Your data matches 19 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001038
(load all 2 compositions to match this statistic)
Mapping
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001038: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [2]
 => [1,0,1,0]
 => 2
([],2)
 => [2,2]
 => [1,1,1,0,0,0]
 => 2
([(0,1)],2)
 => [3]
 => [1,0,1,0,1,0]
 => 3
([],3)
 => [2,2,2,2]
 => [1,1,1,1,0,1,0,0,0,0]
 => 2
([(1,2)],3)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 6
([(0,1),(0,2)],3)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 2
([(0,2),(2,1)],3)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 4
([(0,2),(1,2)],3)
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 2
([(2,3)],4)
 => [6,6]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => 6
([(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
([(0,1),(0,2),(0,3)],4)
 => [3,2,2,2]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 2
([(0,2),(0,3),(3,1)],4)
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => 7
([(0,1),(0,2),(1,3),(2,3)],4)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2
([(1,2),(2,3)],4)
 => [4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => 4
([(0,3),(3,1),(3,2)],4)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2
([(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
([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2
([(0,3),(1,3),(2,3)],4)
 => [3,2,2,2]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 2
([(0,3),(1,2)],4)
 => [3,3,3]
 => [1,1,1,1,1,0,0,0,0,0]
 => 3
([(0,3),(1,2),(1,3)],4)
 => [5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => [3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2
([(0,3),(2,1),(3,2)],4)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
([(0,3),(1,2),(2,3)],4)
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => 7
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(2,3),(3,4)],5)
 => [4,4,4,4]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => 4
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(0,4),(1,4),(2,3),(4,2)],5)
 => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 2
([(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
([(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
([(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
([(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
([(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
([(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
([(0,2),(0,4),(3,1),(4,3)],5)
 => [5,4]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => 4
([(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
([(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
([(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
([(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
Description
The minimal height of a column in the parallelogram polyomino associated with the Dyck path.
Matching statistic: St000297
Mapping
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: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [2]
 => [1,1]
 => 110 => 2
([],2)
 => [2,2]
 => [2,2]
 => 1100 => 2
([(0,1)],2)
 => [3]
 => [1,1,1]
 => 1110 => 3
([],3)
 => [2,2,2,2]
 => [4,4]
 => 110000 => 2
([(1,2)],3)
 => [6]
 => [1,1,1,1,1,1]
 => 1111110 => 6
([(0,1),(0,2)],3)
 => [3,2]
 => [2,2,1]
 => 11010 => 2
([(0,2),(2,1)],3)
 => [4]
 => [1,1,1,1]
 => 11110 => 4
([(0,2),(1,2)],3)
 => [3,2]
 => [2,2,1]
 => 11010 => 2
([(2,3)],4)
 => [6,6]
 => [2,2,2,2,2,2]
 => 11111100 => 6
([(1,2),(1,3)],4)
 => [6,2,2]
 => [3,3,1,1,1,1]
 => 110011110 => 2
([(0,1),(0,2),(0,3)],4)
 => [3,2,2,2]
 => [4,4,1]
 => 1100010 => 2
([(0,2),(0,3),(3,1)],4)
 => [7]
 => [1,1,1,1,1,1,1]
 => 11111110 => 7
([(0,1),(0,2),(1,3),(2,3)],4)
 => [4,2]
 => [2,2,1,1]
 => 110110 => 2
([(1,2),(2,3)],4)
 => [4,4]
 => [2,2,2,2]
 => 111100 => 4
([(0,3),(3,1),(3,2)],4)
 => [4,2]
 => [2,2,1,1]
 => 110110 => 2
([(1,3),(2,3)],4)
 => [6,2,2]
 => [3,3,1,1,1,1]
 => 110011110 => 2
([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => [2,2,1,1]
 => 110110 => 2
([(0,3),(1,3),(2,3)],4)
 => [3,2,2,2]
 => [4,4,1]
 => 1100010 => 2
([(0,3),(1,2)],4)
 => [3,3,3]
 => [3,3,3]
 => 111000 => 3
([(0,3),(1,2),(1,3)],4)
 => [5,3]
 => [2,2,2,1,1]
 => 1110110 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => [3,2,2]
 => [3,3,1]
 => 110010 => 2
([(0,3),(2,1),(3,2)],4)
 => [5]
 => [1,1,1,1,1]
 => 111110 => 5
([(0,3),(1,2),(2,3)],4)
 => [7]
 => [1,1,1,1,1,1,1]
 => 11111110 => 7
([(0,2),(0,3),(0,4),(4,1)],5)
 => [7,6]
 => [2,2,2,2,2,2,1]
 => 111111010 => 6
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [4,2,2,2]
 => [4,4,1,1]
 => 11000110 => 2
([(1,2),(1,3),(2,4),(3,4)],5)
 => [4,4,2,2]
 => [4,4,2,2]
 => 11001100 => 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [5,2]
 => [2,2,1,1,1]
 => 1101110 => 2
([(0,3),(0,4),(3,2),(4,1)],5)
 => [4,3,3]
 => [3,3,3,1]
 => 1110010 => 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [5,4]
 => [2,2,2,2,1]
 => 1111010 => 4
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [4,2,2]
 => [3,3,1,1]
 => 1100110 => 2
([(2,3),(3,4)],5)
 => [4,4,4,4]
 => [4,4,4,4]
 => 11110000 => 4
([(1,4),(4,2),(4,3)],5)
 => [4,4,2,2]
 => [4,4,2,2]
 => 11001100 => 2
([(0,4),(4,1),(4,2),(4,3)],5)
 => [4,2,2,2]
 => [4,4,1,1]
 => 11000110 => 2
([(1,4),(2,4),(4,3)],5)
 => [4,4,2,2]
 => [4,4,2,2]
 => 11001100 => 2
([(0,4),(1,4),(4,2),(4,3)],5)
 => [4,2,2]
 => [3,3,1,1]
 => 1100110 => 2
([(0,4),(1,4),(2,4),(4,3)],5)
 => [4,2,2,2]
 => [4,4,1,1]
 => 11000110 => 2
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5,3,2,2]
 => [4,4,2,1,1]
 => 110010110 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2,2,2,2]
 => [5,5,1]
 => 11000010 => 2
([(0,4),(1,4),(2,3),(4,2)],5)
 => [5,2]
 => [2,2,1,1,1]
 => 1101110 => 2
([(0,4),(1,4),(2,3),(2,4)],5)
 => [6,5,3]
 => [3,3,3,2,2,1]
 => 111011010 => 3
([(0,4),(1,4),(2,3),(3,4)],5)
 => [7,6]
 => [2,2,2,2,2,2,1]
 => 111111010 => 6
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => [5,4]
 => [2,2,2,2,1]
 => 1111010 => 4
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => [7,2]
 => [2,2,1,1,1,1,1]
 => 110111110 => 2
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [4,2,2]
 => [3,3,1,1]
 => 1100110 => 2
([(0,4),(1,2),(1,3),(1,4)],5)
 => [6,5,3]
 => [3,3,3,2,2,1]
 => 111011010 => 3
([(0,2),(0,4),(3,1),(4,3)],5)
 => [5,4]
 => [2,2,2,2,1]
 => 1111010 => 4
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => 111111110 => 8
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [5,3,2,2]
 => [4,4,2,1,1]
 => 110010110 => 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [3,2,2,2,2]
 => [5,5,1]
 => 11000010 => 2
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => [7,2]
 => [2,2,1,1,1,1,1]
 => 110111110 => 2
Description
The number of leading ones in a binary word.
