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Your data matches 126 different statistics following compositions of up to 3 maps.
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Matching statistic: St001588
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
St001588: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 0
[2]
=> 0
[1,1]
=> 0
[3]
=> 0
[2,1]
=> 1
[1,1,1]
=> 0
[4]
=> 0
[3,1]
=> 0
[2,2]
=> 0
[2,1,1]
=> 1
[1,1,1,1]
=> 0
[5]
=> 0
[4,1]
=> 1
[3,2]
=> 0
[3,1,1]
=> 0
[2,2,1]
=> 1
[2,1,1,1]
=> 1
[1,1,1,1,1]
=> 0
[6]
=> 0
[5,1]
=> 0
[4,2]
=> 0
[4,1,1]
=> 1
[3,3]
=> 0
[3,2,1]
=> 1
[3,1,1,1]
=> 0
[2,2,2]
=> 0
[2,2,1,1]
=> 1
[2,1,1,1,1]
=> 1
[1,1,1,1,1,1]
=> 0
[7]
=> 0
[6,1]
=> 1
[5,2]
=> 0
[5,1,1]
=> 0
[4,3]
=> 1
[4,2,1]
=> 1
[4,1,1,1]
=> 1
[3,3,1]
=> 0
[3,2,2]
=> 0
[3,2,1,1]
=> 1
[3,1,1,1,1]
=> 0
[2,2,2,1]
=> 1
[2,2,1,1,1]
=> 1
[2,1,1,1,1,1]
=> 1
[1,1,1,1,1,1,1]
=> 0
[8]
=> 0
[7,1]
=> 0
[6,2]
=> 0
[6,1,1]
=> 1
[5,3]
=> 0
[5,2,1]
=> 1
Description
The number of distinct odd parts smaller than the largest even part in an integer partition.
Matching statistic: St001092
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001092: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001092: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1]
=> []
=> []
=> 0
[2]
=> []
=> []
=> 0
[1,1]
=> [1]
=> [1]
=> 0
[3]
=> []
=> []
=> 0
[2,1]
=> [1]
=> [1]
=> 0
[1,1,1]
=> [1,1]
=> [2]
=> 1
[4]
=> []
=> []
=> 0
[3,1]
=> [1]
=> [1]
=> 0
[2,2]
=> [2]
=> [1,1]
=> 0
[2,1,1]
=> [1,1]
=> [2]
=> 1
[1,1,1,1]
=> [1,1,1]
=> [3]
=> 0
[5]
=> []
=> []
=> 0
[4,1]
=> [1]
=> [1]
=> 0
[3,2]
=> [2]
=> [1,1]
=> 0
[3,1,1]
=> [1,1]
=> [2]
=> 1
[2,2,1]
=> [2,1]
=> [2,1]
=> 1
[2,1,1,1]
=> [1,1,1]
=> [3]
=> 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 1
[6]
=> []
=> []
=> 0
[5,1]
=> [1]
=> [1]
=> 0
[4,2]
=> [2]
=> [1,1]
=> 0
[4,1,1]
=> [1,1]
=> [2]
=> 1
[3,3]
=> [3]
=> [1,1,1]
=> 0
[3,2,1]
=> [2,1]
=> [2,1]
=> 1
[3,1,1,1]
=> [1,1,1]
=> [3]
=> 0
[2,2,2]
=> [2,2]
=> [2,2]
=> 1
[2,2,1,1]
=> [2,1,1]
=> [3,1]
=> 0
[2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> 0
[7]
=> []
=> []
=> 0
[6,1]
=> [1]
=> [1]
=> 0
[5,2]
=> [2]
=> [1,1]
=> 0
[5,1,1]
=> [1,1]
=> [2]
=> 1
[4,3]
=> [3]
=> [1,1,1]
=> 0
[4,2,1]
=> [2,1]
=> [2,1]
=> 1
[4,1,1,1]
=> [1,1,1]
=> [3]
=> 0
[3,3,1]
=> [3,1]
=> [2,1,1]
=> 1
[3,2,2]
=> [2,2]
=> [2,2]
=> 1
[3,2,1,1]
=> [2,1,1]
=> [3,1]
=> 0
[3,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 1
[2,2,2,1]
=> [2,2,1]
=> [3,2]
=> 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [4,1]
=> 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [6]
=> 1
[8]
=> []
=> []
=> 0
[7,1]
=> [1]
=> [1]
=> 0
[6,2]
=> [2]
=> [1,1]
=> 0
[6,1,1]
=> [1,1]
=> [2]
=> 1
[5,3]
=> [3]
=> [1,1,1]
=> 0
[5,2,1]
=> [2,1]
=> [2,1]
=> 1
[2,2,2,2,2,1,1,1,1,1]
=> [2,2,2,2,1,1,1,1,1]
=> [9,4]
=> ? ∊ {1,1}
[2,2,2,1,1,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1,1]
=> [11,2]
=> ? ∊ {1,1}
[3,3,2,2,1,1,1,1,1,1]
=> [3,2,2,1,1,1,1,1,1]
=> [9,3,1]
=> ? ∊ {0,0,0,1,1,2,2}
[3,3,2,1,1,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1,1,1]
=> [10,2,1]
=> ? ∊ {0,0,0,1,1,2,2}
[3,3,1,1,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1,1,1]
=> [11,1,1]
=> ? ∊ {0,0,0,1,1,2,2}
[3,2,2,2,2,1,1,1,1,1]
=> [2,2,2,2,1,1,1,1,1]
=> [9,4]
=> ? ∊ {0,0,0,1,1,2,2}
[3,2,2,1,1,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1,1]
=> [11,2]
=> ? ∊ {0,0,0,1,1,2,2}
[2,2,2,2,2,1,1,1,1,1,1]
=> [2,2,2,2,1,1,1,1,1,1]
=> [10,4]
=> ? ∊ {0,0,0,1,1,2,2}
[2,2,2,2,1,1,1,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1,1,1,1]
=> [11,3]
=> ? ∊ {0,0,0,1,1,2,2}
Description
The number of distinct even parts of a partition.
See Section 3.3.1 of [1].
