Your data matches 8 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000145
Mapping
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000145: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
 => [1,1]
 => -1 = 0 - 1
[1,1]
 => [2]
 => 1 = 2 - 1
[3]
 => [1,1,1]
 => -2 = -1 - 1
[2,1]
 => [2,1]
 => 0 = 1 - 1
[1,1,1]
 => [3]
 => 2 = 3 - 1
[4]
 => [1,1,1,1]
 => -3 = -2 - 1
[3,1]
 => [2,1,1]
 => -1 = 0 - 1
[2,2]
 => [2,2]
 => 0 = 1 - 1
[2,1,1]
 => [3,1]
 => 1 = 2 - 1
[1,1,1,1]
 => [4]
 => 3 = 4 - 1
[5]
 => [1,1,1,1,1]
 => -4 = -3 - 1
[4,1]
 => [2,1,1,1]
 => -2 = -1 - 1
[3,2]
 => [2,2,1]
 => -1 = 0 - 1
[3,1,1]
 => [3,1,1]
 => 0 = 1 - 1
[2,2,1]
 => [3,2]
 => 1 = 2 - 1
[2,1,1,1]
 => [4,1]
 => 2 = 3 - 1
[1,1,1,1,1]
 => [5]
 => 4 = 5 - 1
[6]
 => [1,1,1,1,1,1]
 => -5 = -4 - 1
[5,1]
 => [2,1,1,1,1]
 => -3 = -2 - 1
[4,2]
 => [2,2,1,1]
 => -2 = -1 - 1
[4,1,1]
 => [3,1,1,1]
 => -1 = 0 - 1
[3,3]
 => [2,2,2]
 => -1 = 0 - 1
[3,2,1]
 => [3,2,1]
 => 0 = 1 - 1
[3,1,1,1]
 => [4,1,1]
 => 1 = 2 - 1
[2,2,2]
 => [3,3]
 => 1 = 2 - 1
[2,2,1,1]
 => [4,2]
 => 2 = 3 - 1
[2,1,1,1,1]
 => [5,1]
 => 3 = 4 - 1
[1,1,1,1,1,1]
 => [6]
 => 5 = 6 - 1
[5,2]
 => [2,2,1,1,1]
 => -3 = -2 - 1
[4,3]
 => [2,2,2,1]
 => -2 = -1 - 1
[4,2,1]
 => [3,2,1,1]
 => -1 = 0 - 1
[3,3,1]
 => [3,2,2]
 => 0 = 1 - 1
[3,2,2]
 => [3,3,1]
 => 0 = 1 - 1
[3,2,1,1]
 => [4,2,1]
 => 1 = 2 - 1
[2,2,2,1]
 => [4,3]
 => 2 = 3 - 1
[2,2,1,1,1]
 => [5,2]
 => 3 = 4 - 1
[5,3]
 => [2,2,2,1,1]
 => -3 = -2 - 1
[4,4]
 => [2,2,2,2]
 => -2 = -1 - 1
[4,3,1]
 => [3,2,2,1]
 => -1 = 0 - 1
[3,3,2]
 => [3,3,2]
 => 0 = 1 - 1
[3,3,1,1]
 => [4,2,2]
 => 1 = 2 - 1
[3,2,2,1]
 => [4,3,1]
 => 1 = 2 - 1
[2,2,2,2]
 => [4,4]
 => 2 = 3 - 1
[2,2,2,1,1]
 => [5,3]
 => 3 = 4 - 1
[5,4]
 => [2,2,2,2,1]
 => -3 = -2 - 1
[4,4,1]
 => [3,2,2,2]
 => -1 = 0 - 1
[3,3,3]
 => [3,3,3]
 => 0 = 1 - 1
[3,3,2,1]
 => [4,3,2]
 => 1 = 2 - 1
[2,2,2,2,1]
 => [5,4]
 => 3 = 4 - 1
[5,5]
 => [2,2,2,2,2]
 => -3 = -2 - 1
Description
