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Your data matches 10 different statistics following compositions of up to 3 maps.
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Matching statistic: St000326
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [2]
=> 100 => 001 => 3
([],2)
=> [2,2]
=> 1100 => 0011 => 3
([(0,1)],2)
=> [3]
=> 1000 => 0001 => 4
([],3)
=> [2,2,2,2]
=> 111100 => 001111 => 3
([(1,2)],3)
=> [6]
=> 1000000 => 0000001 => 7
([(0,1),(0,2)],3)
=> [3,2]
=> 10100 => 00101 => 3
([(0,2),(2,1)],3)
=> [4]
=> 10000 => 00001 => 5
([(0,2),(1,2)],3)
=> [3,2]
=> 10100 => 00101 => 3
([(2,3)],4)
=> [6,6]
=> 11000000 => 00000011 => 7
([(1,2),(1,3)],4)
=> [6,2,2]
=> 100001100 => 001100001 => 3
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> 1011100 => 0011101 => 3
([(0,2),(0,3),(3,1)],4)
=> [7]
=> 10000000 => 00000001 => 8
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> 100100 => 001001 => 3
([(1,2),(2,3)],4)
=> [4,4]
=> 110000 => 000011 => 5
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> 100100 => 001001 => 3
([(1,3),(2,3)],4)
=> [6,2,2]
=> 100001100 => 001100001 => 3
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> 100100 => 001001 => 3
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> 1011100 => 0011101 => 3
([(0,3),(1,2)],4)
=> [3,3,3]
=> 111000 => 000111 => 4
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> 1001000 => 0001001 => 4
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> 101100 => 001101 => 3
([(0,3),(2,1),(3,2)],4)
=> [5]
=> 100000 => 000001 => 6
([(0,3),(1,2),(2,3)],4)
=> [7]
=> 10000000 => 00000001 => 8
([(3,4)],5)
=> [6,6,6,6]
=> 1111000000 => 0000001111 => 7
([(0,1),(0,2),(0,3),(0,4)],5)
=> [3,2,2,2,2,2,2,2]
=> 10111111100 => 00111111101 => 3
([(0,2),(0,3),(0,4),(4,1)],5)
=> [7,6]
=> 101000000 => 000000101 => 7
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [4,2,2,2]
=> 10011100 => 00111001 => 3
([(1,2),(1,3),(2,4),(3,4)],5)
=> [4,4,2,2]
=> 11001100 => 00110011 => 3
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> 1000100 => 0010001 => 3
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,3,3]
=> 1011000 => 0001101 => 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [5,4]
=> 1010000 => 0000101 => 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [4,2,2]
=> 1001100 => 0011001 => 3
([(2,3),(3,4)],5)
=> [4,4,4,4]
=> 11110000 => 00001111 => 5
([(1,4),(4,2),(4,3)],5)
=> [4,4,2,2]
=> 11001100 => 00110011 => 3
([(0,4),(4,1),(4,2),(4,3)],5)
=> [4,2,2,2]
=> 10011100 => 00111001 => 3
([(1,4),(2,4),(4,3)],5)
=> [4,4,2,2]
=> 11001100 => 00110011 => 3
([(0,4),(1,4),(4,2),(4,3)],5)
=> [4,2,2]
=> 1001100 => 0011001 => 3
([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> 10011100 => 00111001 => 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> [3,2,2,2,2,2,2,2]
=> 10111111100 => 00111111101 => 3
([(0,4),(1,4),(2,3)],5)
=> [6,3,3,3]
=> 1000111000 => 0001110001 => 4
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5,3,2,2]
=> 100101100 => 001101001 => 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2,2,2,2]
=> 10111100 => 00111101 => 3
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> 1000100 => 0010001 => 3
([(0,4),(1,4),(2,3),(2,4)],5)
=> [6,5,3]
=> 101001000 => 000100101 => 4
([(0,4),(1,4),(2,3),(3,4)],5)
=> [7,6]
=> 101000000 => 000000101 => 7
([(1,4),(2,3)],5)
=> [6,6,6]
=> 111000000 => 000000111 => 7
([(0,4),(1,2),(1,4),(2,3)],5)
=> [8,3]
=> 1000001000 => 0001000001 => 4
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [5,4]
=> 1010000 => 0000101 => 5
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [7,2]
=> 100000100 => 001000001 => 3
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [4,2,2]
=> 1001100 => 0011001 => 3
Description
The position of the first one in a binary word after appending a 1 at the end.
Regarding the binary word as a subset of $\{1,\dots,n,n+1\}$ that contains $n+1$, this is the minimal element of the set.
Matching statistic: St000993
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 78% ●values known / values provided: 89%●distinct values known / distinct values provided: 78%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 78% ●values known / values provided: 89%●distinct values known / distinct values provided: 78%
Values
([],1)
=> [2]
=> []
=> []
=> ? = 3 - 1
([],2)
=> [2,2]
=> [2]
=> [1,1]
=> 2 = 3 - 1
([(0,1)],2)
=> [3]
=> []
=> []
=> ? = 4 - 1
([],3)
=> [2,2,2,2]
=> [2,2,2]
=> [3,3]
=> 2 = 3 - 1
([(1,2)],3)
=> [6]
=> []
=> []
=> ? = 7 - 1
([(0,1),(0,2)],3)
=> [3,2]
=> [2]
=> [1,1]
=> 2 = 3 - 1
([(0,2),(2,1)],3)
=> [4]
=> []
=> []
=> ? = 5 - 1
([(0,2),(1,2)],3)
=> [3,2]
=> [2]
=> [1,1]
=> 2 = 3 - 1
([(2,3)],4)
=> [6,6]
=> [6]
=> [1,1,1,1,1,1]
=> 6 = 7 - 1
([(1,2),(1,3)],4)
=> [6,2,2]
=> [2,2]
=> [2,2]
=> 2 = 3 - 1
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> [2,2,2]
=> [3,3]
=> 2 = 3 - 1
([(0,2),(0,3),(3,1)],4)
=> [7]
=> []
=> []
=> ? = 8 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> [2]
=> [1,1]
=> 2 = 3 - 1
([(1,2),(2,3)],4)
=> [4,4]
=> [4]
=> [1,1,1,1]
=> 4 = 5 - 1
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> [2]
=> [1,1]
=> 2 = 3 - 1
([(1,3),(2,3)],4)
=> [6,2,2]
=> [2,2]
=> [2,2]
=> 2 = 3 - 1
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [2]
=> [1,1]
=> 2 = 3 - 1
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [2,2,2]
=> [3,3]
=> 2 = 3 - 1
([(0,3),(1,2)],4)
=> [3,3,3]
=> [3,3]
=> [2,2,2]
=> 3 = 4 - 1
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> [3]
=> [1,1,1]
=> 3 = 4 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> [2,2]
=> [2,2]
=> 2 = 3 - 1
([(0,3),(2,1),(3,2)],4)
=> [5]
=> []
=> []
=> ? = 6 - 1
([(0,3),(1,2),(2,3)],4)
=> [7]
=> []
=> []
=> ? = 8 - 1
([(3,4)],5)
=> [6,6,6,6]
=> [6,6,6]
=> [3,3,3,3,3,3]
=> ? = 7 - 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> [3,2,2,2,2,2,2,2]
=> [2,2,2,2,2,2,2]
=> [7,7]
=> ? = 3 - 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> [7,6]
=> [6]
=> [1,1,1,1,1,1]
=> 6 = 7 - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [4,2,2,2]
=> [2,2,2]
=> [3,3]
=> 2 = 3 - 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> [4,4,2,2]
=> [4,2,2]
=> [3,3,1,1]
=> 2 = 3 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> [2]
=> [1,1]
=> 2 = 3 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,3,3]
=> [3,3]
=> [2,2,2]
=> 3 = 4 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [5,4]
=> [4]
=> [1,1,1,1]
=> 4 = 5 - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [4,2,2]
=> [2,2]
=> [2,2]
=> 2 = 3 - 1
([(2,3),(3,4)],5)
=> [4,4,4,4]
=> [4,4,4]
=> [3,3,3,3]
=> 4 = 5 - 1
([(1,4),(4,2),(4,3)],5)
=> [4,4,2,2]
=> [4,2,2]
=> [3,3,1,1]
=> 2 = 3 - 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> [4,2,2,2]
=> [2,2,2]
=> [3,3]
=> 2 = 3 - 1
([(1,4),(2,4),(4,3)],5)
=> [4,4,2,2]
=> [4,2,2]
=> [3,3,1,1]
=> 2 = 3 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> [4,2,2]
=> [2,2]
=> [2,2]
=> 2 = 3 - 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> [2,2,2]
=> [3,3]
=> 2 = 3 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [3,2,2,2,2,2,2,2]
=> [2,2,2,2,2,2,2]
=> [7,7]
=> ? = 3 - 1
([(0,4),(1,4),(2,3)],5)
=> [6,3,3,3]
=> [3,3,3]
=> [3,3,3]
=> 3 = 4 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5,3,2,2]
=> [3,2,2]
=> [3,3,1]
=> 2 = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2,2,2,2]
=> [2,2,2,2]
=> [4,4]
=> 2 = 3 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [2]
=> [1,1]
=> 2 = 3 - 1
([(0,4),(1,4),(2,3),(2,4)],5)
=> [6,5,3]
=> [5,3]
=> [2,2,2,1,1]
=> 3 = 4 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [7,6]
=> [6]
=> [1,1,1,1,1,1]
=> 6 = 7 - 1
([(1,4),(2,3)],5)
=> [6,6,6]
=> [6,6]
=> [2,2,2,2,2,2]
=> 6 = 7 - 1
([(0,4),(1,2),(1,4),(2,3)],5)
=> [8,3]
=> [3]
=> [1,1,1]
=> 3 = 4 - 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [5,4]
=> [4]
=> [1,1,1,1]
=> 4 = 5 - 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [7,2]
=> [2]
=> [1,1]
=> 2 = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [4,2,2]
=> [2,2]
=> [2,2]
=> 2 = 3 - 1
([(0,4),(1,2),(1,4),(4,3)],5)
=> [10]
=> []
=> []
=> ? = 11 - 1
([(0,4),(1,2),(1,3)],5)
=> [6,3,3,3]
=> [3,3,3]
=> [3,3,3]
=> 3 = 4 - 1
([(0,4),(1,2),(1,3),(1,4)],5)
=> [6,5,3]
=> [5,3]
=> [2,2,2,1,1]
=> 3 = 4 - 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> [5,4]
=> [4]
=> [1,1,1,1]
=> 4 = 5 - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [8]
=> []
=> []
=> ? = 9 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [5,3,2,2]
=> [3,2,2]
=> [3,3,1]
=> 2 = 3 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [3,2,2,2,2]
=> [2,2,2,2]
=> [4,4]
=> 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [10]
=> []
=> []
=> ? = 11 - 1
([(0,3),(1,2),(1,4),(3,4)],5)
=> [8,3]
=> [3]
=> [1,1,1]
=> 3 = 4 - 1
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [7,2]
=> [2]
=> [1,1]
=> 2 = 3 - 1
([(1,4),(3,2),(4,3)],5)
=> [10]
=> []
=> []
=> ? = 11 - 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> [5,2]
=> [2]
=> [1,1]
=> 2 = 3 - 1
([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> []
=> []
=> ? = 9 - 1
([(0,4),(3,2),(4,1),(4,3)],5)
=> [8]
=> []
=> []
=> ? = 9 - 1
([(0,4),(1,2),(2,3),(2,4)],5)
=> [10]
=> []
=> []
=> ? = 11 - 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> []
=> []
=> ? = 7 - 1
([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> [3,3]
=> [2,2,2]
=> 3 = 4 - 1
([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> [4]
=> [1,1,1,1]
=> 4 = 5 - 1
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> [9]
=> []
=> []
=> ? = 10 - 1
([(0,4),(3,2),(4,5),(5,1),(5,3)],6)
=> [9]
=> []
=> []
=> ? = 10 - 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [7]
=> []
=> []
=> ? = 8 - 1
([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [9]
=> []
=> []
=> ? = 10 - 1
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> [9]
=> []
=> []
=> ? = 10 - 1
([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6)],7)
=> [5,5,4,2,2]
=> [5,4,2,2]
=> [4,4,2,2,1]
=> ? = 3 - 1
([(0,6),(1,6),(2,3),(3,5),(6,4)],7)
=> [4,4,4,4,4,4]
=> [4,4,4,4,4]
=> [5,5,5,5]
=> ? = 5 - 1
([(0,5),(0,6),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(6,4)],7)
=> [7,6,5,2]
=> [6,5,2]
=> [3,3,2,2,2,1]
=> ? = 3 - 1
([(0,5),(1,3),(1,4),(3,6),(4,6),(5,2)],7)
=> [4,4,4,4,4,4]
=> [4,4,4,4,4]
=> [5,5,5,5]
=> ? = 5 - 1
([(0,4),(0,5),(0,6),(1,2),(1,3),(2,6),(3,4),(3,5),(3,6)],7)
=> [7,6,5,2]
=> [6,5,2]
=> [3,3,2,2,2,1]
=> ? = 3 - 1
([(1,6),(2,3),(3,5),(3,6),(6,4)],7)
=> [6,6,6,6]
=> [6,6,6]
=> [3,3,3,3,3,3]
=> ? = 7 - 1
([(2,6),(4,5),(5,3),(6,4)],7)
=> [6,6,6,6]
=> [6,6,6]
=> [3,3,3,3,3,3]
=> ? = 7 - 1
([(0,6),(1,5),(5,4),(6,2),(6,3)],7)
=> [4,4,4,4,4,4]
=> [4,4,4,4,4]
=> [5,5,5,5]
=> ? = 5 - 1
([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> [10]
=> []
=> []
=> ? = 11 - 1
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> [5,5,4,2,2]
=> [5,4,2,2]
=> [4,4,2,2,1]
=> ? = 3 - 1
([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7)
=> [10]
=> []
=> []
=> ? = 11 - 1
([(0,5),(3,6),(4,1),(5,3),(6,2),(6,4)],7)
=> [10]
=> []
=> []
=> ? = 11 - 1
([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> [8]
=> []
=> []
=> ? = 9 - 1
([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> [10]
=> []
=> []
=> ? = 11 - 1
([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7)
=> [10]
=> []
=> []
=> ? = 11 - 1
Description
The multiplicity of the largest part of an integer partition.