Matching statistic: St000291
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
Mp00096: Binary words —Foata bijection⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Mp00105: Binary words —complement⟶ Binary words
Mp00096: Binary words —Foata bijection⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Values
[1]
=> 1 => 0 => 0 => 0
[2]
=> 0 => 1 => 1 => 0
[1,1]
=> 11 => 00 => 00 => 0
[3]
=> 1 => 0 => 0 => 0
[2,1]
=> 01 => 10 => 10 => 1
[1,1,1]
=> 111 => 000 => 000 => 0
[4]
=> 0 => 1 => 1 => 0
[3,1]
=> 11 => 00 => 00 => 0
[2,2]
=> 00 => 11 => 11 => 0
[2,1,1]
=> 011 => 100 => 010 => 1
[1,1,1,1]
=> 1111 => 0000 => 0000 => 0
[5]
=> 1 => 0 => 0 => 0
[4,1]
=> 01 => 10 => 10 => 1
[3,2]
=> 10 => 01 => 01 => 0
[3,1,1]
=> 111 => 000 => 000 => 0
[2,2,1]
=> 001 => 110 => 110 => 1
[2,1,1,1]
=> 0111 => 1000 => 0010 => 1
[1,1,1,1,1]
=> 11111 => 00000 => 00000 => 0
[6]
=> 0 => 1 => 1 => 0
[5,1]
=> 11 => 00 => 00 => 0
[4,2]
=> 00 => 11 => 11 => 0
[4,1,1]
=> 011 => 100 => 010 => 1
[3,3]
=> 11 => 00 => 00 => 0
[3,2,1]
=> 101 => 010 => 100 => 1
[3,1,1,1]
=> 1111 => 0000 => 0000 => 0
[2,2,2]
=> 000 => 111 => 111 => 0
[2,2,1,1]
=> 0011 => 1100 => 0110 => 1
[2,1,1,1,1]
=> 01111 => 10000 => 00010 => 1
[1,1,1,1,1,1]
=> 111111 => 000000 => 000000 => 0
[7]
=> 1 => 0 => 0 => 0
[6,1]
=> 01 => 10 => 10 => 1
[5,2]
=> 10 => 01 => 01 => 0
[5,1,1]
=> 111 => 000 => 000 => 0
[4,3]
=> 01 => 10 => 10 => 1
[4,2,1]
=> 001 => 110 => 110 => 1
[4,1,1,1]
=> 0111 => 1000 => 0010 => 1
[3,3,1]
=> 111 => 000 => 000 => 0
[3,2,2]
=> 100 => 011 => 011 => 0
[3,2,1,1]
=> 1011 => 0100 => 0100 => 1
[3,1,1,1,1]
=> 11111 => 00000 => 00000 => 0
[2,2,2,1]
=> 0001 => 1110 => 1110 => 1
[2,2,1,1,1]
=> 00111 => 11000 => 00110 => 1
[2,1,1,1,1,1]
=> 011111 => 100000 => 000010 => 1
[1,1,1,1,1,1,1]
=> 1111111 => 0000000 => 0000000 => 0
[8]
=> 0 => 1 => 1 => 0
[7,1]
=> 11 => 00 => 00 => 0
[6,2]
=> 00 => 11 => 11 => 0
[6,1,1]
=> 011 => 100 => 010 => 1
[5,3]
=> 11 => 00 => 00 => 0
[5,2,1]
=> 101 => 010 => 100 => 1
[1,1,1,1,1,1,1,1,1,1,1]
=> 11111111111 => 00000000000 => 00000000000 => ? = 0
[1,1,1,1,1,1,1,1,1,1,1,1]
=> 111111111111 => 000000000000 => 000000000000 => ? = 0
[3,1,1,1,1,1,1,1,1,1,1]
=> 11111111111 => 00000000000 => 00000000000 => ? ∊ {0,0,1,1,1}
[2,2,2,1,1,1,1,1,1,1]
=> 0001111111 => 1110000000 => ? => ? ∊ {0,0,1,1,1}
[2,2,1,1,1,1,1,1,1,1,1]
=> 00111111111 => 11000000000 => 00000000110 => ? ∊ {0,0,1,1,1}
[2,1,1,1,1,1,1,1,1,1,1,1]
=> 011111111111 => 100000000000 => ? => ? ∊ {0,0,1,1,1}
[1,1,1,1,1,1,1,1,1,1,1,1,1]
=> 1111111111111 => ? => ? => ? ∊ {0,0,1,1,1}
[3,2,1,1,1,1,1,1,1,1,1]
=> 10111111111 => 01000000000 => 00000000100 => ? ∊ {0,0,1,1,1,1}
[3,1,1,1,1,1,1,1,1,1,1,1]
=> 111111111111 => 000000000000 => 000000000000 => ? ∊ {0,0,1,1,1,1}
[2,2,2,1,1,1,1,1,1,1,1]
=> 00011111111 => 11100000000 => ? => ? ∊ {0,0,1,1,1,1}
[2,2,1,1,1,1,1,1,1,1,1,1]
=> 001111111111 => 110000000000 => 000000000110 => ? ∊ {0,0,1,1,1,1}
[2,1,1,1,1,1,1,1,1,1,1,1,1]
=> 0111111111111 => ? => ? => ? ∊ {0,0,1,1,1,1}
[1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> 11111111111111 => ? => ? => ? ∊ {0,0,1,1,1,1}
[5,1,1,1,1,1,1,1,1,1,1]
=> 11111111111 => 00000000000 => 00000000000 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2}
[4,2,2,1,1,1,1,1,1,1]
=> 0001111111 => 1110000000 => ? => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2}
[4,2,1,1,1,1,1,1,1,1,1]
=> 00111111111 => 11000000000 => 00000000110 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2}
[4,1,1,1,1,1,1,1,1,1,1,1]
=> 011111111111 => 100000000000 => ? => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2}
[3,3,1,1,1,1,1,1,1,1,1]
=> 11111111111 => 00000000000 => 00000000000 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2}
[3,2,2,1,1,1,1,1,1,1,1]
=> 10011111111 => 01100000000 => 00000001010 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2}
[3,2,1,1,1,1,1,1,1,1,1,1]
=> 101111111111 => 010000000000 => ? => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2}
[3,1,1,1,1,1,1,1,1,1,1,1,1]
=> 1111111111111 => ? => ? => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2}
[2,2,2,2,1,1,1,1,1,1,1]
=> 00001111111 => 11110000000 => 00000011110 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2}
[2,2,2,1,1,1,1,1,1,1,1,1]
=> 000111111111 => ? => ? => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2}
[2,2,1,1,1,1,1,1,1,1,1,1,1]
=> 0011111111111 => ? => ? => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2}
[2,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> 01111111111111 => ? => ? => ? ∊ {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]
=> 111111111111111 => ? => ? => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2}
[5,2,1,1,1,1,1,1,1,1,1]
=> 10111111111 => 01000000000 => 00000000100 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2}
[5,1,1,1,1,1,1,1,1,1,1,1]
=> 111111111111 => 000000000000 => 000000000000 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2}
[4,2,2,1,1,1,1,1,1,1,1]
=> 00011111111 => 11100000000 => ? => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2}
[4,2,1,1,1,1,1,1,1,1,1,1]
=> 001111111111 => 110000000000 => 000000000110 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2}
[4,1,1,1,1,1,1,1,1,1,1,1,1]
=> 0111111111111 => ? => ? => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2}
[3,3,2,1,1,1,1,1,1,1,1]
=> 11011111111 => 00100000000 => ? => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2}
[3,3,1,1,1,1,1,1,1,1,1,1]
=> 111111111111 => 000000000000 => 000000000000 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2}
[3,2,2,2,1,1,1,1,1,1,1]
=> 10001111111 => 01110000000 => 00000010110 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2}
[3,2,2,1,1,1,1,1,1,1,1,1]
=> 100111111111 => ? => ? => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2}
[3,2,1,1,1,1,1,1,1,1,1,1,1]
=> 1011111111111 => ? => ? => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2}
[3,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> 11111111111111 => ? => ? => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2}
[2,2,2,2,2,1,1,1,1,1,1]
=> 00000111111 => 11111000000 => ? => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2}
[2,2,2,2,1,1,1,1,1,1,1,1]
=> 000011111111 => 111100000000 => 000000011110 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2}
[2,2,2,1,1,1,1,1,1,1,1,1,1]
=> 0001111111111 => ? => ? => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2}
[2,2,1,1,1,1,1,1,1,1,1,1,1,1]
=> 00111111111111 => ? => ? => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2}
[2,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> 011111111111111 => ? => ? => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> 1111111111111111 => ? => ? => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2}
Description
The number of descents of a binary word.
Matching statistic: St000257
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
St000257: Integer partitions ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
St000257: Integer partitions ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Values
[1]
=> []
=> []
=> ?
=> ? = 0
[2]
=> []
=> []
=> ?
=> ? = 0
[1,1]
=> [1]
=> [1]
=> [1]
=> 0
[3]
=> []
=> []
=> ?
=> ? = 0
[2,1]
=> [1]
=> [1]
=> [1]
=> 0
[1,1,1]
=> [1,1]
=> [2]
=> [1,1]
=> 1
[4]
=> []
=> []
=> ?
=> ? = 0
[3,1]
=> [1]
=> [1]
=> [1]
=> 0
[2,2]
=> [2]
=> [1,1]
=> [2]
=> 0
[2,1,1]
=> [1,1]
=> [2]
=> [1,1]
=> 1
[1,1,1,1]
=> [1,1,1]
=> [3]
=> [3]
=> 0
[5]
=> []
=> []
=> ?
=> ? = 0
[4,1]
=> [1]
=> [1]
=> [1]
=> 0
[3,2]
=> [2]
=> [1,1]
=> [2]
=> 0
[3,1,1]
=> [1,1]
=> [2]
=> [1,1]
=> 1
[2,2,1]
=> [2,1]
=> [2,1]
=> [1,1,1]
=> 1
[2,1,1,1]
=> [1,1,1]
=> [3]
=> [3]
=> 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> [2,2]
=> 1
[6]
=> []
=> []
=> ?
