Your data matches 5 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000290
(load all 3 compositions to match this statistic)
Mapping
Mp00095: Integer partitions —to binary word⟶ Binary words
St000290: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 10 => 1
[2]
 => 100 => 1
[1,1]
 => 110 => 2
[3]
 => 1000 => 1
[2,1]
 => 1010 => 4
[1,1,1]
 => 1110 => 3
[4]
 => 10000 => 1
[3,1]
 => 10010 => 5
[2,2]
 => 1100 => 2
[2,1,1]
 => 10110 => 5
[1,1,1,1]
 => 11110 => 4
[5]
 => 100000 => 1
[4,1]
 => 100010 => 6
[3,2]
 => 10100 => 4
[3,1,1]
 => 100110 => 6
[2,2,1]
 => 11010 => 6
[2,1,1,1]
 => 101110 => 6
[1,1,1,1,1]
 => 111110 => 5
[6]
 => 1000000 => 1
[5,1]
 => 1000010 => 7
[4,2]
 => 100100 => 5
[4,1,1]
 => 1000110 => 7
[3,3]
 => 11000 => 2
[3,2,1]
 => 101010 => 9
[3,1,1,1]
 => 1001110 => 7
[2,2,2]
 => 11100 => 3
[2,2,1,1]
 => 110110 => 7
[2,1,1,1,1]
 => 1011110 => 7
[1,1,1,1,1,1]
 => 1111110 => 6
[7]
 => 10000000 => 1
[6,1]
 => 10000010 => 8
[5,2]
 => 1000100 => 6
[5,1,1]
 => 10000110 => 8
[4,3]
 => 101000 => 4
[4,2,1]
 => 1001010 => 11
[4,1,1,1]
 => 10001110 => 8
[3,3,1]
 => 110010 => 7
[3,2,2]
 => 101100 => 5
[3,2,1,1]
 => 1010110 => 10
[3,1,1,1,1]
 => 10011110 => 8
[2,2,2,1]
 => 111010 => 8
[2,2,1,1,1]
 => 1101110 => 8
[2,1,1,1,1,1]
 => 10111110 => 8
[1,1,1,1,1,1,1]
 => 11111110 => 7
[8]
 => 100000000 => 1
[7,1]
 => 100000010 => 9
[6,2]
 => 10000100 => 7
[6,1,1]
 => 100000110 => 9
[5,3]
 => 1001000 => 5
[5,2,1]
 => 10001010 => 13
Description
The major index of a binary word. This is the sum of the positions of descents, i.e., a one followed by a zero. For words of length $n$ with $a$ zeros, the generating function for the major index is the $q$-binomial coefficient $\binom{n}{a}_q$.
Matching statistic: St000293
Mapping
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00096: Binary words —Foata bijection⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 10 => 10 => 1
[2]
 => 100 => 010 => 1
[1,1]
 => 110 => 110 => 2
[3]
 => 1000 => 0010 => 1
[2,1]
 => 1010 => 1100 => 4
[1,1,1]
 => 1110 => 1110 => 3
[4]
 => 10000 => 00010 => 1
[3,1]
 => 10010 => 10100 => 5
[2,2]
 => 1100 => 0110 => 2
[2,1,1]
 => 10110 => 11010 => 5
[1,1,1,1]
 => 11110 => 11110 => 4
[5]
 => 100000 => 000010 => 1
[4,1]
 => 100010 => 100100 => 6
[3,2]
 => 10100 => 01100 => 4
[3,1,1]
 => 100110 => 101010 => 6
[2,2,1]
 => 11010 => 11100 => 6
[2,1,1,1]
 => 101110 => 110110 => 6