The Dyson rank of a partition. This rank is defined as the largest part minus the number of parts. It was introduced by Dyson [1] in connection to Ramanujan's partition congruences $$p(5n+4) \equiv 0 \pmod 5$$ and $$p(7n+6) \equiv 0 \pmod 7.$$
Matching statistic: St000878
(load all 80 compositions to match this statistic)
Mapping
Mp00095: Integer partitions —to binary word⟶ Binary words
St000878: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
 => 100 => -1 = 0 - 1
[1,1]
 => 110 => 1 = 2 - 1
[3]
 => 1000 => -2 = -1 - 1
[2,1]
 => 1010 => 0 = 1 - 1
[1,1,1]
 => 1110 => 2 = 3 - 1
[4]
 => 10000 => -3 = -2 - 1
[3,1]
 => 10010 => -1 = 0 - 1
[2,2]
 => 1100 => 0 = 1 - 1
[2,1,1]
 => 10110 => 1 = 2 - 1
[1,1,1,1]
 => 11110 => 3 = 4 - 1
[5]
 => 100000 => -4 = -3 - 1
[4,1]
 => 100010 => -2 = -1 - 1
[3,2]
 => 10100 => -1 = 0 - 1
[3,1,1]
 => 100110 => 0 = 1 - 1
[2,2,1]
 => 11010 => 1 = 2 - 1
[2,1,1,1]
 => 101110 => 2 = 3 - 1
[1,1,1,1,1]
 => 111110 => 4 = 5 - 1
[6]
 => 1000000 => -5 = -4 - 1
[5,1]
 => 1000010 => -3 = -2 - 1
[4,2]
 => 100100 => -2 = -1 - 1
[4,1,1]
 => 1000110 => -1 = 0 - 1
[3,3]
 => 11000 => -1 = 0 - 1
[3,2,1]
 => 101010 => 0 = 1 - 1
[3,1,1,1]
 => 1001110 => 1 = 2 - 1
[2,2,2]
 => 11100 => 1 = 2 - 1
[2,2,1,1]
 => 110110 => 2 = 3 - 1
[2,1,1,1,1]
 => 1011110 => 3 = 4 - 1
[1,1,1,1,1,1]
 => 1111110 => 5 = 6 - 1
[5,2]
 => 1000100 => -3 = -2 - 1
[4,3]
 => 101000 => -2 = -1 - 1
[4,2,1]
 => 1001010 => -1 = 0 - 1
[3,3,1]
 => 110010 => 0 = 1 - 1
[3,2,2]
 => 101100 => 0 = 1 - 1
[3,2,1,1]
 => 1010110 => 1 = 2 - 1
[2,2,2,1]
 => 111010 => 2 = 3 - 1
[2,2,1,1,1]
 => 1101110 => 3 = 4 - 1
[5,3]
 => 1001000 => -3 = -2 - 1
[4,4]
 => 110000 => -2 = -1 - 1
[4,3,1]
 => 1010010 => -1 = 0 - 1
[3,3,2]
 => 110100 => 0 = 1 - 1
[3,3,1,1]
 => 1100110 => 1 = 2 - 1
[3,2,2,1]
 => 1011010 => 1 = 2 - 1
[2,2,2,2]
 => 111100 => 2 = 3 - 1
[2,2,2,1,1]
 => 1110110 => 3 = 4 - 1
[5,4]
 => 1010000 => -3 = -2 - 1
[4,4,1]
 => 1100010 => -1 = 0 - 1
[3,3,3]
 => 111000 => 0 = 1 - 1
[3,3,2,1]
 => 1101010 => 1 = 2 - 1
[2,2,2,2,1]
 => 1111010 => 3 = 4 - 1
[5,5]
 => 1100000 => -3 = -2 - 1
Description
The number of ones minus the number of zeros of a binary word.
