Your data matches 6 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000454
Mapping
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => {{1}}
 => [1] => ([],1)
 => 0
[1,2] => {{1},{2}}
 => [1,2] => ([],2)
 => 0
[2,1] => {{1,2}}
 => [2,1] => ([(0,1)],2)
 => 1
[1,2,3] => {{1},{2},{3}}
 => [1,2,3] => ([],3)
 => 0
[1,3,2] => {{1},{2,3}}
 => [1,3,2] => ([(1,2)],3)
 => 1
[2,1,3] => {{1,2},{3}}
 => [2,1,3] => ([(1,2)],3)
 => 1
[3,1,2] => {{1,3},{2}}
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[3,2,1] => {{1,3},{2}}
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,2,3,4] => {{1},{2},{3},{4}}
 => [1,2,3,4] => ([],4)
 => 0
[1,2,4,3] => {{1},{2},{3,4}}
 => [1,2,4,3] => ([(2,3)],4)
 => 1
[1,3,2,4] => {{1},{2,3},{4}}
 => [1,3,2,4] => ([(2,3)],4)
 => 1
[1,4,2,3] => {{1},{2,4},{3}}
 => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[1,4,3,2] => {{1},{2,4},{3}}
 => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[2,1,3,4] => {{1,2},{3},{4}}
 => [2,1,3,4] => ([(2,3)],4)
 => 1
[2,1,4,3] => {{1,2},{3,4}}
 => [2,1,4,3] => ([(0,3),(1,2)],4)
 => 1
[3,1,2,4] => {{1,3},{2},{4}}
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 2
[3,2,1,4] => {{1,3},{2},{4}}
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 2
[3,4,1,2] => {{1,3},{2,4}}
 => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[3,4,2,1] => {{1,3},{2,4}}
 => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[4,3,1,2] => {{1,4},{2,3}}
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[4,3,2,1] => {{1,4},{2,3}}
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => ([],5)
 => 0
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
 => [1,2,3,5,4] => ([(3,4)],5)
 => 1
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
 => [1,2,4,3,5] => ([(3,4)],5)
 => 1
[1,2,5,3,4] => {{1},{2},{3,5},{4}}
 => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 2
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
 => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 2
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
 => [1,3,2,4,5] => ([(3,4)],5)
 => 1
[1,3,2,5,4] => {{1},{2,3},{4,5}}
 => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => 1
[1,4,2,3,5] => {{1},{2,4},{3},{5}}
 => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => 2
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
 => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => 2
[1,4,5,2,3] => {{1},{2,4},{3,5}}
 => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[1,4,5,3,2] => {{1},{2,4},{3,5}}
 => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[1,5,4,2,3] => {{1},{2,5},{3,4}}
 => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,5,4,3,2] => {{1},{2,5},{3,4}}
 => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[2,1,3,4,5] => {{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => ([(3,4)],5)
 => 1
[2,1,3,5,4] => {{1,2},{3},{4,5}}
 => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => 1
[2,1,4,3,5] => {{1,2},{3,4},{5}}
 => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => 1
[2,1,5,3,4] => {{1,2},{3,5},{4}}
 => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => 2
[2,1,5,4,3] => {{1,2},{3,5},{4}}
 => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => 2
[2,3,4,5,1] => {{1,2,3,4,5}}
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[3,1,2,4,5] => {{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => 2
[3,1,2,5,4] => {{1,3},{2},{4,5}}
 => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => 2
[3,2,1,4,5] => {{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => 2
[3,2,1,5,4] => {{1,3},{2},{4,5}}
 => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => 2
[3,4,1,2,5] => {{1,3},{2,4},{5}}
 => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[3,4,2,1,5] => {{1,3},{2,4},{5}}
 => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[3,4,5,1,2] => {{1,3,5},{2,4}}
 => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[3,4,5,2,1] => {{1,3,5},{2,4}}
 => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[4,3,1,2,5] => {{1,4},{2,3},{5}}
 => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[4,3,2,1,5] => {{1,4},{2,3},{5}}
 => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 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.