=> ? = 0
[5,1]
=> [1]
=> [1]
=> [1]
=> 0
[4,2]
=> [2]
=> [1,1]
=> [2]
=> 0
[4,1,1]
=> [1,1]
=> [2]
=> [1,1]
=> 1
[3,3]
=> [3]
=> [1,1,1]
=> [2,1]
=> 0
[3,2,1]
=> [2,1]
=> [2,1]
=> [1,1,1]
=> 1
[3,1,1,1]
=> [1,1,1]
=> [3]
=> [3]
=> 0
[2,2,2]
=> [2,2]
=> [2,2]
=> [1,1,1,1]
=> 1
[2,2,1,1]
=> [2,1,1]
=> [3,1]
=> [3,1]
=> 0
[2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> [2,2]
=> 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> [5]
=> 0
[7]
=> []
=> []
=> ?
=> ? = 0
[6,1]
=> [1]
=> [1]
=> [1]
=> 0
[5,2]
=> [2]
=> [1,1]
=> [2]
=> 0
[5,1,1]
=> [1,1]
=> [2]
=> [1,1]
=> 1
[4,3]
=> [3]
=> [1,1,1]
=> [2,1]
=> 0
[4,2,1]
=> [2,1]
=> [2,1]
=> [1,1,1]
=> 1
[4,1,1,1]
=> [1,1,1]
=> [3]
=> [3]
=> 0
[3,3,1]
=> [3,1]
=> [2,1,1]
=> [2,1,1]
=> 1
[3,2,2]
=> [2,2]
=> [2,2]
=> [1,1,1,1]
=> 1
[3,2,1,1]
=> [2,1,1]
=> [3,1]
=> [3,1]
=> 0
[3,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> [2,2]
=> 1
[2,2,2,1]
=> [2,2,1]
=> [3,2]
=> [3,1,1]
=> 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [4,1]
=> [2,2,1]
=> 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> [5]
=> 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [6]
=> [3,3]
=> 1
[8]
=> []
=> []
=> ?
=> ? = 0
[7,1]
=> [1]
=> [1]
=> [1]
=> 0
[6,2]
=> [2]
=> [1,1]
=> [2]
=> 0
[6,1,1]
=> [1,1]
=> [2]
=> [1,1]
=> 1
[5,3]
=> [3]
=> [1,1,1]
=> [2,1]
=> 0
[5,2,1]
=> [2,1]
=> [2,1]
=> [1,1,1]
=> 1
[5,1,1,1]
=> [1,1,1]
=> [3]
=> [3]
=> 0
[4,4]
=> [4]
=> [1,1,1,1]
=> [4]
=> 0
[4,3,1]
=> [3,1]
=> [2,1,1]
=> [2,1,1]
=> 1
[4,2,2]
=> [2,2]
=> [2,2]
=> [1,1,1,1]
=> 1
[4,2,1,1]
=> [2,1,1]
=> [3,1]
=> [3,1]
=> 0
[4,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> [2,2]
=> 1
[3,3,2]
=> [3,2]
=> [2,2,1]
=> [1,1,1,1,1]
=> 1
[3,3,1,1]
=> [3,1,1]
=> [3,1,1]
=> [3,2]
=> 0
[9]
=> []
=> []
=> ?
=> ? = 0
[10]
=> []
=> []
=> ?
=> ? = 0
[11]
=> []
=> []
=> ?
=> ? = 0
[12]
=> []
=> []
=> ?
=> ? = 0
[13]
=> []
=> []
=> ?
=> ? = 0
[14]
=> []
=> []
=> ?
=> ? ∊ {0,0,0}
[3,3,2,2,1,1,1,1]
=> [3,2,2,1,1,1,1]
=> [7,3,1]
=> [7,3,1]
=> ? ∊ {0,0,0}
[3,3,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> [9,1,1]
=> [9,2]
=> ? ∊ {0,0,0}
[15]
=> []
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,2}
[4,4,2,1,1,1,1,1]
=> [4,2,1,1,1,1,1]
=> [7,2,1,1]
=> [7,2,1,1]
=> ? ∊ {0,0,0,1,1,1,1,1,2}
[4,3,2,2,1,1,1,1]
=> [3,2,2,1,1,1,1]
=> [7,3,1]
=> [7,3,1]
=> ? ∊ {0,0,0,1,1,1,1,1,2}
[4,3,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> [9,1,1]
=> [9,2]
=> ? ∊ {0,0,0,1,1,1,1,1,2}
[3,3,3,2,1,1,1,1]
=> [3,3,2,1,1,1,1]
=> [7,3,2]
=> [7,3,1,1]
=> ? ∊ {0,0,0,1,1,1,1,1,2}
[3,3,3,1,1,1,1,1,1]
=> [3,3,1,1,1,1,1,1]
=> [8,2,2]
=> [4,4,1,1,1,1]
=> ? ∊ {0,0,0,1,1,1,1,1,2}
[3,3,2,2,2,1,1,1]
=> [3,2,2,2,1,1,1]
=> [7,4,1]
=> [7,2,2,1]
=> ? ∊ {0,0,0,1,1,1,1,1,2}
[3,3,2,1,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1,1]
=> [9,2,1]
=> [9,1,1,1]
=> ? ∊ {0,0,0,1,1,1,1,1,2}
[2,2,2,1,1,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1,1]
=> [11,2]
=> [11,1,1]
=> ? ∊ {0,0,0,1,1,1,1,1,2}
[16]
=> []
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[5,5,2,1,1,1,1]
=> [5,2,1,1,1,1]
=> [6,2,1,1,1]
=> [3,3,2,1,1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[5,4,2,1,1,1,1,1]
=> [4,2,1,1,1,1,1]
=> [7,2,1,1]
=> [7,2,1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[5,3,2,2,1,1,1,1]
=> [3,2,2,1,1,1,1]
=> [7,3,1]
=> [7,3,1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[5,3,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> [9,1,1]
=> [9,2]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[4,4,3,2,1,1,1]
=> [4,3,2,1,1,1]
=> [6,3,2,1]
=> [3,3,3,1,1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[4,4,3,1,1,1,1,1]
=> [4,3,1,1,1,1,1]
=> [7,2,2,1]
=> [7,1,1,1,1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[4,4,2,2,1,1,1,1]
=> [4,2,2,1,1,1,1]
=> [7,3,1,1]
=> [7,3,2]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[4,4,1,1,1,1,1,1,1,1]
=> [4,1,1,1,1,1,1,1,1]
=> [9,1,1,1]
=> [9,2,1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[4,3,3,2,1,1,1,1]
=> [3,3,2,1,1,1,1]
=> [7,3,2]
=> [7,3,1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[4,3,3,1,1,1,1,1,1]
=> [3,3,1,1,1,1,1,1]
=> [8,2,2]
=> [4,4,1,1,1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[4,3,2,2,2,1,1,1]
=> [3,2,2,2,1,1,1]
=> [7,4,1]
=> [7,2,2,1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[4,3,2,1,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1,1]
=> [9,2,1]
=> [9,1,1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[3,3,3,3,3,1]
=> [3,3,3,3,1]
=> [5,4,4]
=> [5,2,2,2,2]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[3,3,3,2,2,1,1,1]
=> [3,3,2,2,1,1,1]
=> [7,4,2]
=> [7,2,2,1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[3,3,3,1,1,1,1,1,1,1]
=> [3,3,1,1,1,1,1,1,1]
=> [9,2,2]
=> [9,1,1,1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[3,3,2,2,1,1,1,1,1,1]
=> [3,2,2,1,1,1,1,1,1]
=> [9,3,1]
=> [9,3,1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[3,3,2,1,1,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1,1,1]
=> [10,2,1]
=> [5,5,1,1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[3,3,1,1,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1,1,1]
=> [11,1,1]
=> [11,2]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[3,2,2,1,1,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1,1]
=> [11,2]
=> [11,1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[2,2,2,2,2,1,1,1,1,1,1]
=> [2,2,2,2,1,1,1,1,1,1]
=> [10,4]
=> [5,5,2,2]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[2,2,2,2,1,1,1,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1,1,1,1]
=> [11,3]
=> [11,3]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
Description
The number of distinct parts of a partition that occur at least twice.