Matching statistic: St000297
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [2]
=> [1,1]
=> 110 => 2 = 3 - 1
([],2)
=> [2,2]
=> [2,2]
=> 1100 => 2 = 3 - 1
([(0,1)],2)
=> [3]
=> [1,1,1]
=> 1110 => 3 = 4 - 1
([],3)
=> [2,2,2,2]
=> [4,4]
=> 110000 => 2 = 3 - 1
([(1,2)],3)
=> [6]
=> [1,1,1,1,1,1]
=> 1111110 => 6 = 7 - 1
([(0,1),(0,2)],3)
=> [3,2]
=> [2,2,1]
=> 11010 => 2 = 3 - 1
([(0,2),(2,1)],3)
=> [4]
=> [1,1,1,1]
=> 11110 => 4 = 5 - 1
([(0,2),(1,2)],3)
=> [3,2]
=> [2,2,1]
=> 11010 => 2 = 3 - 1
([(2,3)],4)
=> [6,6]
=> [2,2,2,2,2,2]
=> 11111100 => 6 = 7 - 1
([(1,2),(1,3)],4)
=> [6,2,2]
=> [3,3,1,1,1,1]
=> 110011110 => 2 = 3 - 1
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> [4,4,1]
=> 1100010 => 2 = 3 - 1
([(0,2),(0,3),(3,1)],4)
=> [7]
=> [1,1,1,1,1,1,1]
=> 11111110 => 7 = 8 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> [2,2,1,1]
=> 110110 => 2 = 3 - 1
([(1,2),(2,3)],4)
=> [4,4]
=> [2,2,2,2]
=> 111100 => 4 = 5 - 1
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> [2,2,1,1]
=> 110110 => 2 = 3 - 1
([(1,3),(2,3)],4)
=> [6,2,2]
=> [3,3,1,1,1,1]
=> 110011110 => 2 = 3 - 1
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [2,2,1,1]
=> 110110 => 2 = 3 - 1
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [4,4,1]
=> 1100010 => 2 = 3 - 1
([(0,3),(1,2)],4)
=> [3,3,3]
=> [3,3,3]
=> 111000 => 3 = 4 - 1
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> [2,2,2,1,1]
=> 1110110 => 3 = 4 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> [3,3,1]
=> 110010 => 2 = 3 - 1
([(0,3),(2,1),(3,2)],4)
=> [5]
=> [1,1,1,1,1]
=> 111110 => 5 = 6 - 1
([(0,3),(1,2),(2,3)],4)
=> [7]
=> [1,1,1,1,1,1,1]
=> 11111110 => 7 = 8 - 1
([(3,4)],5)
=> [6,6,6,6]
=> [4,4,4,4,4,4]
=> 1111110000 => ? = 7 - 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> [3,2,2,2,2,2,2,2]
=> [8,8,1]
=> 11000000010 => ? = 3 - 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> [7,6]
=> [2,2,2,2,2,2,1]
=> 111111010 => 6 = 7 - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [4,2,2,2]
=> [4,4,1,1]
=> 11000110 => 2 = 3 - 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> [4,4,2,2]
=> [4,4,2,2]
=> 11001100 => 2 = 3 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> [2,2,1,1,1]
=> 1101110 => 2 = 3 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,3,3]
=> [3,3,3,1]
=> 1110010 => 3 = 4 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [5,4]
=> [2,2,2,2,1]
=> 1111010 => 4 = 5 - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [4,2,2]
=> [3,3,1,1]
=> 1100110 => 2 = 3 - 1
([(2,3),(3,4)],5)
=> [4,4,4,4]
=> [4,4,4,4]
=> 11110000 => 4 = 5 - 1
([(1,4),(4,2),(4,3)],5)
=> [4,4,2,2]
=> [4,4,2,2]
=> 11001100 => 2 = 3 - 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> [4,2,2,2]
=> [4,4,1,1]
=> 11000110 => 2 = 3 - 1
([(1,4),(2,4),(4,3)],5)
=> [4,4,2,2]
=> [4,4,2,2]
=> 11001100 => 2 = 3 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> [4,2,2]
=> [3,3,1,1]
=> 1100110 => 2 = 3 - 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> [4,4,1,1]
=> 11000110 => 2 = 3 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [3,2,2,2,2,2,2,2]
=> [8,8,1]
=> 11000000010 => ? = 3 - 1
([(0,4),(1,4),(2,3)],5)
=> [6,3,3,3]
=> [4,4,4,1,1,1]
=> 1110001110 => ? = 4 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5,3,2,2]
=> [4,4,2,1,1]
=> 110010110 => 2 = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2,2,2,2]
=> [5,5,1]
=> 11000010 => 2 = 3 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [2,2,1,1,1]
=> 1101110 => 2 = 3 - 1
([(0,4),(1,4),(2,3),(2,4)],5)
=> [6,5,3]
=> [3,3,3,2,2,1]
=> 111011010 => 3 = 4 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [7,6]
=> [2,2,2,2,2,2,1]
=> 111111010 => 6 = 7 - 1
([(1,4),(2,3)],5)
=> [6,6,6]
=> [3,3,3,3,3,3]
=> 111111000 => 6 = 7 - 1
([(0,4),(1,2),(1,4),(2,3)],5)
=> [8,3]
=> [2,2,2,1,1,1,1,1]
=> 1110111110 => 3 = 4 - 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [5,4]
=> [2,2,2,2,1]
=> 1111010 => 4 = 5 - 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [7,2]
=> [2,2,1,1,1,1,1]
=> 110111110 => 2 = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [4,2,2]
=> [3,3,1,1]
=> 1100110 => 2 = 3 - 1
([(0,4),(1,2),(1,4),(4,3)],5)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1]
=> 11111111110 => 10 = 11 - 1
([(0,4),(1,2),(1,3)],5)
=> [6,3,3,3]
=> [4,4,4,1,1,1]
=> 1110001110 => ? = 4 - 1
([(0,4),(1,2),(1,3),(1,4)],5)
=> [6,5,3]
=> [3,3,3,2,2,1]
=> 111011010 => 3 = 4 - 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> [5,4]
=> [2,2,2,2,1]
=> 1111010 => 4 = 5 - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 111111110 => 8 = 9 - 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
=> [6,4,3,3]
=> [4,4,4,2,1,1]
=> 1110010110 => ? = 4 - 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5)],6)
=> [8,4,2]
=> [3,3,2,2,1,1,1,1]
=> 11011011110 => ? = 3 - 1
([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
=> [7,6,6]
=> [3,3,3,3,3,3,1]
=> 1111110010 => ? = 7 - 1
([(0,4),(0,5),(4,3),(5,1),(5,2)],6)
=> [6,4,3,3]
=> [4,4,4,2,1,1]
=> 1110010110 => ? = 4 - 1
([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
=> [8,4,2]
=> [3,3,2,2,1,1,1,1]
=> 11011011110 => ? = 3 - 1
([(0,5),(1,4),(2,4),(2,5),(5,3)],6)
=> [8,6,2]
=> [3,3,2,2,2,2,1,1]
=> 11011110110 => ? = 3 - 1
([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> [8,4,2]
=> [3,3,2,2,1,1,1,1]
=> 11011011110 => ? = 3 - 1
([(0,4),(1,4),(2,3),(2,5),(4,5)],6)
=> [8,6,3]
=> [3,3,3,2,2,2,1,1]
=> 11101110110 => ? = 4 - 1
([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> [8,6]
=> [2,2,2,2,2,2,1,1]
=> 1111110110 => ? = 7 - 1
([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> [6,4,3,3]
=> [4,4,4,2,1,1]
=> 1110010110 => ? = 4 - 1
([(0,5),(1,4),(3,5),(4,2),(4,3)],6)
=> [8,4,2]
=> [3,3,2,2,1,1,1,1]
=> 11011011110 => ? = 3 - 1
([(0,5),(1,4),(1,5),(4,2),(4,3)],6)
=> [8,6,3]
=> [3,3,3,2,2,2,1,1]
=> 11101110110 => ? = 4 - 1
([(0,4),(1,2),(1,4),(2,5),(4,3),(4,5)],6)
=> [8,4]
=> [2,2,2,2,1,1,1,1]
=> 1111011110 => ? = 5 - 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(3,5)],6)
=> [8,6,3]
=> [3,3,3,2,2,2,1,1]
=> 11101110110 => ? = 4 - 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(3,4),(3,5)],6)
=> [7,6,2]
=> [3,3,2,2,2,2,1]
=> 1101111010 => ? = 3 - 1
([(0,4),(1,2),(1,3),(1,4),(2,5),(3,5)],6)
=> [8,6,3]
=> [3,3,3,2,2,2,1,1]
=> 11101110110 => ? = 4 - 1
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> [8,6]
=> [2,2,2,2,2,2,1,1]
=> 1111110110 => ? = 7 - 1
([(0,5),(1,2),(1,3),(3,5),(5,4)],6)
=> [8,4,2]
=> [3,3,2,2,1,1,1,1]
=> 11011011110 => ? = 3 - 1
([(1,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [8,8]
=> [2,2,2,2,2,2,2,2]
=> 1111111100 => ? = 9 - 1
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> [6,4,3,3]
=> [4,4,4,2,1,1]
=> 1110010110 => ? = 4 - 1
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(5,2)],6)
=> [7,6,2]
=> [3,3,2,2,2,2,1]
=> 1101111010 => ? = 3 - 1
([(0,4),(0,5),(1,2),(1,4),(1,5),(2,3)],6)
=> [8,6,3]
=> [3,3,3,2,2,2,1,1]
=> 11101110110 => ? = 4 - 1
([(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(5,3)],6)
=> [7,5,2]
=> [3,3,2,2,2,1,1]
=> 1101110110 => ? = 3 - 1
([(0,4),(0,5),(1,3),(1,5),(5,2)],6)
=> [7,5,5]
=> [3,3,3,3,3,1,1]
=> 1111100110 => ? = 6 - 1
([(0,2),(0,5),(1,4),(1,5),(2,3),(2,4),(5,3)],6)
=> [8,4]
=> [2,2,2,2,1,1,1,1]
=> 1111011110 => ? = 5 - 1
([(0,3),(0,4),(1,2),(1,4),(2,5),(3,5)],6)
=> [8,7]
=> [2,2,2,2,2,2,2,1]
=> 1111111010 => ? = 8 - 1
([(0,3),(0,4),(1,2),(1,4),(2,5),(3,5),(4,5)],6)
=> [8,4,2]
=> [3,3,2,2,1,1,1,1]
=> 11011011110 => ? = 3 - 1
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,5)],6)