Matching statistic: St000894
(load all 4 compositions to match this statistic)
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
St000894: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
 => [1,0,1,0]
 => [1,1,0,0]
 => [[0,1],[1,0]]
 => 0
[1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => [[1,0],[0,1]]
 => 2
[3]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => [[0,1,0],[1,-1,1],[0,1,0]]
 => -1
[2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [[0,1,0],[1,0,0],[0,0,1]]
 => 1
[1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => [[1,0,0],[0,1,0],[0,0,1]]
 => 3
[4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
 => -2
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
 => 0
[2,2]
 => [1,1,1,0,0,0]
 => [1,0,1,1,0,0]
 => [[1,0,0],[0,0,1],[0,1,0]]
 => 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
 => 2
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
 => 4
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => -3
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => -1
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,0,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
 => 0
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => 1
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
 => 2
[2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => 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]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => 5
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
 => -4
[5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
 => -2
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => -1
[4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
 => 0
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
 => 0
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => 1
[3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
 => 2
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
 => 2
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => 3
[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]
 => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
 => 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]
 => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
 => 6
[5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
 => -2
[4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => -1
[4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
 => 0
[3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => 1
[3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
 => 1
[3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
 => 2
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => 3
[2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
 => 4
[5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
 => -2
[4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => -1
[4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
 => 0
[3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
 => 1
[3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
 => 2
[3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
 => 2
[2,2,2,2]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0]]
 => 3
[2,2,2,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
 => 4
[5,4]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0]
 => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
 => -2
[4,4,1]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
 => 0
[3,3,3]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
 => 1
[3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
 => 2
[2,2,2,2,1]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0,1,0]
 => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
 => 4
[5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
 => -2
Description
The trace of an alternating sign matrix.
Matching statistic: St000090
Mapping
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000090: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
 => 100 => 001 => [2,1] => -1 = 0 - 1
[1,1]
 => 110 => 011 => [1,2] => 1 = 2 - 1
[3]
 => 1000 => 0001 => [3,1] => -2 = -1 - 1
[2,1]
 => 1010 => 0011 => [2,2] => 0 = 1 - 1
[1,1,1]
 => 1110 => 0111 => [1,3] => 2 = 3 - 1
[4]
 => 10000 => 00001 => [4,1] => -3 = -2 - 1
[3,1]
 => 10010 => 00011 => [3,2] => -1 = 0 - 1
[2,2]
 => 1100 => 0011 => [2,2] => 0 = 1 - 1
[2,1,1]
 => 10110 => 00111 => [2,3] => 1 = 2 - 1
[1,1,1,1]
 => 11110 => 01111 => [1,4] => 3 = 4 - 1
[5]
 => 100000 => 000001 => [5,1] => -4 = -3 - 1
[4,1]
 => 100010 => 000011 => [4,2] => -2 = -1 - 1
[3,2]
 => 10100 => 00011 => [3,2] => -1 = 0 - 1
[3,1,1]
 => 100110 => 000111 => [3,3] => 0 = 1 - 1
[2,2,1]
 => 11010 => 00111 => [2,3] => 1 = 2 - 1
[2,1,1,1]
 => 101110 => 001111 => [2,4] => 2 = 3 - 1
[1,1,1,1,1]
 => 111110 => 011111 => [1,5] => 4 = 5 - 1
[6]
 => 1000000 => 0000001 => [6,1] => -5 = -4 - 1
[5,1]
