Your data matches 13 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000326
Mapping
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00280: Binary words —path rowmotion⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1]
 => 10 => 11 => 1
[2]
 => [1,1]
 => 110 => 111 => 1
[1,1]
 => [2]
 => 100 => 011 => 2
[3]
 => [1,1,1]
 => 1110 => 1111 => 1
[2,1]
 => [2,1]
 => 1010 => 1101 => 1
[1,1,1]
 => [3]
 => 1000 => 0011 => 3
[4]
 => [1,1,1,1]
 => 11110 => 11111 => 1
[3,1]
 => [2,1,1]
 => 10110 => 11011 => 1
[2,2]
 => [2,2]
 => 1100 => 0111 => 2
[2,1,1]
 => [3,1]
 => 10010 => 01101 => 2
[1,1,1,1]
 => [4]
 => 10000 => 00011 => 4
[5]
 => [1,1,1,1,1]
 => 111110 => 111111 => 1
[4,1]
 => [2,1,1,1]
 => 101110 => 110111 => 1
[3,2]
 => [2,2,1]
 => 11010 => 11101 => 1
[3,1,1]
 => [3,1,1]
 => 100110 => 011011 => 2
[2,2,1]
 => [3,2]
 => 10100 => 11001 => 1
[2,1,1,1]
 => [4,1]
 => 100010 => 001101 => 3
[1,1,1,1,1]
 => [5]
 => 100000 => 000011 => 5
[6]
 => [1,1,1,1,1,1]
 => 1111110 => 1111111 => 1
[5,1]
 => [2,1,1,1,1]
 => 1011110 => 1101111 => 1
[4,2]
 => [2,2,1,1]
 => 110110 => 111011 => 1
[4,1,1]
 => [3,1,1,1]
 => 1001110 => 0110111 => 2
[3,3]
 => [2,2,2]
 => 11100 => 01111 => 2
[3,2,1]
 => [3,2,1]
 => 101010 => 110101 => 1
[3,1,1,1]
 => [4,1,1]
 => 1000110 => 0011011 => 3
[2,2,2]
 => [3,3]
 => 11000 => 00111 => 3
[2,2,1,1]
 => [4,2]
 => 100100 => 011001 => 2
[2,1,1,1,1]
 => [5,1]
 => 1000010 => 0001101 => 4
[1,1,1,1,1,1]
 => [6]
 => 1000000 => 0000011 => 6
[7]
 => [1,1,1,1,1,1,1]
 => 11111110 => 11111111 => 1
[6,1]
 => [2,1,1,1,1,1]
 => 10111110 => 11011111 => 1
[5,2]
 => [2,2,1,1,1]
 => 1101110 => 1110111 => 1
[5,1,1]
 => [3,1,1,1,1]
 => 10011110 => 01101111 => 2
[4,3]
 => [2,2,2,1]
 => 111010 => 111101 => 1
[4,2,1]
 => [3,2,1,1]
 => 1010110 => 1101011 => 1
[4,1,1,1]
 => [4,1,1,1]
 => 10001110 => 00110111 => 3
[3,3,1]
 => [3,2,2]
 => 101100 => 110011 => 1
[3,2,2]
 => [3,3,1]
 => 110010 => 011101 => 2
[3,2,1,1]
 => [4,2,1]
 => 1001010 => 0110101 => 2
[3,1,1,1,1]
 => [5,1,1]
 => 10000110 => 00011011 => 4
[2,2,2,1]
 => [4,3]
 => 101000 => 110001 => 1
[2,2,1,1,1]
 => [5,2]
 => 1000100 => 0011001 => 3
[2,1,1,1,1,1]
 => [6,1]
 => 10000010 => 00001101 => 5
[1,1,1,1,1,1,1]
 => [7]
 => 10000000 => 00000011 => 7
[8]
 => [1,1,1,1,1,1,1,1]
 => 111111110 => 111111111 => 1
[6,2]
 => [2,2,1,1,1,1]
 => 11011110 => 11101111 => 1
[5,3]
 => [2,2,2,1,1]
 => 1110110 => 1111011 => 1
[5,2,1]
 => [3,2,1,1,1]
 => 10101110 => 11010111 => 1
[4,4]
 => [2,2,2,2]
 => 111100 => 011111 => 2
[4,3,1]
 => [3,2,2,1]
 => 1011010 => 1101101 => 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: St000382
(load all 2 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [1,1] => [1,1] => 1
[2]
 => [1,1,0,0,1,0]
 => [2,1] => [1,2] => 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,2] => [2,1] => 2
[3]
 => [1,1,1,0,0,0,1,0]
 => [3,1] => [1,3] => 1
[2,1]
 => [1,0,1,0,1,0]
 => [1,1,1] => [1,1,1] => 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,3] => [3,1] => 3
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1] => [1,4] => 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [2,1,1] => [1,1,2] => 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,2] => [2,2] => 2
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,2,1] => [2,1,1] => 2
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,4] => [4,1] => 4
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [5,1] => [1,5] => 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [3,1,1] => [1,1,3] => 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,1,1] => [1,1,2] => 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,2,1] => [2,1,1] => 2
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,2] => [1,2,1] => 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,3,1] => [3,1,1] => 3
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,5] => [5,1] => 5
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [6,1] => [1,6] => 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [4,1,1] => [1,1,4] => 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,1,1] => [1,1,3] => 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,2,1] => [2,1,2] => 2
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,2] => [2,3] => 2
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1] => [1,1,1,1] => 1
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,3,1] => [3,1,1] => 3
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,3] => [3,2] => 3
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,2,2] => [2,2,1] => 2
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,4,1] => [4,1,1] => 4
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,6] => [6,1] => 6
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [7,1] => [1,7] => 1
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [5,1,1] => [1,1,5] => 1
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [4,1,1] => [1,1,4] => 1
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [3,2,1] => [2,1,3] => 2
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,1,1] => [1,1,3] => 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [2,1,1,1] => [1,1,1,2] => 1
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => [3,1,1] => 3
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [2,1,2] => [1,2,2] => 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,2,1] => [2,1,2] => 2
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,2,1,1] => [2,1,1,1] => 2
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,4,1] => [4,1,1] => 4
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => [1,3,1] => 1
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,3,2] => [3,2,1] => 3
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,5,1] => [5,1,1] => 5
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,7] => [7,1] => 7
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [8,1] => [1,8] => 1
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [5,1,1] => [1,1,5] => 1
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [4,1,1] => [1,1,4] => 1
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [3,1,1,1] => [1,1,1,3] => 1
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [4,2] => [2,4] => 2
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [2,1,1,1] => [1,1,1,2] => 1
Description
The first part of an integer composition.
Matching statistic: St000907
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St000907: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0]
 => [1,0]
 => ([],1)
 => 1
[2]
 => [1,0,1,0]
 => [1,1,0,0]
 => ([],2)
 => 1
[1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => ([(0,1)],2)
 => 2
[3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => ([],3)
 => 1
[2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => ([(0,1),(0,2)],3)
 => 1
[1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => 3
[4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => ([],4)
 => 1
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3)],4)
 => 1
[2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => ([(1,2)],3)
 => 2
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => ([(0,3),(3,1),(3,2)],4)
 => 2
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ([],5)
 => 1
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(0,4)],5)
 => 1
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => ([(1,2),(1,3)],4)
 => 1
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ([(0,4),(4,1),(4,2),(4,3)],5)
 => 2
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(3,1)],4)
 => 1
[2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => ([(0,3),(3,4),(4,1),(4,2)],5)
 => 3
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 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)
 => 1
[5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
 => 1
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => ([(1,2),(1,3),(1,4)],5)
 => 1
[4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => ([(0,5),(5,1),(5,2),(5,3),(5,4)],6)
 => 2
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => ([(2,3)],4)
 => 2
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => ([(0,3),(0,4),(4,1),(4,2)],5)
 => 1
[3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => ([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
 => 3
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3)],4)
 => 3
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(3,2),(4,1),(4,3)],5)
 => 2
[2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => ([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
 => 4
[1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[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]
 => ([],7)
 => 1
[6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6)],7)
 => 1
[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]
 => ([(1,2),(1,3),(1,4),(1,5)],6)
 => 1
[5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => ([(0,6),(6,1),(6,2),(6,3),(6,4),(6,5)],7)
 => 2
[4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => ([(2,3),(2,4)],5)
 => 1
[4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => ([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
 => 1
[4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => ([(0,5),(5,6),(6,1),(6,2),(6,3),(6,4)],7)
 => 3
[3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => ([(0,2),(0,3),(0,4),(4,1)],5)
 => 1
[3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => 2
[3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => ([(0,5),(4,2),(4,3),(5,1),(5,4)],6)
 => 2
[3,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => ([(0,5),(4,6),(5,4),(6,1),(6,2),(6,3)],7)
 => 4
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => 1
[2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => ([(0,4),(3,2),(4,5),(5,1),(5,3)],6)
 => 3
[2,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => ([(0,5),(3,4),(4,6),(5,3),(6,1),(6,2)],7)
 => 5
[1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7
[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]
 => ([],8)
 => 1
[6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => ([(1,2),(1,3),(1,4),(1,5),(1,6)],7)
 => 1
[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]
 => ([(2,3),(2,4),(2,5)],6)
 => 1
[5,2,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
 => ([(0,5),(0,6),(6,1),(6,2),(6,3),(6,4)],7)
 => 1
[4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => ([(3,4)],5)
 => 2
[4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => ([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
 => 1
Description
The number of maximal antichains of minimal length in a poset.