=> [8,6,3]
=> [3,3,3,2,2,2,1,1]
=> 11101110110 => ? = 4 - 1
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> [8,7]
=> [2,2,2,2,2,2,2,1]
=> 1111111010 => ? = 8 - 1
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4),(3,5)],6)
=> [7,5,2]
=> [3,3,2,2,2,1,1]
=> 1101110110 => ? = 3 - 1
([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
=> [7,4,4]
=> [3,3,3,3,1,1,1]
=> 1111001110 => ? = 5 - 1
([(0,4),(1,3),(1,5),(4,2),(4,5)],6)
=> [8,6,2]
=> [3,3,2,2,2,2,1,1]
=> 11011110110 => ? = 3 - 1
([(1,5),(2,3),(3,5),(5,4)],6)
=> [8,8]
=> [2,2,2,2,2,2,2,2]
=> 1111111100 => ? = 9 - 1
([(1,5),(4,3),(5,2),(5,4)],6)
=> [8,8]
=> [2,2,2,2,2,2,2,2]
=> 1111111100 => ? = 9 - 1
([(0,5),(4,3),(5,1),(5,2),(5,4)],6)
=> [8,6]
=> [2,2,2,2,2,2,1,1]
=> 1111110110 => ? = 7 - 1
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [3,2,2,2,2,2,2]
=> [7,7,1]
=> 1100000010 => ? = 3 - 1
([(0,5),(1,4),(2,3),(3,4),(3,5)],6)
=> [7,5,5]
=> [3,3,3,3,3,1,1]
=> 1111100110 => ? = 6 - 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [7,6,6]
=> [3,3,3,3,3,3,1]
=> 1111110010 => ? = 7 - 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> [8,5,2]
=> [3,3,2,2,2,1,1,1]
=> 11011101110 => ? = 3 - 1
([(0,2),(0,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(6,1)],7)
=> [8,6,2]
=> [3,3,2,2,2,2,1,1]
=> 11011110110 => ? = 3 - 1
([(0,3),(0,4),(3,5),(3,6),(4,2),(4,5),(4,6),(6,1)],7)
=> [9,8]
=> [2,2,2,2,2,2,2,2,1]
=> 11111111010 => ? = 9 - 1
([(0,5),(1,4),(2,4),(2,5),(4,6),(5,6),(6,3)],7)
=> [8,5,2]
=> [3,3,2,2,2,1,1,1]
=> 11011101110 => ? = 3 - 1
([(0,6),(1,6),(5,2),(6,3),(6,4),(6,5)],7)
=> [8,6,2]
=> [3,3,2,2,2,2,1,1]
=> 11011110110 => ? = 3 - 1
([(0,6),(1,6),(2,3),(2,6),(3,4),(3,5),(6,4),(6,5)],7)
=> [6,5,4,2]
=> [4,4,3,3,2,1]
=> 1101101010 => ? = 3 - 1
([(0,3),(0,4),(1,5),(2,5),(3,2),(3,6),(4,1),(4,6)],7)
=> [8,8]
=> [2,2,2,2,2,2,2,2]
=> 1111111100 => ? = 9 - 1
Description
The number of leading ones in a binary word.
Matching statistic: St000642
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
([],1)
=> ? = 3 - 1
([],2)
=> 2 = 3 - 1
([(0,1)],2)
=> 3 = 4 - 1
([],3)
=> 2 = 3 - 1
([(1,2)],3)
=> 6 = 7 - 1
([(0,1),(0,2)],3)
=> 2 = 3 - 1
([(0,2),(2,1)],3)
=> 4 = 5 - 1
([(0,2),(1,2)],3)
=> 2 = 3 - 1
([(2,3)],4)
=> 6 = 7 - 1
([(1,2),(1,3)],4)
=> 2 = 3 - 1
([(0,1),(0,2),(0,3)],4)
=> 2 = 3 - 1
([(0,2),(0,3),(3,1)],4)
=> 7 = 8 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
([(1,2),(2,3)],4)
=> 4 = 5 - 1
([(0,3),(3,1),(3,2)],4)
=> 2 = 3 - 1
([(1,3),(2,3)],4)
=> 2 = 3 - 1
([(0,3),(1,3),(3,2)],4)
=> 2 = 3 - 1
([(0,3),(1,3),(2,3)],4)
=> 2 = 3 - 1
([(0,3),(1,2)],4)
=> 3 = 4 - 1
([(0,3),(1,2),(1,3)],4)
=> 3 = 4 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 3 - 1
([(0,3),(2,1),(3,2)],4)
=> 5 = 6 - 1
([(0,3),(1,2),(2,3)],4)
=> 7 = 8 - 1
([(3,4)],5)
=> 6 = 7 - 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> 2 = 3 - 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> 6 = 7 - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 3 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> 3 = 4 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 4 = 5 - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2 = 3 - 1
([(2,3),(3,4)],5)
=> 4 = 5 - 1
([(1,4),(4,2),(4,3)],5)
=> 2 = 3 - 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> 2 = 3 - 1
([(1,4),(2,4),(4,3)],5)
=> 2 = 3 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> 2 = 3 - 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> 2 = 3 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
([(0,4),(1,4),(2,3)],5)
=> 3 = 4 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2 = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2 = 3 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> 2 = 3 - 1
([(0,4),(1,4),(2,3),(2,4)],5)
=> 3 = 4 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> 6 = 7 - 1
([(1,4),(2,3)],5)
=> 6 = 7 - 1
([(0,4),(1,2),(1,4),(2,3)],5)
=> 3 = 4 - 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> 2 = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> 2 = 3 - 1
([(0,4),(1,2),(1,4),(4,3)],5)
=> 10 = 11 - 1
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,1),(4,6),(6,5)],7)
=> ? = 6 - 1
([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
=> ? = 4 - 1
([(0,3),(1,2),(1,5),(1,6),(3,5),(3,6),(5,4),(6,4)],7)
=> ? = 3 - 1
([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
=> ? = 4 - 1
([(0,2),(0,3),(1,5),(1,6),(2,4),(3,1),(3,4),(4,5),(4,6)],7)
=> ? = 3 - 1
([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 6 - 1
([(0,3),(0,4),(2,5),(3,6),(4,2),(4,6),(6,1),(6,5)],7)
=> ? = 6 - 1
([(0,3),(0,5),(3,6),(4,1),(4,6),(5,4),(6,2)],7)
=> ? = 7 - 1
([(0,2),(0,3),(1,4),(1,6),(2,4),(2,5),(3,1),(3,5),(5,6)],7)
=> ? = 6 - 1
([(0,3),(0,4),(2,6),(3,5),(3,6),(4,2),(4,5),(6,1)],7)
=> ? = 7 - 1
([(0,3),(0,4),(2,5),(3,5),(3,6),(4,2),(4,6),(6,1)],7)
=> ? = 4 - 1
([(0,4),(0,5),(2,6),(4,2),(5,1),(5,6),(6,3)],7)
=> ? = 4 - 1
([(0,3),(1,2),(1,5),(2,6),(3,5),(3,6),(5,4),(6,4)],7)
=> ? = 4 - 1
([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7)
=> ? = 5 - 1
([(0,2),(0,4),(1,5),(1,6),(2,5),(2,6),(3,1),(4,3)],7)
=> ? = 3 - 1
([(0,2),(0,5),(2,6),(3,4),(4,1),(4,6),(5,3)],7)
=> ? = 7 - 1
([(0,6),(1,2),(1,3),(2,5),(2,6),(3,5),(3,6),(5,4)],7)
=> ? = 3 - 1
([(0,5),(1,2),(1,3),(2,6),(3,6),(5,6),(6,4)],7)
=> ? = 4 - 1
([(0,5),(1,3),(1,4),(3,6),(4,6),(5,2)],7)
=> ? = 5 - 1
([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? = 3 - 1
([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(5,6)],7)
=> ? = 3 - 1
([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 3 - 1
([(0,2),(0,3),(2,5),(2,6),(3,5),(3,6),(4,1),(6,4)],7)
=> ? = 3 - 1
([(0,3),(0,5),(3,6),(4,2),(5,1),(5,6),(6,4)],7)
=> ? = 7 - 1
([(0,4),(0,5),(1,6),(2,6),(4,2),(5,1),(6,3)],7)
=> ? = 4 - 1
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(3,6),(4,6),(5,2),(5,6)],7)
=> ? = 3 - 1
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,6),(3,6),(4,6),(5,2)],7)
=> ? = 3 - 1
([(0,4),(0,5),(1,2),(1,4),(1,5),(2,6),(4,6),(5,6),(6,3)],7)
=> ? = 3 - 1
([(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,6),(4,6),(5,3),(5,6)],7)
=> ? = 3 - 1
([(0,4),(0,5),(1,3),(1,5),(3,6),(4,6),(5,6),(6,2)],7)
=> ? = 3 - 1
([(0,3),(0,6),(1,2),(1,6),(2,5),(3,5),(5,4),(6,4)],7)
=> ? = 9 - 1
([(0,5),(0,6),(1,2),(1,3),(1,5),(2,6),(3,6),(6,4)],7)
=> ? = 6 - 1
([(0,2),(0,3),(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,1)],7)
=> ? = 3 - 1
([(0,2),(0,3),(0,4),(1,5),(1,6),(2,6),(3,5),(3,6),(4,1)],7)
=> ? = 9 - 1
([(0,4),(0,5),(0,6),(1,2),(1,3),(2,6),(3,4),(3,5),(3,6)],7)
=> ? = 3 - 1
([(0,5),(0,6),(1,2),(1,3),(2,6),(3,5),(3,6),(5,4),(6,4)],7)
=> ? = 3 - 1
([(0,2),(0,3),(1,5),(1,6),(2,6),(3,5),(5,4),(6,4)],7)
=> ? = 9 - 1
([(0,2),(0,5),(2,6),(3,1),(4,3),(4,6),(5,4)],7)
=> ? = 4 - 1
([(0,2),(0,4),(1,5),(2,5),(2,6),(3,1),(3,6),(4,3)],7)
=> ? = 7 - 1
([(0,3),(0,4),(3,6),(4,6),(5,1),(5,2),(6,5)],7)
=> ? = 3 - 1
([(0,5),(1,4),(1,6),(5,6),(6,2),(6,3)],7)
=> ? = 3 - 1
([(0,3),(1,5),(1,6),(3,5),(3,6),(5,4),(6,2),(6,4)],7)
=> ? = 6 - 1
([(0,3),(1,5),(1,6),(2,6),(3,2),(3,5),(5,4),(6,4)],7)
=> ? = 7 - 1
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 6 - 1
([(0,5),(0,6),(1,4),(3,5),(3,6),(4,3),(6,2)],7)
=> ? = 5 - 1
([(0,2),(1,5),(1,6),(2,3),(3,5),(3,6),(5,4),(6,4)],7)
=> ? = 3 - 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2),(4,6),(5,6)],7)
=> ? = 3 - 1
([(0,5),(1,6),(2,6),(5,1),(5,2),(6,3),(6,4)],7)
=> ? = 3 - 1
([(1,6),(2,3),(3,5),(3,6),(6,4)],7)
=> ? = 7 - 1
Description
The size of the smallest orbit of antichains under Panyushev complementation.