 => 1000010 => 0000011 => [5,2] => -3 = -2 - 1
[4,2]
 => 100100 => 000011 => [4,2] => -2 = -1 - 1
[4,1,1]
 => 1000110 => 0000111 => [4,3] => -1 = 0 - 1
[3,3]
 => 11000 => 00011 => [3,2] => -1 = 0 - 1
[3,2,1]
 => 101010 => 001011 => [2,1,1,2] => 0 = 1 - 1
[3,1,1,1]
 => 1001110 => 0001111 => [3,4] => 1 = 2 - 1
[2,2,2]
 => 11100 => 00111 => [2,3] => 1 = 2 - 1
[2,2,1,1]
 => 110110 => 001111 => [2,4] => 2 = 3 - 1
[2,1,1,1,1]
 => 1011110 => 0011111 => [2,5] => 3 = 4 - 1
[1,1,1,1,1,1]
 => 1111110 => 0111111 => [1,6] => 5 = 6 - 1
[5,2]
 => 1000100 => 0000011 => [5,2] => -3 = -2 - 1
[4,3]
 => 101000 => 000011 => [4,2] => -2 = -1 - 1
[4,2,1]
 => 1001010 => 0001011 => [3,1,1,2] => -1 = 0 - 1
[3,3,1]
 => 110010 => 000111 => [3,3] => 0 = 1 - 1
[3,2,2]
 => 101100 => 000111 => [3,3] => 0 = 1 - 1
[3,2,1,1]
 => 1010110 => 0010111 => [2,1,1,3] => 1 = 2 - 1
[2,2,2,1]
 => 111010 => 001111 => [2,4] => 2 = 3 - 1
[2,2,1,1,1]
 => 1101110 => 0011111 => [2,5] => 3 = 4 - 1
[5,3]
 => 1001000 => 0000011 => [5,2] => -3 = -2 - 1
[4,4]
 => 110000 => 000011 => [4,2] => -2 = -1 - 1
[4,3,1]
 => 1010010 => 0001011 => [3,1,1,2] => -1 = 0 - 1
[3,3,2]
 => 110100 => 000111 => [3,3] => 0 = 1 - 1
[3,3,1,1]
 => 1100110 => 0001111 => [3,4] => 1 = 2 - 1
[3,2,2,1]
 => 1011010 => 0010111 => [2,1,1,3] => 1 = 2 - 1
[2,2,2,2]
 => 111100 => 001111 => [2,4] => 2 = 3 - 1
[2,2,2,1,1]
 => 1110110 => 0011111 => [2,5] => 3 = 4 - 1
[5,4]
 => 1010000 => 0000011 => [5,2] => -3 = -2 - 1
[4,4,1]
 => 1100010 => 0000111 => [4,3] => -1 = 0 - 1
[3,3,3]
 => 111000 => 000111 => [3,3] => 0 = 1 - 1
[3,3,2,1]
 => 1101010 => 0010111 => [2,1,1,3] => 1 = 2 - 1
[2,2,2,2,1]
 => 1111010 => 0011111 => [2,5] => 3 = 4 - 1
[5,5]
 => 1100000 => 0000011 => [5,2] => -3 = -2 - 1
Description
The variation of a composition.
Matching statistic: St001232
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 45%
Values
[2]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => ? = 0
[1,1]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
[3]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => ? = -1
[2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => ? = 1
[1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
[4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => ? = -2
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => ? = 0
[2,2]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => ? = 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => ? = 2
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = -3
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = -1
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => ? = 0
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => ? = 1
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => ? = 2
[2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => ? = 3
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = -4
[5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = -2
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = -1
[4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => ? = 0
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => ? = 0
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => ? = 1
[3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => ? = 2
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => ? = 2
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => ? = 3
[2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => ? = 4
[1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => 6
[5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = -2
[4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => ? = -1
[4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => ? = 0
[3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => ? = 1
[3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => ? = 1
[3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
 => ? = 2
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => ? = 3
[2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => ? = 4
[5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => ? = -2
[4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = -1
[4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,1,0,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,1,0,1,0,0,0]
 => ? = 0
[3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => ? = 1
[3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
 => ? = 2
[3,2,2,1]
 => [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]
 => [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => ? = 2
[2,2,2,2]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => ? = 3
[2,2,2,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
 => [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => ? = 4
[5,4]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = -2
[4,4,1]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