Matching statistic: St000983
(load all 2 compositions to match this statistic)
Mapping
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00158: Binary words —alternating inverse⟶ Binary words
St000983: Binary words ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 57%
Values
[1] => [1,0]
 => 10 => 11 => 1 = 0 + 1
[1,2] => [1,0,1,0]
 => 1010 => 1111 => 1 = 0 + 1
[2,1] => [1,1,0,0]
 => 1100 => 1001 => 2 = 1 + 1
[1,2,3] => [1,0,1,0,1,0]
 => 101010 => 111111 => 1 = 0 + 1
[1,3,2] => [1,0,1,1,0,0]
 => 101100 => 111001 => 2 = 1 + 1
[2,1,3] => [1,1,0,0,1,0]
 => 110010 => 100111 => 2 = 1 + 1
[3,1,2] => [1,1,1,0,0,0]
 => 111000 => 101101 => 3 = 2 + 1
[3,2,1] => [1,1,1,0,0,0]
 => 111000 => 101101 => 3 = 2 + 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 10101010 => 11111111 => 1 = 0 + 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
 => 10101100 => 11111001 => 2 = 1 + 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
 => 10110010 => 11100111 => 2 = 1 + 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
 => 10111000 => 11101101 => 3 = 2 + 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 10111000 => 11101101 => 3 = 2 + 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
 => 11001010 => 10011111 => 2 = 1 + 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 11001100 => 10011001 => 2 = 1 + 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
 => 11100010 => 10110111 => 3 = 2 + 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 11100010 => 10110111 => 3 = 2 + 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 11101000 => 10111101 => 3 = 2 + 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 11101000 => 10111101 => 3 = 2 + 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
 => 11110000 => 10100101 => 4 = 3 + 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 11110000 => 10100101 => 4 = 3 + 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => 1010101010 => 1111111111 => ? = 0 + 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => 1010101100 => 1111111001 => ? = 1 + 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => 1010110010 => 1111100111 => ? = 1 + 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => 1010111000 => 1111101101 => ? = 2 + 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => 1010111000 => 1111101101 => ? = 2 + 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => 1011001010 => 1110011111 => ? = 1 + 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => 1011001100 => 1110011001 => ? = 1 + 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => 1011100010 => 1110110111 => ? = 2 + 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => 1011100010 => 1110110111 => ? = 2 + 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => 1110111101 => ? = 2 + 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => 1110111101 => ? = 2 + 1
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => 1110100101 => ? = 3 + 1
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => 1110100101 => ? = 3 + 1
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => 1100101010 => 1001111111 => ? = 1 + 1
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
 => 1100101100 => 1001111001 => ? = 1 + 1
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
 => 1100110010 => 1001100111 => ? = 1 + 1
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
 => 1100111000 => 1001101101 => ? = 2 + 1
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1100111000 => 1001101101 => ? = 2 + 1
[2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => 1101010100 => 1000000001 => ? = 2 + 1
[3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => 1110001010 => 1011011111 => ? = 2 + 1
[3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => 1011011001 => ? = 2 + 1
[3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => 1110001010 => 1011011111 => ? = 2 + 1
[3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => 1011011001 => ? = 2 + 1
[3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
 => 1110100010 => 1011110111 => ? = 2 + 1
[3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0]
 => 1110100010 => 1011110111 => ? = 2 + 1
[3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
 => 1110101000 => 1011111101 => ? = 3 + 1
[3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => 1110101000 => 1011111101 => ? = 3 + 1
[4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
 => 1111000010 => 1010010111 => ? = 3 + 1
[4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => 1111000010 => 1010010111 => ? = 3 + 1
[5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => 1111100000 => 1010110101 => ? = 3 + 1
[5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
 => 1111100000 => 1010110101 => ? = 3 + 1
[5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
 => 1111100000 => 1010110101 => ? = 3 + 1
[5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
 => 1111100000 => 1010110101 => ? = 3 + 1
[5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => 1111100000 => 1010110101 => ? = 3 + 1
[5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
 => 1111100000 => 1010110101 => ? = 3 + 1
[5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => 1111100000 => 1010110101 => ? = 3 + 1
[5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => 1111100000 => 1010110101 => ? = 3 + 1
[5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => 1111100000 => 1010110101 => ? = 4 + 1
[5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => 1111100000 => 1010110101 => ? = 4 + 1
[5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => 1111100000 => 1010110101 => ? = 4 + 1
[5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => 1111100000 => 1010110101 => ? = 4 + 1
[5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => 1111100000 => 1010110101 => ? = 4 + 1
[5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 1111100000 => 1010110101 => ? = 4 + 1
[1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 101010101010 => 111111111111 => ? = 0 + 1
[1,2,3,4,6,5] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 101010101100 => 111111111001 => ? = 1 + 1
[1,2,3,5,4,6] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 101010110010 => 111111100111 => ? = 1 + 1
[1,2,3,6,4,5] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 101010111000 => 111111101101 => ? = 2 + 1
[1,2,3,6,5,4] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 101010111000 => 111111101101 => ? = 2 + 1
[1,2,4,3,5,6] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => 101011001010 => 111110011111 => ? = 1 + 1
[1,2,4,3,6,5] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 101011001100 => 111110011001 => ? = 1 + 1
Description
The length of the longest alternating subword. This is the length of the longest consecutive subword of the form $010...$ or of the form $101...$.