Matching statistic: St000326
Mapping
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%
Values
([],1)
 => [2]
 => 100 => 001 => 3 = 2 + 1
([],2)
 => [2,2]
 => 1100 => 0011 => 3 = 2 + 1
([(0,1)],2)
 => [3]
 => 1000 => 0001 => 4 = 3 + 1
([],3)
 => [2,2,2,2]
 => 111100 => 001111 => 3 = 2 + 1
([(1,2)],3)
 => [6]
 => 1000000 => 0000001 => 7 = 6 + 1
([(0,1),(0,2)],3)
 => [3,2]
 => 10100 => 00101 => 3 = 2 + 1
([(0,2),(2,1)],3)
 => [4]
 => 10000 => 00001 => 5 = 4 + 1
([(0,2),(1,2)],3)
 => [3,2]
 => 10100 => 00101 => 3 = 2 + 1
([(2,3)],4)
 => [6,6]
 => 11000000 => 00000011 => 7 = 6 + 1
([(1,2),(1,3)],4)
 => [6,2,2]
 => 100001100 => 001100001 => 3 = 2 + 1
([(0,1),(0,2),(0,3)],4)
 => [3,2,2,2]
 => 1011100 => 0011101 => 3 = 2 + 1
([(0,2),(0,3),(3,1)],4)
 => [7]
 => 10000000 => 00000001 => 8 = 7 + 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => [4,2]
 => 100100 => 001001 => 3 = 2 + 1
([(1,2),(2,3)],4)
 => [4,4]
 => 110000 => 000011 => 5 = 4 + 1
([(0,3),(3,1),(3,2)],4)
 => [4,2]
 => 100100 => 001001 => 3 = 2 + 1
([(1,3),(2,3)],4)
 => [6,2,2]
 => 100001100 => 001100001 => 3 = 2 + 1
([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => 100100 => 001001 => 3 = 2 + 1
([(0,3),(1,3),(2,3)],4)
 => [3,2,2,2]
 => 1011100 => 0011101 => 3 = 2 + 1
([(0,3),(1,2)],4)
 => [3,3,3]
 => 111000 => 000111 => 4 = 3 + 1
([(0,3),(1,2),(1,3)],4)
 => [5,3]
 => 1001000 => 0001001 => 4 = 3 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [3,2,2]
 => 101100 => 001101 => 3 = 2 + 1
([(0,3),(2,1),(3,2)],4)
 => [5]
 => 100000 => 000001 => 6 = 5 + 1
([(0,3),(1,2),(2,3)],4)
 => [7]
 => 10000000 => 00000001 => 8 = 7 + 1
([(0,2),(0,3),(0,4),(4,1)],5)
 => [7,6]
 => 101000000 => 000000101 => 7 = 6 + 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [4,2,2,2]
 => 10011100 => 00111001 => 3 = 2 + 1
([(1,2),(1,3),(2,4),(3,4)],5)
 => [4,4,2,2]
 => 11001100 => 00110011 => 3 = 2 + 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [5,2]
 => 1000100 => 0010001 => 3 = 2 + 1
([(0,3),(0,4),(3,2),(4,1)],5)
 => [4,3,3]
 => 1011000 => 0001101 => 4 = 3 + 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [5,4]
 => 1010000 => 0000101 => 5 = 4 + 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [4,2,2]
 => 1001100 => 0011001 => 3 = 2 + 1
([(2,3),(3,4)],5)
 => [4,4,4,4]
 => 11110000 => 00001111 => 5 = 4 + 1
([(1,4),(4,2),(4,3)],5)
 => [4,4,2,2]
 => 11001100 => 00110011 => 3 = 2 + 1
([(0,4),(4,1),(4,2),(4,3)],5)
 => [4,2,2,2]
 => 10011100 => 00111001 => 3 = 2 + 1
([(1,4),(2,4),(4,3)],5)
 => [4,4,2,2]
 => 11001100 => 00110011 => 3 = 2 + 1
([(0,4),(1,4),(4,2),(4,3)],5)
 => [4,2,2]
 => 1001100 => 0011001 => 3 = 2 + 1
([(0,4),(1,4),(2,4),(4,3)],5)
 => [4,2,2,2]
 => 10011100 => 00111001 => 3 = 2 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5,3,2,2]
 => 100101100 => 001101001 => 3 = 2 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2,2,2,2]
 => 10111100 => 00111101 => 3 = 2 + 1
([(0,4),(1,4),(2,3),(4,2)],5)
 => [5,2]
 => 1000100 => 0010001 => 3 = 2 + 1
([(0,4),(1,4),(2,3),(2,4)],5)
 => [6,5,3]
 => 101001000 => 000100101 => 4 = 3 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [7,6]
 => 101000000 => 000000101 => 7 = 6 + 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => [5,4]
 => 1010000 => 0000101 => 5 = 4 + 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => [7,2]
 => 100000100 => 001000001 => 3 = 2 + 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [4,2,2]
 => 1001100 => 0011001 => 3 = 2 + 1
([(0,4),(1,2),(1,3),(1,4)],5)
 => [6,5,3]
 => 101001000 => 000100101 => 4 = 3 + 1
([(0,2),(0,4),(3,1),(4,3)],5)
 => [5,4]
 => 1010000 => 0000101 => 5 = 4 + 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [8]
 => 100000000 => 000000001 => 9 = 8 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [5,3,2,2]
 => 100101100 => 001101001 => 3 = 2 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [3,2,2,2,2]
 => 10111100 => 00111101 => 3 = 2 + 1
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => [7,2]
 => 100000100 => 001000001 => 3 = 2 + 1
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)
Mapping
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [2]
 => [1,1]
 => [1]
 => ? = 2 - 1
([],2)
 => [2,2]
 => [2,2]
 => [2]
 => 1 = 2 - 1
([(0,1)],2)
 => [3]
 => [1,1,1]
 => [1,1]
 => 2 = 3 - 1
([],3)
 => [2,2,2,2]
 => [4,4]
 => [4]
 => 1 = 2 - 1
([(1,2)],3)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 5 = 6 - 1
([(0,1),(0,2)],3)
 => [3,2]
 => [2,2,1]
 => [2,1]
 => 1 = 2 - 1
([(0,2),(2,1)],3)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 3 = 4 - 1
([(0,2),(1,2)],3)
 => [3,2]
 => [2,2,1]
 => [2,1]
 => 1 = 2 - 1
([(2,3)],4)
 => [6,6]
 => [2,2,2,2,2,2]
 => [2,2,2,2,2]
 => 5 = 6 - 1
([(1,2),(1,3)],4)
 => [6,2,2]
 => [3,3,1,1,1,1]
 => [3,1,1,1,1]
 => 1 = 2 - 1
([(0,1),(0,2),(0,3)],4)
 => [3,2,2,2]
 => [4,4,1]
 => [4,1]
 => 1 = 2 - 1
([(0,2),(0,3),(3,1)],4)