 => [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => ? = 0
[3,3,3]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => ? = 1
[3,3,2,1]
 => [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]
 => [1,1,0,1,1,0,1,1,0,1,0,0,0,0]
 => ? = 2
[2,2,2,2,1]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => ? = 4
[5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = -2
[3,3,3,1]
 => [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]
 => [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => ? = 2
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001645
Mapping
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 45%
Values
[2]
 => [[1,2]]
 => [2] => ([],2)
 => ? = 0
[1,1]
 => [[1],[2]]
 => [1,1] => ([(0,1)],2)
 => 2
[3]
 => [[1,2,3]]
 => [3] => ([],3)
 => ? = -1
[2,1]
 => [[1,3],[2]]
 => [1,2] => ([(1,2)],3)
 => ? = 1
[1,1,1]
 => [[1],[2],[3]]
 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[4]
 => [[1,2,3,4]]
 => [4] => ([],4)
 => ? = -2
[3,1]
 => [[1,3,4],[2]]
 => [1,3] => ([(2,3)],4)
 => ? = 0
[2,2]
 => [[1,2],[3,4]]
 => [2,2] => ([(1,3),(2,3)],4)
 => ? = 1
[2,1,1]
 => [[1,4],[2],[3]]
 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 2
[1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[5]
 => [[1,2,3,4,5]]
 => [5] => ([],5)
 => ? = -3
[4,1]
 => [[1,3,4,5],[2]]
 => [1,4] => ([(3,4)],5)
 => ? = -1
[3,2]
 => [[1,2,5],[3,4]]
 => [2,3] => ([(2,4),(3,4)],5)
 => ? = 0
[3,1,1]
 => [[1,4,5],[2],[3]]
 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ? = 1
[2,2,1]
 => [[1,3],[2,5],[4]]
 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2
[2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
[1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[6]
 => [[1,2,3,4,5,6]]
 => [6] => ([],6)
 => ? = -4
[5,1]
 => [[1,3,4,5,6],[2]]
 => [1,5] => ([(4,5)],6)
 => ? = -2
[4,2]
 => [[1,2,5,6],[3,4]]
 => [2,4] => ([(3,5),(4,5)],6)
 => ? = -1
[4,1,1]
 => [[1,4,5,6],[2],[3]]
 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => ? = 0
[3,3]
 => [[1,2,3],[4,5,6]]
 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ? = 0
[3,2,1]
 => [[1,3,6],[2,5],[4]]
 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
[3,1,1,1]
 => [[1,5,6],[2],[3],[4]]
 => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2
[2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2
[2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
[2,1,1,1,1]
 => [[1,6],[2],[3],[4],[5]]
 => [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
[1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[5,2]
 => [[1,2,5,6,7],[3,4]]
 => [2,5] => ([(4,6),(5,6)],7)
 => ? = -2
[4,3]
 => [[1,2,3,7],[4,5,6]]
 => [3,4] => ([(3,6),(4,6),(5,6)],7)
 => ? = -1
[4,2,1]
 => [[1,3,6,7],[2,5],[4]]
 => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[3,3,1]
 => [[1,3,4],[2,6,7],[5]]
 => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[3,2,2]
 => [[1,2,7],[3,4],[5,6]]
 => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[3,2,1,1]
 => [[1,4,7],[2,6],[3],[5]]
 => [1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[2,2,2,1]
 => [[1,3],[2,5],[4,7],[6]]
 => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[2,2,1,1,1]
 => [[1,5],[2,7],[3],[4],[6]]
 => [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)
 => ? = 4
[5,3]
 => [[1,2,3,7,8],[4,5,6]]
 => [3,5] => ([(4,7),(5,7),(6,7)],8)
 => ? = -2
[4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
 => ? = -1
[4,3,1]
 => [[1,3,4,8],[2,6,7],[5]]
 => [1,3,4] => ([(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 0
[3,3,2]
 => [[1,2,5],[3,4,8],[6,7]]
 => [2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1
[3,3,1,1]
 => [[1,4,5],[2,7,8],[3],[6]]
 => [1,1,3,3] => ([(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[3,2,2,1]
 => [[1,3,8],[2,5],[4,7],[6]]
 => [1,2,2,3] => ([(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[2,2,2,1,1]
 => [[1,4],[2,6],[3,8],[5],[7]]
 => [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)
 => ? = 4
[5,4]
 => [[1,2,3,4,9],[5,6,7,8]]
 => [4,5] => ([(4,8),(5,8),(6,8),(7,8)],9)
 => ? = -2
[4,4,1]
 => [[1,3,4,5],[2,7,8,9],[6]]
 => [1,4,4] => ([(3,8),(4,8),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 0
[3,3,3]
 => [[1,2,3],[4,5,6],[7,8,9]]
 => [3,3,3] => ([(2,8),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 1
[3,3,2,1]
 => [[1,3,6],[2,5,9],[4,8],[7]]
 => [1,2,3,3] => ([(2,8),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2
[2,2,2,2,1]
 => [[1,3],[2,5],[4,7],[6,9],[8]]
 => [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)
 => ? = 4
[5,5]
 => [[1,2,3,4,5],[6,7,8,9,10]]
 => [5,5] => ([(4,9),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = -2
[3,3,3,1]
 => [[1,3,4],[2,6,7],[5,9,10],[8]]
 => [1,3,3,3] => ([(2,9),(3,9),(4,8),(4,9),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 2
Description
The pebbling number of a connected graph.