Matching statistic: St000381
Mapping
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 57%
Values
[1] => [1,0]
 => 10 => [1,2] => 2 = 0 + 2
[1,2] => [1,0,1,0]
 => 1010 => [1,2,2] => 2 = 0 + 2
[2,1] => [1,1,0,0]
 => 1100 => [1,1,3] => 3 = 1 + 2
[1,2,3] => [1,0,1,0,1,0]
 => 101010 => [1,2,2,2] => 2 = 0 + 2
[1,3,2] => [1,0,1,1,0,0]
 => 101100 => [1,2,1,3] => 3 = 1 + 2
[2,1,3] => [1,1,0,0,1,0]
 => 110010 => [1,1,3,2] => 3 = 1 + 2
[3,1,2] => [1,1,1,0,0,0]
 => 111000 => [1,1,1,4] => 4 = 2 + 2
[3,2,1] => [1,1,1,0,0,0]
 => 111000 => [1,1,1,4] => 4 = 2 + 2
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 10101010 => [1,2,2,2,2] => 2 = 0 + 2
[1,2,4,3] => [1,0,1,0,1,1,0,0]
 => 10101100 => [1,2,2,1,3] => 3 = 1 + 2
[1,3,2,4] => [1,0,1,1,0,0,1,0]
 => 10110010 => [1,2,1,3,2] => 3 = 1 + 2
[1,4,2,3] => [1,0,1,1,1,0,0,0]
 => 10111000 => [1,2,1,1,4] => 4 = 2 + 2
[1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 10111000 => [1,2,1,1,4] => 4 = 2 + 2
[2,1,3,4] => [1,1,0,0,1,0,1,0]
 => 11001010 => [1,1,3,2,2] => 3 = 1 + 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 11001100 => [1,1,3,1,3] => 3 = 1 + 2
[3,1,2,4] => [1,1,1,0,0,0,1,0]
 => 11100010 => [1,1,1,4,2] => 4 = 2 + 2
[3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 11100010 => [1,1,1,4,2] => 4 = 2 + 2
[3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 11101000 => [1,1,1,2,4] => 4 = 2 + 2
[3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 11101000 => [1,1,1,2,4] => 4 = 2 + 2
[4,3,1,2] => [1,1,1,1,0,0,0,0]
 => 11110000 => [1,1,1,1,5] => 5 = 3 + 2
[4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 11110000 => [1,1,1,1,5] => 5 = 3 + 2
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => 1010101010 => [1,2,2,2,2,2] => ? = 0 + 2
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => 1010101100 => [1,2,2,2,1,3] => ? = 1 + 2
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => 1010110010 => [1,2,2,1,3,2] => ? = 1 + 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => 1010111000 => [1,2,2,1,1,4] => ? = 2 + 2
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => 1010111000 => [1,2,2,1,1,4] => ? = 2 + 2
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => 1011001010 => [1,2,1,3,2,2] => ? = 1 + 2
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => 1011001100 => [1,2,1,3,1,3] => ? = 1 + 2
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => 1011100010 => [1,2,1,1,4,2] => ? = 2 + 2
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => 1011100010 => [1,2,1,1,4,2] => ? = 2 + 2
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => [1,2,1,1,2,4] => ? = 2 + 2
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => [1,2,1,1,2,4] => ? = 2 + 2
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => [1,2,1,1,1,5] => ? = 3 + 2
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => [1,2,1,1,1,5] => ? = 3 + 2
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => 1100101010 => [1,1,3,2,2,2] => ? = 1 + 2
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
 => 1100101100 => [1,1,3,2,1,3] => ? = 1 + 2
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
 => 1100110010 => [1,1,3,1,3,2] => ? = 1 + 2
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
 => 1100111000 => [1,1,3,1,1,4] => ? = 2 + 2
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1100111000 => [1,1,3,1,1,4] => ? = 2 + 2
[2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => 1101010100 => [1,1,2,2,2,3] => ? = 2 + 2
[3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => 1110001010 => [1,1,1,4,2,2] => ? = 2 + 2
[3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => [1,1,1,4,1,3] => ? = 2 + 2
[3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => 1110001010 => [1,1,1,4,2,2] => ? = 2 + 2
[3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => [1,1,1,4,1,3] => ? = 2 + 2
[3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
 => 1110100010 => [1,1,1,2,4,2] => ? = 2 + 2
[3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0]
 => 1110100010 => [1,1,1,2,4,2] => ? = 2 + 2
[3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
 => 1110101000 => [1,1,1,2,2,4] => ? = 3 + 2
[3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => 1110101000 => [1,1,1,2,2,4] => ? = 3 + 2
[4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
 => 1111000010 => [1,1,1,1,5,2] => ? = 3 + 2
[4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => 1111000010 => [1,1,1,1,5,2] => ? = 3 + 2
[5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => 1111100000 => [1,1,1,1,1,6] => ? = 3 + 2
[5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
 => 1111100000 => [1,1,1,1,1,6] => ? = 3 + 2
[5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
 => 1111100000 => [1,1,1,1,1,6] => ? = 3 + 2
[5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
 => 1111100000 => [1,1,1,1,1,6] => ? = 3 + 2
