1,1,1,2,2,2,2,3} - 1
[6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
=> [1,2,3,4,6,5,7,8] => ? ∊ {1,1,1,2,2,2,2,3} - 1
[6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
=> [1,2,3,4,5,7,8,9,6] => ? ∊ {1,1,1,2,2,2,2,3} - 1
[5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,2,3,4,6,7,8,9,5] => ? ∊ {1,1,1,2,2,2,2,3} - 1
[4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,2,4,3,5,6,7,8] => ? ∊ {1,1,1,2,2,2,2,3} - 1
[4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,2,3,5,6,7,8,9,4] => ? ∊ {1,1,1,2,2,2,2,3} - 1
[3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,2,4,5,6,7,8,9,3] => ? ∊ {1,1,1,2,2,2,2,3} - 1
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,3,4,5,6,7,8,9,2] => ? ∊ {1,1,1,2,2,2,2,3} - 1
[8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,8,7,9] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3} - 1
[8,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7,9,10,8] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3} - 1
[7,3]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
=> [1,2,3,4,6,8,5,7] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3} - 1
[7,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0,0]
=> [1,2,3,4,5,7,6,8,9] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3} - 1
[7,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,8,9,10,7] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3} - 1
[6,3,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,1,1,0,0,0,0,0]
=> [1,2,3,5,8,4,6,7] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3} - 1
[6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,2,3,4,6,5,8,7] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3} - 1
[6,2,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,2,3,4,6,5,7,8,9] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3} - 1
[6,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,2,3,4,5,7,8,9,10,6] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3} - 1
[5,3,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,1,0,0,0,0]
=> [1,2,4,8,3,5,6,7] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3} - 1
[5,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [1,2,3,5,4,6,7,8,9] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3} - 1
[5,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,2,3,4,6,7,8,9,10,5] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3} - 1
[4,3,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,3,8,2,4,5,6,7] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3} - 1
[4,2,2,1,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,1,0,0,0]
=> [1,2,4,3,8,5,6,7] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3} - 1
[4,2,1,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [1,2,4,3,5,6,7,8,9] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3} - 1
[4,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,2,3,5,6,7,8,9,10,4] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3} - 1
[3,2,1,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,3,2,4,5,6,7,8,9] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3} - 1
[3,1,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,2,4,5,6,7,8,9,10,3] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3} - 1
[2,2,2,2,1,1]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,6,1,3,7,4,5] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3} - 1
[2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,3,4,5,6,7,8,9,10,2] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3} - 1
Description
The number of adjacent transpositions in the cycle decomposition of a permutation.
Mp00202: Integer partitions first row removalInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
Mp00044: Integer partitions conjugateInteger partitions
St000755: Integer partitions ⟶ ℤResult quality: 67% values known / values provided: 75%distinct values known / distinct values provided: 67%
Values
[2]
=> []
=> ?
=> ?
=> ? ∊ {1,2}
[1,1]
=> [1]
=> []
=> []
=> ? ∊ {1,2}
[3]
=> []
=> ?
=> ?
=> ? ∊ {1,2}
[2,1]
=> [1]
=> []
=> []
=> ? ∊ {1,2}
[1,1,1]
=> [1,1]
=> [1]
=> [1]
=> 1
[4]
=> []
=> ?
=> ?
=> ? ∊ {1,1,2}
[3,1]
=> [1]
=> []
=> []
=> ? ∊ {1,1,2}
[2,2]
=> [2]
=> []
=> []
=> ? ∊ {1,1,2}
[2,1,1]
=> [1,1]
=> [1]
=> [1]
=> 1
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
[5]
=> []
=> ?
=> ?
=> ? ∊ {1,2,2}
[4,1]
=> [1]
=> []
=> []
=> ? ∊ {1,2,2}
[3,2]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2}
[3,1,1]
=> [1,1]
=> [1]
=> [1]
=> 1
[2,2,1]
=> [2,1]
=> [1]
=> [1]
=> 1
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[6]
=> []
=> ?
=> ?
=> ? ∊ {1,1,2,3}
[5,1]
=> [1]
=> []
=> []
=> ? ∊ {1,1,2,3}
[4,2]
=> [2]
=> []
=> []
=> ? ∊ {1,1,2,3}
[4,1,1]
=> [1,1]
=> [1]
=> [1]
=> 1
[3,3]
=> [3]
=> []
=> []
=> ? ∊ {1,1,2,3}
[3,2,1]
=> [2,1]
=> [1]
=> [1]
=> 1
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
[2,2,2]
=> [2,2]
=> [2]
=> [1,1]
=> 1
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [2]
=> 2
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 2
[7]
=> []
=> ?
=> ?
=> ? ∊ {1,2,2,3}
[6,1]
=> [1]
=> []
=> []
=> ? ∊ {1,2,2,3}
[5,2]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2,3}
[5,1,1]
=> [1,1]
=> [1]
=> [1]
=> 1
[4,3]
=> [3]
=> []
=> []
=> ? ∊ {1,2,2,3}
[4,2,1]
=> [2,1]
=> [1]
=> [1]
=> 1
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
[3,3,1]
=> [3,1]
=> [1]
=> [1]
=> 1
[3,2,2]
=> [2,2]
=> [2]
=> [1,1]
=> 1
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [2]
=> 2
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [2,1]
=> 2
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 2
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> 1
[8]
=> []
=> ?
=> ?