Matching statistic: St000771
Mapping
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 45%
Values
[2]
 => [[1,2]]
 => [2] => ([],2)
 => ? = 0 - 1
[1,1]
 => [[1],[2]]
 => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
[3]
 => [[1,2,3]]
 => [3] => ([],3)
 => ? = -1 - 1
[2,1]
 => [[1,3],[2]]
 => [1,2] => ([(1,2)],3)
 => ? = 1 - 1
[1,1,1]
 => [[1],[2],[3]]
 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
[4]
 => [[1,2,3,4]]
 => [4] => ([],4)
 => ? = -2 - 1
[3,1]
 => [[1,3,4],[2]]
 => [1,3] => ([(2,3)],4)
 => ? = 0 - 1
[2,2]
 => [[1,2],[3,4]]
 => [2,2] => ([(1,3),(2,3)],4)
 => ? = 1 - 1
[2,1,1]
 => [[1,4],[2],[3]]
 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 2 - 1
[1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
[5]
 => [[1,2,3,4,5]]
 => [5] => ([],5)
 => ? = -3 - 1
[4,1]
 => [[1,3,4,5],[2]]
 => [1,4] => ([(3,4)],5)
 => ? = -1 - 1
[3,2]
 => [[1,2,5],[3,4]]
 => [2,3] => ([(2,4),(3,4)],5)
 => ? = 0 - 1
[3,1,1]
 => [[1,4,5],[2],[3]]
 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ? = 1 - 1
[2,2,1]
 => [[1,3],[2,5],[4]]
 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 - 1
[1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 5 - 1
[6]
 => [[1,2,3,4,5,6]]
 => [6] => ([],6)
 => ? = -4 - 1
[5,1]
 => [[1,3,4,5,6],[2]]
 => [1,5] => ([(4,5)],6)
 => ? = -2 - 1
[4,2]
 => [[1,2,5,6],[3,4]]
 => [2,4] => ([(3,5),(4,5)],6)
 => ? = -1 - 1
[4,1,1]
 => [[1,4,5,6],[2],[3]]
 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => ? = 0 - 1
[3,3]
 => [[1,2,3],[4,5,6]]
 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ? = 0 - 1
[3,2,1]
 => [[1,3,6],[2,5],[4]]
 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
[3,1,1,1]
 => [[1,5,6],[2],[3],[4]]
 => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
[2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
[2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 - 1
[2,1,1,1,1]
 => [[1,6],[2],[3],[4],[5]]
 => [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 - 1
[1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5 = 6 - 1
[5,2]
 => [[1,2,5,6,7],[3,4]]
 => [2,5] => ([(4,6),(5,6)],7)
 => ? = -2 - 1
[4,3]
 => [[1,2,3,7],[4,5,6]]
 => [3,4] => ([(3,6),(4,6),(5,6)],7)
 => ? = -1 - 1
[4,2,1]
 => [[1,3,6,7],[2,5],[4]]
 => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0 - 1
[3,3,1]
 => [[1,3,4],[2,6,7],[5]]
 => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[3,2,2]
 => [[1,2,7],[3,4],[5,6]]
 => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[3,2,1,1]
 => [[1,4,7],[2,6],[3],[5]]
 => [1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 - 1
[2,2,2,1]
 => [[1,3],[2,5],[4,7],[6]]
 => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 - 1
[2,2,1,1,1]
 => [[1,5],[2,7],[3],[4],[6]]
 => [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)
 => ? = 4 - 1
[5,3]
 => [[1,2,3,7,8],[4,5,6]]
 => [3,5] => ([(4,7),(5,7),(6,7)],8)
 => ? = -2 - 1
[4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
 => ? = -1 - 1
[4,3,1]
 => [[1,3,4,8],[2,6,7],[5]]