See Section 3.3.1 of [2].
Matching statistic: St000875
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00096: Binary words —Foata bijection⟶ Binary words
Mp00136: Binary words —rotate back-to-front⟶ Binary words
St000875: Binary words ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Mp00096: Binary words —Foata bijection⟶ Binary words
Mp00136: Binary words —rotate back-to-front⟶ Binary words
St000875: Binary words ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Values
[1]
=> 1 => 1 => 1 => 0
[2]
=> 0 => 0 => 0 => 0
[1,1]
=> 11 => 11 => 11 => 0
[3]
=> 1 => 1 => 1 => 0
[2,1]
=> 01 => 01 => 10 => 1
[1,1,1]
=> 111 => 111 => 111 => 0
[4]
=> 0 => 0 => 0 => 0
[3,1]
=> 11 => 11 => 11 => 0
[2,2]
=> 00 => 00 => 00 => 0
[2,1,1]
=> 011 => 011 => 101 => 1
[1,1,1,1]
=> 1111 => 1111 => 1111 => 0
[5]
=> 1 => 1 => 1 => 0
[4,1]
=> 01 => 01 => 10 => 1
[3,2]
=> 10 => 10 => 01 => 0
[3,1,1]
=> 111 => 111 => 111 => 0
[2,2,1]
=> 001 => 001 => 100 => 1
[2,1,1,1]
=> 0111 => 0111 => 1011 => 1
[1,1,1,1,1]
=> 11111 => 11111 => 11111 => 0
[6]
=> 0 => 0 => 0 => 0
[5,1]
=> 11 => 11 => 11 => 0
[4,2]
=> 00 => 00 => 00 => 0
[4,1,1]
=> 011 => 011 => 101 => 1
[3,3]
=> 11 => 11 => 11 => 0
[3,2,1]
=> 101 => 101 => 110 => 1
[3,1,1,1]
=> 1111 => 1111 => 1111 => 0
[2,2,2]
=> 000 => 000 => 000 => 0
[2,2,1,1]
=> 0011 => 0011 => 1001 => 1
[2,1,1,1,1]
=> 01111 => 01111 => 10111 => 1
[1,1,1,1,1,1]
=> 111111 => 111111 => 111111 => 0
[7]
=> 1 => 1 => 1 => 0
[6,1]
=> 01 => 01 => 10 => 1
[5,2]
=> 10 => 10 => 01 => 0
[5,1,1]
=> 111 => 111 => 111 => 0
[4,3]
=> 01 => 01 => 10 => 1
[4,2,1]
=> 001 => 001 => 100 => 1
[4,1,1,1]
=> 0111 => 0111 => 1011 => 1
[3,3,1]
=> 111 => 111 => 111 => 0
[3,2,2]
=> 100 => 010 => 001 => 0
[3,2,1,1]
=> 1011 => 1011 => 1101 => 1
[3,1,1,1,1]
=> 11111 => 11111 => 11111 => 0
[2,2,2,1]
=> 0001 => 0001 => 1000 => 1
[2,2,1,1,1]
=> 00111 => 00111 => 10011 => 1
[2,1,1,1,1,1]
=> 011111 => 011111 => 101111 => 1
[1,1,1,1,1,1,1]
=> 1111111 => 1111111 => 1111111 => 0
[8]
=> 0 => 0 => 0 => 0
[7,1]
=> 11 => 11 => 11 => 0
[6,2]
=> 00 => 00 => 00 => 0
[6,1,1]
=> 011 => 011 => 101 => 1
[5,3]
=> 11 => 11 => 11 => 0
[5,2,1]
=> 101 => 101 => 110 => 1
[1,1,1,1,1,1,1,1,1,1]
=> 1111111111 => 1111111111 => 1111111111 => ? = 0
[2,1,1,1,1,1,1,1,1,1]
=> 0111111111 => 0111111111 => 1011111111 => ? ∊ {0,2}
[1,1,1,1,1,1,1,1,1,1,1]
=> 11111111111 => 11111111111 => 11111111111 => ? ∊ {0,2}
[3,1,1,1,1,1,1,1,1,1]
=> 1111111111 => 1111111111 => 1111111111 => ? ∊ {0,0,2,2}
[2,2,1,1,1,1,1,1,1,1]
=> 0011111111 => 0011111111 => 1001111111 => ? ∊ {0,0,2,2}
[2,1,1,1,1,1,1,1,1,1,1]
=> 01111111111 => 01111111111 => 10111111111 => ? ∊ {0,0,2,2}
[1,1,1,1,1,1,1,1,1,1,1,1]
=> 111111111111 => 111111111111 => 111111111111 => ? ∊ {0,0,2,2}
[4,1,1,1,1,1,1,1,1,1]
=> 0111111111 => 0111111111 => 1011111111 => ? ∊ {0,0,1,2,2,2,2}
[3,2,1,1,1,1,1,1,1,1]
=> 1011111111 => 1011111111 => 1101111111 => ? ∊ {0,0,1,2,2,2,2}
[3,1,1,1,1,1,1,1,1,1,1]
=> 11111111111 => 11111111111 => 11111111111 => ? ∊ {0,0,1,2,2,2,2}
[2,2,2,1,1,1,1,1,1,1]
=> 0001111111 => 0001111111 => 1000111111 => ? ∊ {0,0,1,2,2,2,2}
[2,2,1,1,1,1,1,1,1,1,1]
=> 00111111111 => 00111111111 => 10011111111 => ? ∊ {0,0,1,2,2,2,2}
[2,1,1,1,1,1,1,1,1,1,1,1]
=> 011111111111 => 011111111111 => 101111111111 => ? ∊ {0,0,1,2,2,2,2}
[1,1,1,1,1,1,1,1,1,1,1,1,1]
=> 1111111111111 => ? => ? => ? ∊ {0,0,1,2,2,2,2}
[5,1,1,1,1,1,1,1,1,1]
=> 1111111111 => 1111111111 => 1111111111 => ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2}
[4,2,1,1,1,1,1,1,1,1]
=> 0011111111 => 0011111111 => 1001111111 => ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2}
[4,1,1,1,1,1,1,1,1,1,1]
=> 01111111111 => 01111111111 => 10111111111 => ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2}
[3,3,1,1,1,1,1,1,1,1]
=> 1111111111 => 1111111111 => 1111111111 => ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2}
[3,2,2,1,1,1,1,1,1,1]
=> 1001111111 => 0101111111 => 1010111111 => ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2}
[3,2,1,1,1,1,1,1,1,1,1]
=> 10111111111 => 10111111111 => 11011111111 => ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2}
[3,1,1,1,1,1,1,1,1,1,1,1]
=> 111111111111 => 111111111111 => 111111111111 => ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2}
[2,2,2,2,1,1,1,1,1,1]
=> 0000111111 => 0000111111 => 1000011111 => ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2}
[2,2,2,1,1,1,1,1,1,1,1]
=> 00011111111 => 00011111111 => 10001111111 => ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2}
[2,2,1,1,1,1,1,1,1,1,1,1]
=> 001111111111 => 001111111111 => 100111111111 => ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2}
[2,1,1,1,1,1,1,1,1,1,1,1,1]
=> 0111111111111 => ? => ? => ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2}
[1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> 11111111111111 => 11111111111111 => 11111111111111 => ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2}
[6,1,1,1,1,1,1,1,1,1]
=> 0111111111 => 0111111111 => 1011111111 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[5,2,1,1,1,1,1,1,1,1]
=> 1011111111 => 1011111111 => 1101111111 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[5,1,1,1,1,1,1,1,1,1,1]
=> 11111111111 => 11111111111 => 11111111111 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[4,3,1,1,1,1,1,1,1,1]
=> 0111111111 => 0111111111 => 1011111111 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[4,2,2,1,1,1,1,1,1,1]
=> 0001111111 => 0001111111 => 1000111111 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[4,2,1,1,1,1,1,1,1,1,1]
=> 00111111111 => 00111111111 => 10011111111 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[4,1,1,1,1,1,1,1,1,1,1,1]
=> 011111111111 => 011111111111 => 101111111111 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[3,3,2,1,1,1,1,1,1,1]
=> 1101111111 => 1101111111 => 1110111111 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[3,3,1,1,1,1,1,1,1,1,1]
=> 11111111111 => 11111111111 => 11111111111 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[3,2,2,2,1,1,1,1,1,1]
=> 1000111111 => 0010111111 => 1001011111 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[3,2,2,1,1,1,1,1,1,1,1]
=> 10011111111 => 01011111111 => 10101111111 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[3,2,1,1,1,1,1,1,1,1,1,1]
=> 101111111111 => ? => ? => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[3,1,1,1,1,1,1,1,1,1,1,1,1]
=> 1111111111111 => ? => ? => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[2,2,2,2,2,1,1,1,1,1]
=> 0000011111 => 0000011111 => 1000001111 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[2,2,2,2,1,1,1,1,1,1,1]
=> 00001111111 => 00001111111 => 10000111111 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[2,2,2,1,1,1,1,1,1,1,1,1]
=> 000111111111 => 000111111111 => 100011111111 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[2,2,1,1,1,1,1,1,1,1,1,1,1]
=> 0011111111111 => ? => ? => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[2,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> 01111111111111 => ? => ? => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> 111111111111111 => ? => ? => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[7,1,1,1,1,1,1,1,1,1]
=> 1111111111 => 1111111111 => 1111111111 => ? ∊ {0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[6,2,1,1,1,1,1,1,1,1]
=> 0011111111 => 0011111111 => 1001111111 => ? ∊ {0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[6,1,1,1,1,1,1,1,1,1,1]
=> 01111111111 => 01111111111 => 10111111111 => ? ∊ {0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,3,1,1,1,1,1,1,1,1]
=> 1111111111 => 1111111111 => 1111111111 => ? ∊ {0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[5,2,2,1,1,1,1,1,1,1]
=> 1001111111 => 0101111111 => 1010111111 => ? ∊ {0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
Description
The semilength of the longest Dyck word in the Catalan factorisation of a binary word.