Matching statistic: St001038
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001038: Dyck paths ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 78%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001038: Dyck paths ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 78%
Values
([],1)
=> [2]
=> [1,0,1,0]
=> 2 = 3 - 1
([],2)
=> [2,2]
=> [1,1,1,0,0,0]
=> 2 = 3 - 1
([(0,1)],2)
=> [3]
=> [1,0,1,0,1,0]
=> 3 = 4 - 1
([],3)
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2 = 3 - 1
([(1,2)],3)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 7 - 1
([(0,1),(0,2)],3)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
([(0,2),(2,1)],3)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
([(0,2),(1,2)],3)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
([(2,3)],4)
=> [6,6]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> 6 = 7 - 1
([(1,2),(1,3)],4)
=> [6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 2 = 3 - 1
([(0,2),(0,3),(3,1)],4)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 7 = 8 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
([(1,2),(2,3)],4)
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> 4 = 5 - 1
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
([(1,3),(2,3)],4)
=> [6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 2 = 3 - 1
([(0,3),(1,2)],4)
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> 3 = 4 - 1
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 3 = 4 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
([(0,3),(2,1),(3,2)],4)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 6 - 1
([(0,3),(1,2),(2,3)],4)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 7 = 8 - 1
([(3,4)],5)
=> [6,6,6,6]
=> [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> ? = 7 - 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> [3,2,2,2,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 3 - 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> [7,6]
=> [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> 6 = 7 - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> 2 = 3 - 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> [4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> 2 = 3 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 3 = 4 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 4 = 5 - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
([(2,3),(3,4)],5)
=> [4,4,4,4]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 4 = 5 - 1
([(1,4),(4,2),(4,3)],5)
=> [4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> 2 = 3 - 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> [4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> 2 = 3 - 1
([(1,4),(2,4),(4,3)],5)
=> [4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> 2 = 3 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> 2 = 3 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [3,2,2,2,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 3 - 1
([(0,4),(1,4),(2,3)],5)
=> [6,3,3,3]
=> [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 4 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5,3,2,2]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> 2 = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> 2 = 3 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
([(0,4),(1,4),(2,3),(2,4)],5)
=> [6,5,3]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> 3 = 4 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [7,6]
=> [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> 6 = 7 - 1
([(1,4),(2,3)],5)
=> [6,6,6]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 7 - 1
([(0,4),(1,2),(1,4),(2,3)],5)
=> [8,3]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 4 - 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 4 = 5 - 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
([(0,4),(1,2),(1,4),(4,3)],5)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 11 - 1
([(0,4),(1,2),(1,3)],5)
=> [6,3,3,3]
=> [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 4 - 1
([(0,4),(1,2),(1,3),(1,4)],5)
=> [6,5,3]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> 3 = 4 - 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> [5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 4 = 5 - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 8 = 9 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [5,3,2,2]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> 2 = 3 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [3,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 11 - 1
([(0,3),(1,2),(1,4),(3,4)],5)
=> [8,3]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 4 - 1
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
([(1,4),(3,2),(4,3)],5)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 11 - 1
([(0,4),(1,2),(2,3),(2,4)],5)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 11 - 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
=> [6,4,3,3]
=> [1,0,1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 4 - 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5)],6)
=> [8,4,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 3 - 1
([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
=> [7,6,6]
=> [1,0,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 7 - 1
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
=> [8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3 - 1
([(0,4),(0,5),(4,3),(5,1),(5,2)],6)
=> [6,4,3,3]
=> [1,0,1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 4 - 1
([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
=> [8,4,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 3 - 1
([(0,5),(1,4),(2,4),(2,5),(5,3)],6)
=> [8,6,2]
=> [1,0,1,0,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> ? = 3 - 1
([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> [8,4,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 3 - 1
([(0,5),(1,5),(4,2),(5,3),(5,4)],6)
=> [8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3 - 1
([(0,4),(1,4),(2,3),(2,5),(4,5)],6)
=> [8,6,3]
=> [1,0,1,0,1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 4 - 1
([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> [8,6]
=> [1,0,1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 7 - 1
([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> [6,4,3,3]
=> [1,0,1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 4 - 1
([(0,5),(1,4),(3,5),(4,2),(4,3)],6)
=> [8,4,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 3 - 1
([(0,5),(1,4),(1,5),(4,2),(4,3)],6)
=> [8,6,3]
=> [1,0,1,0,1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 4 - 1
([(0,4),(1,2),(1,4),(2,5),(4,3),(4,5)],6)
=> [8,4]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> ? = 5 - 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(4,2),(5,3)],6)
=> [8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3 - 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(3,5)],6)
=> [8,6,3]
=> [1,0,1,0,1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 4 - 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(3,4),(3,5)],6)
=> [7,6,2]
=> [1,0,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> ? = 3 - 1
([(0,4),(1,2),(1,3),(1,4),(2,5),(3,5)],6)
=> [8,6,3]
=> [1,0,1,0,1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 4 - 1
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> [8,6]
=> [1,0,1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 7 - 1
([(0,5),(1,2),(1,3),(3,5),(5,4)],6)
=> [8,4,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 3 - 1
([(1,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [8,8]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 9 - 1
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10 - 1
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> [6,4,3,3]
=> [1,0,1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 4 - 1
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(5,2)],6)
=> [7,6,2]
=> [1,0,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> ? = 3 - 1
([(0,4),(0,5),(1,2),(1,4),(1,5),(2,3)],6)
=> [8,6,3]
=> [1,0,1,0,1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 4 - 1
([(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(5,3)],6)
=> [7,5,2]
=> [1,0,1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> ? = 3 - 1
([(0,4),(0,5),(1,3),(1,5),(5,2)],6)
=> [7,5,5]
=> [1,0,1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 6 - 1
([(0,2),(0,5),(1,4),(1,5),(2,3),(2,4),(5,3)],6)
=> [8,4]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> ? = 5 - 1
([(0,3),(0,4),(1,2),(1,4),(2,5),(3,5)],6)
=> [8,7]
=> [1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 8 - 1
([(0,3),(0,4),(1,2),(1,4),(2,5),(3,5),(4,5)],6)
=> [8,4,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 3 - 1
([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1)],6)
=> [8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3 - 1
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,5)],6)
=> [8,6,3]
=> [1,0,1,0,1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 4 - 1
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> [8,7]
=> [1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 8 - 1
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4),(3,5)],6)
=> [7,5,2]
=> [1,0,1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> ? = 3 - 1
([(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3 - 1
([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
=> [7,4,4]
=> [1,0,1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 5 - 1
([(0,4),(1,3),(1,5),(4,2),(4,5)],6)
=> [8,6,2]
=> [1,0,1,0,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> ? = 3 - 1
Description
The minimal height of a column in the parallelogram polyomino associated with the Dyck path.
Matching statistic: St000733
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 41% ●values known / values provided: 41%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 41% ●values known / values provided: 41%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [2]
=> [1,1]
=> [[1],[2]]
=> 2 = 3 - 1
([],2)
=> [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 2 = 3 - 1
([(0,1)],2)
=> [3]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3 = 4 - 1
([],3)
=> [2,2,2,2]
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> 2 = 3 - 1
([(1,2)],3)
=> [6]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> 6 = 7 - 1
([(0,1),(0,2)],3)
=> [3,2]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 2 = 3 - 1
([(0,2),(2,1)],3)
=> [4]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4 = 5 - 1
([(0,2),(1,2)],3)
=> [3,2]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 2 = 3 - 1
([(2,3)],4)
=> [6,6]
=> [2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]
=> 6 = 7 - 1
([(1,2),(1,3)],4)
=> [6,2,2]
=> [3,3,1,1,1,1]
=> [[1,6,7],[2,9,10],[3],[4],[5],[8]]
=> 2 = 3 - 1
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> [4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6]]
=> 2 = 3 - 1
([(0,2),(0,3),(3,1)],4)
=> [7]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> 7 = 8 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> 2 = 3 - 1
([(1,2),(2,3)],4)