Matching statistic: St000160
Mapping
St000160: Integer partitions ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Values
[1]
 => 1
[2]
 => 1
[1,1]
 => 2
[3]
 => 1
[2,1]
 => 1
[1,1,1]
 => 3
[4]
 => 1
[3,1]
 => 1
[2,2]
 => 2
[2,1,1]
 => 2
[1,1,1,1]
 => 4
[5]
 => 1
[4,1]
 => 1
[3,2]
 => 1
[3,1,1]
 => 2
[2,2,1]
 => 1
[2,1,1,1]
 => 3
[1,1,1,1,1]
 => 5
[6]
 => 1
[5,1]
 => 1
[4,2]
 => 1
[4,1,1]
 => 2
[3,3]
 => 2
[3,2,1]
 => 1
[3,1,1,1]
 => 3
[2,2,2]
 => 3
[2,2,1,1]
 => 2
[2,1,1,1,1]
 => 4
[1,1,1,1,1,1]
 => 6
[7]
 => 1
[6,1]
 => 1
[5,2]
 => 1
[5,1,1]
 => 2
[4,3]
 => 1
[4,2,1]
 => 1
[4,1,1,1]
 => 3
[3,3,1]
 => 1
[3,2,2]
 => 2
[3,2,1,1]
 => 2
[3,1,1,1,1]
 => 4
[2,2,2,1]
 => 1
[2,2,1,1,1]
 => 3
[2,1,1,1,1,1]
 => 5
[1,1,1,1,1,1,1]
 => 7
[8]
 => 1
[6,2]
 => 1
[5,3]
 => 1
[5,2,1]
 => 1
[4,4]
 => 2
[4,3,1]
 => 1
[5,5,3]
 => ? = 1
[4,4,4,1]
 => ? = 1
[3,3,3,3,1]
 => ? = 1
[5,5,4]
 => ? = 1
[5,5,5]
 => ? = 3
Description
The multiplicity of the smallest part of a partition. This counts the number of occurrences of the smallest part $spt(\lambda)$ of a partition $\lambda$. The sum $spt(n) = \sum_{\lambda \vdash n} spt(\lambda)$ satisfies the congruences \begin{align*} spt(5n+4) &\equiv 0\quad \pmod{5}\\\ spt(7n+5) &\equiv 0\quad \pmod{7}\\\ spt(13n+6) &\equiv 0\quad \pmod{13}, \end{align*} analogous to those of the counting function of partitions, see [1] and [2].
Matching statistic: St000383
(load all 2 compositions to match this statistic)
Mapping
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Values
[1]
 => 10 => [1,1] => [1,1] => 1
[2]
 => 100 => [1,2] => [2,1] => 1
[1,1]
 => 110 => [2,1] => [1,2] => 2
[3]
 => 1000 => [1,3] => [3,1] => 1
[2,1]
 => 1010 => [1,1,1,1] => [1,1,1,1] => 1
[1,1,1]
 => 1110 => [3,1] => [1,3] => 3
[4]
 => 10000 => [1,4] => [4,1] => 1
[3,1]
 => 10010 => [1,2,1,1] => [1,1,2,1] => 1
[2,2]
 => 1100 => [2,2] => [2,2] => 2
[2,1,1]
 => 10110 => [1,1,2,1] => [1,1,1,2] => 2
[1,1,1,1]
 => 11110 => [4,1] => [1,4] => 4
[5]
 => 100000 => [1,5] => [5,1] => 1
[4,1]
 => 100010 => [1,3,1,1] => [1,1,3,1] => 1
[3,2]
 => 10100 => [1,1,1,2] => [2,1,1,1] => 1
[3,1,1]
 => 100110 => [1,2,2,1] => [1,1,2,2] => 2
[2,2,1]
 => 11010 => [2,1,1,1] => [1,2,1,1] => 1
[2,1,1,1]
 => 101110 => [1,1,3,1] => [1,1,1,3] => 3
[1,1,1,1,1]
 => 111110 => [5,1] => [1,5] => 5
[6]
 => 1000000 => [1,6] => [6,1] => 1
[5,1]
 => 1000010 => [1,4,1,1] => [1,1,4,1] => 1
[4,2]
 => 100100 => [1,2,1,2] => [2,1,2,1] => 1
[4,1,1]
 => 1000110 => [1,3,2,1] => [1,1,3,2] => 2
[3,3]
 => 11000 => [2,3] => [3,2] => 2
[3,2,1]
 => 101010 => [1,1,1,1,1,1] => [1,1,1,1,1,1] => 1
[3,1,1,1]
 => 1001110 => [1,2,3,1] => [1,1,2,3] => 3
[2,2,2]
 => 11100 => [3,2] => [2,3] => 3
[2,2,1,1]
 => 110110 => [2,1,2,1] => [1,2,1,2] => 2
[2,1,1,1,1]
 => 1011110 => [1,1,4,1] => [1,1,1,4] => 4
[1,1,1,1,1,1]
 => 1111110 => [6,1] => [1,6] => 6
[7]
 => 10000000 => [1,7] => [7,1] => 1
[6,1]
 => 10000010 => [1,5,1,1] => [1,1,5,1] => 1
[5,2]
 => 1000100 => [1,3,1,2] => [2,1,3,1] => 1
[5,1,1]
 => 10000110 => [1,4,2,1] => [1,1,4,2] => ? = 2
[4,3]
 => 101000 => [1,1,1,3] => [3,1,1,1] => 1
[4,2,1]
 => 1001010 => [1,2,1,1,1,1] => [1,1,2,1,1,1] => 1
[4,1,1,1]
 => 10001110 => [1,3,3,1] => [1,1,3,3] => 3
[3,3,1]
 => 110010 => [2,2,1,1] => [1,2,2,1] => 1
[3,2,2]
 => 101100 => [1,1,2,2] => [2,1,1,2] => 2
[3,2,1,1]
 => 1010110 => [1,1,1,1,2,1] => [1,1,1,1,1,2] => 2
[3,1,1,1,1]
 => 10011110 => [1,2,4,1] => [1,1,2,4] => ? = 4
[2,2,2,1]
 => 111010 => [3,1,1,1] => [1,3,1,1] => 1
[2,2,1,1,1]
 => 1101110 => [2,1,3,1] => [1,2,1,3] => 3
[2,1,1,1,1,1]
 => 10111110 => [1,1,5,1] => [1,1,1,5] => ? = 5
[1,1,1,1,1,1,1]
 => 11111110 => [7,1] => [1,7] => 7
[8]
 => 100000000 => [1,8] => [8,1] => 1
[6,2]
 => 10000100 => [1,4,1,2] => [2,1,4,1] => 1
[5,3]
 => 1001000 => [1,2,1,3] => [3,1,2,1] => 1
[5,2,1]
 => 10001010 => [1,3,1,1,1,1] => [1,1,3,1,1,1] => 1
[4,4]
 => 110000 => [2,4] => [4,2] => 2
[4,3,1]
 => 1010010 => [1,1,1,2,1,1] => [1,1,1,1,2,1] => 1
[4,2,2]
 => 1001100 => [1,2,2,2] => [2,1,2,2] => 2
[4,2,1,1]
 => 10010110 => [1,2,1,1,2,1] => [1,1,2,1,1,2] => 2
[3,3,2]
 => 110100 => [2,1,1,2] => [2,2,1,1] => 1
[3,2,1,1,1]
 => 10101110 => [1,1,1,1,3,1] => [1,1,1,1,1,3] => ? = 3
[6,6]
 => 11000000 => [2,6] => [6,2] => ? = 2
[5,5,4]
 => 11010000 => [2,1,1,4] => [4,2,1,1] => ? = 1
[5,5,5]
 => 11100000 => [3,5] => [5,3] => ? = 3
Description
The last part of an integer composition.