=> ? ∊ {1,1,2,3,3}
[7,1]
=> [1]
=> []
=> []
=> ? ∊ {1,1,2,3,3}
[6,2]
=> [2]
=> []
=> []
=> ? ∊ {1,1,2,3,3}
[6,1,1]
=> [1,1]
=> [1]
=> [1]
=> 1
[5,3]
=> [3]
=> []
=> []
=> ? ∊ {1,1,2,3,3}
[5,2,1]
=> [2,1]
=> [1]
=> [1]
=> 1
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
[4,4]
=> [4]
=> []
=> []
=> ? ∊ {1,1,2,3,3}
[4,3,1]
=> [3,1]
=> [1]
=> [1]
=> 1
[4,2,2]
=> [2,2]
=> [2]
=> [1,1]
=> 1
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [2]
=> 2
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[3,3,2]
=> [3,2]
=> [2]
=> [1,1]
=> 1
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> [2,1]
=> 2
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 2
[2,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2,2]
=> 2
[2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [3,1]
=> 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 2
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> 1
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [6]
=> 2
[9]
=> []
=> ?
=> ?
=> ? ∊ {2,2,3,3,3}
[8,1]
=> [1]
=> []
=> []
=> ? ∊ {2,2,3,3,3}
[7,2]
=> [2]
=> []
=> []
=> ? ∊ {2,2,3,3,3}
[7,1,1]
=> [1,1]
=> [1]
=> [1]
=> 1
[6,3]
=> [3]
=> []
=> []
=> ? ∊ {2,2,3,3,3}
[6,2,1]
=> [2,1]
=> [1]
=> [1]
=> 1
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
[5,4]
=> [4]
=> []
=> []
=> ? ∊ {2,2,3,3,3}
[5,3,1]
=> [3,1]
=> [1]
=> [1]
=> 1
[5,2,2]
=> [2,2]
=> [2]
=> [1,1]
=> 1
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> [2]
=> 2
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[4,4,1]
=> [4,1]
=> [1]
=> [1]
=> 1
[10]
=> []
=> ?
=> ?
=> ? ∊ {1,3,3,3,3,3}
[9,1]
=> [1]
=> []
=> []
=> ? ∊ {1,3,3,3,3,3}
[8,2]
=> [2]
=> []
=> []
=> ? ∊ {1,3,3,3,3,3}
[7,3]
=> [3]
=> []
=> []
=> ? ∊ {1,3,3,3,3,3}
[6,4]
=> [4]
=> []
=> []
=> ? ∊ {1,3,3,3,3,3}
[5,5]
=> [5]
=> []
=> []
=> ? ∊ {1,3,3,3,3,3}
Description
The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. Consider the recurrence f(n)=pλf(np). This statistic returns the number of distinct real roots of the associated characteristic polynomial. For example, the partition (2,1) corresponds to the recurrence f(n)=f(n1)+f(n2) with associated characteristic polynomial x2x1, which has two real roots.
The following 103 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001432The order dimension of the partition. St000678The number of up steps after the last double rise of a Dyck path. St001955The number of natural descents for set-valued two row standard Young tableaux. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001597The Frobenius rank of a skew partition. St001199The dominant dimension of eAe for the corresponding Nakayama algebra A with minimal faithful projective-injective module eA. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series L=[c0,c1,...,cn1] such that n=c0<ci for all i>0 a special CNakayama algebra. St001061The number of indices that are both descents and recoils of a permutation. St000214The number of adjacencies of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000990The first ascent of a permutation. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000668The least common multiple of the parts of the partition. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000654The first descent of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St001201The grade of the simple module S0 in the special CNakayama algebra corresponding to the Dyck path. St001732The number of peaks visible from the left. St000542The number of left-to-right-minima of a permutation. St000663The number of right floats of a permutation. St000779The tier of a permutation. St000989The number of final rises of a permutation. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001728The number of invisible descents of a permutation. St000756The sum of the positions of the left to right maxima of a permutation. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000665The number of rafts of a permutation. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000764The number of strong records in an integer composition. St000761The number of ascents in an integer composition. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001569The maximal modular displacement of a permutation. St001128The exponens consonantiae of a partition. St001487The number of inner corners of a skew partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000708The product of the parts of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St000993The multiplicity of the largest part of an integer partition. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000260The radius of a connected graph. St001549The number of restricted non-inversions between exceedances. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000237The number of small exceedances. St000352The Elizalde-Pak rank of a permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001722The number of minimal chains with small intervals between a binary word and the top element. St001737The number of descents of type 2 in a permutation. St000007The number of saliances of the permutation. St000754The Grundy value for the game of removing nestings in a perfect matching. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St000245The number of ascents of a permutation. St000834The number of right outer peaks of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000646The number of big ascents of a permutation. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001192The maximal dimension of Ext2A(S,A) for a simple module S over the corresponding Nakayama algebra A. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001481The minimal height of a peak of a Dyck path. St001665The number of pure excedances of a permutation. St000117The number of centered tunnels of a Dyck path. St000241The number of cyclical small excedances. St000664The number of right ropes of a permutation. St000740The last entry of a permutation. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001557The number of inversions of the second entry of a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. St001948The number of augmented double ascents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000741The Colin de Verdière graph invariant. St000259The diameter of a connected graph. St000461The rix statistic of a permutation. St001195The global dimension of the algebra A/AfA of the corresponding Nakayama algebra A with minimal left faithful projective-injective module Af. St001435The number of missing boxes in the first row. St000782The indicator function of whether a given perfect matching is an L & P matching. St000768The number of peaks in an integer composition. St001198The number of simple modules in the algebra eAe with projective dimension at most 1 in the corresponding Nakayama algebra A with minimal faithful projective-injective module eA. St001206The maximal dimension of an indecomposable projective eAe-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module eA. St000326The position of the first one in a binary word after appending a 1 at the end. St000383The last part of an integer composition. St000657The smallest part of an integer composition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.