=> [4,4]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> 4 = 5 - 1
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> 2 = 3 - 1
([(1,3),(2,3)],4)
=> [6,2,2]
=> [3,3,1,1,1,1]
=> [[1,6,7],[2,9,10],[3],[4],[5],[8]]
=> 2 = 3 - 1
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> 2 = 3 - 1
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6]]
=> 2 = 3 - 1
([(0,3),(1,2)],4)
=> [3,3,3]
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> 3 = 4 - 1
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> [2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5],[7]]
=> 3 = 4 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> [3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> 2 = 3 - 1
([(0,3),(2,1),(3,2)],4)
=> [5]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 5 = 6 - 1
([(0,3),(1,2),(2,3)],4)
=> [7]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> 7 = 8 - 1
([(3,4)],5)
=> [6,6,6,6]
=> [4,4,4,4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16],[17,18,19,20],[21,22,23,24]]
=> ? = 7 - 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> [3,2,2,2,2,2,2,2]
=> [8,8,1]
=> [[1,3,4,5,6,7,8,9],[2,11,12,13,14,15,16,17],[10]]
=> ? = 3 - 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> [7,6]
=> [2,2,2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8,11],[10,13],[12]]
=> ? = 7 - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [4,2,2,2]
=> [4,4,1,1]
=> [[1,4,5,6],[2,8,9,10],[3],[7]]
=> 2 = 3 - 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> [4,4,2,2]
=> [4,4,2,2]
=> [[1,2,7,8],[3,4,11,12],[5,6],[9,10]]
=> 2 = 3 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> [2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> 2 = 3 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,3,3]
=> [3,3,3,1]
=> [[1,3,4],[2,6,7],[5,9,10],[8]]
=> 3 = 4 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [5,4]
=> [2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8]]
=> 4 = 5 - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [4,2,2]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> 2 = 3 - 1
([(2,3),(3,4)],5)
=> [4,4,4,4]
=> [4,4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16]]
=> ? = 5 - 1
([(1,4),(4,2),(4,3)],5)
=> [4,4,2,2]
=> [4,4,2,2]
=> [[1,2,7,8],[3,4,11,12],[5,6],[9,10]]
=> 2 = 3 - 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> [4,2,2,2]
=> [4,4,1,1]
=> [[1,4,5,6],[2,8,9,10],[3],[7]]
=> 2 = 3 - 1
([(1,4),(2,4),(4,3)],5)
=> [4,4,2,2]
=> [4,4,2,2]
=> [[1,2,7,8],[3,4,11,12],[5,6],[9,10]]
=> 2 = 3 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> [4,2,2]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> 2 = 3 - 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> [4,4,1,1]
=> [[1,4,5,6],[2,8,9,10],[3],[7]]
=> 2 = 3 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [3,2,2,2,2,2,2,2]
=> [8,8,1]
=> [[1,3,4,5,6,7,8,9],[2,11,12,13,14,15,16,17],[10]]
=> ? = 3 - 1
([(0,4),(1,4),(2,3)],5)
=> [6,3,3,3]
=> [4,4,4,1,1,1]
=> [[1,5,6,7],[2,9,10,11],[3,13,14,15],[4],[8],[12]]
=> ? = 4 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5,3,2,2]
=> [4,4,2,1,1]
=> [[1,4,7,8],[2,6,11,12],[3,10],[5],[9]]
=> 2 = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2,2,2,2]
=> [5,5,1]
=> [[1,3,4,5,6],[2,8,9,10,11],[7]]
=> ? = 3 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> 2 = 3 - 1
([(0,4),(1,4),(2,3),(2,4)],5)
=> [6,5,3]
=> [3,3,3,2,2,1]
=> [[1,3,8],[2,5,11],[4,7,14],[6,10],[9,13],[12]]
=> ? = 4 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [7,6]
=> [2,2,2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8,11],[10,13],[12]]
=> ? = 7 - 1
([(1,4),(2,3)],5)
=> [6,6,6]
=> [3,3,3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12],[13,14,15],[16,17,18]]
=> ? = 7 - 1
([(0,4),(1,2),(1,4),(2,3)],5)
=> [8,3]
=> [2,2,2,1,1,1,1,1]
=> [[1,7],[2,9],[3,11],[4],[5],[6],[8],[10]]
=> ? = 4 - 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [5,4]
=> [2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8]]
=> 4 = 5 - 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [7,2]
=> [2,2,1,1,1,1,1]
=> [[1,7],[2,9],[3],[4],[5],[6],[8]]
=> 2 = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [4,2,2]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> 2 = 3 - 1
([(0,4),(1,2),(1,4),(4,3)],5)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> 10 = 11 - 1
([(0,4),(1,2),(1,3)],5)
=> [6,3,3,3]
=> [4,4,4,1,1,1]
=> [[1,5,6,7],[2,9,10,11],[3,13,14,15],[4],[8],[12]]
=> ? = 4 - 1
([(0,4),(1,2),(1,3),(1,4)],5)
=> [6,5,3]
=> [3,3,3,2,2,1]
=> [[1,3,8],[2,5,11],[4,7,14],[6,10],[9,13],[12]]
=> ? = 4 - 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> [5,4]
=> [2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8]]
=> 4 = 5 - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> 8 = 9 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [5,3,2,2]
=> [4,4,2,1,1]
=> [[1,4,7,8],[2,6,11,12],[3,10],[5],[9]]
=> 2 = 3 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [3,2,2,2,2]
=> [5,5,1]
=> [[1,3,4,5,6],[2,8,9,10,11],[7]]
=> ? = 3 - 1
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> 10 = 11 - 1
([(0,3),(1,2),(1,4),(3,4)],5)
=> [8,3]
=> [2,2,2,1,1,1,1,1]
=> [[1,7],[2,9],[3,11],[4],[5],[6],[8],[10]]
=> ? = 4 - 1
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [7,2]
=> [2,2,1,1,1,1,1]
=> [[1,7],[2,9],[3],[4],[5],[6],[8]]
=> 2 = 3 - 1
([(1,4),(3,2),(4,3)],5)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> 10 = 11 - 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> [5,2]
=> [2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> 2 = 3 - 1
([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> 8 = 9 - 1
([(0,4),(3,2),(4,1),(4,3)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> 8 = 9 - 1
([(0,4),(1,2),(2,3),(2,4)],5)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> 10 = 11 - 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
=> [6,4,3,3]
=> [4,4,4,2,1,1]
=> [[1,4,7,8],[2,6,11,12],[3,10,15,16],[5,14],[9],[13]]
=> ? = 4 - 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5)],6)
=> [8,4,2]
=> [3,3,2,2,1,1,1,1]
=> [[1,6,11],[2,8,14],[3,10],[4,13],[5],[7],[9],[12]]
=> ? = 3 - 1
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2,2,2,2]
=> [5,5,1,1]
=> [[1,4,5,6,7],[2,9,10,11,12],[3],[8]]
=> ? = 3 - 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
=> [6,5,4]
=> [3,3,3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8,12],[7,11,15],[10,14],[13]]
=> ? = 5 - 1
([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
=> [7,6,6]
=> [3,3,3,3,3,3,1]
=> [[1,3,4],[2,6,7],[5,9,10],[8,12,13],[11,15,16],[14,18,19],[17]]
=> ? = 7 - 1
([(0,4),(0,5),(4,3),(5,1),(5,2)],6)
=> [6,4,3,3]
=> [4,4,4,2,1,1]
=> [[1,4,7,8],[2,6,11,12],[3,10,15,16],[5,14],[9],[13]]
=> ? = 4 - 1
([(0,3),(0,4),(3,5),(4,1),(4,2),(4,5)],6)
=> [6,5,4]
=> [3,3,3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8,12],[7,11,15],[10,14],[13]]
=> ? = 5 - 1
([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
=> [8,4,2]
=> [3,3,2,2,1,1,1,1]
=> [[1,6,11],[2,8,14],[3,10],[4,13],[5],[7],[9],[12]]
=> ? = 3 - 1
([(0,1),(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [4,2,2,2,2]
=> [5,5,1,1]
=> [[1,4,5,6,7],[2,9,10,11,12],[3],[8]]
=> ? = 3 - 1
([(0,5),(1,5),(5,2),(5,3),(5,4)],6)
=> [4,2,2,2,2]
=> [5,5,1,1]
=> [[1,4,5,6,7],[2,9,10,11,12],[3],[8]]
=> ? = 3 - 1
([(0,5),(1,5),(2,5),(5,3),(5,4)],6)
=> [4,2,2,2,2]
=> [5,5,1,1]
=> [[1,4,5,6,7],[2,9,10,11,12],[3],[8]]
=> ? = 3 - 1
([(0,5),(1,4),(2,4),(2,5),(5,3)],6)
=> [8,6,2]
=> [3,3,2,2,2,2,1,1]
=> [[1,4,13],[2,6,16],[3,8],[5,10],[7,12],[9,15],[11],[14]]
=> ? = 3 - 1
([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> [8,4,2]
=> [3,3,2,2,1,1,1,1]
=> [[1,6,11],[2,8,14],[3,10],[4,13],[5],[7],[9],[12]]
=> ? = 3 - 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [4,2,2,2,2]
=> [5,5,1,1]
=> [[1,4,5,6,7],[2,9,10,11,12],[3],[8]]
=> ? = 3 - 1
([(0,5),(1,5),(2,3),(2,5),(3,4),(5,4)],6)
=> [6,5,4]
=> [3,3,3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8,12],[7,11,15],[10,14],[13]]
=> ? = 5 - 1
([(0,4),(1,4),(2,3),(2,5),(4,5)],6)
=> [8,6,3]
=> [3,3,3,2,2,2,1,1]
=> [[1,4,11],[2,6,14],[3,8,17],[5,10],[7,13],[9,16],[12],[15]]
=> ? = 4 - 1
([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> [8,6]
=> [2,2,2,2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5,10],[7,12],[9,14],[11],[13]]
=> ? = 7 - 1
([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> [6,4,3,3]
=> [4,4,4,2,1,1]
=> [[1,4,7,8],[2,6,11,12],[3,10,15,16],[5,14],[9],[13]]
=> ? = 4 - 1
([(0,5),(1,4),(3,5),(4,2),(4,3)],6)
=> [8,4,2]
=> [3,3,2,2,1,1,1,1]
=> [[1,6,11],[2,8,14],[3,10],[4,13],[5],[7],[9],[12]]
=> ? = 3 - 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [5,4,4,4]
=> [4,4,4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6,11,12,13],[10,15,16,17],[14]]
=> ? = 5 - 1
([(0,5),(1,4),(1,5),(4,2),(4,3)],6)
=> [8,6,3]
=> [3,3,3,2,2,2,1,1]
=> [[1,4,11],[2,6,14],[3,8,17],[5,10],[7,13],[9,16],[12],[15]]
=> ? = 4 - 1
([(0,4),(1,2),(1,4),(2,3),(2,5),(4,5)],6)
=> [6,4,3]
=> [3,3,3,2,1,1]
=> [[1,4,7],[2,6,10],[3,9,13],[5,12],[8],[11]]
=> ? = 4 - 1
([(0,4),(1,2),(1,4),(2,5),(4,3),(4,5)],6)
=> [8,4]
=> [2,2,2,2,1,1,1,1]
=> [[1,6],[2,8],[3,10],[4,12],[5],[7],[9],[11]]
=> ? = 5 - 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(3,5)],6)
=> [8,6,3]
=> [3,3,3,2,2,2,1,1]
=> [[1,4,11],[2,6,14],[3,8,17],[5,10],[7,13],[9,16],[12],[15]]
=> ? = 4 - 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(3,4),(3,5)],6)
=> [7,6,2]
=> [3,3,2,2,2,2,1]
=> [[1,3,12],[2,5,15],[4,7],[6,9],[8,11],[10,14],[13]]
=> ? = 3 - 1
([(0,4),(1,2),(1,3),(1,4),(2,5),(3,5)],6)
=> [8,6,3]
=> [3,3,3,2,2,2,1,1]
=> [[1,4,11],[2,6,14],[3,8,17],[5,10],[7,13],[9,16],[12],[15]]
=> ? = 4 - 1
([(0,4),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> [6,5,4]
=> [3,3,3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8,12],[7,11,15],[10,14],[13]]
=> ? = 5 - 1
([(0,2),(0,3),(0,5),(4,1),(5,4)],6)
=> [5,4,4,4]
=> [4,4,4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6,11,12,13],[10,15,16,17],[14]]
=> ? = 5 - 1
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> [8,6]
=> [2,2,2,2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5,10],[7,12],[9,14],[11],[13]]
=> ? = 7 - 1
([(0,5),(1,2),(1,3),(3,5),(5,4)],6)
=> [8,4,2]
=> [3,3,2,2,1,1,1,1]
=> [[1,6,11],[2,8,14],[3,10],[4,13],[5],[7],[9],[12]]
=> ? = 3 - 1
([(1,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [8,8]
=> [2,2,2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14],[15,16]]
=> ? = 9 - 1
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> [6,4,3,3]
=> [4,4,4,2,1,1]
=> [[1,4,7,8],[2,6,11,12],[3,10,15,16],[5,14],[9],[13]]
=> ? = 4 - 1
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(5,2)],6)
=> [7,6,2]
=> [3,3,2,2,2,2,1]
=> [[1,3,12],[2,5,15],[4,7],[6,9],[8,11],[10,14],[13]]
=> ? = 3 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> [4,2,2,2,2]
=> [5,5,1,1]
=> [[1,4,5,6,7],[2,9,10,11,12],[3],[8]]
=> ? = 3 - 1
([(0,4),(0,5),(1,2),(1,4),(1,5),(2,3)],6)
=> [8,6,3]
=> [3,3,3,2,2,2,1,1]
=> [[1,4,11],[2,6,14],[3,8,17],[5,10],[7,13],[9,16],[12],[15]]
=> ? = 4 - 1
Description
The row containing the largest entry of a standard tableau.