Matching statistic: St000993
Mapping
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00189: Skew partitions —rotate⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 71%
Values
[1]
 => [[1],[]]
 => [[1],[]]
 => []
 => ? = 1
[2]
 => [[2],[]]
 => [[2],[]]
 => []
 => ? = 1
[1,1]
 => [[1,1],[]]
 => [[1,1],[]]
 => []
 => ? = 2
[3]
 => [[3],[]]
 => [[3],[]]
 => []
 => ? = 1
[2,1]
 => [[2,1],[]]
 => [[2,2],[1]]
 => [1]
 => ? = 1
[1,1,1]
 => [[1,1,1],[]]
 => [[1,1,1],[]]
 => []
 => ? = 3
[4]
 => [[4],[]]
 => [[4],[]]
 => []
 => ? = 1
[3,1]
 => [[3,1],[]]
 => [[3,3],[2]]
 => [2]
 => 1
[2,2]
 => [[2,2],[]]
 => [[2,2],[]]
 => []
 => ? = 2
[2,1,1]
 => [[2,1,1],[]]
 => [[2,2,2],[1,1]]
 => [1,1]
 => 2
[1,1,1,1]
 => [[1,1,1,1],[]]
 => [[1,1,1,1],[]]
 => []
 => ? = 4
[5]
 => [[5],[]]
 => [[5],[]]
 => []
 => ? = 1
[4,1]
 => [[4,1],[]]
 => [[4,4],[3]]
 => [3]
 => 1
[3,2]
 => [[3,2],[]]
 => [[3,3],[1]]
 => [1]
 => ? = 1
[3,1,1]
 => [[3,1,1],[]]
 => [[3,3,3],[2,2]]
 => [2,2]
 => 2
[2,2,1]
 => [[2,2,1],[]]
 => [[2,2,2],[1]]
 => [1]
 => ? = 1
[2,1,1,1]
 => [[2,1,1,1],[]]
 => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => 3
[1,1,1,1,1]
 => [[1,1,1,1,1],[]]
 => [[1,1,1,1,1],[]]
 => []
 => ? = 5
[6]
 => [[6],[]]
 => [[6],[]]
 => []
 => ? = 1
[5,1]
 => [[5,1],[]]
 => [[5,5],[4]]
 => [4]
 => 1
[4,2]
 => [[4,2],[]]
 => [[4,4],[2]]
 => [2]
 => 1
[4,1,1]
 => [[4,1,1],[]]
 => [[4,4,4],[3,3]]
 => [3,3]
 => 2
[3,3]
 => [[3,3],[]]
 => [[3,3],[]]
 => []
 => ? = 2
[3,2,1]
 => [[3,2,1],[]]
 => [[3,3,3],[2,1]]
 => [2,1]
 => 1
[3,1,1,1]
 => [[3,1,1,1],[]]
 => [[3,3,3,3],[2,2,2]]
 => [2,2,2]
 => 3
[2,2,2]
 => [[2,2,2],[]]
 => [[2,2,2],[]]
 => []
 => ? = 3
[2,2,1,1]
 => [[2,2,1,1],[]]
 => [[2,2,2,2],[1,1]]
 => [1,1]
 => 2
[2,1,1,1,1]
 => [[2,1,1,1,1],[]]
 => [[2,2,2,2,2],[1,1,1,1]]
 => [1,1,1,1]
 => 4
[1,1,1,1,1,1]
 => [[1,1,1,1,1,1],[]]
 => [[1,1,1,1,1,1],[]]
 => []
 => ? = 6
[7]
 => [[7],[]]
 => [[7],[]]
 => []
 => ? = 1
[6,1]
 => [[6,1],[]]
 => [[6,6],[5]]
 => [5]
 => 1
[5,2]
 => [[5,2],[]]
 => [[5,5],[3]]
 => [3]
 => 1
[5,1,1]
 => [[5,1,1],[]]
 => [[5,5,5],[4,4]]
 => [4,4]
 => 2
[4,3]
 => [[4,3],[]]
 => [[4,4],[1]]
 => [1]
 => ? = 1
[4,2,1]
 => [[4,2,1],[]]
 => [[4,4,4],[3,2]]
 => [3,2]
 => 1
[4,1,1,1]
 => [[4,1,1,1],[]]
 => [[4,4,4,4],[3,3,3]]
 => [3,3,3]
 => 3
[3,3,1]
 => [[3,3,1],[]]
 => [[3,3,3],[2]]
 => [2]
 => 1
[3,2,2]
 => [[3,2,2],[]]
 => [[3,3,3],[1,1]]
 => [1,1]
 => 2
[3,2,1,1]
 => [[3,2,1,1],[]]
 => [[3,3,3,3],[2,2,1]]
 => [2,2,1]
 => 2
[3,1,1,1,1]
 => [[3,1,1,1,1],[]]
 => [[3,3,3,3,3],[2,2,2,2]]
 => [2,2,2,2]
 => 4
[2,2,2,1]
 => [[2,2,2,1],[]]
 => [[2,2,2,2],[1]]
 => [1]
 => ? = 1
[2,2,1,1,1]
 => [[2,2,1,1,1],[]]
 => [[2,2,2,2,2],[1,1,1]]
 => [1,1,1]
 => 3
[2,1,1,1,1,1]
 => [[2,1,1,1,1,1],[]]
 => [[2,2,2,2,2,2],[1,1,1,1,1]]
 => [1,1,1,1,1]
 => 5
[1,1,1,1,1,1,1]
 => [[1,1,1,1,1,1,1],[]]
 => [[1,1,1,1,1,1,1],[]]
 => []
 => ? = 7
[8]
 => [[8],[]]
 => [[8],[]]
 => []
 => ? = 1
[6,2]
 => [[6,2],[]]
 => [[6,6],[4]]
 => [4]
 => 1
[5,3]
 => [[5,3],[]]
 => [[5,5],[2]]
 => [2]
 => 1
[5,2,1]
 => [[5,2,1],[]]
 => [[5,5,5],[4,3]]
 => [4,3]
 => 1
[4,4]
 => [[4,4],[]]
 => [[4,4],[]]
 => []
 => ? = 2
[4,3,1]
 => [[4,3,1],[]]
 => [[4,4,4],[3,1]]
 => [3,1]
 => 1
[4,2,2]
 => [[4,2,2],[]]
 => [[4,4,4],[2,2]]
 => [2,2]
 => 2
[4,2,1,1]
 => [[4,2,1,1],[]]
 => [[4,4,4,4],[3,3,2]]
 => [3,3,2]
 => 2
[3,3,2]
 => [[3,3,2],[]]
 => [[3,3,3],[1]]
 => [1]
 => ? = 1
[3,3,1,1]
 => [[3,3,1,1],[]]
 => [[3,3,3,3],[2,2]]
 => [2,2]
 => 2
[3,2,2,1]
 => [[3,2,2,1],[]]
 => [[3,3,3,3],[2,1,1]]
 => [2,1,1]
 => 1
[3,2,1,1,1]
 => [[3,2,1,1,1],[]]
 => [[3,3,3,3,3],[2,2,2,1]]
 => [2,2,2,1]
 => 3
[2,2,2,2]
 => [[2,2,2,2],[]]
 => [[2,2,2,2],[]]
 => []
 => ? = 4
[2,2,2,1,1]
 => [[2,2,2,1,1],[]]
 => [[2,2,2,2,2],[1,1]]
 => [1,1]
 => 2
[2,2,1,1,1,1]
 => [[2,2,1,1,1,1],[]]
 => [[2,2,2,2,2,2],[1,1,1,1]]
 => [1,1,1,1]
 => 4
[6,3]
 => [[6,3],[]]
 => [[6,6],[3]]
 => [3]
 => 1
[5,4]
 => [[5,4],[]]
 => [[5,5],[1]]
 => [1]
 => ? = 1
[5,3,1]
 => [[5,3,1],[]]
 => [[5,5,5],[4,2]]
 => [4,2]
 => 1
[4,4,1]
 => [[4,4,1],[]]
 => [[4,4,4],[3]]
 => [3]
 => 1
[4,3,2]
 => [[4,3,2],[]]
 => [[4,4,4],[2,1]]
 => [2,1]
 => 1
[4,3,1,1]
 => [[4,3,1,1],[]]
 => [[4,4,4,4],[3,3,1]]
 => [3,3,1]
 => 2
[4,2,2,1]
 => [[4,2,2,1],[]]
 => [[4,4,4,4],[3,2,2]]
 => [3,2,2]
 => 1
[3,3,3]
 => [[3,3,3],[]]
 => [[3,3,3],[]]
 => []
 => ? = 3
[3,3,2,1]
 => [[3,3,2,1],[]]
 => [[3,3,3,3],[2,1]]
 => [2,1]
 => 1
[3,3,1,1,1]
 => [[3,3,1,1,1],[]]