[1,1,1,1,1]
 => 111110 => 111110 => 5
[6]
 => 1000000 => 0000010 => 1
[5,1]
 => 1000010 => 1000100 => 7
[4,2]
 => 100100 => 010100 => 5
[4,1,1]
 => 1000110 => 1001010 => 7
[3,3]
 => 11000 => 00110 => 2
[3,2,1]
 => 101010 => 111000 => 9
[3,1,1,1]
 => 1001110 => 1010110 => 7
[2,2,2]
 => 11100 => 01110 => 3
[2,2,1,1]
 => 110110 => 111010 => 7
[2,1,1,1,1]
 => 1011110 => 1101110 => 7
[1,1,1,1,1,1]
 => 1111110 => 1111110 => 6
[7]
 => 10000000 => 00000010 => 1
[6,1]
 => 10000010 => 10000100 => 8
[5,2]
 => 1000100 => 0100100 => 6
[5,1,1]
 => 10000110 => 10001010 => 8
[4,3]
 => 101000 => 001100 => 4
[4,2,1]
 => 1001010 => 1101000 => 11
[4,1,1,1]
 => 10001110 => 10010110 => 8
[3,3,1]
 => 110010 => 101100 => 7
[3,2,2]
 => 101100 => 011010 => 5
[3,2,1,1]
 => 1010110 => 1110010 => 10
[3,1,1,1,1]
 => 10011110 => 10101110 => 8
[2,2,2,1]
 => 111010 => 111100 => 8
[2,2,1,1,1]
 => 1101110 => 1110110 => 8
[2,1,1,1,1,1]
 => 10111110 => 11011110 => 8
[1,1,1,1,1,1,1]
 => 11111110 => 11111110 => 7
[8]
 => 100000000 => 000000010 => 1
[7,1]
 => 100000010 => 100000100 => 9
[6,2]
 => 10000100 => 01000100 => 7
[6,1,1]
 => 100000110 => 100001010 => 9
[5,3]
 => 1001000 => 0010100 => 5
[5,2,1]
 => 10001010 => 11001000 => 13
Description
The number of inversions of a binary word.
Matching statistic: St000770
Mapping
St000770: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1]
 => ? = 1
[2]
 => 2
[1,1]
 => 1
[3]
 => 3
[2,1]
 => 4
[1,1,1]
 => 1
[4]
 => 4
[3,1]
 => 5
[2,2]
 => 2
[2,1,1]
 => 5
[1,1,1,1]
 => 1
[5]
 => 5
[4,1]
 => 6
[3,2]
 => 6
[3,1,1]
 => 6
[2,2,1]
 => 4
[2,1,1,1]
 => 6
[1,1,1,1,1]
 => 1
[6]
 => 6
[5,1]
 => 7
[4,2]
 => 7
[4,1,1]
 => 7
[3,3]
 => 3
[3,2,1]
 => 9
[3,1,1,1]
 => 7
[2,2,2]
 => 2
[2,2,1,1]
 => 5
[2,1,1,1,1]
 => 7
[1,1,1,1,1,1]
 => 1
[7]
 => 7
[6,1]
 => 8
[5,2]
 => 8
[5,1,1]
 => 8
[4,3]
 => 8
[4,2,1]
 => 10
[4,1,1,1]
 => 8
[3,3,1]
 => 5
[3,2,2]
 => 7
[3,2,1,1]
 => 11
[3,1,1,1,1]
 => 8
[2,2,2,1]
 => 4
[2,2,1,1,1]
 => 6
[2,1,1,1,1,1]
 => 8
[1,1,1,1,1,1,1]
 => 1
[8]
 => 8
[7,1]
 => 9
[6,2]
 => 9
[6,1,1]
 => 9
[5,3]
 => 9
[5,2,1]
 => 11
[5,1,1,1]
 => 9
Description
The major index of an integer partition when read from bottom to top. This is the sum of the positions of the corners of the shape of an integer partition when reading from bottom to top. For example, the partition $\lambda = (8,6,6,4,3,3)$ has corners at positions 3,6,9, and 13, giving a major index of 31.