Matching statistic: St000745
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 100%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [2]
=> [[1,2]]
=> [[1],[2]]
=> 2 = 3 - 1
([],2)
=> [2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 2 = 3 - 1
([(0,1)],2)
=> [3]
=> [[1,2,3]]
=> [[1],[2],[3]]
=> 3 = 4 - 1
([],3)
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 3 - 1
([(1,2)],3)
=> [6]
=> [[1,2,3,4,5,6]]
=> [[1],[2],[3],[4],[5],[6]]
=> 6 = 7 - 1
([(0,1),(0,2)],3)
=> [3,2]
=> [[1,2,5],[3,4]]
=> [[1,3],[2,4],[5]]
=> 2 = 3 - 1
([(0,2),(2,1)],3)
=> [4]
=> [[1,2,3,4]]
=> [[1],[2],[3],[4]]
=> 4 = 5 - 1
([(0,2),(1,2)],3)
=> [3,2]
=> [[1,2,5],[3,4]]
=> [[1,3],[2,4],[5]]
=> 2 = 3 - 1
([(2,3)],4)
=> [6,6]
=> [[1,2,3,4,5,6],[7,8,9,10,11,12]]
=> [[1,7],[2,8],[3,9],[4,10],[5,11],[6,12]]
=> 6 = 7 - 1
([(1,2),(1,3)],4)
=> [6,2,2]
=> [[1,2,7,8,9,10],[3,4],[5,6]]
=> [[1,3,5],[2,4,6],[7],[8],[9],[10]]
=> 2 = 3 - 1
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> [[1,2,9],[3,4],[5,6],[7,8]]
=> [[1,3,5,7],[2,4,6,8],[9]]
=> 2 = 3 - 1
([(0,2),(0,3),(3,1)],4)
=> [7]
=> [[1,2,3,4,5,6,7]]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> 7 = 8 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> [[1,3],[2,4],[5],[6]]
=> 2 = 3 - 1
([(1,2),(2,3)],4)
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [[1,5],[2,6],[3,7],[4,8]]
=> 4 = 5 - 1
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> [[1,3],[2,4],[5],[6]]
=> 2 = 3 - 1
([(1,3),(2,3)],4)
=> [6,2,2]
=> [[1,2,7,8,9,10],[3,4],[5,6]]
=> [[1,3,5],[2,4,6],[7],[8],[9],[10]]
=> 2 = 3 - 1
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> [[1,3],[2,4],[5],[6]]
=> 2 = 3 - 1
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [[1,2,9],[3,4],[5,6],[7,8]]
=> [[1,3,5,7],[2,4,6,8],[9]]
=> 2 = 3 - 1
([(0,3),(1,2)],4)
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [[1,4,7],[2,5,8],[3,6,9]]
=> 3 = 4 - 1
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> [[1,4],[2,5],[3,6],[7],[8]]
=> 3 = 4 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [[1,3,5],[2,4,6],[7]]
=> 2 = 3 - 1
([(0,3),(2,1),(3,2)],4)
=> [5]
=> [[1,2,3,4,5]]
=> [[1],[2],[3],[4],[5]]
=> 5 = 6 - 1
([(0,3),(1,2),(2,3)],4)
=> [7]
=> [[1,2,3,4,5,6,7]]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> 7 = 8 - 1
([(3,4)],5)
=> [6,6,6,6]
=> [[1,2,3,4,5,6],[7,8,9,10,11,12],[13,14,15,16,17,18],[19,20,21,22,23,24]]
=> [[1,7,13,19],[2,8,14,20],[3,9,15,21],[4,10,16,22],[5,11,17,23],[6,12,18,24]]
=> ? = 7 - 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> [3,2,2,2,2,2,2,2]
=> [[1,2,17],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14],[15,16]]
=> [[1,3,5,7,9,11,13,15],[2,4,6,8,10,12,14,16],[17]]
=> ? = 3 - 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> [7,6]
=> [[1,2,3,4,5,6,13],[7,8,9,10,11,12]]
=> [[1,7],[2,8],[3,9],[4,10],[5,11],[6,12],[13]]
=> ? = 7 - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [4,2,2,2]
=> [[1,2,9,10],[3,4],[5,6],[7,8]]
=> [[1,3,5,7],[2,4,6,8],[9],[10]]
=> 2 = 3 - 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> [4,4,2,2]
=> [[1,2,7,8],[3,4,11,12],[5,6],[9,10]]
=> [[1,3,5,9],[2,4,6,10],[7,11],[8,12]]
=> ? = 3 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> [[1,2,5,6,7],[3,4]]
=> [[1,3],[2,4],[5],[6],[7]]
=> 2 = 3 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,3,3]
=> [[1,2,3,10],[4,5,6],[7,8,9]]
=> [[1,4,7],[2,5,8],[3,6,9],[10]]
=> 3 = 4 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> [[1,5],[2,6],[3,7],[4,8],[9]]
=> 4 = 5 - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> [[1,3,5],[2,4,6],[7],[8]]
=> 2 = 3 - 1
([(2,3),(3,4)],5)
=> [4,4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16]]
=> [[1,5,9,13],[2,6,10,14],[3,7,11,15],[4,8,12,16]]
=> ? = 5 - 1
([(1,4),(4,2),(4,3)],5)
=> [4,4,2,2]
=> [[1,2,7,8],[3,4,11,12],[5,6],[9,10]]
=> [[1,3,5,9],[2,4,6,10],[7,11],[8,12]]
=> ? = 3 - 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> [4,2,2,2]
=> [[1,2,9,10],[3,4],[5,6],[7,8]]
=> [[1,3,5,7],[2,4,6,8],[9],[10]]
=> 2 = 3 - 1
([(1,4),(2,4),(4,3)],5)
=> [4,4,2,2]
=> [[1,2,7,8],[3,4,11,12],[5,6],[9,10]]
=> [[1,3,5,9],[2,4,6,10],[7,11],[8,12]]
=> ? = 3 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> [[1,3,5],[2,4,6],[7],[8]]
=> 2 = 3 - 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> [[1,2,9,10],[3,4],[5,6],[7,8]]
=> [[1,3,5,7],[2,4,6,8],[9],[10]]
=> 2 = 3 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [3,2,2,2,2,2,2,2]
=> [[1,2,17],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14],[15,16]]
=> [[1,3,5,7,9,11,13,15],[2,4,6,8,10,12,14,16],[17]]
=> ? = 3 - 1
([(0,4),(1,4),(2,3)],5)
=> [6,3,3,3]
=> [[1,2,3,13,14,15],[4,5,6],[7,8,9],[10,11,12]]
=> [[1,4,7,10],[2,5,8,11],[3,6,9,12],[13],[14],[15]]
=> ? = 4 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5,3,2,2]
=> [[1,2,7,11,12],[3,4,10],[5,6],[8,9]]
=> [[1,3,5,8],[2,4,6,9],[7,10],[11],[12]]
=> ? = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2,2,2,2]
=> [[1,2,11],[3,4],[5,6],[7,8],[9,10]]
=> [[1,3,5,7,9],[2,4,6,8,10],[11]]
=> ? = 3 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [[1,2,5,6,7],[3,4]]
=> [[1,3],[2,4],[5],[6],[7]]
=> 2 = 3 - 1
([(0,4),(1,4),(2,3),(2,4)],5)
=> [6,5,3]
=> [[1,2,3,7,8,14],[4,5,6,12,13],[9,10,11]]
=> [[1,4,9],[2,5,10],[3,6,11],[7,12],[8,13],[14]]
=> ? = 4 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [7,6]
=> [[1,2,3,4,5,6,13],[7,8,9,10,11,12]]
=> [[1,7],[2,8],[3,9],[4,10],[5,11],[6,12],[13]]
=> ? = 7 - 1
([(1,4),(2,3)],5)
=> [6,6,6]
=> [[1,2,3,4,5,6],[7,8,9,10,11,12],[13,14,15,16,17,18]]
=> [[1,7,13],[2,8,14],[3,9,15],[4,10,16],[5,11,17],[6,12,18]]
=> ? = 7 - 1
([(0,4),(1,2),(1,4),(2,3)],5)
=> [8,3]
=> [[1,2,3,7,8,9,10,11],[4,5,6]]
=> [[1,4],[2,5],[3,6],[7],[8],[9],[10],[11]]
=> ? = 4 - 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> [[1,5],[2,6],[3,7],[4,8],[9]]
=> 4 = 5 - 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> [[1,3],[2,4],[5],[6],[7],[8],[9]]
=> 2 = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> [[1,3,5],[2,4,6],[7],[8]]
=> 2 = 3 - 1
([(0,4),(1,2),(1,4),(4,3)],5)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> 10 = 11 - 1
([(0,4),(1,2),(1,3)],5)
=> [6,3,3,3]
=> [[1,2,3,13,14,15],[4,5,6],[7,8,9],[10,11,12]]
=> [[1,4,7,10],[2,5,8,11],[3,6,9,12],[13],[14],[15]]
=> ? = 4 - 1
([(0,4),(1,2),(1,3),(1,4)],5)
=> [6,5,3]
=> [[1,2,3,7,8,14],[4,5,6,12,13],[9,10,11]]
=> [[1,4,9],[2,5,10],[3,6,11],[7,12],[8,13],[14]]
=> ? = 4 - 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> [5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> [[1,5],[2,6],[3,7],[4,8],[9]]
=> 4 = 5 - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [8]
=> [[1,2,3,4,5,6,7,8]]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> 8 = 9 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [5,3,2,2]
=> [[1,2,7,11,12],[3,4,10],[5,6],[8,9]]
=> [[1,3,5,8],[2,4,6,9],[7,10],[11],[12]]
=> ? = 3 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [3,2,2,2,2]
=> [[1,2,11],[3,4],[5,6],[7,8],[9,10]]
=> [[1,3,5,7,9],[2,4,6,8,10],[11]]
=> ? = 3 - 1
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> 10 = 11 - 1
([(0,3),(1,2),(1,4),(3,4)],5)
=> [8,3]
=> [[1,2,3,7,8,9,10,11],[4,5,6]]
=> [[1,4],[2,5],[3,6],[7],[8],[9],[10],[11]]
=> ? = 4 - 1
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> [[1,3],[2,4],[5],[6],[7],[8],[9]]
=> 2 = 3 - 1
([(1,4),(3,2),(4,3)],5)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> 10 = 11 - 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> [5,2]
=> [[1,2,5,6,7],[3,4]]
=> [[1,3],[2,4],[5],[6],[7]]
=> 2 = 3 - 1
([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> [[1,2,3,4,5,6,7,8]]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> 8 = 9 - 1
([(0,4),(3,2),(4,1),(4,3)],5)
=> [8]
=> [[1,2,3,4,5,6,7,8]]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> 8 = 9 - 1
([(0,4),(1,2),(2,3),(2,4)],5)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> 10 = 11 - 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> [[1,2,3,4,5,6]]
=> [[1],[2],[3],[4],[5],[6]]
=> 6 = 7 - 1
([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> [[1,2,3,10],[4,5,6],[7,8,9]]
=> [[1,4,7],[2,5,8],[3,6,9],[10]]
=> 3 = 4 - 1
([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> [[1,5],[2,6],[3,7],[4,8],[9]]
=> 4 = 5 - 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [5,2]
=> [[1,2,5,6,7],[3,4]]
=> [[1,3],[2,4],[5],[6],[7]]
=> 2 = 3 - 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> [5,2,2,2]
=> [[1,2,9,10,11],[3,4],[5,6],[7,8]]
=> [[1,3,5,7],[2,4,6,8],[9],[10],[11]]
=> ? = 3 - 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
=> [6,4,3,3]
=> [[1,2,3,10,15,16],[4,5,6,14],[7,8,9],[11,12,13]]
=> ?
=> ? = 4 - 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5)],6)
=> [8,4,2]
=> [[1,2,5,6,11,12,13,14],[3,4,9,10],[7,8]]
=> ?
=> ? = 3 - 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,4,2,2]
=> [[1,2,7,8,13],[3,4,11,12],[5,6],[9,10]]
=> [[1,3,5,9],[2,4,6,10],[7,11],[8,12],[13]]
=> ? = 3 - 1
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2,2,2,2]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ?
=> ? = 3 - 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
=> [6,5,4]
=> [[1,2,3,4,9,15],[5,6,7,8,14],[10,11,12,13]]
=> [[1,5,10],[2,6,11],[3,7,12],[4,8,13],[9,14],[15]]
=> ? = 5 - 1
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
=> [5,4,2,2]
=> [[1,2,7,8,13],[3,4,11,12],[5,6],[9,10]]
=> [[1,3,5,9],[2,4,6,10],[7,11],[8,12],[13]]
=> ? = 3 - 1
([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
=> [7,6,6]
=> [[1,2,3,4,5,6,19],[7,8,9,10,11,12],[13,14,15,16,17,18]]
=> ?
=> ? = 7 - 1
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> [5,2,2]
=> [[1,2,7,8,9],[3,4],[5,6]]
=> [[1,3,5],[2,4,6],[7],[8],[9]]
=> 2 = 3 - 1
([(0,4),(0,5),(4,3),(5,1),(5,2)],6)
=> [6,4,3,3]
=> [[1,2,3,10,15,16],[4,5,6,14],[7,8,9],[11,12,13]]
=> ?
=> ? = 4 - 1
([(0,3),(0,4),(3,5),(4,1),(4,2),(4,5)],6)
=> [6,5,4]
=> [[1,2,3,4,9,15],[5,6,7,8,14],[10,11,12,13]]
=> [[1,5,10],[2,6,11],[3,7,12],[4,8,13],[9,14],[15]]
=> ? = 5 - 1
([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
=> [8,4,2]
=> [[1,2,5,6,11,12,13,14],[3,4,9,10],[7,8]]
=> ?