 => [[3,3,3,3,3],[2,2,2]]
 => [2,2,2]
 => 3
[3,2,2,2]
 => [[3,2,2,2],[]]
 => [[3,3,3,3],[1,1,1]]
 => [1,1,1]
 => 3
[2,2,2,2,1]
 => [[2,2,2,2,1],[]]
 => [[2,2,2,2,2],[1]]
 => [1]
 => ? = 1
[6,4]
 => [[6,4],[]]
 => [[6,6],[2]]
 => [2]
 => 1
[5,5]
 => [[5,5],[]]
 => [[5,5],[]]
 => []
 => ? = 2
[5,4,1]
 => [[5,4,1],[]]
 => [[5,5,5],[4,1]]
 => [4,1]
 => 1
[4,4,2]
 => [[4,4,2],[]]
 => [[4,4,4],[2]]
 => [2]
 => 1
[4,4,1,1]
 => [[4,4,1,1],[]]
 => [[4,4,4,4],[3,3]]
 => [3,3]
 => 2
[4,3,3]
 => [[4,3,3],[]]
 => [[4,4,4],[1,1]]
 => [1,1]
 => 2
[4,3,2,1]
 => [[4,3,2,1],[]]
 => [[4,4,4,4],[3,2,1]]
 => [3,2,1]
 => 1
[3,3,3,1]
 => [[3,3,3,1],[]]
 => [[3,3,3,3],[2]]
 => [2]
 => 1
[2,2,2,2,2]
 => [[2,2,2,2,2],[]]
 => [[2,2,2,2,2],[]]
 => []
 => ? = 5
[6,5]
 => [[6,5],[]]
 => [[6,6],[1]]
 => [1]
 => ? = 1
[4,4,3]
 => [[4,4,3],[]]
 => [[4,4,4],[1]]
 => [1]
 => ? = 1
[3,3,3,2]
 => [[3,3,3,2],[]]
 => [[3,3,3,3],[1]]
 => [1]
 => ? = 1
[6,6]
 => [[6,6],[]]
 => [[6,6],[]]
 => []
 => ? = 2
[4,4,4]
 => [[4,4,4],[]]
 => [[4,4,4],[]]
 => []
 => ? = 3
[3,3,3,3]
 => [[3,3,3,3],[]]
 => [[3,3,3,3],[]]
 => []
 => ? = 4
[5,5,4]
 => [[5,5,4],[]]
 => [[5,5,5],[1]]
 => [1]
 => ? = 1
[5,5,5]
 => [[5,5,5],[]]
 => [[5,5,5],[]]
 => []
 => ? = 3
Description
The multiplicity of the largest part of an integer partition.
Matching statistic: St000153
Mapping
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St000153: Permutations ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 100%
Values
[1]
 => [[1]]
 => {{1}}
 => [1] => 1
[2]
 => [[1,2]]
 => {{1,2}}
 => [2,1] => 1
[1,1]
 => [[1],[2]]
 => {{1},{2}}
 => [1,2] => 2
[3]
 => [[1,2,3]]
 => {{1,2,3}}
 => [2,3,1] => 1
[2,1]
 => [[1,3],[2]]
 => {{1,3},{2}}
 => [3,2,1] => 1
[1,1,1]
 => [[1],[2],[3]]
 => {{1},{2},{3}}
 => [1,2,3] => 3
[4]
 => [[1,2,3,4]]
 => {{1,2,3,4}}
 => [2,3,4,1] => 1
[3,1]
 => [[1,3,4],[2]]
 => {{1,3,4},{2}}
 => [3,2,4,1] => 1
[2,2]
 => [[1,2],[3,4]]
 => {{1,2},{3,4}}
 => [2,1,4,3] => 2
[2,1,1]
 => [[1,4],[2],[3]]
 => {{1,4},{2},{3}}
 => [4,2,3,1] => 2
[1,1,1,1]
 => [[1],[2],[3],[4]]
 => {{1},{2},{3},{4}}
 => [1,2,3,4] => 4
[5]
 => [[1,2,3,4,5]]
 => {{1,2,3,4,5}}
 => [2,3,4,5,1] => 1
[4,1]
 => [[1,3,4,5],[2]]
 => {{1,3,4,5},{2}}
 => [3,2,4,5,1] => 1
[3,2]
 => [[1,2,5],[3,4]]
 => {{1,2,5},{3,4}}
 => [2,5,4,3,1] => 1
[3,1,1]
 => [[1,4,5],[2],[3]]
 => {{1,4,5},{2},{3}}
 => [4,2,3,5,1] => 2
[2,2,1]
 => [[1,3],[2,5],[4]]
 => {{1,3},{2,5},{4}}
 => [3,5,1,4,2] => 1
[2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => {{1,5},{2},{3},{4}}
 => [5,2,3,4,1] => 3
[1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => {{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => 5
[6]
 => [[1,2,3,4,5,6]]
 => {{1,2,3,4,5,6}}
 => [2,3,4,5,6,1] => 1
[5,1]
 => [[1,3,4,5,6],[2]]
 => {{1,3,4,5,6},{2}}
 => [3,2,4,5,6,1] => 1
[4,2]
 => [[1,2,5,6],[3,4]]
 => {{1,2,5,6},{3,4}}
 => [2,5,4,3,6,1] => 1
[4,1,1]
 => [[1,4,5,6],[2],[3]]
 => {{1,4,5,6},{2},{3}}
 => [4,2,3,5,6,1] => 2
[3,3]
 => [[1,2,3],[4,5,6]]
 => {{1,2,3},{4,5,6}}
 => [2,3,1,5,6,4] => 2
[3,2,1]
 => [[1,3,6],[2,5],[4]]
 => {{1,3,6},{2,5},{4}}
 => [3,5,6,4,2,1] => 1
[3,1,1,1]
 => [[1,5,6],[2],[3],[4]]
 => {{1,5,6},{2},{3},{4}}
 => [5,2,3,4,6,1] => 3
[2,2,2]
 => [[1,2],[3,4],[5,6]]
 => {{1,2},{3,4},{5,6}}
 => [2,1,4,3,6,5] => 3
[2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => {{1,4},{2,6},{3},{5}}
 => [4,6,3,1,5,2] => 2
[2,1,1,1,1]
 => [[1,6],[2],[3],[4],[5]]
 => {{1,6},{2},{3},{4},{5}}
 => [6,2,3,4,5,1] => 4
[1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => {{1},{2},{3},{4},{5},{6}}
 => [1,2,3,4,5,6] => 6
[7]
 => [[1,2,3,4,5,6,7]]
 => {{1,2,3,4,5,6,7}}
 => [2,3,4,5,6,7,1] => 1
[6,1]
 => [[1,3,4,5,6,7],[2]]
 => {{1,3,4,5,6,7},{2}}
 => [3,2,4,5,6,7,1] => 1
[5,2]
 => [[1,2,5,6,7],[3,4]]
 => {{1,2,5,6,7},{3,4}}
 => [2,5,4,3,6,7,1] => ? = 1
[5,1,1]
 => [[1,4,5,6,7],[2],[3]]
 => {{1,4,5,6,7},{2},{3}}
 => [4,2,3,5,6,7,1] => 2
[4,3]
 => [[1,2,3,7],[4,5,6]]
 => {{1,2,3,7},{4,5,6}}
 => [2,3,7,5,6,4,1] => ? = 1
[4,2,1]
 => [[1,3,6,7],[2,5],[4]]
 => {{1,3,6,7},{2,5},{4}}
 => [3,5,6,4,2,7,1] => ? = 1
[4,1,1,1]
 => [[1,5,6,7],[2],[3],[4]]
 => {{1,5,6,7},{2},{3},{4}}
 => [5,2,3,4,6,7,1] => 3
[3,3,1]
 => [[1,3,4],[2,6,7],[5]]
 => {{1,3,4},{2,6,7},{5}}
 => [3,6,4,1,5,7,2] => ? = 1
[3,2,2]
 => [[1,2,7],[3,4],[5,6]]
 => {{1,2,7},{3,4},{5,6}}
 => [2,7,4,3,6,5,1] => ? = 2
[3,2,1,1]
 => [[1,4,7],[2,6],[3],[5]]
 => {{1,4,7},{2,6},{3},{5}}
 => [4,6,3,7,5,2,1] => ? = 2
[3,1,1,1,1]