Every binary word can be written in a unique way as (D0)ℓD(1D)m, where D is the set of Dyck words. This is the Catalan factorisation, see [1, sec.9.1.2].
This statistic records the semilength of the longest Dyck word in this factorisation.
Matching statistic: St001115
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001115: Permutations ⟶ ℤResult quality: 75% ●values known / values provided: 84%●distinct values known / distinct values provided: 75%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001115: Permutations ⟶ ℤResult quality: 75% ●values known / values provided: 84%●distinct values known / distinct values provided: 75%
Values
[1]
=> []
=> []
=> [] => 0
[2]
=> []
=> []
=> [] => 0
[1,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 0
[3]
=> []
=> []
=> [] => 0
[2,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 0
[1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 1
[4]
=> []
=> []
=> [] => 0
[3,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 0
[2,2]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 0
[2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 1
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 0
[5]
=> []
=> []
=> [] => 0
[4,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 0
[3,2]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 0
[3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 1
[2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 1
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1
[6]
=> []
=> []
=> [] => 0
[5,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 0
[4,2]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 0
[4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 1
[3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 0
[3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 1
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 0
[2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 1
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 0
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 0
[7]
=> []
=> []
=> [] => 0
[6,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 0
[5,2]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 0
[5,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 1
[4,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 0
[4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 1
[4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 0
[3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 1
[3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 1
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 0
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 0
[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]
=> [2,3,4,5,6,7,1] => 1
[8]
=> []
=> []
=> [] => 0
[7,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 0
[6,2]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 0
[6,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 1
[5,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 0
[5,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 1
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> ? => ? = 0
[2,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,2,1] => ? ∊ {0,1,1,1,1,1,1}
[2,2,2,2,2,1,1,1]
=> [2,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [3,4,5,6,2,7,8,1] => ? ∊ {0,1,1,1,1,1,1}
[2,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [3,4,5,2,6,7,8,9,1] => ? ∊ {0,1,1,1,1,1,1}
[2,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [3,4,2,5,6,7,8,9,10,1] => ? ∊ {0,1,1,1,1,1,1}
[2,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,9,10,11,1] => ? ∊ {0,1,1,1,1,1,1}
[2,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> ? => ? ∊ {0,1,1,1,1,1,1}
[1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? => ? ∊ {0,1,1,1,1,1,1}
[3,3,3,2,1,1,1]
=> [3,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [4,5,3,2,6,7,1] => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[3,3,3,1,1,1,1,1]
=> [3,3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [4,5,2,3,6,7,8,1] => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[3,3,2,2,2,2]
=> [3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,7,1,2] => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[3,3,2,2,2,1,1]
=> [3,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,2,7,1] => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[3,3,2,2,1,1,1,1]
=> [3,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [4,3,5,2,6,7,8,1] => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[3,3,2,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [4,3,2,5,6,7,8,9,1] => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[3,3,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [4,2,3,5,6,7,8,9,10,1] => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[3,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,2,1] => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[3,2,2,2,2,1,1,1]
=> [2,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [3,4,5,6,2,7,8,1] => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[3,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [3,4,5,2,6,7,8,9,1] => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[3,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [3,4,2,5,6,7,8,9,10,1] => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[3,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,9,10,11,1] => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[3,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> ? => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[2,2,2,2,2,2,1,1]
=> [2,2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [3,4,5,6,7,2,8,1] => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[2,2,2,2,2,1,1,1,1]
=> [2,2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [3,4,5,6,2,7,8,9,1] => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[2,2,2,2,1,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
=> [3,4,5,2,6,7,8,9,10,1] => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[2,2,2,1,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0]
=> [3,4,2,5,6,7,8,9,10,11,1] => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[2,2,1,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,9,10,11,12,1] => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[2,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[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]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[4,4,3,1,1,1,1]
=> [4,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [5,4,2,3,6,7,1] => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[4,4,2,2,1,1,1]
=> [4,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [5,3,4,2,6,7,1] => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[4,4,2,1,1,1,1,1]
=> [4,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [5,3,2,4,6,7,8,1] => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[4,4,1,1,1,1,1,1,1]
=> [4,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [5,2,3,4,6,7,8,9,1] => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[4,3,3,2,1,1,1]
=> [3,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [4,5,3,2,6,7,1] => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[4,3,3,1,1,1,1,1]
=> [3,3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [4,5,2,3,6,7,8,1] => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[4,3,2,2,2,2]
=> [3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,7,1,2] => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[4,3,2,2,2,1,1]
=> [3,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,2,7,1] => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[4,3,2,2,1,1,1,1]
=> [3,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [4,3,5,2,6,7,8,1] => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[4,3,2,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [4,3,2,5,6,7,8,9,1] => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[4,3,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [4,2,3,5,6,7,8,9,10,1] => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[4,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,2,1] => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[4,2,2,2,2,1,1,1]
=> [2,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [3,4,5,6,2,7,8,1] => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[4,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [3,4,5,2,6,7,8,9,1] => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[4,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [3,4,2,5,6,7,8,9,10,1] => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[4,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,9,10,11,1] => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[4,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> ? => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[3,3,3,3,3]
=> [3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,1,2,3] => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[3,3,3,3,1,1,1]
=> [3,3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [4,5,6,2,3,7,1] => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[3,3,3,2,2,2]
=> [3,3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [4,5,3,6,7,1,2] => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[3,3,3,2,2,1,1]
=> [3,3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [4,5,3,6,2,7,1] => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
[3,3,3,2,1,1,1,1]
=> [3,3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [4,5,3,2,6,7,8,1] => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3}
Description
The number of even descents of a permutation.