[5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => 1111100000 => [1,1,1,1,1,6] => ? = 3 + 2
[5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
 => 1111100000 => [1,1,1,1,1,6] => ? = 3 + 2
[5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => 1111100000 => [1,1,1,1,1,6] => ? = 3 + 2
[5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => 1111100000 => [1,1,1,1,1,6] => ? = 3 + 2
[5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => 1111100000 => [1,1,1,1,1,6] => ? = 4 + 2
[5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => 1111100000 => [1,1,1,1,1,6] => ? = 4 + 2
[5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => 1111100000 => [1,1,1,1,1,6] => ? = 4 + 2
[5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => 1111100000 => [1,1,1,1,1,6] => ? = 4 + 2
[5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => 1111100000 => [1,1,1,1,1,6] => ? = 4 + 2
[5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 1111100000 => [1,1,1,1,1,6] => ? = 4 + 2
[1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 101010101010 => [1,2,2,2,2,2,2] => ? = 0 + 2
[1,2,3,4,6,5] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 101010101100 => [1,2,2,2,2,1,3] => ? = 1 + 2
[1,2,3,5,4,6] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 101010110010 => [1,2,2,2,1,3,2] => ? = 1 + 2
[1,2,3,6,4,5] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 101010111000 => [1,2,2,2,1,1,4] => ? = 2 + 2
[1,2,3,6,5,4] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 101010111000 => [1,2,2,2,1,1,4] => ? = 2 + 2
[1,2,4,3,5,6] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => 101011001010 => [1,2,2,1,3,2,2] => ? = 1 + 2
[1,2,4,3,6,5] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 101011001100 => [1,2,2,1,3,1,3] => ? = 1 + 2
Description
The largest part of an integer composition.
Matching statistic: St001526
(load all 3 compositions to match this statistic)
Mapping
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001526: Dyck paths ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 57%
Values
[1] => [1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2 = 0 + 2
[1,2] => [1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 2 = 0 + 2
[2,1] => [1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 3 = 1 + 2
[1,2,3] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 2 = 0 + 2
[1,3,2] => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => 3 = 1 + 2
[2,1,3] => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 3 = 1 + 2
[3,1,2] => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 4 = 2 + 2
[3,2,1] => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 4 = 2 + 2
[1,2,3,4] => [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 = 0 + 2
[1,2,4,3] => [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]
 => 3 = 1 + 2
[1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 3 = 1 + 2
[1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 4 = 2 + 2
[1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 4 = 2 + 2
[2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 3 = 1 + 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 3 = 1 + 2
[3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 4 = 2 + 2
[3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 4 = 2 + 2
[3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 4 = 2 + 2
[3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 4 = 2 + 2
[4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 5 = 3 + 2
[4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 5 = 3 + 2
[1,2,3,4,5] => [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]
 => ? = 0 + 2
[1,2,3,5,4] => [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 + 2
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => ? = 1 + 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => ? = 2 + 2
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => ? = 2 + 2
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => ? = 1 + 2
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => ? = 1 + 2
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0,1,0]
 => ? = 2 + 2
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0,1,0]
 => ? = 2 + 2
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => ? = 2 + 2
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => ? = 2 + 2
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => ? = 3 + 2
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => ? = 3 + 2
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => ? = 1 + 2
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => ? = 1 + 2
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => ? = 1 + 2
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => ? = 2 + 2
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => ? = 2 + 2
[2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 2 + 2