 => [[1,6,7],[2],[3],[4],[5]]
 => {{1,6,7},{2},{3},{4},{5}}
 => [6,2,3,4,5,7,1] => 4
[2,2,2,1]
 => [[1,3],[2,5],[4,7],[6]]
 => {{1,3},{2,5},{4,7},{6}}
 => [3,5,1,7,2,6,4] => ? = 1
[2,2,1,1,1]
 => [[1,5],[2,7],[3],[4],[6]]
 => {{1,5},{2,7},{3},{4},{6}}
 => [5,7,3,4,1,6,2] => ? = 3
[2,1,1,1,1,1]
 => [[1,7],[2],[3],[4],[5],[6]]
 => {{1,7},{2},{3},{4},{5},{6}}
 => [7,2,3,4,5,6,1] => ? = 5
[1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => {{1},{2},{3},{4},{5},{6},{7}}
 => [1,2,3,4,5,6,7] => 7
[8]
 => [[1,2,3,4,5,6,7,8]]
 => {{1,2,3,4,5,6,7,8}}
 => [2,3,4,5,6,7,8,1] => 1
[6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => {{1,2,5,6,7,8},{3,4}}
 => [2,5,4,3,6,7,8,1] => ? = 1
[5,3]
 => [[1,2,3,7,8],[4,5,6]]
 => {{1,2,3,7,8},{4,5,6}}
 => [2,3,7,5,6,4,8,1] => ? = 1
[5,2,1]
 => [[1,3,6,7,8],[2,5],[4]]
 => {{1,3,6,7,8},{2,5},{4}}
 => [3,5,6,4,2,7,8,1] => ? = 1
[4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => {{1,2,3,4},{5,6,7,8}}
 => [2,3,4,1,6,7,8,5] => 2
[4,3,1]
 => [[1,3,4,8],[2,6,7],[5]]
 => {{1,3,4,8},{2,6,7},{5}}
 => [3,6,4,8,5,7,2,1] => ? = 1
[4,2,2]
 => [[1,2,7,8],[3,4],[5,6]]
 => {{1,2,7,8},{3,4},{5,6}}
 => [2,7,4,3,6,5,8,1] => ? = 2
[4,2,1,1]
 => [[1,4,7,8],[2,6],[3],[5]]
 => {{1,4,7,8},{2,6},{3},{5}}
 => [4,6,3,7,5,2,8,1] => ? = 2
[3,3,2]
 => [[1,2,5],[3,4,8],[6,7]]
 => {{1,2,5},{3,4,8},{6,7}}
 => [2,5,4,8,1,7,6,3] => ? = 1
[3,3,1,1]
 => [[1,4,5],[2,7,8],[3],[6]]
 => {{1,4,5},{2,7,8},{3},{6}}
 => [4,7,3,5,1,6,8,2] => ? = 2
[3,2,2,1]
 => [[1,3,8],[2,5],[4,7],[6]]
 => {{1,3,8},{2,5},{4,7},{6}}
 => [3,5,8,7,2,6,4,1] => ? = 1
[3,2,1,1,1]
 => [[1,5,8],[2,7],[3],[4],[6]]
 => {{1,5,8},{2,7},{3},{4},{6}}
 => [5,7,3,4,8,6,2,1] => ? = 3
[2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => {{1,2},{3,4},{5,6},{7,8}}
 => [2,1,4,3,6,5,8,7] => 4
[2,2,2,1,1]
 => [[1,4],[2,6],[3,8],[5],[7]]
 => {{1,4},{2,6},{3,8},{5},{7}}
 => [4,6,8,1,5,2,7,3] => ? = 2
[2,2,1,1,1,1]
 => [[1,6],[2,8],[3],[4],[5],[7]]
 => {{1,6},{2,8},{3},{4},{5},{7}}
 => [6,8,3,4,5,1,7,2] => ? = 4
[6,3]
 => [[1,2,3,7,8,9],[4,5,6]]
 => {{1,2,3,7,8,9},{4,5,6}}
 => [2,3,7,5,6,4,8,9,1] => ? = 1
[5,4]
 => [[1,2,3,4,9],[5,6,7,8]]
 => {{1,2,3,4,9},{5,6,7,8}}
 => [2,3,4,9,6,7,8,5,1] => ? = 1
[5,3,1]
 => [[1,3,4,8,9],[2,6,7],[5]]
 => {{1,3,4,8,9},{2,6,7},{5}}
 => [3,6,4,8,5,7,2,9,1] => ? = 1
[4,4,1]
 => [[1,3,4,5],[2,7,8,9],[6]]
 => {{1,3,4,5},{2,7,8,9},{6}}
 => [3,7,4,5,1,6,8,9,2] => ? = 1
[4,3,2]
 => [[1,2,5,9],[3,4,8],[6,7]]
 => {{1,2,5,9},{3,4,8},{6,7}}
 => [2,5,4,8,9,7,6,3,1] => ? = 1
[4,3,1,1]
 => [[1,4,5,9],[2,7,8],[3],[6]]
 => {{1,4,5,9},{2,7,8},{3},{6}}
 => [4,7,3,5,9,6,8,2,1] => ? = 2
[4,2,2,1]
 => [[1,3,8,9],[2,5],[4,7],[6]]
 => {{1,3,8,9},{2,5},{4,7},{6}}
 => [3,5,8,7,2,6,4,9,1] => ? = 1
[3,3,3]
 => [[1,2,3],[4,5,6],[7,8,9]]
 => {{1,2,3},{4,5,6},{7,8,9}}
 => [2,3,1,5,6,4,8,9,7] => ? = 3
[3,3,2,1]
 => [[1,3,6],[2,5,9],[4,8],[7]]
 => {{1,3,6},{2,5,9},{4,8},{7}}
 => [3,5,6,8,9,1,7,4,2] => ? = 1
[3,3,1,1,1]
 => [[1,5,6],[2,8,9],[3],[4],[7]]
 => {{1,5,6},{2,8,9},{3},{4},{7}}
 => [5,8,3,4,6,1,7,9,2] => ? = 3
[3,2,2,2]
 => [[1,2,9],[3,4],[5,6],[7,8]]
 => {{1,2,9},{3,4},{5,6},{7,8}}
 => [2,9,4,3,6,5,8,7,1] => ? = 3
[2,2,2,2,1]
 => [[1,3],[2,5],[4,7],[6,9],[8]]
 => {{1,3},{2,5},{4,7},{6,9},{8}}
 => [3,5,1,7,2,9,4,8,6] => ? = 1
[6,4]
 => [[1,2,3,4,9,10],[5,6,7,8]]
 => {{1,2,3,4,9,10},{5,6,7,8}}
 => [2,3,4,9,6,7,8,5,10,1] => ? = 1
[5,5]
 => [[1,2,3,4,5],[6,7,8,9,10]]
 => {{1,2,3,4,5},{6,7,8,9,10}}
 => [2,3,4,5,1,7,8,9,10,6] => ? = 2
[5,4,1]
 => [[1,3,4,5,10],[2,7,8,9],[6]]
 => {{1,3,4,5,10},{2,7,8,9},{6}}
 => [3,7,4,5,10,6,8,9,2,1] => ? = 1
[4,4,2]
 => [[1,2,5,6],[3,4,9,10],[7,8]]
 => {{1,2,5,6},{3,4,9,10},{7,8}}
 => [2,5,4,9,6,1,8,7,10,3] => ? = 1
[4,4,1,1]
 => [[1,4,5,6],[2,8,9,10],[3],[7]]
 => {{1,4,5,6},{2,8,9,10},{3},{7}}
 => [4,8,3,5,6,1,7,9,10,2] => ? = 2
[4,3,3]
 => [[1,2,3,10],[4,5,6],[7,8,9]]
 => {{1,2,3,10},{4,5,6},{7,8,9}}
 => [2,3,10,5,6,4,8,9,7,1] => ? = 2
[4,3,2,1]
 => [[1,3,6,10],[2,5,9],[4,8],[7]]
 => {{1,3,6,10},{2,5,9},{4,8},{7}}
 => [3,5,6,8,9,10,7,4,2,1] => ? = 1
[3,3,3,1]
 => [[1,3,4],[2,6,7],[5,9,10],[8]]
 => {{1,3,4},{2,6,7},{5,9,10},{8}}
 => [3,6,4,1,9,7,2,8,10,5] => ? = 1
[3,3,2,2]
 => [[1,2,7],[3,4,10],[5,6],[8,9]]
 => {{1,2,7},{3,4,10},{5,6},{8,9}}
 => [2,7,4,10,6,5,1,9,8,3] => ? = 2
[2,2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8],[9,10]]
 => {{1,2},{3,4},{5,6},{7,8},{9,10}}
 => [2,1,4,3,6,5,8,7,10,9] => 5
[6,5]
 => [[1,2,3,4,5,11],[6,7,8,9,10]]
 => ?