Matching statistic: St001232
(load all 4 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00124: Dyck paths —Adin-Bagno-Roichman transformation⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 47%
Values
[1]
 => [1,0,1,0]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 2 = 1 + 1
[2]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3 = 2 + 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 1 + 1
[2,1]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => ? = 4 + 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4 = 3 + 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 2 = 1 + 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => ? ∊ {5,5} + 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 3 = 2 + 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => ? ∊ {5,5} + 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5 = 4 + 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => 2 = 1 + 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => ? ∊ {4,6,6,6,6} + 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => ? ∊ {4,6,6,6,6} + 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => ? ∊ {4,6,6,6,6} + 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => ? ∊ {4,6,6,6,6} + 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => ? ∊ {4,6,6,6,6} + 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => 6 = 5 + 1
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
 => ? ∊ {1,5,6,7,7,7,7,9} + 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
 => ? ∊ {1,5,6,7,7,7,7,9} + 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => ? ∊ {1,5,6,7,7,7,7,9} + 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => 8 = 7 + 1
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 3 = 2 + 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => ? ∊ {1,5,6,7,7,7,7,9} + 1
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => ? ∊ {1,5,6,7,7,7,7,9} + 1
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 4 = 3 + 1
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => ? ∊ {1,5,6,7,7,7,7,9} + 1
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => ? ∊ {1,5,6,7,7,7,7,9} + 1
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => ? ∊ {1,5,6,7,7,7,7,9} + 1
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
 => ? ∊ {1,4,5,6,7,7,8,8,8,8,8,8,8,11} + 1
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,1,0,0]
 => ? ∊ {1,4,5,6,7,7,8,8,8,8,8,8,8,11} + 1
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,1,1,0,0,0,1,0,0]
 => ? ∊ {1,4,5,6,7,7,8,8,8,8,8,8,8,11} + 1
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
 => 11 = 10 + 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => ? ∊ {1,4,5,6,7,7,8,8,8,8,8,8,8,11} + 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => ? ∊ {1,4,5,6,7,7,8,8,8,8,8,8,8,11} + 1
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => ? ∊ {1,4,5,6,7,7,8,8,8,8,8,8,8,11} + 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => ? ∊ {1,4,5,6,7,7,8,8,8,8,8,8,8,11} + 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => ? ∊ {1,4,5,6,7,7,8,8,8,8,8,8,8,11} + 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => ? ∊ {1,4,5,6,7,7,8,8,8,8,8,8,8,11} + 1
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => ? ∊ {1,4,5,6,7,7,8,8,8,8,8,8,8,11} + 1
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => ? ∊ {1,4,5,6,7,7,8,8,8,8,8,8,8,11} + 1
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => ? ∊ {1,4,5,6,7,7,8,8,8,8,8,8,8,11} + 1
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
 => ? ∊ {1,4,5,6,7,7,8,8,8,8,8,8,8,11} + 1
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => ? ∊ {1,4,5,6,7,7,8,8,8,8,8,8,8,11} + 1
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,0]
 => ? ∊ {1,5,6,6,7,8,8,9,9,9,9,9,9,9,10,11,11,12,13} + 1
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0,0]
 => ? ∊ {1,5,6,6,7,8,8,9,9,9,9,9,9,9,10,11,11,12,13} + 1
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,1,0,0]
 => ? ∊ {1,5,6,6,7,8,8,9,9,9,9,9,9,9,10,11,11,12,13} + 1
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,1,0,0]