Matching statistic: St000256
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000256: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 78%●distinct values known / distinct values provided: 75%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000256: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 78%●distinct values known / distinct values provided: 75%
Values
[1]
=> []
=> []
=> ?
=> ? = 0
[2]
=> []
=> []
=> ?
=> ? = 0
[1,1]
=> [1]
=> [1]
=> [1]
=> 0
[3]
=> []
=> []
=> ?
=> ? = 0
[2,1]
=> [1]
=> [1]
=> [1]
=> 0
[1,1,1]
=> [1,1]
=> [2]
=> [2]
=> 1
[4]
=> []
=> []
=> ?
=> ? = 0
[3,1]
=> [1]
=> [1]
=> [1]
=> 0
[2,2]
=> [2]
=> [1,1]
=> [1,1]
=> 0
[2,1,1]
=> [1,1]
=> [2]
=> [2]
=> 1
[1,1,1,1]
=> [1,1,1]
=> [3]
=> [2,1]
=> 0
[5]
=> []
=> []
=> ?
=> ? = 0
[4,1]
=> [1]
=> [1]
=> [1]
=> 0
[3,2]
=> [2]
=> [1,1]
=> [1,1]
=> 0
[3,1,1]
=> [1,1]
=> [2]
=> [2]
=> 1
[2,2,1]
=> [2,1]
=> [2,1]
=> [3]
=> 1
[2,1,1,1]
=> [1,1,1]
=> [3]
=> [2,1]
=> 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> [2,2]
=> 1
[6]
=> []
=> []
=> ?
=> ? = 0
[5,1]
=> [1]
=> [1]
=> [1]
=> 0
[4,2]
=> [2]
=> [1,1]
=> [1,1]
=> 0
[4,1,1]
=> [1,1]
=> [2]
=> [2]
=> 1
[3,3]
=> [3]
=> [1,1,1]
=> [1,1,1]
=> 0
[3,2,1]
=> [2,1]
=> [2,1]
=> [3]
=> 1
[3,1,1,1]
=> [1,1,1]
=> [3]
=> [2,1]
=> 0
[2,2,2]
=> [2,2]
=> [2,2]
=> [4]
=> 1
[2,2,1,1]
=> [2,1,1]
=> [3,1]
=> [2,1,1]
=> 0
[2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> [2,2]
=> 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> [2,2,1]
=> 0
[7]
=> []
=> []
=> ?
=> ? = 0
[6,1]
=> [1]
=> [1]
=> [1]
=> 0
[5,2]
=> [2]
=> [1,1]
=> [1,1]
=> 0
[5,1,1]
=> [1,1]
=> [2]
=> [2]
=> 1
[4,3]
=> [3]
=> [1,1,1]
=> [1,1,1]
=> 0
[4,2,1]
=> [2,1]
=> [2,1]
=> [3]
=> 1
[4,1,1,1]
=> [1,1,1]
=> [3]
=> [2,1]
=> 0
[3,3,1]
=> [3,1]
=> [2,1,1]
=> [3,1]
=> 1
[3,2,2]
=> [2,2]
=> [2,2]
=> [4]
=> 1
[3,2,1,1]
=> [2,1,1]
=> [3,1]
=> [2,1,1]
=> 0
[3,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> [2,2]
=> 1
[2,2,2,1]
=> [2,2,1]
=> [3,2]
=> [4,1]
=> 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [4,1]
=> [3,2]
=> 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> [2,2,1]
=> 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [6]
=> [2,2,2]
=> 1
[8]
=> []
=> []
=> ?
=> ? = 0
[7,1]
=> [1]
=> [1]
=> [1]
=> 0
[6,2]
=> [2]
=> [1,1]
=> [1,1]
=> 0
[6,1,1]
=> [1,1]
=> [2]
=> [2]
=> 1
[5,3]
=> [3]
=> [1,1,1]
=> [1,1,1]
=> 0
[5,2,1]
=> [2,1]
=> [2,1]
=> [3]
=> 1
[5,1,1,1]
=> [1,1,1]
=> [3]
=> [2,1]
=> 0
[4,4]
=> [4]
=> [1,1,1,1]
=> [1,1,1,1]
=> 0
[4,3,1]
=> [3,1]
=> [2,1,1]
=> [3,1]
=> 1
[4,2,2]
=> [2,2]
=> [2,2]
=> [4]
=> 1
[4,2,1,1]
=> [2,1,1]
=> [3,1]
=> [2,1,1]
=> 0
[4,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> [2,2]
=> 1
[3,3,2]
=> [3,2]
=> [2,2,1]
=> [5]
=> 1
[3,3,1,1]
=> [3,1,1]
=> [3,1,1]
=> [2,1,1,1]
=> 0
[9]
=> []
=> []
=> ?
=> ? = 0
[10]
=> []
=> []
=> ?
=> ? = 0
[11]
=> []
=> []
=> ?
=> ? = 0
[12]
=> []
=> []
=> ?
=> ? ∊ {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]
=> [11]
=> [2,2,2,2,2,1]
=> ? ∊ {0,0}
[13]
=> []
=> []
=> ?
=> ? ∊ {0,0,1,1,1,1,1,1}
[2,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [6,5]
=> [4,4,3]
=> ? ∊ {0,0,1,1,1,1,1,1}
[2,2,2,2,2,1,1,1]
=> [2,2,2,2,1,1,1]
=> [7,4]
=> [4,4,2,1]
=> ? ∊ {0,0,1,1,1,1,1,1}
[2,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> [8,3]
=> [4,3,2,2]
=> ? ∊ {0,0,1,1,1,1,1,1}
[2,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [9,2]
=> [4,2,2,2,1]
=> ? ∊ {0,0,1,1,1,1,1,1}
[2,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [10,1]
=> [3,2,2,2,2]
=> ? ∊ {0,0,1,1,1,1,1,1}
[2,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [11]
=> [2,2,2,2,2,1]
=> ? ∊ {0,0,1,1,1,1,1,1}
[1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [12]
=> [2,2,2,2,2,2]
=> ? ∊ {0,0,1,1,1,1,1,1}
[14]
=> []
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[3,3,3,3,2]
=> [3,3,3,2]
=> [4,4,3]
=> [6,5]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[3,3,3,3,1,1]
=> [3,3,3,1,1]
=> [5,3,3]
=> [3,3,2,2,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[3,3,3,2,2,1]
=> [3,3,2,2,1]
=> [5,4,2]
=> [6,4,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[3,3,3,2,1,1,1]
=> [3,3,2,1,1,1]
=> [6,3,2]
=> [6,3,2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[3,3,3,1,1,1,1,1]
=> [3,3,1,1,1,1,1]
=> [7,2,2]
=> [6,2,2,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[3,3,2,2,2,2]
=> [3,2,2,2,2]
=> [5,5,1]
=> [4,3,2,1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[3,3,2,2,2,1,1]
=> [3,2,2,2,1,1]
=> [6,4,1]
=> [5,4,2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[3,3,2,2,1,1,1,1]
=> [3,2,2,1,1,1,1]
=> [7,3,1]
=> [3,2,2,2,1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[3,3,2,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1]
=> [8,2,1]
=> [5,2,2,2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[3,3,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> [9,1,1]
=> [2,2,2,2,1,1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[3,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [6,5]
=> [4,4,3]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[3,2,2,2,2,1,1,1]
=> [2,2,2,2,1,1,1]
=> [7,4]
=> [4,4,2,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[3,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> [8,3]
=> [4,3,2,2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[3,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [9,2]
=> [4,2,2,2,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[3,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [10,1]
=> [3,2,2,2,2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[3,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [11]
=> [2,2,2,2,2,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[2,2,2,2,2,2,2]
=> [2,2,2,2,2,2]
=> [6,6]
=> [4,4,4]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[2,2,2,2,2,2,1,1]
=> [2,2,2,2,2,1,1]
=> [7,5]
=> [4,3,2,2,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[2,2,2,2,2,1,1,1,1]
=> [2,2,2,2,1,1,1,1]
=> [8,4]
=> [4,4,2,2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[2,2,2,2,1,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1,1]
=> [9,3]
=> [3,2,2,2,2,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[2,2,2,1,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1]
=> [10,2]
=> [4,2,2,2,2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[2,2,1,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1,1]
=> [11,1]
=> [2,2,2,2,2,1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[2,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [12]
=> [2,2,2,2,2,2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[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]
=> [13]
=> [2,2,2,2,2,2,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2}
[15]
=> []
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3}
[4,4,4,3]
=> [4,4,3]
=> [3,3,3,2]
=> [5,3,2,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3}
[4,4,4,2,1]
=> [4,4,2,1]
=> [4,3,2,2]
=> [8,3]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3}
[4,4,4,1,1,1]
=> [4,4,1,1,1]
=> [5,2,2,2]
=> [8,2,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3}
Description
The number of parts from which one can substract 2 and still get an integer partition.