 => ? => ? = 1
[5,5,1]
 => [[1,3,4,5,6],[2,8,9,10,11],[7]]
 => ?
 => ? => ? = 1
[4,4,3]
 => [[1,2,3,7],[4,5,6,11],[8,9,10]]
 => {{1,2,3,7},{4,5,6,11},{8,9,10}}
 => ? => ? = 1
[4,4,2,1]
 => [[1,3,6,7],[2,5,10,11],[4,9],[8]]
 => {{1,3,6,7},{2,5,10,11},{4,9},{8}}
 => ? => ? = 1
[4,3,3,1]
 => [[1,3,4,11],[2,6,7],[5,9,10],[8]]
 => {{1,3,4,11},{2,6,7},{5,9,10},{8}}
 => ? => ? = 1
[3,3,3,2]
 => [[1,2,5],[3,4,8],[6,7,11],[9,10]]
 => {{1,2,5},{3,4,8},{6,7,11},{9,10}}
 => ? => ? = 1
[6,6]
 => [[1,2,3,4,5,6],[7,8,9,10,11,12]]
 => {{1,2,3,4,5,6},{7,8,9,10,11,12}}
 => [2,3,4,5,6,1,8,9,10,11,12,7] => ? = 2
[5,5,2]
 => [[1,2,5,6,7],[3,4,10,11,12],[8,9]]
 => ?
 => ? => ? = 1
Description
The number of adjacent cycles of a permutation. This is the number of cycles of the permutation of the form (i,i+1,i+2,...i+k) which includes the fixed points (i).
Matching statistic: St000910
Mapping
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000910: Posets ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 100%
Values
[1]
 => [[1]]
 => [1] => ([],1)
 => ? = 1
[2]
 => [[1,2]]
 => [1,2] => ([(0,1)],2)
 => 1
[1,1]
 => [[1],[2]]
 => [2,1] => ([],2)
 => 2
[3]
 => [[1,2,3]]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => 1
[2,1]
 => [[1,2],[3]]
 => [3,1,2] => ([(1,2)],3)
 => 1
[1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => ([],3)
 => 3
[4]
 => [[1,2,3,4]]
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1
[3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => ([(1,2),(2,3)],4)
 => 1
[2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => 2
[2,1,1]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => ([(2,3)],4)
 => 2
[1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => ([],4)
 => 4
[5]
 => [[1,2,3,4,5]]
 => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[4,1]
 => [[1,2,3,4],[5]]
 => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
 => 1
[3,2]
 => [[1,2,3],[4,5]]
 => [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
 => 1
[3,1,1]
 => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => ([(2,3),(3,4)],5)
 => 2
[2,2,1]
 => [[1,2],[3,4],[5]]
 => [5,3,4,1,2] => ([(1,4),(2,3)],5)
 => 1
[2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => ([(3,4)],5)
 => 3
[1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => ([],5)
 => 5
[6]
 => [[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[5,1]
 => [[1,2,3,4,5],[6]]
 => [6,1,2,3,4,5] => ([(1,5),(3,4),(4,2),(5,3)],6)
 => 1
[4,2]
 => [[1,2,3,4],[5,6]]
 => [5,6,1,2,3,4] => ([(0,5),(1,3),(4,2),(5,4)],6)
 => 1
[4,1,1]
 => [[1,2,3,4],[5],[6]]
 => [6,5,1,2,3,4] => ([(2,3),(3,5),(5,4)],6)
 => 2
[3,3]
 => [[1,2,3],[4,5,6]]
 => [4,5,6,1,2,3] => ([(0,5),(1,4),(4,2),(5,3)],6)
 => 2
[3,2,1]
 => [[1,2,3],[4,5],[6]]
 => [6,4,5,1,2,3] => ([(1,3),(2,4),(4,5)],6)
 => 1
[3,1,1,1]
 => [[1,2,3],[4],[5],[6]]
 => [6,5,4,1,2,3] => ([(3,4),(4,5)],6)
 => 3
[2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [5,6,3,4,1,2] => ([(0,5),(1,4),(2,3)],6)
 => 3
[2,2,1,1]
 => [[1,2],[3,4],[5],[6]]
 => [6,5,3,4,1,2] => ([(2,5),(3,4)],6)
 => 2
[2,1,1,1,1]
 => [[1,2],[3],[4],[5],[6]]
 => [6,5,4,3,1,2] => ([(4,5)],6)
 => 4
[1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => ([],6)
 => 6
[7]
 => [[1,2,3,4,5,6,7]]
 => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ? = 1
[6,1]
 => [[1,2,3,4,5,6],[7]]
 => [7,1,2,3,4,5,6] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
 => ? = 1
[5,2]
 => [[1,2,3,4,5],[6,7]]
 => [6,7,1,2,3,4,5] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
 => ? = 1
[5,1,1]
 => [[1,2,3,4,5],[6],[7]]
 => [7,6,1,2,3,4,5] => ([(2,6),(4,5),(5,3),(6,4)],7)
 => ? = 2
[4,3]
 => [[1,2,3,4],[5,6,7]]
 => [5,6,7,1,2,3,4] => ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
 => ? = 1
[4,2,1]
 => [[1,2,3,4],[5,6],[7]]
 => [7,5,6,1,2,3,4] => ([(1,6),(2,4),(5,3),(6,5)],7)
 => ? = 1
[4,1,1,1]
 => [[1,2,3,4],[5],[6],[7]]
 => [7,6,5,1,2,3,4] => ([(3,4),(4,6),(6,5)],7)
 => 3
[3,3,1]
 => [[1,2,3],[4,5,6],[7]]
 => [7,4,5,6,1,2,3] => ([(1,6),(2,5),(5,3),(6,4)],7)
 => ? = 1
[3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ([(0,5),(1,4),(2,6),(6,3)],7)
 => ? = 2
[3,2,1,1]
 => [[1,2,3],[4,5],[6],[7]]
 => [7,6,4,5,1,2,3] => ([(2,4),(3,5),(5,6)],7)
 => ? = 2
[3,1,1,1,1]
 => [[1,2,3],[4],[5],[6],[7]]
 => [7,6,5,4,1,2,3] => ([(4,5),(5,6)],7)
 => 4
[2,2,2,1]
 => [[1,2],[3,4],[5,6],[7]]
 => [7,5,6,3,4,1,2] => ([(1,6),(2,5),(3,4)],7)
 => ? = 1
[2,2,1,1,1]
 => [[1,2],[3,4],[5],[6],[7]]
 => [7,6,5,3,4,1,2] => ([(3,6),(4,5)],7)
 => 3
[2,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,1,2] => ([(5,6)],7)