 => [1,3,4] => ([(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 0 - 1
[3,3,2]
 => [[1,2,5],[3,4,8],[6,7]]
 => [2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1 - 1
[3,3,1,1]
 => [[1,4,5],[2,7,8],[3],[6]]
 => [1,1,3,3] => ([(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 - 1
[3,2,2,1]
 => [[1,3,8],[2,5],[4,7],[6]]
 => [1,2,2,3] => ([(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 - 1
[2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 - 1
[2,2,2,1,1]
 => [[1,4],[2,6],[3,8],[5],[7]]
 => [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)
 => ? = 4 - 1
[5,4]
 => [[1,2,3,4,9],[5,6,7,8]]
 => [4,5] => ([(4,8),(5,8),(6,8),(7,8)],9)
 => ? = -2 - 1
[4,4,1]
 => [[1,3,4,5],[2,7,8,9],[6]]
 => [1,4,4] => ([(3,8),(4,8),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 0 - 1
[3,3,3]
 => [[1,2,3],[4,5,6],[7,8,9]]
 => [3,3,3] => ([(2,8),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 1 - 1
[3,3,2,1]
 => [[1,3,6],[2,5,9],[4,8],[7]]
 => [1,2,3,3] => ([(2,8),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2 - 1
[2,2,2,2,1]
 => [[1,3],[2,5],[4,7],[6,9],[8]]
 => [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)
 => ? = 4 - 1
[5,5]
 => [[1,2,3,4,5],[6,7,8,9,10]]
 => [5,5] => ([(4,9),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = -2 - 1
[3,3,3,1]
 => [[1,3,4],[2,6,7],[5,9,10],[8]]
 => [1,3,3,3] => ([(2,9),(3,9),(4,8),(4,9),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 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
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 45%
Values
[2]
 => [[1,2]]
 => [2] => ([],2)
 => ? = 0 - 1
[1,1]
 => [[1],[2]]
 => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
[3]
 => [[1,2,3]]
 => [3] => ([],3)
 => ? = -1 - 1
[2,1]
 => [[1,3],[2]]
 => [1,2] => ([(1,2)],3)
 => ? = 1 - 1
[1,1,1]
 => [[1],[2],[3]]
 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
[4]
 => [[1,2,3,4]]
 => [4] => ([],4)
 => ? = -2 - 1
[3,1]
 => [[1,3,4],[2]]
 => [1,3] => ([(2,3)],4)
 => ? = 0 - 1
[2,2]
 => [[1,2],[3,4]]
 => [2,2] => ([(1,3),(2,3)],4)
 => ? = 1 - 1
[2,1,1]
 => [[1,4],[2],[3]]
 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 2 - 1
[1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
[5]
 => [[1,2,3,4,5]]
 => [5] => ([],5)
 => ? = -3 - 1
[4,1]
 => [[1,3,4,5],[2]]
 => [1,4] => ([(3,4)],5)
 => ? = -1 - 1
[3,2]
 => [[1,2,5],[3,4]]
 => [2,3] => ([(2,4),(3,4)],5)
 => ? = 0 - 1
[3,1,1]
 => [[1,4,5],[2],[3]]
 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ? = 1 - 1
[2,2,1]
 => [[1,3],[2,5],[4]]
 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
[2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 - 1
[1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 5 - 1
[6]
 => [[1,2,3,4,5,6]]
 => [6] => ([],6)
 => ? = -4 - 1
[5,1]
 => [[1,3,4,5,6],[2]]
 => [1,5] => ([(4,5)],6)
 => ? = -2 - 1
[4,2]
 => [[1,2,5,6],[3,4]]
 => [2,4] => ([(3,5),(4,5)],6)
 => ? = -1 - 1
[4,1,1]
 => [[1,4,5,6],[2],[3]]
 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => ? = 0 - 1