Matching statistic: St001195
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001195: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 52%●distinct values known / distinct values provided: 50%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001195: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 52%●distinct values known / distinct values provided: 50%
Values
[1]
=> []
=> ?
=> ?
=> ? = 0
[2]
=> []
=> ?
=> ?
=> ? ∊ {0,0}
[1,1]
=> [1]
=> []
=> []
=> ? ∊ {0,0}
[3]
=> []
=> ?
=> ?
=> ? ∊ {0,0,1}
[2,1]
=> [1]
=> []
=> []
=> ? ∊ {0,0,1}
[1,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> ? ∊ {0,0,1}
[4]
=> []
=> ?
=> ?
=> ? ∊ {0,0,0,0}
[3,1]
=> [1]
=> []
=> []
=> ? ∊ {0,0,0,0}
[2,2]
=> [2]
=> []
=> []
=> ? ∊ {0,0,0,0}
[2,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> ? ∊ {0,0,0,0}
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[5]
=> []
=> ?
=> ?
=> ? ∊ {0,0,0,0,1}
[4,1]
=> [1]
=> []
=> []
=> ? ∊ {0,0,0,0,1}
[3,2]
=> [2]
=> []
=> []
=> ? ∊ {0,0,0,0,1}
[3,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> ? ∊ {0,0,0,0,1}
[2,2,1]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> ? ∊ {0,0,0,0,1}
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[6]
=> []
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0}
[5,1]
=> [1]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0}
[4,2]
=> [2]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0}
[4,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> ? ∊ {0,0,0,0,0,0}
[3,3]
=> [3]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0}
[3,2,1]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> ? ∊ {0,0,0,0,0,0}
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[2,2,2]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[7]
=> []
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1}
[6,1]
=> [1]
=> []
=> []
=> ? ∊ {0,0,0,0,0,1,1,1}
[5,2]
=> [2]
=> []
=> []
=> ? ∊ {0,0,0,0,0,1,1,1}
[5,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> ? ∊ {0,0,0,0,0,1,1,1}
[4,3]
=> [3]
=> []
=> []
=> ? ∊ {0,0,0,0,0,1,1,1}
[4,2,1]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> ? ∊ {0,0,0,0,0,1,1,1}
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[3,3,1]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> ? ∊ {0,0,0,0,0,1,1,1}
[3,2,2]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,1,1,1}
[8]
=> []
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,2}
[7,1]
=> [1]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,2}
[6,2]
=> [2]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,2}
[6,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,2}
[5,3]
=> [3]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,2}
[5,2,1]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,2}
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[4,4]
=> [4]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,2}
[4,3,1]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,2}
[4,2,2]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[3,3,2]
=> [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[2,2,2,2]
=> [2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1
[2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,2}
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,2}
[9]
=> []
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2}
[8,1]
=> [1]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2}
[7,2]
=> [2]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2}
[7,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2}
[6,3]
=> [3]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2}
[6,2,1]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2}
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[5,4]
=> [4]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2}
[5,3,1]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2}
[5,2,2]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[4,4,1]
=> [4,1]
=> [1]
=> [1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2}
[4,3,2]
=> [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[4,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[4,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[3,3,3]
=> [3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[3,3,2,1]
=> [3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0
[3,3,1,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[3,2,2,2]
=> [2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1
[3,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2}
[2,2,2,2,1]
=> [2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2}
[7,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[6,2,2]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[6,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[6,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[5,3,2]
=> [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[5,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 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: St000259
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 50% ●values known / values provided: 52%●distinct values known / distinct values provided: 50%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 50% ●values known / values provided: 52%●distinct values known / distinct values provided: 50%
Values
[1]
=> 1 => [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
[2]
=> 0 => [2] => ([],2)
=> ? = 0 + 1
[1,1]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[3]
=> 1 => [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
[2,1]
=> 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,1,1]
=> 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[4]
=> 0 => [2] => ([],2)
=> ? ∊ {0,0} + 1
[3,1]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[2,2]
=> 00 => [3] => ([],3)
=> ? ∊ {0,0} + 1
[2,1,1]
=> 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,1,1]
=> 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[5]
=> 1 => [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
[4,1]
=> 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[3,2]
=> 10 => [1,2] => ([(1,2)],3)
=> ? = 0 + 1
[3,1,1]
=> 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[2,2,1]
=> 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,1,1,1]
=> 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,1,1,1,1]
=> 11111 => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[6]
=> 0 => [2] => ([],2)
=> ? ∊ {0,0,0} + 1
[5,1]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[4,2]
=> 00 => [3] => ([],3)
=> ? ∊ {0,0,0} + 1
[4,1,1]
=> 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,3]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[3,2,1]
=> 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,1,1,1]
=> 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[2,2,2]
=> 000 => [4] => ([],4)
=> ? ∊ {0,0,0} + 1
[2,2,1,1]
=> 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[2,1,1,1,1]
=> 01111 => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,1,1,1,1,1]
=> 111111 => [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 0 + 1
[7]
=> 1 => [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
[6,1]
=> 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[5,2]
=> 10 => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0} + 1
[5,1,1]
=> 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[4,3]
=> 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[4,2,1]
=> 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,1,1,1]
=> 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,3,1]
=> 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[3,2,2]
=> 100 => [1,3] => ([(2,3)],4)
=> ? ∊ {0,0,0} + 1
[3,2,1,1]
=> 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,1,1,1,1]
=> 11111 => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[2,2,2,1]
=> 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[2,2,1,1,1]
=> 00111 => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[2,1,1,1,1,1]
=> 011111 => [2,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 1 + 1
[1,1,1,1,1,1,1]
=> 1111111 => [1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,0} + 1
[8]
=> 0 => [2] => ([],2)
=> ? ∊ {0,0,0,0,0,0,0,2} + 1
[7,1]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[6,2]
=> 00 => [3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,0,2} + 1
[6,1,1]
=> 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[5,3]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[5,2,1]
=> 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[5,1,1,1]
=> 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[4,4]
=> 00 => [3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,0,2} + 1
[4,3,1]
=> 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,2,2]
=> 000 => [4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,2} + 1
[4,2,1,1]
=> 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[4,1,1,1,1]
=> 01111 => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[3,3,2]
=> 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,2} + 1
[3,3,1,1]
=> 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[3,2,2,1]
=> 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,2,1,1,1]
=> 10111 => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[3,1,1,1,1,1]
=> 111111 => [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 0 + 1
[2,2,2,2]
=> 0000 => [5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,2} + 1
[2,2,2,1,1]
=> 00011 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[2,2,1,1,1,1]
=> 001111 => [3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 1 + 1
[2,1,1,1,1,1,1]
=> 0111111 => [2,1,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,0,2} + 1
[1,1,1,1,1,1,1,1]
=> 11111111 => [1,1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,0,2} + 1
[9]
=> 1 => [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
[8,1]
=> 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[7,2]
=> 10 => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,1,1,2} + 1
[5,4]
=> 10 => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,1,1,2} + 1
[5,2,2]
=> 100 => [1,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,1,1,2} + 1
[4,3,2]
=> 010 => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,1,1,2} + 1
[3,2,2,2]
=> 1000 => [1,4] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,2} + 1
[3,1,1,1,1,1,1]
=> 1111111 => [1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,1,1,2} + 1
[2,2,1,1,1,1,1]
=> 0011111 => [3,1,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,1,1,2} + 1
[2,1,1,1,1,1,1,1]
=> 01111111 => [2,1,1,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,1,1,2} + 1
[1,1,1,1,1,1,1,1,1]
=> 111111111 => [1,1,1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? ∊ {0,0,0,0,0,0,1,1,2} + 1
[10]
=> 0 => [2] => ([],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2} + 1
[8,2]
=> 00 => [3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2} + 1
[6,4]
=> 00 => [3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2} + 1
[6,2,2]
=> 000 => [4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2} + 1
[5,3,2]
=> 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2} + 1
[4,4,2]
=> 000 => [4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2} + 1
[4,2,2,2]
=> 0000 => [5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2} + 1
[4,1,1,1,1,1,1]
=> 0111111 => [2,1,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2} + 1
[3,3,2,2]
=> 1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2} + 1
[3,2,1,1,1,1,1]
=> 1011111 => [1,2,1,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2} + 1
[3,1,1,1,1,1,1,1]
=> 11111111 => [1,1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2} + 1
[2,2,2,2,2]
=> 00000 => [6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2} + 1
[2,2,2,1,1,1,1]
=> 0001111 => [4,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2} + 1
[2,2,1,1,1,1,1,1]
=> 00111111 => [3,1,1,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2} + 1
[2,1,1,1,1,1,1,1,1]
=> 011111111 => [2,1,1,1,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2} + 1
[1,1,1,1,1,1,1,1,1,1]