 => 5
[1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,2,1] => ([],7)
 => 7
[8]
 => [[1,2,3,4,5,6,7,8]]
 => [1,2,3,4,5,6,7,8] => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
 => ? = 1
[6,2]
 => [[1,2,3,4,5,6],[7,8]]
 => [7,8,1,2,3,4,5,6] => ([(0,7),(1,3),(4,6),(5,4),(6,2),(7,5)],8)
 => ? = 1
[5,3]
 => [[1,2,3,4,5],[6,7,8]]
 => [6,7,8,1,2,3,4,5] => ([(0,7),(1,6),(4,5),(5,3),(6,4),(7,2)],8)
 => ? = 1
[5,2,1]
 => [[1,2,3,4,5],[6,7],[8]]
 => [8,6,7,1,2,3,4,5] => ([(1,7),(2,4),(5,6),(6,3),(7,5)],8)
 => ? = 1
[4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => [5,6,7,8,1,2,3,4] => ([(0,7),(1,6),(4,2),(5,3),(6,4),(7,5)],8)
 => ? = 2
[4,3,1]
 => [[1,2,3,4],[5,6,7],[8]]
 => [8,5,6,7,1,2,3,4] => ([(1,6),(2,7),(5,4),(6,5),(7,3)],8)
 => ? = 1
[4,2,2]
 => [[1,2,3,4],[5,6],[7,8]]
 => [7,8,5,6,1,2,3,4] => ([(0,5),(1,4),(2,7),(6,3),(7,6)],8)
 => ? = 2
[4,2,1,1]
 => [[1,2,3,4],[5,6],[7],[8]]
 => [8,7,5,6,1,2,3,4] => ([(2,4),(3,5),(5,6),(6,7)],8)
 => ? = 2
[3,3,2]
 => [[1,2,3],[4,5,6],[7,8]]
 => [7,8,4,5,6,1,2,3] => ([(0,5),(1,7),(2,6),(6,3),(7,4)],8)
 => ? = 1
[3,3,1,1]
 => [[1,2,3],[4,5,6],[7],[8]]
 => [8,7,4,5,6,1,2,3] => ([(2,5),(3,4),(4,6),(5,7)],8)
 => ? = 2
[3,2,2,1]
 => [[1,2,3],[4,5],[6,7],[8]]
 => [8,6,7,4,5,1,2,3] => ([(1,5),(2,4),(3,6),(6,7)],8)
 => ? = 1
[3,2,1,1,1]
 => [[1,2,3],[4,5],[6],[7],[8]]
 => [8,7,6,4,5,1,2,3] => ([(3,5),(4,6),(6,7)],8)
 => ? = 3
[2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => [7,8,5,6,3,4,1,2] => ([(0,7),(1,6),(2,5),(3,4)],8)
 => ? = 4
[2,2,2,1,1]
 => [[1,2],[3,4],[5,6],[7],[8]]
 => [8,7,5,6,3,4,1,2] => ([(2,7),(3,6),(4,5)],8)
 => ? = 2
[2,2,1,1,1,1]
 => [[1,2],[3,4],[5],[6],[7],[8]]
 => [8,7,6,5,3,4,1,2] => ([(4,7),(5,6)],8)
 => ? = 4
[6,3]
 => [[1,2,3,4,5,6],[7,8,9]]
 => [7,8,9,1,2,3,4,5,6] => ([(0,8),(1,7),(4,6),(5,4),(6,3),(7,5),(8,2)],9)
 => ? = 1
[5,4]
 => [[1,2,3,4,5],[6,7,8,9]]
 => [6,7,8,9,1,2,3,4,5] => ([(0,7),(1,8),(4,5),(5,2),(6,3),(7,6),(8,4)],9)
 => ? = 1
[5,3,1]
 => [[1,2,3,4,5],[6,7,8],[9]]
 => [9,6,7,8,1,2,3,4,5] => ([(1,8),(2,7),(5,6),(6,4),(7,5),(8,3)],9)
 => ? = 1
[4,4,1]
 => [[1,2,3,4],[5,6,7,8],[9]]
 => [9,5,6,7,8,1,2,3,4] => ([(1,8),(2,7),(5,3),(6,4),(7,5),(8,6)],9)
 => ? = 1
[4,3,2]
 => [[1,2,3,4],[5,6,7],[8,9]]
 => [8,9,5,6,7,1,2,3,4] => ([(0,5),(1,7),(2,8),(6,3),(7,4),(8,6)],9)
 => ? = 1
[4,3,1,1]
 => [[1,2,3,4],[5,6,7],[8],[9]]
 => [9,8,5,6,7,1,2,3,4] => ([(2,7),(3,8),(6,5),(7,6),(8,4)],9)
 => ? = 2
[4,2,2,1]
 => [[1,2,3,4],[5,6],[7,8],[9]]
 => [9,7,8,5,6,1,2,3,4] => ([(1,6),(2,5),(3,8),(7,4),(8,7)],9)
 => ? = 1
[3,3,3]
 => [[1,2,3],[4,5,6],[7,8,9]]
 => [7,8,9,4,5,6,1,2,3] => ([(0,8),(1,7),(2,6),(6,3),(7,4),(8,5)],9)
 => ? = 3
[3,3,2,1]
 => [[1,2,3],[4,5,6],[7,8],[9]]
 => [9,7,8,4,5,6,1,2,3] => ([(1,6),(2,8),(3,7),(7,4),(8,5)],9)
 => ? = 1
[3,3,1,1,1]
 => [[1,2,3],[4,5,6],[7],[8],[9]]
 => [9,8,7,4,5,6,1,2,3] => ([(3,6),(4,5),(5,7),(6,8)],9)
 => ? = 3
[3,2,2,2]
 => [[1,2,3],[4,5],[6,7],[8,9]]
 => [8,9,6,7,4,5,1,2,3] => ([(0,7),(1,6),(2,5),(3,8),(8,4)],9)
 => ? = 3
[2,2,2,2,1]
 => [[1,2],[3,4],[5,6],[7,8],[9]]
 => [9,7,8,5,6,3,4,1,2] => ([(1,8),(2,7),(3,6),(4,5)],9)
 => ? = 1
[6,4]
 => [[1,2,3,4,5,6],[7,8,9,10]]
 => [7,8,9,10,1,2,3,4,5,6] => ([(0,8),(1,9),(4,6),(5,4),(6,3),(7,2),(8,7),(9,5)],10)
 => ? = 1
[5,5]
 => [[1,2,3,4,5],[6,7,8,9,10]]
 => [6,7,8,9,10,1,2,3,4,5] => ([(0,9),(1,8),(4,6),(5,7),(6,2),(7,3),(8,4),(9,5)],10)
 => ? = 2
[5,4,1]
 => [[1,2,3,4,5],[6,7,8,9],[10]]
 => [10,6,7,8,9,1,2,3,4,5] => ([(1,8),(2,9),(5,6),(6,3),(7,4),(8,7),(9,5)],10)
 => ? = 1
[4,4,2]
 => [[1,2,3,4],[5,6,7,8],[9,10]]
 => [9,10,5,6,7,8,1,2,3,4] => ([(0,9),(1,8),(2,5),(6,3),(7,4),(8,6),(9,7)],10)
 => ? = 1
[4,4,1,1]
 => [[1,2,3,4],[5,6,7,8],[9],[10]]
 => [10,9,5,6,7,8,1,2,3,4] => ([(2,9),(3,8),(6,4),(7,5),(8,6),(9,7)],10)
 => ? = 2
[4,3,3]
 => [[1,2,3,4],[5,6,7],[8,9,10]]
 => [8,9,10,5,6,7,1,2,3,4] => ([(0,9),(1,8),(2,7),(6,5),(7,6),(8,3),(9,4)],10)
 => ? = 2
[4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [10,8,9,5,6,7,1,2,3,4] => ([(1,6),(2,8),(3,9),(7,4),(8,5),(9,7)],10)
 => ? = 1
[3,3,3,1]
 => [[1,2,3],[4,5,6],[7,8,9],[10]]
 => [10,7,8,9,4,5,6,1,2,3] => ([(1,9),(2,8),(3,7),(7,4),(8,5),(9,6)],10)
 => ? = 1
[3,3,2,2]
 => [[1,2,3],[4,5,6],[7,8],[9,10]]
 => [9,10,7,8,4,5,6,1,2,3] => ([(0,7),(1,6),(2,9),(3,8),(8,4),(9,5)],10)
 => ? = 2
[2,2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8],[9,10]]
 => [9,10,7,8,5,6,3,4,1,2] => ([(0,9),(1,8),(2,7),(3,6),(4,5)],10)
 => ? = 5
[6,5]
 => [[1,2,3,4,5,6],[7,8,9,10,11]]
 => ? => ?