 => [7]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 6 = 7 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => [4,2]
 => [2,2,1,1]
 => [2,1,1]
 => 1 = 2 - 1
([(1,2),(2,3)],4)
 => [4,4]
 => [2,2,2,2]
 => [2,2,2]
 => 3 = 4 - 1
([(0,3),(3,1),(3,2)],4)
 => [4,2]
 => [2,2,1,1]
 => [2,1,1]
 => 1 = 2 - 1
([(1,3),(2,3)],4)
 => [6,2,2]
 => [3,3,1,1,1,1]
 => [3,1,1,1,1]
 => 1 = 2 - 1
([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => [2,2,1,1]
 => [2,1,1]
 => 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
 => [3,2,2,2]
 => [4,4,1]
 => [4,1]
 => 1 = 2 - 1
([(0,3),(1,2)],4)
 => [3,3,3]
 => [3,3,3]
 => [3,3]
 => 2 = 3 - 1
([(0,3),(1,2),(1,3)],4)
 => [5,3]
 => [2,2,2,1,1]
 => [2,2,1,1]
 => 2 = 3 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [3,2,2]
 => [3,3,1]
 => [3,1]
 => 1 = 2 - 1
([(0,3),(2,1),(3,2)],4)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 4 = 5 - 1
([(0,3),(1,2),(2,3)],4)
 => [7]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 6 = 7 - 1
([(0,2),(0,3),(0,4),(4,1)],5)
 => [7,6]
 => [2,2,2,2,2,2,1]
 => [2,2,2,2,2,1]
 => 5 = 6 - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [4,2,2,2]
 => [4,4,1,1]
 => [4,1,1]
 => 1 = 2 - 1
([(1,2),(1,3),(2,4),(3,4)],5)
 => [4,4,2,2]
 => [4,4,2,2]
 => [4,2,2]
 => 1 = 2 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [5,2]
 => [2,2,1,1,1]
 => [2,1,1,1]
 => 1 = 2 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
 => [4,3,3]
 => [3,3,3,1]
 => [3,3,1]
 => 2 = 3 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [5,4]
 => [2,2,2,2,1]
 => [2,2,2,1]
 => 3 = 4 - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [4,2,2]
 => [3,3,1,1]
 => [3,1,1]
 => 1 = 2 - 1
([(2,3),(3,4)],5)
 => [4,4,4,4]
 => [4,4,4,4]
 => [4,4,4]
 => 3 = 4 - 1
([(1,4),(4,2),(4,3)],5)
 => [4,4,2,2]
 => [4,4,2,2]
 => [4,2,2]
 => 1 = 2 - 1
([(0,4),(4,1),(4,2),(4,3)],5)
 => [4,2,2,2]
 => [4,4,1,1]
 => [4,1,1]
 => 1 = 2 - 1
([(1,4),(2,4),(4,3)],5)
 => [4,4,2,2]
 => [4,4,2,2]
 => [4,2,2]
 => 1 = 2 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
 => [4,2,2]
 => [3,3,1,1]
 => [3,1,1]
 => 1 = 2 - 1
([(0,4),(1,4),(2,4),(4,3)],5)
 => [4,2,2,2]
 => [4,4,1,1]
 => [4,1,1]
 => 1 = 2 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5,3,2,2]
 => [4,4,2,1,1]
 => [4,2,1,1]
 => 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2,2,2,2]
 => [5,5,1]
 => [5,1]
 => 1 = 2 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
 => [5,2]
 => [2,2,1,1,1]
 => [2,1,1,1]
 => 1 = 2 - 1
([(0,4),(1,4),(2,3),(2,4)],5)
 => [6,5,3]
 => [3,3,3,2,2,1]
 => [3,3,2,2,1]
 => 2 = 3 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [7,6]
 => [2,2,2,2,2,2,1]
 => [2,2,2,2,2,1]
 => 5 = 6 - 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => [5,4]
 => [2,2,2,2,1]
 => [2,2,2,1]
 => 3 = 4 - 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => [7,2]
 => [2,2,1,1,1,1,1]
 => [2,1,1,1,1,1]
 => 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [4,2,2]
 => [3,3,1,1]
 => [3,1,1]
 => 1 = 2 - 1
([(0,4),(1,2),(1,3),(1,4)],5)
 => [6,5,3]
 => [3,3,3,2,2,1]
 => [3,3,2,2,1]
 => 2 = 3 - 1
([(0,2),(0,4),(3,1),(4,3)],5)
 => [5,4]
 => [2,2,2,2,1]
 => [2,2,2,1]
 => 3 = 4 - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => 7 = 8 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [5,3,2,2]
 => [4,4,2,1,1]
 => [4,2,1,1]
 => 1 = 2 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [3,2,2,2,2]
 => [5,5,1]
 => [5,1]
 => 1 = 2 - 1
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => [7,2]
 => [2,2,1,1,1,1,1]
 => [2,1,1,1,1,1]
 => 1 = 2 - 1
([(0,3),(3,4),(4,1),(4,2)],5)
 => [5,2]
 => [2,2,1,1,1]
 => [2,1,1,1]
 => 1 = 2 - 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => [5,4,4,4]
 => [4,4,4,4,1]
 => [4,4,4,1]
 => ? = 4 - 1
([(0,2),(0,3),(0,5),(4,1),(5,4)],6)
 => [5,4,4,4]
 => [4,4,4,4,1]
 => [4,4,4,1]
 => ? = 4 - 1
([(0,6),(1,6),(2,3),(2,6),(3,5),(5,4),(6,5)],7)
 => [6,5,5]
 => [3,3,3,3,3,1]
 => [3,3,3,3,1]
 => ? = 5 - 1
([(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,2),(4,5),(5,6)],7)
 => [6,5,5]
 => [3,3,3,3,3,1]
 => [3,3,3,3,1]
 => ? = 5 - 1
([(0,5),(1,2),(1,3),(1,5),(2,6),(3,6),(5,6),(6,4)],7)
 => [6,5,5]
 => [3,3,3,3,3,1]
 => [3,3,3,3,1]
 => ? = 5 - 1
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,1),(4,6),(6,5)],7)
 => [6,5,5]
 => [3,3,3,3,3,1]
 => [3,3,3,3,1]
 => ? = 5 - 1
([(0,5),(2,6),(3,6),(4,1),(4,6),(5,2),(5,3),(5,4)],7)
 => [6,5,5]
 => [3,3,3,3,3,1]
 => [3,3,3,3,1]
 => ? = 5 - 1
([(0,5),(3,6),(4,1),(4,2),(4,6),(5,3),(5,4)],7)
 => [6,5,5]
 => [3,3,3,3,3,1]
 => [3,3,3,3,1]
 => ? = 5 - 1
([(0,5),(0,6),(3,2),(4,1),(5,3),(6,4)],7)
 => [5,4,4,4]
 => [4,4,4,4,1]
 => [4,4,4,1]
 => ? = 4 - 1
([(0,5),(0,6),(1,4),(1,6),(2,5),(3,2),(4,3)],7)
 => [6,5,5]
 => [3,3,3,3,3,1]
 => [3,3,3,3,1]
 => ? = 5 - 1
([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
 => [5,4,4,4]
 => [4,4,4,4,1]
 => [4,4,4,1]
 => ? = 4 - 1
Description
The multiplicity of the largest part of an integer partition.