 => ? ∊ {1,5,6,6,7,8,8,9,9,9,9,9,9,9,10,11,11,12,13} + 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]
 => [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
 => ? ∊ {1,5,6,6,7,8,8,9,9,9,9,9,9,9,10,11,11,12,13} + 1
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,1,1,0,0,0,1,0,0]
 => ? ∊ {1,5,6,6,7,8,8,9,9,9,9,9,9,9,10,11,11,12,13} + 1
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
 => 10 = 9 + 1
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => 3 = 2 + 1
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => ? ∊ {1,5,6,6,7,8,8,9,9,9,9,9,9,9,10,11,11,12,13} + 1
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => ? ∊ {1,5,6,6,7,8,8,9,9,9,9,9,9,9,10,11,11,12,13} + 1
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => ? ∊ {1,5,6,6,7,8,8,9,9,9,9,9,9,9,10,11,11,12,13} + 1
[4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
 => ? ∊ {1,5,6,6,7,8,8,9,9,9,9,9,9,9,10,11,11,12,13} + 1
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => ? ∊ {1,5,6,6,7,8,8,9,9,9,9,9,9,9,10,11,11,12,13} + 1
[3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => ? ∊ {1,5,6,6,7,8,8,9,9,9,9,9,9,9,10,11,11,12,13} + 1
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => ? ∊ {1,5,6,6,7,8,8,9,9,9,9,9,9,9,10,11,11,12,13} + 1
[3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => ? ∊ {1,5,6,6,7,8,8,9,9,9,9,9,9,9,10,11,11,12,13} + 1
[3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
 => ? ∊ {1,5,6,6,7,8,8,9,9,9,9,9,9,9,10,11,11,12,13} + 1
[2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => 5 = 4 + 1
[2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => ? ∊ {1,5,6,6,7,8,8,9,9,9,9,9,9,9,10,11,11,12,13} + 1
[2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0]
 => ? ∊ {1,5,6,6,7,8,8,9,9,9,9,9,9,9,10,11,11,12,13} + 1
[2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0]
 => ? ∊ {1,5,6,6,7,8,8,9,9,9,9,9,9,9,10,11,11,12,13} + 1
[1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
 => ? ∊ {1,5,6,6,7,8,8,9,9,9,9,9,9,9,10,11,11,12,13} + 1
[9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0,0]
 => ? ∊ {1,4,6,6,8,8,9,9,9,10,10,10,10,10,10,10,10,10,10,11,12,12,12,12,13,13,14,15} + 1
[5,2,2]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
 => 8 = 7 + 1
[3,3,3]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => 4 = 3 + 1
[4,4,1,1]
 => [1,1,0,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
 => 11 = 10 + 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000454
Mapping
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 29%
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)
 => ? = 4
[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)
 => ? ∊ {5,5}
[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)
 => ? ∊ {5,5}
[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)
 => ? ∊ {4,6,6,6,6}
[3,2]
 => 10100 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {4,6,6,6,6}
[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)
 => ? ∊ {4,6,6,6,6}
[2,2,1]
 => 11010 => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {4,6,6,6,6}
[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)
 => ? ∊ {4,6,6,6,6}
[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)
 => ? ∊ {1,5,6,7,7,7,7,7,9}
[5,1]
 => 1000010 => [1,5,2] => ([(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {1,5,6,7,7,7,7,7,9}
[4,2]
 => 100100 => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,5,6,7,7,7,7,7,9}
[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)
 => ? ∊ {1,5,6,7,7,7,7,7,9}
[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)
 => ? ∊ {1,5,6,7,7,7,7,7,9}
[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)
 => ? ∊ {1,5,6,7,7,7,7,7,9}
[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)
 => ? ∊ {1,5,6,7,7,7,7,7,9}
[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)
 => ? ∊ {1,5,6,7,7,7,7,7,9}
[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)
 => ? ∊ {1,5,6,7,7,7,7,7,9}
[7]
 => 10000000 => [1,8] => ([(7,8)],9)
 => ? ∊ {1,4,5,6,7,7,8,8,8,8,8,8,8,10,11}
[6,1]
 => 10000010 => [1,6,2] => ([(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(6,8),(7,8)],9)