=> ? = 3 - 1
([(0,2),(0,3),(2,4),(2,5),(3,1),(3,4),(3,5)],6)
=> [5,4,2,2]
=> [[1,2,7,8,13],[3,4,11,12],[5,6],[9,10]]
=> [[1,3,5,9],[2,4,6,10],[7,11],[8,12],[13]]
=> ? = 3 - 1
([(0,1),(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [4,2,2,2,2]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ?
=> ? = 3 - 1
([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
=> [5,2,2,2]
=> [[1,2,9,10,11],[3,4],[5,6],[7,8]]
=> [[1,3,5,7],[2,4,6,8],[9],[10],[11]]
=> ? = 3 - 1
([(0,5),(1,5),(5,2),(5,3),(5,4)],6)
=> [4,2,2,2,2]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ?
=> ? = 3 - 1
([(0,5),(1,5),(2,5),(5,3),(5,4)],6)
=> [4,2,2,2,2]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ?
=> ? = 3 - 1
([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
=> [5,2,2,2]
=> [[1,2,9,10,11],[3,4],[5,6],[7,8]]
=> [[1,3,5,7],[2,4,6,8],[9],[10],[11]]
=> ? = 3 - 1
([(0,5),(1,4),(2,4),(2,5),(5,3)],6)
=> [8,6,2]
=> [[1,2,5,6,7,8,15,16],[3,4,11,12,13,14],[9,10]]
=> ?
=> ? = 3 - 1
([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> [8,4,2]
=> [[1,2,5,6,11,12,13,14],[3,4,9,10],[7,8]]
=> ?
=> ? = 3 - 1
([(0,5),(1,3),(1,5),(2,3),(2,5),(3,4),(5,4)],6)
=> [5,4,2,2]
=> [[1,2,7,8,13],[3,4,11,12],[5,6],[9,10]]
=> [[1,3,5,9],[2,4,6,10],[7,11],[8,12],[13]]
=> ? = 3 - 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [4,2,2,2,2]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ?
=> ? = 3 - 1
([(0,5),(1,5),(2,3),(2,5),(3,4),(5,4)],6)
=> [6,5,4]
=> [[1,2,3,4,9,15],[5,6,7,8,14],[10,11,12,13]]
=> [[1,5,10],[2,6,11],[3,7,12],[4,8,13],[9,14],[15]]
=> ? = 5 - 1
([(0,4),(1,4),(2,3),(2,5),(4,5)],6)
=> [8,6,3]
=> [[1,2,3,7,8,9,16,17],[4,5,6,13,14,15],[10,11,12]]
=> ?
=> ? = 4 - 1
([(0,3),(0,4),(1,5),(2,5),(4,1),(4,2)],6)
=> [5,4,2,2]
=> [[1,2,7,8,13],[3,4,11,12],[5,6],[9,10]]
=> [[1,3,5,9],[2,4,6,10],[7,11],[8,12],[13]]
=> ? = 3 - 1
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> [5,3,3]
=> [[1,2,3,10,11],[4,5,6],[7,8,9]]
=> [[1,4,7],[2,5,8],[3,6,9],[10],[11]]
=> ? = 4 - 1
([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> [8,6]
=> [[1,2,3,4,5,6,13,14],[7,8,9,10,11,12]]
=> ?
=> ? = 7 - 1
([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> [6,4,3,3]
=> [[1,2,3,10,15,16],[4,5,6,14],[7,8,9],[11,12,13]]
=> ?
=> ? = 4 - 1
([(0,5),(1,4),(3,5),(4,2),(4,3)],6)
=> [8,4,2]
=> [[1,2,5,6,11,12,13,14],[3,4,9,10],[7,8]]
=> ?
=> ? = 3 - 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [5,4,4,4]
=> [[1,2,3,4,17],[5,6,7,8],[9,10,11,12],[13,14,15,16]]
=> ?
=> ? = 5 - 1
([(0,5),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> [5,4,2,2]
=> [[1,2,7,8,13],[3,4,11,12],[5,6],[9,10]]
=> [[1,3,5,9],[2,4,6,10],[7,11],[8,12],[13]]
=> ? = 3 - 1
Description
The index of the last row whose first entry is the row number in a standard Young tableau.
Matching statistic: St001184
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001184: Dyck paths ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 56%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001184: Dyck paths ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 56%
Values
([],1)
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2 = 3 - 1
([],2)
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
([(0,1)],2)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3 = 4 - 1
([],3)
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2 = 3 - 1
([(1,2)],3)
=> [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]
=> 6 = 7 - 1
([(0,1),(0,2)],3)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
([(0,2),(2,1)],3)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
([(0,2),(1,2)],3)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
([(2,3)],4)
=> [6,6]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 7 - 1
([(1,2),(1,3)],4)
=> [6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 2 = 3 - 1
([(0,2),(0,3),(3,1)],4)
=> [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]
=> ? = 8 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 3 - 1
([(1,2),(2,3)],4)
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 5 - 1
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 3 - 1
([(1,3),(2,3)],4)
=> [6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 3 - 1
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 2 = 3 - 1
([(0,3),(1,2)],4)
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3 = 4 - 1
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3 = 4 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2 = 3 - 1
([(0,3),(2,1),(3,2)],4)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
([(0,3),(1,2),(2,3)],4)
=> [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]
=> ? = 8 - 1
([(3,4)],5)
=> [6,6,6,6]
=> [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 7 - 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> [3,2,2,2,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> [7,6]
=> [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 7 - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> [4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2 = 3 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> 3 = 4 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 4 = 5 - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 2 = 3 - 1
([(2,3),(3,4)],5)
=> [4,4,4,4]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 5 - 1
([(1,4),(4,2),(4,3)],5)
=> [4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> [4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(1,4),(2,4),(4,3)],5)
=> [4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 2 = 3 - 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [3,2,2,2,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,4),(1,4),(2,3)],5)
=> [6,3,3,3]
=> [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0,0]
=> ? = 4 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5,3,2,2]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2 = 3 - 1
([(0,4),(1,4),(2,3),(2,4)],5)
=> [6,5,3]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0]
=> ? = 4 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [7,6]
=> [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 7 - 1
([(1,4),(2,3)],5)
=> [6,6,6]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 7 - 1
([(0,4),(1,2),(1,4),(2,3)],5)
=> [8,3]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0]
=> ? = 4 - 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 4 = 5 - 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 2 = 3 - 1
([(0,4),(1,2),(1,4),(4,3)],5)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 11 - 1
([(0,4),(1,2),(1,3)],5)
=> [6,3,3,3]
=> [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0,0]
=> ? = 4 - 1
([(0,4),(1,2),(1,3),(1,4)],5)
=> [6,5,3]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0]
=> ? = 4 - 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> [5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 4 = 5 - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [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]
=> ? = 9 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [5,3,2,2]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [3,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 11 - 1
([(0,3),(1,2),(1,4),(3,4)],5)
=> [8,3]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0]
=> ? = 4 - 1
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 3 - 1
([(1,4),(3,2),(4,3)],5)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 11 - 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2 = 3 - 1
([(0,4),(1,2),(2,4),(4,3)],5)
=> [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]
=> ? = 9 - 1
([(0,4),(3,2),(4,1),(4,3)],5)
=> [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]
=> ? = 9 - 1
([(0,4),(1,2),(2,3),(2,4)],5)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 11 - 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> [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]
=> 6 = 7 - 1
([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> 3 = 4 - 1
([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 4 = 5 - 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2 = 3 - 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> [5,2,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
=> [6,4,3,3]
=> [1,0,1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ?
=> ? = 4 - 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5)],6)
=> [8,4,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,1,0,0]
=> ? = 3 - 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,4,2,2]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
=> [6,5,4]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> ? = 5 - 1
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
=> [5,4,2,2]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
=> [7,6,6]
=> [1,0,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0]
=> ? = 7 - 1
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> [5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
=> [8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> ? = 3 - 1
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,4),(0,5),(4,3),(5,1),(5,2)],6)
=> [6,4,3,3]
=> [1,0,1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ?
=> ? = 4 - 1
([(0,3),(0,4),(3,5),(4,1),(4,2),(4,5)],6)
=> [6,5,4]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> ? = 5 - 1
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
=> [5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5 = 6 - 1
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5 = 6 - 1
([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
=> [5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5 = 6 - 1
Description
Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra.
Matching statistic: St001481
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001481: Dyck paths ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 56%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001481: Dyck paths ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 56%
Values
([],1)
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2 = 3 - 1
([],2)
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
([(0,1)],2)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3 = 4 - 1
([],3)
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2 = 3 - 1
([(1,2)],3)
=> [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]
=> 6 = 7 - 1
([(0,1),(0,2)],3)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
([(0,2),(2,1)],3)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
([(0,2),(1,2)],3)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
([(2,3)],4)
=> [6,6]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 7 - 1
([(1,2),(1,3)],4)
=> [6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 2 = 3 - 1
([(0,2),(0,3),(3,1)],4)
=> [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]
=> ? = 8 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 3 - 1
([(1,2),(2,3)],4)
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 5 - 1
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 3 - 1
([(1,3),(2,3)],4)
=> [6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 3 - 1
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 2 = 3 - 1
([(0,3),(1,2)],4)
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3 = 4 - 1
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3 = 4 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2 = 3 - 1
([(0,3),(2,1),(3,2)],4)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
([(0,3),(1,2),(2,3)],4)
=> [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]
=> ? = 8 - 1
([(3,4)],5)
=> [6,6,6,6]
=> [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 7 - 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> [3,2,2,2,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> [7,6]
=> [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 7 - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> [4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2 = 3 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> 3 = 4 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 4 = 5 - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 2 = 3 - 1
([(2,3),(3,4)],5)
=> [4,4,4,4]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 5 - 1
([(1,4),(4,2),(4,3)],5)
=> [4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> [4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(1,4),(2,4),(4,3)],5)
=> [4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 2 = 3 - 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [3,2,2,2,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,4),(1,4),(2,3)],5)
=> [6,3,3,3]
=> [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0,0]
=> ? = 4 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5,3,2,2]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2 = 3 - 1
([(0,4),(1,4),(2,3),(2,4)],5)
=> [6,5,3]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0]
=> ? = 4 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [7,6]
=> [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 7 - 1
([(1,4),(2,3)],5)
=> [6,6,6]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 7 - 1
([(0,4),(1,2),(1,4),(2,3)],5)
=> [8,3]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0]
=> ? = 4 - 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 4 = 5 - 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 2 = 3 - 1
([(0,4),(1,2),(1,4),(4,3)],5)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 11 - 1
([(0,4),(1,2),(1,3)],5)
=> [6,3,3,3]
=> [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0,0]
=> ? = 4 - 1
([(0,4),(1,2),(1,3),(1,4)],5)
=> [6,5,3]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0]
=> ? = 4 - 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> [5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 4 = 5 - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [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]
=> ? = 9 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [5,3,2,2]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [3,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 11 - 1
([(0,3),(1,2),(1,4),(3,4)],5)
=> [8,3]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0]
=> ? = 4 - 1
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 3 - 1
([(1,4),(3,2),(4,3)],5)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 11 - 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2 = 3 - 1
([(0,4),(1,2),(2,4),(4,3)],5)
=> [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]
=> ? = 9 - 1
([(0,4),(3,2),(4,1),(4,3)],5)
=> [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]
=> ? = 9 - 1
([(0,4),(1,2),(2,3),(2,4)],5)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 11 - 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> [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]
=> 6 = 7 - 1
([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> 3 = 4 - 1
([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 4 = 5 - 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2 = 3 - 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> [5,2,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
=> [6,4,3,3]
=> [1,0,1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ?
=> ? = 4 - 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5)],6)
=> [8,4,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,1,0,0]
=> ? = 3 - 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,4,2,2]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
=> [6,5,4]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> ? = 5 - 1
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
=> [5,4,2,2]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
=> [7,6,6]
=> [1,0,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0]
=> ? = 7 - 1
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> [5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
=> [8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> ? = 3 - 1
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> ? = 3 - 1
([(0,4),(0,5),(4,3),(5,1),(5,2)],6)
=> [6,4,3,3]
=> [1,0,1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ?