Matching statistic: St000642
(load all 2 compositions to match this statistic)
Mapping
St000642: Posets ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ? = 2
([],2)
 => 2
([(0,1)],2)
 => 3
([],3)
 => 2
([(1,2)],3)
 => 6
([(0,1),(0,2)],3)
 => 2
([(0,2),(2,1)],3)
 => 4
([(0,2),(1,2)],3)
 => 2
([(2,3)],4)
 => 6
([(1,2),(1,3)],4)
 => 2
([(0,1),(0,2),(0,3)],4)
 => 2
([(0,2),(0,3),(3,1)],4)
 => 7
([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
([(1,2),(2,3)],4)
 => 4
([(0,3),(3,1),(3,2)],4)
 => 2
([(1,3),(2,3)],4)
 => 2
([(0,3),(1,3),(3,2)],4)
 => 2
([(0,3),(1,3),(2,3)],4)
 => 2
([(0,3),(1,2)],4)
 => 3
([(0,3),(1,2),(1,3)],4)
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
([(0,3),(2,1),(3,2)],4)
 => 5
([(0,3),(1,2),(2,3)],4)
 => 7
([(0,2),(0,3),(0,4),(4,1)],5)
 => 6
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 2
([(1,2),(1,3),(2,4),(3,4)],5)
 => 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 2
([(0,3),(0,4),(3,2),(4,1)],5)
 => 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => 4
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
([(2,3),(3,4)],5)
 => 4
([(1,4),(4,2),(4,3)],5)
 => 2
([(0,4),(4,1),(4,2),(4,3)],5)
 => 2
([(1,4),(2,4),(4,3)],5)
 => 2
([(0,4),(1,4),(4,2),(4,3)],5)
 => 2
([(0,4),(1,4),(2,4),(4,3)],5)
 => 2
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
([(0,4),(1,4),(2,3),(4,2)],5)
 => 2
([(0,4),(1,4),(2,3),(2,4)],5)
 => 3
([(0,4),(1,4),(2,3),(3,4)],5)
 => 6
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => 4
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => 2
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => 2
([(0,4),(1,2),(1,3),(1,4)],5)
 => 3
([(0,2),(0,4),(3,1),(4,3)],5)
 => 4
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 8
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => 2
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => 2
([(0,3),(3,4),(4,1),(4,2)],5)
 => 2
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,1),(4,6),(6,5)],7)
 => ? = 5
([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
 => ? = 3
([(0,2),(0,3),(1,5),(1,6),(2,4),(3,1),(3,4),(4,5),(4,6)],7)
 => ? = 2
([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => ? = 5
([(0,3),(0,5),(3,6),(4,1),(4,6),(5,4),(6,2)],7)
 => ? = 6
([(0,3),(0,4),(2,6),(3,5),(3,6),(4,2),(4,5),(6,1)],7)
 => ? = 6
([(0,3),(0,4),(2,5),(3,5),(3,6),(4,2),(4,6),(6,1)],7)
 => ? = 3
([(0,4),(0,5),(2,6),(4,2),(5,1),(5,6),(6,3)],7)
 => ? = 3
([(0,3),(1,2),(1,5),(2,6),(3,5),(3,6),(5,4),(6,4)],7)
 => ? = 3
([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7)
 => ? = 4
([(0,2),(0,4),(1,5),(1,6),(2,5),(2,6),(3,1),(4,3)],7)
 => ? = 2
([(0,2),(0,5),(2,6),(3,4),(4,1),(4,6),(5,3)],7)
 => ? = 6
([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
 => ? = 2
([(0,2),(0,3),(2,5),(2,6),(3,5),(3,6),(4,1),(6,4)],7)
 => ? = 2
([(0,3),(0,5),(3,6),(4,2),(5,1),(5,6),(6,4)],7)
 => ? = 6
([(0,4),(0,5),(1,6),(2,6),(4,2),(5,1),(6,3)],7)
 => ? = 3
([(0,2),(0,5),(2,6),(3,1),(4,3),(4,6),(5,4)],7)
 => ? = 3
([(0,2),(0,4),(1,5),(2,5),(2,6),(3,1),(3,6),(4,3)],7)
 => ? = 6
([(0,3),(0,4),(3,6),(4,6),(5,1),(5,2),(6,5)],7)
 => ? = 2
([(0,3),(1,5),(1,6),(2,6),(3,2),(3,5),(5,4),(6,4)],7)
 => ? = 6
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ? = 5
([(0,2),(1,5),(1,6),(2,3),(3,5),(3,6),(5,4),(6,4)],7)
 => ? = 2
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2),(4,6),(5,6)],7)
 => ? = 2
([(0,5),(1,6),(2,6),(5,1),(5,2),(6,3),(6,4)],7)
 => ? = 2
([(0,5),(1,4),(4,6),(5,6),(6,2),(6,3)],7)
 => ? = 2
([(0,3),(1,2),(2,5),(2,6),(3,5),(3,6),(5,4),(6,4)],7)
 => ? = 2
([(0,6),(1,5),(2,6),(5,2),(6,3),(6,4)],7)
 => ? = 2
([(0,6),(1,3),(1,6),(2,4),(3,2),(3,5),(5,4),(6,5)],7)
 => ? = 3
([(0,6),(1,2),(1,6),(2,4),(2,5),(4,3),(5,3),(6,4),(6,5)],7)
 => ? = 2
([(0,3),(0,4),(1,5),(2,5),(2,6),(3,2),(4,1),(4,6)],7)
 => ? = 3
([(0,6),(1,3),(1,6),(3,5),(4,2),(5,4),(6,5)],7)
 => ? = 5
([(0,6),(1,3),(1,6),(2,5),(3,5),(4,2),(6,4)],7)
 => ? = 6
([(0,5),(2,6),(3,6),(4,1),(4,6),(5,2),(5,3),(5,4)],7)
 => ? = 5
([(0,5),(3,6),(4,1),(4,2),(4,6),(5,3),(5,4)],7)
 => ? = 5
([(0,5),(0,6),(3,2),(4,1),(5,3),(6,4)],7)
 => ? = 4
([(0,3),(0,5),(1,5),(1,6),(2,4),(3,6),(5,2),(6,4)],7)
 => ? = 3
([(0,3),(0,6),(1,4),(1,6),(2,5),(3,4),(4,2),(6,5)],7)
 => ? = 6
([(0,5),(0,6),(1,4),(1,6),(2,5),(3,2),(4,3)],7)
 => ? = 5
([(0,3),(1,5),(1,6),(2,5),(2,6),(3,4),(4,1),(4,2)],7)
 => ? = 2
([(0,4),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3)],7)
 => ? = 5
([(0,3),(1,4),(1,6),(2,5),(3,6),(4,5),(6,2)],7)
 => ? = 3
([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7)
 => ? = 6
([(0,5),(3,4),(4,6),(5,3),(6,1),(6,2)],7)
 => ? = 2
([(0,5),(3,4),(4,1),(5,6),(6,2),(6,3)],7)
 => ? = 4
([(0,6),(1,4),(2,5),(3,5),(4,3),(4,6),(6,2)],7)
 => ? = 6
([(0,5),(3,2),(4,1),(5,6),(6,3),(6,4)],7)
 => ? = 3
([(0,6),(1,4),(2,5),(3,2),(3,6),(4,3),(6,5)],7)
 => ? = 6
([(0,6),(1,4),(2,5),(3,2),(4,3),(4,6),(6,5)],7)
 => ? = 3
([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ? = 8
Description
The size of the smallest orbit of antichains under Panyushev complementation.