 => ? ∊ {1,4,5,6,7,7,8,8,8,8,8,8,8,10,11}
[5,2]
 => 1000100 => [1,4,3] => ([(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {1,4,5,6,7,7,8,8,8,8,8,8,8,10,11}
[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)
 => ? ∊ {1,4,5,6,7,7,8,8,8,8,8,8,8,10,11}
[4,3]
 => 101000 => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,4,5,6,7,7,8,8,8,8,8,8,8,10,11}
[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)
 => ? ∊ {1,4,5,6,7,7,8,8,8,8,8,8,8,10,11}
[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)
 => ? ∊ {1,4,5,6,7,7,8,8,8,8,8,8,8,10,11}
[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)
 => ? ∊ {1,4,5,6,7,7,8,8,8,8,8,8,8,10,11}
[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)
 => ? ∊ {1,4,5,6,7,7,8,8,8,8,8,8,8,10,11}
[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)
 => ? ∊ {1,4,5,6,7,7,8,8,8,8,8,8,8,10,11}
[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)
 => ? ∊ {1,4,5,6,7,7,8,8,8,8,8,8,8,10,11}
[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)
 => ? ∊ {1,4,5,6,7,7,8,8,8,8,8,8,8,10,11}
[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)
 => ? ∊ {1,4,5,6,7,7,8,8,8,8,8,8,8,10,11}
[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)
 => ? ∊ {1,4,5,6,7,7,8,8,8,8,8,8,8,10,11}
[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)
 => ? ∊ {1,4,5,6,7,7,8,8,8,8,8,8,8,10,11}
[8]
 => 100000000 => [1,9] => ([(8,9)],10)
 => ? ∊ {1,5,6,6,7,8,8,9,9,9,9,9,9,9,9,10,11,11,12,13}
[7,1]
 => 100000010 => [1,7,2] => ([(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,8),(7,9),(8,9)],10)
 => ? ∊ {1,5,6,6,7,8,8,9,9,9,9,9,9,9,9,10,11,11,12,13}
[6,2]
 => 10000100 => [1,5,3] => ([(2,8),(3,8),(4,8),(5,8),(6,7),(6,8),(7,8)],9)
 => ? ∊ {1,5,6,6,7,8,8,9,9,9,9,9,9,9,9,10,11,11,12,13}
[6,1,1]
 => 100000110 => [1,6,1,2] => ([(1,8),(1,9),(2,8),(2,9),(3,8),(3,9),(4,8),(4,9),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? ∊ {1,5,6,6,7,8,8,9,9,9,9,9,9,9,9,10,11,11,12,13}
[5,3]
 => 1001000 => [1,3,4] => ([(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {1,5,6,6,7,8,8,9,9,9,9,9,9,9,9,10,11,11,12,13}
[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)
 => ? ∊ {1,5,6,6,7,8,8,9,9,9,9,9,9,9,9,10,11,11,12,13}
[5,1,1,1]
 => 100001110 => [1,5,1,1,2] => ([(1,7),(1,8),(1,9),(2,7),(2,8),(2,9),(3,7),(3,8),(3,9),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? ∊ {1,5,6,6,7,8,8,9,9,9,9,9,9,9,9,10,11,11,12,13}
[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)
 => ? ∊ {1,5,6,6,7,8,8,9,9,9,9,9,9,9,9,10,11,11,12,13}
[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)
 => ? ∊ {1,5,6,6,7,8,8,9,9,9,9,9,9,9,9,10,11,11,12,13}
[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)
 => ? ∊ {1,5,6,6,7,8,8,9,9,9,9,9,9,9,9,10,11,11,12,13}
[4,1,1,1,1]
 => 100011110 => [1,4,1,1,1,2] => ([(1,6),(1,7),(1,8),(1,9),(2,6),(2,7),(2,8),(2,9),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? ∊ {1,5,6,6,7,8,8,9,9,9,9,9,9,9,9,10,11,11,12,13}
[3,3,2]
 => 110100 => [1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,5,6,6,7,8,8,9,9,9,9,9,9,9,9,10,11,11,12,13}
[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)
 => ? ∊ {1,5,6,6,7,8,8,9,9,9,9,9,9,9,9,10,11,11,12,13}
[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)
 => ? ∊ {1,5,6,6,7,8,8,9,9,9,9,9,9,9,9,10,11,11,12,13}
[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)
 => ? ∊ {1,5,6,6,7,8,8,9,9,9,9,9,9,9,9,10,11,11,12,13}
[3,1,1,1,1,1]
 => 100111110 => [1,3,1,1,1,1,2] => ([(1,5),(1,6),(1,7),(1,8),(1,9),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? ∊ {1,5,6,6,7,8,8,9,9,9,9,9,9,9,9,10,11,11,12,13}
[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)
 => ? ∊ {1,5,6,6,7,8,8,9,9,9,9,9,9,9,9,10,11,11,12,13}
[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)
 => ? ∊ {1,5,6,6,7,8,8,9,9,9,9,9,9,9,9,10,11,11,12,13}
[3,3,3]
 => 111000 => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3
Description
The largest eigenvalue of a graph if it is integral. If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree. This statistic is undefined if the largest eigenvalue of the graph is not integral.