=> ? = 4 - 1
([(0,3),(0,4),(3,5),(4,1),(4,2),(4,5)],6)
=> [6,5,4]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> ? = 5 - 1
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
=> [5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5 = 6 - 1
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5 = 6 - 1
([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
=> [5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5 = 6 - 1
Description
The minimal height of a peak of a Dyck path.
Matching statistic: St001803
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St001803: Standard tableaux ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 78%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St001803: Standard tableaux ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 78%
Values
([],1)
=> [2]
=> [1,1]
=> [[1],[2]]
=> 1 = 3 - 2
([],2)
=> [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 1 = 3 - 2
([(0,1)],2)
=> [3]
=> [1,1,1]
=> [[1],[2],[3]]
=> 2 = 4 - 2
([],3)
=> [2,2,2,2]
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> 1 = 3 - 2
([(1,2)],3)
=> [6]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> 5 = 7 - 2
([(0,1),(0,2)],3)
=> [3,2]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 1 = 3 - 2
([(0,2),(2,1)],3)
=> [4]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 3 = 5 - 2
([(0,2),(1,2)],3)
=> [3,2]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 1 = 3 - 2
([(2,3)],4)
=> [6,6]
=> [2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]
=> ? = 7 - 2
([(1,2),(1,3)],4)
=> [6,2,2]
=> [3,3,1,1,1,1]
=> [[1,6,7],[2,9,10],[3],[4],[5],[8]]
=> ? = 3 - 2
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> [4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6]]
=> ? = 3 - 2
([(0,2),(0,3),(3,1)],4)
=> [7]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> 6 = 8 - 2
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> 1 = 3 - 2
([(1,2),(2,3)],4)
=> [4,4]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> 3 = 5 - 2
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> 1 = 3 - 2
([(1,3),(2,3)],4)
=> [6,2,2]
=> [3,3,1,1,1,1]
=> [[1,6,7],[2,9,10],[3],[4],[5],[8]]
=> ? = 3 - 2
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> 1 = 3 - 2
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6]]
=> ? = 3 - 2
([(0,3),(1,2)],4)
=> [3,3,3]
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> ? = 4 - 2
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> [2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5],[7]]
=> 2 = 4 - 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> [3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> 1 = 3 - 2
([(0,3),(2,1),(3,2)],4)
=> [5]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 4 = 6 - 2
([(0,3),(1,2),(2,3)],4)
=> [7]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> 6 = 8 - 2
([(3,4)],5)
=> [6,6,6,6]
=> [4,4,4,4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16],[17,18,19,20],[21,22,23,24]]
=> ? = 7 - 2
([(0,1),(0,2),(0,3),(0,4)],5)
=> [3,2,2,2,2,2,2,2]
=> [8,8,1]
=> [[1,3,4,5,6,7,8,9],[2,11,12,13,14,15,16,17],[10]]
=> ? = 3 - 2
([(0,2),(0,3),(0,4),(4,1)],5)
=> [7,6]
=> [2,2,2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8,11],[10,13],[12]]
=> ? = 7 - 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [4,2,2,2]
=> [4,4,1,1]
=> [[1,4,5,6],[2,8,9,10],[3],[7]]
=> ? = 3 - 2
([(1,2),(1,3),(2,4),(3,4)],5)
=> [4,4,2,2]
=> [4,4,2,2]
=> [[1,2,7,8],[3,4,11,12],[5,6],[9,10]]
=> ? = 3 - 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> [2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> 1 = 3 - 2
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,3,3]
=> [3,3,3,1]
=> [[1,3,4],[2,6,7],[5,9,10],[8]]
=> ? = 4 - 2
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [5,4]
=> [2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8]]
=> ? = 5 - 2
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [4,2,2]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> 1 = 3 - 2
([(2,3),(3,4)],5)
=> [4,4,4,4]
=> [4,4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16]]
=> ? = 5 - 2
([(1,4),(4,2),(4,3)],5)
=> [4,4,2,2]
=> [4,4,2,2]
=> [[1,2,7,8],[3,4,11,12],[5,6],[9,10]]
=> ? = 3 - 2
([(0,4),(4,1),(4,2),(4,3)],5)
=> [4,2,2,2]
=> [4,4,1,1]
=> [[1,4,5,6],[2,8,9,10],[3],[7]]
=> ? = 3 - 2
([(1,4),(2,4),(4,3)],5)
=> [4,4,2,2]
=> [4,4,2,2]
=> [[1,2,7,8],[3,4,11,12],[5,6],[9,10]]
=> ? = 3 - 2
([(0,4),(1,4),(4,2),(4,3)],5)
=> [4,2,2]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> 1 = 3 - 2
([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> [4,4,1,1]
=> [[1,4,5,6],[2,8,9,10],[3],[7]]
=> ? = 3 - 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [3,2,2,2,2,2,2,2]
=> [8,8,1]
=> [[1,3,4,5,6,7,8,9],[2,11,12,13,14,15,16,17],[10]]
=> ? = 3 - 2
([(0,4),(1,4),(2,3)],5)
=> [6,3,3,3]
=> [4,4,4,1,1,1]
=> [[1,5,6,7],[2,9,10,11],[3,13,14,15],[4],[8],[12]]
=> ? = 4 - 2
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5,3,2,2]
=> [4,4,2,1,1]
=> [[1,4,7,8],[2,6,11,12],[3,10],[5],[9]]
=> ? = 3 - 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2,2,2,2]
=> [5,5,1]
=> [[1,3,4,5,6],[2,8,9,10,11],[7]]
=> ? = 3 - 2
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> 1 = 3 - 2
([(0,4),(1,4),(2,3),(2,4)],5)
=> [6,5,3]
=> [3,3,3,2,2,1]
=> [[1,3,8],[2,5,11],[4,7,14],[6,10],[9,13],[12]]
=> ? = 4 - 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [7,6]
=> [2,2,2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8,11],[10,13],[12]]
=> ? = 7 - 2
([(1,4),(2,3)],5)
=> [6,6,6]
=> [3,3,3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12],[13,14,15],[16,17,18]]
=> ? = 7 - 2
([(0,4),(1,2),(1,4),(2,3)],5)
=> [8,3]
=> [2,2,2,1,1,1,1,1]
=> [[1,7],[2,9],[3,11],[4],[5],[6],[8],[10]]
=> ? = 4 - 2
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [5,4]
=> [2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8]]
=> ? = 5 - 2
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [7,2]
=> [2,2,1,1,1,1,1]
=> [[1,7],[2,9],[3],[4],[5],[6],[8]]
=> ? = 3 - 2
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [4,2,2]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> 1 = 3 - 2
([(0,4),(1,2),(1,4),(4,3)],5)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? = 11 - 2
([(0,4),(1,2),(1,3)],5)
=> [6,3,3,3]
=> [4,4,4,1,1,1]
=> [[1,5,6,7],[2,9,10,11],[3,13,14,15],[4],[8],[12]]
=> ? = 4 - 2
([(0,4),(1,2),(1,3),(1,4)],5)
=> [6,5,3]
=> [3,3,3,2,2,1]
=> [[1,3,8],[2,5,11],[4,7,14],[6,10],[9,13],[12]]
=> ? = 4 - 2
([(0,2),(0,4),(3,1),(4,3)],5)
=> [5,4]
=> [2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8]]
=> ? = 5 - 2
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> 7 = 9 - 2
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [5,3,2,2]
=> [4,4,2,1,1]
=> [[1,4,7,8],[2,6,11,12],[3,10],[5],[9]]
=> ? = 3 - 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [3,2,2,2,2]
=> [5,5,1]
=> [[1,3,4,5,6],[2,8,9,10,11],[7]]
=> ? = 3 - 2
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? = 11 - 2
([(0,3),(1,2),(1,4),(3,4)],5)
=> [8,3]
=> [2,2,2,1,1,1,1,1]
=> [[1,7],[2,9],[3,11],[4],[5],[6],[8],[10]]
=> ? = 4 - 2
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [7,2]
=> [2,2,1,1,1,1,1]
=> [[1,7],[2,9],[3],[4],[5],[6],[8]]
=> ? = 3 - 2
([(1,4),(3,2),(4,3)],5)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? = 11 - 2
([(0,3),(3,4),(4,1),(4,2)],5)
=> [5,2]
=> [2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> 1 = 3 - 2
([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> 7 = 9 - 2
([(0,4),(3,2),(4,1),(4,3)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> 7 = 9 - 2
([(0,4),(1,2),(2,3),(2,4)],5)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? = 11 - 2
([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> 5 = 7 - 2
([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> [3,3,3,1]
=> [[1,3,4],[2,6,7],[5,9,10],[8]]
=> ? = 4 - 2
([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> [2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8]]
=> ? = 5 - 2
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [5,2]
=> [2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> 1 = 3 - 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> [5,2,2,2]
=> [4,4,1,1,1]
=> [[1,5,6,7],[2,9,10,11],[3],[4],[8]]
=> ? = 3 - 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
=> [6,4,3,3]
=> [4,4,4,2,1,1]
=> [[1,4,7,8],[2,6,11,12],[3,10,15,16],[5,14],[9],[13]]
=> ? = 4 - 2
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5)],6)
=> [8,4,2]
=> [3,3,2,2,1,1,1,1]
=> [[1,6,11],[2,8,14],[3,10],[4,13],[5],[7],[9],[12]]
=> ? = 3 - 2
([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,4,2,2]
=> [4,4,2,2,1]
=> [[1,3,8,9],[2,5,12,13],[4,7],[6,11],[10]]
=> ? = 3 - 2
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2,2,2,2]
=> [5,5,1,1]
=> [[1,4,5,6,7],[2,9,10,11,12],[3],[8]]
=> ? = 3 - 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
=> [6,5,4]
=> [3,3,3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8,12],[7,11,15],[10,14],[13]]
=> ? = 5 - 2
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
=> [5,4,2,2]
=> [4,4,2,2,1]
=> [[1,3,8,9],[2,5,12,13],[4,7],[6,11],[10]]
=> ? = 3 - 2
([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
=> [7,6,6]
=> [3,3,3,3,3,3,1]
=> [[1,3,4],[2,6,7],[5,9,10],[8,12,13],[11,15,16],[14,18,19],[17]]
=> ? = 7 - 2
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> [5,2,2]
=> [3,3,1,1,1]
=> [[1,5,6],[2,8,9],[3],[4],[7]]
=> ? = 3 - 2
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [6,2]
=> [2,2,1,1,1,1]
=> [[1,6],[2,8],[3],[4],[5],[7]]
=> 1 = 3 - 2
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> [6,2]
=> [2,2,1,1,1,1]
=> [[1,6],[2,8],[3],[4],[5],[7]]
=> 1 = 3 - 2
([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
=> [6,2]
=> [2,2,1,1,1,1]
=> [[1,6],[2,8],[3],[4],[5],[7]]
=> 1 = 3 - 2
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [6,2]
=> [2,2,1,1,1,1]
=> [[1,6],[2,8],[3],[4],[5],[7]]
=> 1 = 3 - 2
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [7]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> 6 = 8 - 2
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [6,2]
=> [2,2,1,1,1,1]
=> [[1,6],[2,8],[3],[4],[5],[7]]
=> 1 = 3 - 2
([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> 7 = 9 - 2
Description
The maximal overlap of the cylindrical tableau associated with a tableau.
A cylindrical tableau associated with a standard Young tableau $T$ is the skew row-strict tableau obtained by gluing two copies of $T$ such that the inner shape is a rectangle.
The overlap, recorded in this statistic, equals $\max_C\big(2\ell(T) - \ell(C)\big)$, where $\ell$ denotes the number of rows of a tableau and the maximum is taken over all cylindrical tableaux.
In particular, the statistic equals $0$, if and only if the last entry of the first row is larger than or equal to the first entry of the last row. Moreover, the statistic attains its maximal value, the number of rows of the tableau minus 1, if and only if the tableau consists of a single column.
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