Matching statistic: St000733
(load all 2 compositions to match this statistic)
Mapping
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: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [2]
 => [1,1]
 => [[1],[2]]
 => 2
([],2)
 => [2,2]
 => [2,2]
 => [[1,2],[3,4]]
 => 2
([(0,1)],2)
 => [3]
 => [1,1,1]
 => [[1],[2],[3]]
 => 3
([],3)
 => [2,2,2,2]
 => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => 2
([(1,2)],3)
 => [6]
 => [1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => 6
([(0,1),(0,2)],3)
 => [3,2]
 => [2,2,1]
 => [[1,3],[2,5],[4]]
 => 2
([(0,2),(2,1)],3)
 => [4]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 4
([(0,2),(1,2)],3)
 => [3,2]
 => [2,2,1]
 => [[1,3],[2,5],[4]]
 => 2
([(2,3)],4)
 => [6,6]
 => [2,2,2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]
 => 6
([(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
([(0,1),(0,2),(0,3)],4)
 => [3,2,2,2]
 => [4,4,1]
 => [[1,3,4,5],[2,7,8,9],[6]]
 => 2
([(0,2),(0,3),(3,1)],4)
 => [7]
 => [1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => 7
([(0,1),(0,2),(1,3),(2,3)],4)
 => [4,2]
 => [2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => 2
([(1,2),(2,3)],4)
 => [4,4]
 => [2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => 4
([(0,3),(3,1),(3,2)],4)
 => [4,2]
 => [2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => 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]]
 => 2
([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => [2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => 2
([(0,3),(1,3),(2,3)],4)
 => [3,2,2,2]
 => [4,4,1]
 => [[1,3,4,5],[2,7,8,9],[6]]
 => 2
([(0,3),(1,2)],4)
 => [3,3,3]
 => [3,3,3]
 => [[1,2,3],[4,5,6],[7,8,9]]
 => 3
([(0,3),(1,2),(1,3)],4)
 => [5,3]
 => [2,2,2,1,1]
 => [[1,4],[2,6],[3,8],[5],[7]]
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => [3,2,2]
 => [3,3,1]
 => [[1,3,4],[2,6,7],[5]]
 => 2
([(0,3),(2,1),(3,2)],4)
 => [5]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 5
([(0,3),(1,2),(2,3)],4)
 => [7]
 => [1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => 7
([(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]]
 => ? = 6
([(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
([(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
([(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
([(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
([(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
([(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
([(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]]
 => ? = 4
([(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
([(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
([(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
([(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
([(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
([(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
([(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]]
 => ? = 2
([(0,4),(1,4),(2,3),(4,2)],5)
 => [5,2]
 => [2,2,1,1,1]
 => [[1,5],[2,7],[3],[4],[6]]
 => 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]]
 => ? = 3
([(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]]
 => ? = 6
([(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
([(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
([(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
([(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]]
 => ? = 3
([(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
([(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
([(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
([(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]]
 => ? = 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]]
 => 2
([(0,3),(3,4),(4,1),(4,2)],5)
 => [5,2]
 => [2,2,1,1,1]
 => [[1,5],[2,7],[3],[4],[6]]
 => 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]]
 => 8
([(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
([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => [1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => 6
([(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]]
 => 3
([(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]]
 => 4
([(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]]
 => 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]]
 => ? = 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]]
 => ? = 4
([(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]]
 => ? = 4
([(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]]
 => ? = 2
([(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]]
 => ? = 2
([(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]]
 => ? = 2
([(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]]
 => ? = 2
([(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]]
 => ? = 4
([(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]]
 => ? = 4
([(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]]
 => ? = 3
([(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]]
 => ? = 4
([(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]]
 => ? = 4
([(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]]
 => ? = 2
([(0,4),(0,5),(1,2),(1,4),(2,5),(4,3)],6)
 => [6,4,3]
 => [3,3,3,2,1,1]
 => [[1,4,7],[2,6,10],[3,9,13],[5,12],[8],[11]]
 => ? = 3
([(0,4),(0,5),(1,2),(1,4),(2,5),(5,3)],6)
 => [7,5]
 => [2,2,2,2,2,1,1]
 => [[1,4],[2,6],[3,8],[5,10],[7,12],[9],[11]]
 => ? = 5
([(0,4),(1,3),(1,5),(4,5),(5,2)],6)
 => [5,5,3]
 => [3,3,3,2,2]
 => [[1,2,7],[3,4,10],[5,6,13],[8,9],[11,12]]
 => ? = 3
([(0,3),(1,2),(1,4),(2,5),(3,4),(3,5)],6)
 => [6,4,3]
 => [3,3,3,2,1,1]
 => [[1,4,7],[2,6,10],[3,9,13],[5,12],[8],[11]]
 => ? = 3
([(0,4),(0,5),(1,2),(2,3),(2,5),(3,4)],6)
 => [7,5]
 => [2,2,2,2,2,1,1]
 => [[1,4],[2,6],[3,8],[5,10],[7,12],[9],[11]]
 => ? = 5
([(0,5),(1,4),(4,2),(5,3)],6)
 => [4,4,4,4]
 => [4,4,4,4]
 => [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16]]
 => ? = 4
([(0,5),(1,3),(1,5),(4,2),(5,4)],6)
 => [7,5]
 => [2,2,2,2,2,1,1]
 => [[1,4],[2,6],[3,8],[5,10],[7,12],[9],[11]]
 => ? = 5
([(0,5),(1,4),(3,2),(4,3),(4,5)],6)
 => [5,5,3]
 => [3,3,3,2,2]
 => [[1,2,7],[3,4,10],[5,6,13],[8,9],[11,12]]
 => ? = 3
([(0,5),(1,3),(3,4),(4,2),(4,5)],6)
 => [7,5]
 => [2,2,2,2,2,1,1]
 => [[1,4],[2,6],[3,8],[5,10],[7,12],[9],[11]]
 => ? = 5
([(0,2),(0,3),(2,4),(2,6),(3,4),(3,6),(4,5),(6,1),(6,5)],7)
 => [5,5,2]
 => [3,3,2,2,2]
 => [[1,2,9],[3,4,12],[5,6],[7,8],[10,11]]
 => ? = 2
([(0,6),(1,6),(2,3),(2,6),(3,5),(5,4),(6,5)],7)
 => [6,5,5]
 => [3,3,3,3,3,1]
 => [[1,3,4],[2,6,7],[5,9,10],[8,12,13],[11,15,16],[14]]
 => ? = 5
([(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,2),(4,5),(5,6)],7)
 => [6,5,5]
 => [3,3,3,3,3,1]
 => [[1,3,4],[2,6,7],[5,9,10],[8,12,13],[11,15,16],[14]]
 => ? = 5
([(0,6),(1,6),(3,5),(4,2),(4,5),(6,3),(6,4)],7)
 => [5,5,2]
 => [3,3,2,2,2]