 => ? = 1
[5,5,1]
 => [[1,2,3,4,5],[6,7,8,9,10],[11]]
 => ? => ?
 => ? = 1
Description
The number of maximal chains of minimal length in a poset.
Matching statistic: St000773
Mapping
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000773: Graphs ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 71%
Values
[1]
 => 10 => [1,2] => ([(1,2)],3)
 => 1
[2]
 => 100 => [1,3] => ([(2,3)],4)
 => 1
[1,1]
 => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[3]
 => 1000 => [1,4] => ([(3,4)],5)
 => 1
[2,1]
 => 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,1,1]
 => 1110 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[4]
 => 10000 => [1,5] => ([(4,5)],6)
 => 1
[3,1]
 => 10010 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[2,2]
 => 1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 2
[2,1,1]
 => 10110 => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[1,1,1,1]
 => 11110 => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[5]
 => 100000 => [1,6] => ([(5,6)],7)
 => 1
[4,1]
 => 100010 => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => 1
[3,2]
 => 10100 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[3,1,1]
 => 100110 => [1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 2
[2,2,1]
 => 11010 => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[2,1,1,1]
 => 101110 => [1,2,1,1,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3
[1,1,1,1,1]
 => 111110 => [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[6]
 => 1000000 => [1,7] => ([(6,7)],8)
 => ? = 1
[5,1]
 => 1000010 => [1,5,2] => ([(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1
[4,2]
 => 100100 => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => 1
[4,1,1]
 => 1000110 => [1,4,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[3,3]
 => 11000 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => 2
[3,2,1]
 => 101010 => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1
[3,1,1,1]
 => 1001110 => [1,3,1,1,2] => ([(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[2,2,2]
 => 11100 => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[2,2,1,1]
 => 110110 => [1,1,2,1,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 2
[2,1,1,1,1]
 => 1011110 => [1,2,1,1,1,2] => ([(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[1,1,1,1,1,1]
 => 1111110 => [1,1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6
[7]
 => 10000000 => [1,8] => ([(7,8)],9)
 => ? = 1
[6,1]
 => 10000010 => [1,6,2] => ([(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 1
[5,2]
 => 1000100 => [1,4,3] => ([(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1
[5,1,1]
 => 10000110 => [1,5,1,2] => ([(1,7),(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2
[4,3]
 => 101000 => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
 => 1
[4,2,1]
 => 1001010 => [1,3,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1
[4,1,1,1]
 => 10001110 => [1,4,1,1,2] => ([(1,6),(1,7),(1,8),(2,6),(2,7),(2,8),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 3
[3,3,1]
 => 110010 => [1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1
[3,2,2]
 => 101100 => [1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 2
[3,2,1,1]
 => 1010110 => [1,2,2,1,2] => ([(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[3,1,1,1,1]
 => 10011110 => [1,3,1,1,1,2] => ([(1,5),(1,6),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 4
[2,2,2,1]
 => 111010 => [1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1
[2,2,1,1,1]
 => 1101110 => [1,1,2,1,1,2] => ([(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[2,1,1,1,1,1]
 => 10111110 => [1,2,1,1,1,1,2] => ([(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 5
[1,1,1,1,1,1,1]
 => 11111110 => [1,1,1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 7
[8]
 => 100000000 => [1,9] => ([(8,9)],10)
 => ? = 1
[6,2]
 => 10000100 => [1,5,3] => ([(2,8),(3,8),(4,8),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 1
[5,3]
 => 1001000 => [1,3,4] => ([(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1
[5,2,1]
 => 10001010 => [1,4,2,2] => ([(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 1
[4,4]
 => 110000 => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
 => 2
[4,3,1]
 => 1010010 => [1,2,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1
[4,2,2]
 => 1001100 => [1,3,1,3] => ([(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[4,2,1,1]
 => 10010110 => [1,3,2,1,2] => ([(1,7),(1,8),(2,6),(2,7),(2,8),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2
[3,3,2]
 => 110100 => [1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1
[3,3,1,1]
 => 1100110 => [1,1,3,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[3,2,2,1]
 => 1011010 => [1,2,1,2,2] => ([(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1
[3,2,1,1,1]
 => 10101110 => [1,2,2,1,1,2] => ([(1,6),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 3
[2,2,2,2]
 => 111100 => [1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 4
[2,2,2,1,1]
 => 1110110 => [1,1,1,2,1,2] => ([(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[2,2,1,1,1,1]
 => 11011110 => [1,1,2,1,1,1,2] => ([(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 4
[6,3]
 => 10001000 => [1,4,4] => ([(3,8),(4,8),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 1
[5,4]
 => 1010000 => [1,2,5] => ([(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1
[5,3,1]
 => 10010010 => [1,3,3,2] => ([(1,8),(2,8),(3,7),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 1
[4,4,1]
 => 1100010 => [1,1,4,2] => ([(1,7),(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1
[4,3,2]
 => 1010100 => [1,2,2,3] => ([(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1
[4,3,1,1]
 => 10100110 => [1,2,3,1,2] => ([(1,7),(1,8),(2,7),(2,8),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2
[4,2,2,1]
 => 10011010 => [1,3,1,2,2] => ([(1,8),(2,6),(2,7),(2,8),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 1
[3,3,3]
 => 111000 => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3
[3,3,2,1]
 => 1101010 => [1,1,2,2,2] => ([(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1
[3,3,1,1,1]
 => 11001110 => [1,1,3,1,1,2] => ([(1,6),(1,7),(1,8),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 3
[3,2,2,2]
 => 1011100 => [1,2,1,1,3] => ([(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[2,2,2,2,1]
 => 1111010 => [1,1,1,1,2,2] => ([(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1
[6,4]
 => 10010000 => [1,3,5] => ([(4,8),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 1
[5,5]
 => 1100000 => [1,1,6] => ([(5,6),(5,7),(6,7)],8)
 => ? = 2
[5,4,1]
 => 10100010 => [1,2,4,2] => ([(1,8),(2,8),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 1
[4,4,2]
 => 1100100 => [1,1,3,3] => ([(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1
[4,4,1,1]
 => 11000110 => [1,1,4,1,2] => ([(1,7),(1,8),(2,7),(2,8),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2
[4,3,3]
 => 1011000 => [1,2,1,4] => ([(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[4,3,2,1]
 => 10101010 => [1,2,2,2,2] => ([(1,8),(2,7),(2,8),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 1
[3,3,3,1]
 => 1110010 => [1,1,1,3,2] => ([(1,7),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1
[3,3,2,2]
 => 1101100 => [1,1,2,1,3] => ([(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[2,2,2,2,2]
 => 1111100 => [1,1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
Description
The multiplicity of the largest Laplacian eigenvalue in a graph.
Matching statistic: St001232
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 86%
Values
[1]
 => [1,0,1,0]
 => [1,0,1,0]
 => [1,0,1,0]
 => 1
[2]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 2
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 1
[2,1]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => ? = 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => ? = 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => ? = 2
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => ? = 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => ? = 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => ? = 2
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => ? = 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => ? = 3
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => ? = 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => ? = 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => ? = 2
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => ? = 1
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => ? = 3
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 3
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => ? = 2
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => ? = 4
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => 6
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => ? = 1
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => ? = 1
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => ? = 2
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ? = 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => ? = 1
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => ? = 3
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => ? = 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => ? = 2
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => ? = 2
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => ? = 4
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => ? = 1
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => ? = 3
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => ? = 5
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => ? = 7
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => ? = 1
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => ? = 1
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => ? = 1
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 2
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => ? = 1
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => ? = 2
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => ? = 2
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => ? = 1
[3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => ? = 2
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => ? = 1
[3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => ? = 3
[2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 4
[2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => ? = 2
[2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => ? = 4
[6,3]
 => [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
 => ? = 1
[5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => ? = 1
[5,3,1]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => ? = 1
[4,4,1]
 => [1,1,1,0,1,0,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 1
[4,3,2]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => ? = 1
[4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => ? = 2
[4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => ? = 1
[3,3,3]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 3
[5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => 2
[2,2,2,2,2]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => 5
[4,4,4]
 => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => 3
[3,3,3,3]
 => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => 4
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
The following 3 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000454The largest eigenvalue of a graph if it is integral. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001330The hat guessing number of a graph.