[3,3]
 => [[1,2,3],[4,5,6]]
 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ? = 0 - 1
[3,2,1]
 => [[1,3,6],[2,5],[4]]
 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
[3,1,1,1]
 => [[1,5,6],[2],[3],[4]]
 => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
[2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
[2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 - 1
[2,1,1,1,1]
 => [[1,6],[2],[3],[4],[5]]
 => [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 - 1
[1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5 = 6 - 1
[5,2]
 => [[1,2,5,6,7],[3,4]]
 => [2,5] => ([(4,6),(5,6)],7)
 => ? = -2 - 1
[4,3]
 => [[1,2,3,7],[4,5,6]]
 => [3,4] => ([(3,6),(4,6),(5,6)],7)
 => ? = -1 - 1
[4,2,1]
 => [[1,3,6,7],[2,5],[4]]
 => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0 - 1
[3,3,1]
 => [[1,3,4],[2,6,7],[5]]
 => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[3,2,2]
 => [[1,2,7],[3,4],[5,6]]
 => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[3,2,1,1]
 => [[1,4,7],[2,6],[3],[5]]
 => [1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 - 1
[2,2,2,1]
 => [[1,3],[2,5],[4,7],[6]]
 => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 - 1
[2,2,1,1,1]
 => [[1,5],[2,7],[3],[4],[6]]
 => [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)
 => ? = 4 - 1
[5,3]
 => [[1,2,3,7,8],[4,5,6]]
 => [3,5] => ([(4,7),(5,7),(6,7)],8)
 => ? = -2 - 1
[4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
 => ? = -1 - 1
[4,3,1]
 => [[1,3,4,8],[2,6,7],[5]]
 => [1,3,4] => ([(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 0 - 1
[3,3,2]
 => [[1,2,5],[3,4,8],[6,7]]
 => [2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1 - 1
[3,3,1,1]
 => [[1,4,5],[2,7,8],[3],[6]]
 => [1,1,3,3] => ([(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 - 1
[3,2,2,1]
 => [[1,3,8],[2,5],[4,7],[6]]
 => [1,2,2,3] => ([(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 - 1
[2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 - 1
[2,2,2,1,1]
 => [[1,4],[2,6],[3,8],[5],[7]]
 => [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)
 => ? = 4 - 1
[5,4]
 => [[1,2,3,4,9],[5,6,7,8]]
 => [4,5] => ([(4,8),(5,8),(6,8),(7,8)],9)
 => ? = -2 - 1
[4,4,1]
 => [[1,3,4,5],[2,7,8,9],[6]]
 => [1,4,4] => ([(3,8),(4,8),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 0 - 1
[3,3,3]
 => [[1,2,3],[4,5,6],[7,8,9]]
 => [3,3,3] => ([(2,8),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 1 - 1
[3,3,2,1]
 => [[1,3,6],[2,5,9],[4,8],[7]]
 => [1,2,3,3] => ([(2,8),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2 - 1
[2,2,2,2,1]
 => [[1,3],[2,5],[4,7],[6,9],[8]]
 => [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)
 => ? = 4 - 1
[5,5]
 => [[1,2,3,4,5],[6,7,8,9,10]]
 => [5,5] => ([(4,9),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = -2 - 1
[3,3,3,1]
 => [[1,3,4],[2,6,7],[5,9,10],[8]]
 => [1,3,3,3] => ([(2,9),(3,9),(4,8),(4,9),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 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].