 => [[1,2,9],[3,4,12],[5,6],[7,8],[10,11]]
 => ? = 2
([(0,5),(0,6),(1,5),(1,6),(2,3),(4,3),(5,4),(6,2),(6,4)],7)
 => [5,5,2]
 => [3,3,2,2,2]
 => [[1,2,9],[3,4,12],[5,6],[7,8],[10,11]]
 => ? = 2
([(0,5),(1,4),(1,5),(4,6),(5,6),(6,2),(6,3)],7)
 => [5,5,2]
 => [3,3,2,2,2]
 => [[1,2,9],[3,4,12],[5,6],[7,8],[10,11]]
 => ? = 2
([(0,5),(1,2),(1,3),(1,5),(2,6),(3,6),(5,6),(6,4)],7)
 => [6,5,5]
 => [3,3,3,3,3,1]
 => [[1,3,4],[2,6,7],[5,9,10],[8,12,13],[11,15,16],[14]]
 => ? = 5
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,1),(4,6),(6,5)],7)
 => [6,5,5]
 => [3,3,3,3,3,1]
 => [[1,3,4],[2,6,7],[5,9,10],[8,12,13],[11,15,16],[14]]
 => ? = 5
([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
 => [6,5,3]
 => [3,3,3,2,2,1]
 => [[1,3,8],[2,5,11],[4,7,14],[6,10],[9,13],[12]]
 => ? = 3
([(0,2),(0,3),(1,5),(1,6),(2,4),(3,1),(3,4),(4,5),(4,6)],7)
 => [5,5,2]
 => [3,3,2,2,2]
 => [[1,2,9],[3,4,12],[5,6],[7,8],[10,11]]
 => ? = 2
([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => [6,5]
 => [2,2,2,2,2,1]
 => [[1,3],[2,5],[4,7],[6,9],[8,11],[10]]
 => ? = 5
([(0,3),(0,5),(3,6),(4,1),(4,6),(5,4),(6,2)],7)
 => [7,6]
 => [2,2,2,2,2,2,1]
 => [[1,3],[2,5],[4,7],[6,9],[8,11],[10,13],[12]]
 => ? = 6
([(0,3),(0,4),(2,6),(3,5),(3,6),(4,2),(4,5),(6,1)],7)
 => [7,6]
 => [2,2,2,2,2,2,1]
 => [[1,3],[2,5],[4,7],[6,9],[8,11],[10,13],[12]]
 => ? = 6
([(0,3),(0,4),(2,5),(3,5),(3,6),(4,2),(4,6),(6,1)],7)
 => [6,5,3]
 => [3,3,3,2,2,1]
 => [[1,3,8],[2,5,11],[4,7,14],[6,10],[9,13],[12]]
 => ? = 3
([(0,4),(0,5),(2,6),(4,2),(5,1),(5,6),(6,3)],7)
 => [6,5,3]
 => [3,3,3,2,2,1]
 => [[1,3,8],[2,5,11],[4,7,14],[6,10],[9,13],[12]]
 => ? = 3
([(0,3),(1,2),(1,5),(2,6),(3,5),(3,6),(5,4),(6,4)],7)
 => [6,5,3]
 => [3,3,3,2,2,1]
 => [[1,3,8],[2,5,11],[4,7,14],[6,10],[9,13],[12]]
 => ? = 3
([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7)
 => [7,4]
 => [2,2,2,2,1,1,1]
 => [[1,5],[2,7],[3,9],[4,11],[6],[8],[10]]
 => ? = 4
([(0,2),(0,5),(2,6),(3,4),(4,1),(4,6),(5,3)],7)
 => [7,6]
 => [2,2,2,2,2,2,1]
 => [[1,3],[2,5],[4,7],[6,9],[8,11],[10,13],[12]]
 => ? = 6
([(0,3),(0,5),(3,6),(4,2),(5,1),(5,6),(6,4)],7)
 => [7,6]
 => [2,2,2,2,2,2,1]
 => [[1,3],[2,5],[4,7],[6,9],[8,11],[10,13],[12]]
 => ? = 6
([(0,4),(0,5),(1,6),(2,6),(4,2),(5,1),(6,3)],7)
 => [6,3,3]
 => [3,3,3,1,1,1]
 => [[1,5,6],[2,8,9],[3,11,12],[4],[7],[10]]
 => ? = 3
([(0,2),(0,5),(2,6),(3,1),(4,3),(4,6),(5,4)],7)
 => [6,5,3]
 => [3,3,3,2,2,1]
 => [[1,3,8],[2,5,11],[4,7,14],[6,10],[9,13],[12]]
 => ? = 3
Description
The row containing the largest entry of a standard tableau.
Matching statistic: St000745
Mapping
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: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [2]
 => [[1,2]]
 => [[1],[2]]
 => 2
([],2)
 => [2,2]
 => [[1,2],[3,4]]
 => [[1,3],[2,4]]
 => 2
([(0,1)],2)
 => [3]
 => [[1,2,3]]
 => [[1],[2],[3]]
 => 3
([],3)
 => [2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => [[1,3,5,7],[2,4,6,8]]
 => 2
([(1,2)],3)
 => [6]
 => [[1,2,3,4,5,6]]
 => [[1],[2],[3],[4],[5],[6]]
 => 6
([(0,1),(0,2)],3)
 => [3,2]
 => [[1,2,5],[3,4]]
 => [[1,3],[2,4],[5]]
 => 2
([(0,2),(2,1)],3)
 => [4]
 => [[1,2,3,4]]
 => [[1],[2],[3],[4]]
 => 4
([(0,2),(1,2)],3)
 => [3,2]
 => [[1,2,5],[3,4]]
 => [[1,3],[2,4],[5]]
 => 2
([(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
([(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
([(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
([(0,2),(0,3),(3,1)],4)
 => [7]
 => [[1,2,3,4,5,6,7]]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => 7
([(0,1),(0,2),(1,3),(2,3)],4)
 => [4,2]
 => [[1,2,5,6],[3,4]]
 => [[1,3],[2,4],[5],[6]]
 => 2
([(1,2),(2,3)],4)
 => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => [[1,5],[2,6],[3,7],[4,8]]
 => 4
([(0,3),(3,1),(3,2)],4)
 => [4,2]
 => [[1,2,5,6],[3,4]]
 => [[1,3],[2,4],[5],[6]]
 => 2
([(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
([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => [[1,2,5,6],[3,4]]
 => [[1,3],[2,4],[5],[6]]
 => 2
([(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
([(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
([(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
([(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
([(0,3),(2,1),(3,2)],4)
 => [5]
 => [[1,2,3,4,5]]
 => [[1],[2],[3],[4],[5]]
 => 5
([(0,3),(1,2),(2,3)],4)
 => [7]
 => [[1,2,3,4,5,6,7]]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => 7
([(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]]
 => ? = 6
([(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
([(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]]
 => ? = 2
([(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
([(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
([(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
([(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
([(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]]
 => ? = 4
([(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]]
 => ? = 2
([(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
([(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]]
 => ? = 2
([(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
([(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
([(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]]
 => ? = 2
([(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]]
 => ? = 2
([(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
([(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]]
 => ? = 3
([(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]]
 => ? = 6
([(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
([(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
([(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
([(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]]
 => ? = 3
([(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
([(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
([(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]]
 => ? = 2
([(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]]
 => ? = 2
([(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
([(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
([(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
([(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
([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => [[1,2,3,4,5,6]]
 => [[1],[2],[3],[4],[5],[6]]
 => 6
([(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
([(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
([(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
([(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]]
 => ? = 2
([(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]]
 => ? = 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]
 => [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
 => ?