=> 1111111111 => [1,1,1,1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(1,10),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2} + 1
[9,2]
=> 10 => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2} + 1
[7,4]
=> 10 => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2} + 1
[7,2,2]
=> 100 => [1,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2} + 1
[6,3,2]
=> 010 => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2} + 1
[5,4,2]
=> 100 => [1,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2} + 1
[5,2,2,2]
=> 1000 => [1,4] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2} + 1
[5,1,1,1,1,1,1]
=> 1111111 => [1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2} + 1
Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
Matching statistic: St001732
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00228: Dyck paths —reflect parallelogram polyomino⟶ Dyck paths
St001732: Dyck paths ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 75%
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00228: Dyck paths —reflect parallelogram polyomino⟶ Dyck paths
St001732: Dyck paths ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 75%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[2]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 0 + 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 1 = 0 + 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 0 + 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [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 = 0 + 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> 1 = 0 + 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> 1 = 0 + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1 = 0 + 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [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 = 0 + 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 1 + 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> 1 = 0 + 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 0 + 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1 = 0 + 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 2 = 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]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? ∊ {0,0,2} + 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
=> ? ∊ {0,0,2} + 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> 1 = 0 + 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1 = 0 + 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1 = 0 + 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]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,2} + 1
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? ∊ {0,0,1,1,1,1} + 1
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0,0]
=> ? ∊ {0,0,1,1,1,1} + 1
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> ? ∊ {0,0,1,1,1,1} + 1
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,1,0,0]
=> ? ∊ {0,0,1,1,1,1} + 1
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,1,1,1,1} + 1
[1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,1,1,1,1} + 1
[10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2} + 1
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2} + 1
[8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2} + 1
[8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2} + 1
[7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2} + 1
[7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0,1,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2} + 1
[7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2} + 1
[3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2} + 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]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2} + 1
[2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2} + 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]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2} + 1
[11]
=> [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2} + 1
[10,1]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> ?
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2} + 1
[9,2]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2} + 1
[9,1,1]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
=> ?
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2} + 1
[8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2} + 1
[8,2,1]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2} + 1
[8,1,1,1]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2} + 1
[7,4]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2} + 1
[7,3,1]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2} + 1
[7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2} + 1
[7,2,1,1]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2} + 1
[7,1,1,1,1]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2} + 1
[4,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2} + 1
[3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2} + 1
[3,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2} + 1
[2,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2} + 1
[2,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2} + 1
[2,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2} + 1
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2} + 1
[12]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2} + 1
[11,1]
=> [1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2} + 1
[10,2]
=> [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2} + 1
[10,1,1]
=> [1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2} + 1
[9,3]
=> [1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2} + 1
[9,2,1]
=> [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,1,1,0,0]
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2} + 1
[9,1,1,1]
=> [1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0,1,0,1,0]
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2} + 1
[8,4]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2} + 1
[8,3,1]
=> [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2} + 1
[8,2,2]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2} + 1
Description
The number of peaks visible from the left.
This is, the number of left-to-right maxima of the heights of the peaks of a Dyck path.
The following 116 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001487The number of inner corners of a skew partition. St000754The Grundy value for the game of removing nestings in a perfect matching. St000298The order dimension or Dushnik-Miller dimension of a poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001632The number of indecomposable injective modules I with dimExt1(I,A)=1 for the incidence algebra A of a poset. St000779The tier of a permutation. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001960The number of descents of a permutation minus one if its first entry is not one. St000640The rank of the largest boolean interval in a poset. St000486The number of cycles of length at least 3 of a permutation. St000538The number of even inversions of a permutation. St000710The number of big deficiencies of a permutation. St000836The number of descents of distance 2 of a permutation. St000260The radius of a connected graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St000664The number of right ropes of a permutation. St001728The number of invisible descents of a permutation. St000872The number of very big descents of a permutation. St001737The number of descents of type 2 in a permutation. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St000252The number of nodes of degree 3 of a binary tree. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. 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. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001174The Gorenstein dimension of the algebra A/I when I is the tilting module corresponding to the permutation in the Auslander algebra of K[x]/(xn). St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series L=[c0,c1,...,cn−1] such that n=c0<ci for all i>0 a special CNakayama algebra. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001549The number of restricted non-inversions between exceedances. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001188The number of simple modules S with grade inf at least two in the Nakayama algebra A corresponding to the Dyck path. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001928The number of non-overlapping descents in a permutation. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000058The order of a permutation. St000842The breadth of a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St000007The number of saliances of the permutation. St001095The number of non-isomorphic posets with precisely one further covering relation. St001399The distinguishing number of a poset. St000663The number of right floats of a permutation. St000850The number of 1/2-balanced pairs in a poset. St001199The dominant dimension of eAe for the corresponding Nakayama algebra A with minimal faithful projective-injective module eA. St000633The size of the automorphism group of a poset. St000649The number of 3-excedences of a permutation. St000454The largest eigenvalue of a graph if it is integral. St000352The Elizalde-Pak rank of a permutation. St001964The interval resolution global dimension of a poset. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000741The Colin de Verdière graph invariant. St000782The indicator function of whether a given perfect matching is an L & P matching. St000617The number of global maxima of a Dyck path. St001162The minimum jump of a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length 3. St001513The number of nested exceedences of a permutation. St001665The number of pure excedances of a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001530The depth of a Dyck path. St000488The number of cycles of a permutation of length at most 2. St000650The number of 3-rises of a permutation. St000954Number of times the corresponding LNakayama algebra has Ext^i(D(A),A)=0 for i>0. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001520The number of strict 3-descents. St001556The number of inversions of the third entry 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. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000353The number of inner valleys of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series L=[c_0,c_1,...,c_{n−1}] such that n=c_0 < c_i for all i > 0 a special CNakayama algebra. St001208The 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). St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001481The minimal height of a peak of a Dyck path. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001729The number of visible descents of a permutation. St000092The number of outer peaks of a permutation. St000542The number of left-to-right-minima of a permutation. 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. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St001200The number of simple modules in eAe with projective dimension at most 2 in the corresponding Nakayama algebra A with minimal faithful projective-injective module eA. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001820The size of the image of the pop stack sorting operator. St000317The cycle descent number of a permutation. St000552The number of cut vertices of a graph. St001577The minimal number of edges to add or remove to make a graph a cograph. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001330The hat guessing number of a graph. St000181The number of connected components of the Hasse diagram for the poset. St001490The number of connected components of a skew partition. St001890The maximum magnitude of the Möbius function of a poset. St000455The second largest eigenvalue of a graph if it is integral. St001846The number of elements which do not have a complement in the lattice.
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