 => ? = 2
([(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]]
 => ? = 4
([(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]]
 => ? = 2
([(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
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => [5,2,2]
 => [[1,2,7,8,9],[3,4],[5,6]]
 => [[1,3,5],[2,4,6],[7],[8],[9]]
 => 2
([(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]]
 => ? = 4
([(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]]
 => ? = 2
([(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]]
 => ?
 => ? = 2
([(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]]
 => ? = 2
([(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]]
 => ?
 => ? = 2
([(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]]
 => ?
 => ? = 2
([(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]]
 => ? = 2
([(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]]
 => ? = 2
([(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]]
 => ?
 => ? = 2
([(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]]
 => ? = 4
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
 => [5,2,2]
 => [[1,2,7,8,9],[3,4],[5,6]]
 => [[1,3,5],[2,4,6],[7],[8],[9]]
 => 2
([(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]]
 => ? = 2
([(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]]
 => ? = 3
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => [[1,3],[2,4],[5],[6],[7],[8]]
 => 2
([(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]]
 => ?
 => ? = 4
([(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]]
 => ? = 2
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],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]]
 => ? = 2
([(0,5),(1,4),(2,4),(3,5),(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]]
 => ? = 2
([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
 => [5,2,2]
 => [[1,2,7,8,9],[3,4],[5,6]]
 => [[1,3,5],[2,4,6],[7],[8],[9]]
 => 2
([(0,4),(1,2),(1,4),(2,3),(2,5),(4,5)],6)
 => [6,4,3]
 => [[1,2,3,7,12,13],[4,5,6,11],[8,9,10]]
 => ?
 => ? = 3
([(0,5),(1,2),(1,5),(2,3),(2,4),(5,3),(5,4)],6)
 => [5,4,2]
 => [[1,2,5,6,11],[3,4,9,10],[7,8]]
 => [[1,3,7],[2,4,8],[5,9],[6,10],[11]]
 => ? = 2
([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2)],6)
 => [4,3,3,2]
 => [[1,2,5,12],[3,4,8],[6,7,11],[9,10]]
 => [[1,3,6,9],[2,4,7,10],[5,8,11],[12]]
 => ? = 2
([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2),(5,3)],6)
 => [5,4,2]
 => [[1,2,5,6,11],[3,4,9,10],[7,8]]
 => [[1,3,7],[2,4,8],[5,9],[6,10],[11]]
 => ? = 2
([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6)
 => [5,4,2]
 => [[1,2,5,6,11],[3,4,9,10],[7,8]]
 => [[1,3,7],[2,4,8],[5,9],[6,10],[11]]
 => ? = 2
([(0,4),(1,2),(1,3),(1,4),(2,5),(3,5),(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]]
 => ? = 4
([(0,2),(0,3),(0,5),(4,1),(5,4)],6)
 => [5,4,4,4]
 => [[1,2,3,4,17],[5,6,7,8],[9,10,11,12],[13,14,15,16]]
 => ?
 => ? = 4
([(0,3),(0,4),(4,5),(5,1),(5,2)],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]]
 => ? = 2
([(0,4),(1,2),(1,3),(2,5),(3,5),(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]]
 => ? = 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
 => [4,2,2,2,2]
 => [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
 => ?
 => ? = 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],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]]
 => ? = 2
([(0,4),(0,5),(1,2),(1,4),(2,5),(4,3)],6)
 => [6,4,3]
 => [[1,2,3,7,12,13],[4,5,6,11],[8,9,10]]
 => ?
 => ? = 3
([(0,4),(0,5),(1,2),(1,4),(2,5),(5,3)],6)
 => [7,5]
 => [[1,2,3,4,5,11,12],[6,7,8,9,10]]
 => ?
 => ? = 5
([(0,4),(1,3),(1,5),(4,5),(5,2)],6)
 => [5,5,3]
 => [[1,2,3,7,8],[4,5,6,12,13],[9,10,11]]
 => [[1,4,9],[2,5,10],[3,6,11],[7,12],[8,13]]
 => ? = 3
([(0,3),(1,2),(1,4),(2,5),(3,4),(3,5)],6)
 => [6,4,3]
 => [[1,2,3,7,12,13],[4,5,6,11],[8,9,10]]
 => ?
 => ? = 3
([(0,4),(0,5),(1,2),(2,3),(2,5),(3,4)],6)
 => [7,5]
 => [[1,2,3,4,5,11,12],[6,7,8,9,10]]
 => ?
 => ? = 5
([(0,4),(1,3),(3,5),(4,5),(5,2)],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]]
 => ? = 3
Description
The index of the last row whose first entry is the row number in a standard Young tableau.
Matching statistic: St000771
Mapping
Mp00206: Posets —antichains of maximal size⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 14% ●values known / values provided: 28%●distinct values known / distinct values provided: 14%
Values
([],1)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([],2)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 3 - 1
([],3)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(1,2)],3)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 6 - 1
([(0,1),(0,2)],3)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 4 - 1
([(0,2),(1,2)],3)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 6 - 1
([(1,2),(1,3)],4)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,1),(0,2),(0,3)],4)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 7 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 4 - 1
([(0,3),(3,1),(3,2)],4)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,3),(1,3),(3,2)],4)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => ? = 3 - 1
([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 3 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 2 - 1
([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 5 - 1
([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 7 - 1
([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 6 - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(1,2),(1,3),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => ? = 3 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 4 - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 2 - 1
([(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 4 - 1
([(1,4),(4,2),(4,3)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,4),(4,1),(4,2),(4,3)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(1,4),(2,4),(4,3)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 2 - 1
([(0,4),(1,4),(2,4),(4,3)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,4),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 3 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 6 - 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 4 - 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 2 - 1
([(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 3 - 1
([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 4 - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 8 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 2 - 1
([(0,3),(3,4),(4,1),(4,2)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 8 - 1
([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 8 - 1
([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ? = 6 - 1
([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => ? = 3 - 1
([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 4 - 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 4 - 1
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 2 - 1
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 2 - 1
([(0,3),(0,4),(3,5),(4,1),(4,2),(4,5)],6)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 4 - 1
([(0,2),(0,3),(2,4),(2,5),(3,1),(3,4),(3,5)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,1),(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,5),(1,5),(5,2),(5,3),(5,4)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,5),(1,5),(2,5),(5,3),(5,4)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,5),(1,3),(1,5),(2,3),(2,5),(3,4),(5,4)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,5),(1,5),(2,3),(2,5),(3,4),(5,4)],6)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 4 - 1
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 2 - 1
([(0,3),(0,4),(1,5),(2,5),(4,1),(4,2)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => ? = 3 - 1
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 4 - 1
([(0,5),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 2 - 1
([(0,4),(1,2),(1,4),(2,3),(2,5),(4,5)],6)
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ? = 3 - 1
([(0,5),(1,2),(1,5),(2,3),(2,4),(5,3),(5,4)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 2 - 1
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 5 - 1
([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2)],6)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ? = 2 - 1
([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2),(5,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 2 - 1
([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 2 - 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 2 - 1
([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 2 - 1
([(0,4),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 4 - 1
([(0,2),(0,3),(0,5),(4,1),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 4 - 1
([(0,3),(0,4),(4,5),(5,1),(5,2)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,5),(3,5),(5,4)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
Description
The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue. For example, the cycle on four vertices has distance Laplacian $$ \left(\begin{array}{rrrr} 4 & -1 & -2 & -1 \\ -1 & 4 & -1 & -2 \\ -2 & -1 & 4 & -1 \\ -1 & -2 & -1 & 4 \end{array}\right). $$ Its eigenvalues are $0,4,4,6$, so the statistic is $2$. The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore statistic $1$.
Matching statistic: St000772
Mapping
Mp00206: Posets —antichains of maximal size⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 14% ●values known / values provided: 28%●distinct values known / distinct values provided: 14%
Values
([],1)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([],2)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 3 - 1
([],3)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(1,2)],3)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 6 - 1
([(0,1),(0,2)],3)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 4 - 1
([(0,2),(1,2)],3)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 6 - 1
([(1,2),(1,3)],4)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,1),(0,2),(0,3)],4)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 7 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 4 - 1
([(0,3),(3,1),(3,2)],4)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,3),(1,3),(3,2)],4)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => ? = 3 - 1
([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 3 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 2 - 1
([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 5 - 1
([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 7 - 1
([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 6 - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(1,2),(1,3),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => ? = 3 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 4 - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 2 - 1
([(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 4 - 1
([(1,4),(4,2),(4,3)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,4),(4,1),(4,2),(4,3)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(1,4),(2,4),(4,3)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 2 - 1
([(0,4),(1,4),(2,4),(4,3)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,4),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 3 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 6 - 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 4 - 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 2 - 1
([(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 3 - 1
([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 4 - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 8 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 2 - 1
([(0,3),(3,4),(4,1),(4,2)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 8 - 1
([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 8 - 1
([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ? = 6 - 1
([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => ? = 3 - 1
([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 4 - 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 4 - 1
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 2 - 1
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 2 - 1
([(0,3),(0,4),(3,5),(4,1),(4,2),(4,5)],6)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 4 - 1
([(0,2),(0,3),(2,4),(2,5),(3,1),(3,4),(3,5)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,1),(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,5),(1,5),(5,2),(5,3),(5,4)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,5),(1,5),(2,5),(5,3),(5,4)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,5),(1,3),(1,5),(2,3),(2,5),(3,4),(5,4)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,5),(1,5),(2,3),(2,5),(3,4),(5,4)],6)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 4 - 1
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 2 - 1
([(0,3),(0,4),(1,5),(2,5),(4,1),(4,2)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => ? = 3 - 1
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 4 - 1
([(0,5),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 2 - 1
([(0,4),(1,2),(1,4),(2,3),(2,5),(4,5)],6)
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ? = 3 - 1
([(0,5),(1,2),(1,5),(2,3),(2,4),(5,3),(5,4)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 2 - 1
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 5 - 1
([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2)],6)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ? = 2 - 1
([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2),(5,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 2 - 1
([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 2 - 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 2 - 1
([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 2 - 1
([(0,4),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 4 - 1
([(0,2),(0,3),(0,5),(4,1),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 4 - 1
([(0,3),(0,4),(4,5),(5,1),(5,2)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,5),(3,5),(5,4)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
Description
The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue. For example, the cycle on four vertices has distance Laplacian $$ \left(\begin{array}{rrrr} 4 & -1 & -2 & -1 \\ -1 & 4 & -1 & -2 \\ -2 & -1 & 4 & -1 \\ -1 & -2 & -1 & 4 \end{array}\right). $$ Its eigenvalues are $0,4,4,6$, so the statistic is $1$. The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore also statistic $1$. The graphs with statistic $n-1$, $n-2$ and $n-3$ have been characterised, see [1].
Matching statistic: St000777
Mapping
Mp00206: Posets —antichains of maximal size⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 14% ●values known / values provided: 28%●distinct values known / distinct values provided: 14%
Values
([],1)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([],2)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 3 - 1
([],3)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(1,2)],3)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 6 - 1
([(0,1),(0,2)],3)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 4 - 1
([(0,2),(1,2)],3)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 6 - 1
([(1,2),(1,3)],4)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,1),(0,2),(0,3)],4)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 7 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 4 - 1
([(0,3),(3,1),(3,2)],4)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,3),(1,3),(3,2)],4)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => ? = 3 - 1
([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 3 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 2 - 1
([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 5 - 1
([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 7 - 1
([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 6 - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(1,2),(1,3),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => ? = 3 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 4 - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 2 - 1
([(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 4 - 1
([(1,4),(4,2),(4,3)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,4),(4,1),(4,2),(4,3)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(1,4),(2,4),(4,3)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 2 - 1
([(0,4),(1,4),(2,4),(4,3)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,4),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 3 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 6 - 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 4 - 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 2 - 1
([(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 3 - 1
([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 4 - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 8 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 2 - 1
([(0,3),(3,4),(4,1),(4,2)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 8 - 1
([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 8 - 1
([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ? = 6 - 1
([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => ? = 3 - 1
([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 4 - 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 4 - 1
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 2 - 1
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 2 - 1
([(0,3),(0,4),(3,5),(4,1),(4,2),(4,5)],6)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 4 - 1
([(0,2),(0,3),(2,4),(2,5),(3,1),(3,4),(3,5)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,1),(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,5),(1,5),(5,2),(5,3),(5,4)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,5),(1,5),(2,5),(5,3),(5,4)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,5),(1,3),(1,5),(2,3),(2,5),(3,4),(5,4)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,5),(1,5),(2,3),(2,5),(3,4),(5,4)],6)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 4 - 1
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 2 - 1
([(0,3),(0,4),(1,5),(2,5),(4,1),(4,2)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => ? = 3 - 1
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 4 - 1
([(0,5),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 2 - 1
([(0,4),(1,2),(1,4),(2,3),(2,5),(4,5)],6)
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => ? = 3 - 1
([(0,5),(1,2),(1,5),(2,3),(2,4),(5,3),(5,4)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 2 - 1
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 5 - 1
([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2)],6)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ? = 2 - 1
([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2),(5,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 2 - 1
([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 2 - 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 2 - 1
([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 2 - 1
([(0,4),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 4 - 1
([(0,2),(0,3),(0,5),(4,1),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 4 - 1
([(0,3),(0,4),(4,5),(5,1),(5,2)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,5),(3,5),(5,4)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1 = 2 - 1
Description
The number of distinct eigenvalues of the distance Laplacian of a connected graph.
The following 9 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001645The pebbling number of a connected graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001481The minimal height of a peak of a Dyck path. St001803The maximal overlap of the cylindrical tableau associated with a tableau.