[3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => ? = 2 + 2
[3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => ? = 2 + 2
[3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => ? = 2 + 2
[3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => ? = 2 + 2
[3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => ? = 2 + 2
[3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => ? = 2 + 2
[3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => ? = 3 + 2
[3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => ? = 3 + 2
[4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => ? = 3 + 2
[4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => ? = 3 + 2
[5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => ? = 3 + 2
[5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => ? = 3 + 2
[5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => ? = 3 + 2
[5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => ? = 3 + 2
[5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => ? = 3 + 2
[5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => ? = 3 + 2
[5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => ? = 3 + 2
[5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => ? = 3 + 2
[5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => ? = 4 + 2
[5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => ? = 4 + 2
[5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => ? = 4 + 2
[5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => ? = 4 + 2
[5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => ? = 4 + 2
[5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => ? = 4 + 2
[1,2,3,4,5,6] => [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]
 => ? = 0 + 2
[1,2,3,4,6,5] => [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]
 => ? = 1 + 2
[1,2,3,5,4,6] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => ? = 1 + 2
[1,2,3,6,4,5] => [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]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => ? = 2 + 2
[1,2,3,6,5,4] => [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]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => ? = 2 + 2
[1,2,4,3,5,6] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => ? = 1 + 2
[1,2,4,3,6,5] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => ? = 1 + 2
Description
The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001589
Mapping
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St001589: Perfect matchings ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 57%
Values
[1] => [1,0]
 => [1,1,0,0]
 => [(1,4),(2,3)]
 => 2 = 0 + 2
[1,2] => [1,0,1,0]
 => [1,1,0,1,0,0]
 => [(1,6),(2,3),(4,5)]
 => 2 = 0 + 2
[2,1] => [1,1,0,0]
 => [1,1,1,0,0,0]
 => [(1,6),(2,5),(3,4)]
 => 3 = 1 + 2
[1,2,3] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [(1,8),(2,3),(4,5),(6,7)]
 => 2 = 0 + 2
[1,3,2] => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [(1,8),(2,3),(4,7),(5,6)]
 => 3 = 1 + 2
[2,1,3] => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [(1,8),(2,5),(3,4),(6,7)]
 => 3 = 1 + 2
[3,1,2] => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [(1,8),(2,7),(3,6),(4,5)]
 => 4 = 2 + 2
[3,2,1] => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [(1,8),(2,7),(3,6),(4,5)]
 => 4 = 2 + 2
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [(1,10),(2,3),(4,5),(6,7),(8,9)]
 => 2 = 0 + 2
[1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [(1,10),(2,3),(4,5),(6,9),(7,8)]
 => 3 = 1 + 2
[1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [(1,10),(2,3),(4,7),(5,6),(8,9)]
 => 3 = 1 + 2
[1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [(1,10),(2,3),(4,9),(5,8),(6,7)]
 => 4 = 2 + 2
[1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [(1,10),(2,3),(4,9),(5,8),(6,7)]
 => 4 = 2 + 2
[2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [(1,10),(2,5),(3,4),(6,7),(8,9)]
 => 3 = 1 + 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [(1,10),(2,5),(3,4),(6,9),(7,8)]
 => 3 = 1 + 2
[3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [(1,10),(2,7),(3,6),(4,5),(8,9)]
 => 4 = 2 + 2
[3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [(1,10),(2,7),(3,6),(4,5),(8,9)]
 => 4 = 2 + 2
[3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [(1,10),(2,9),(3,8),(4,5),(6,7)]
 => 4 = 2 + 2
[3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [(1,10),(2,9),(3,8),(4,5),(6,7)]
 => 4 = 2 + 2
[4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [(1,10),(2,9),(3,8),(4,7),(5,6)]
 => 5 = 3 + 2
[4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [(1,10),(2,9),(3,8),(4,7),(5,6)]
 => 5 = 3 + 2
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [(1,12),(2,3),(4,5),(6,7),(8,9),(10,11)]
 => ? = 0 + 2
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [(1,12),(2,3),(4,5),(6,7),(8,11),(9,10)]
 => ? = 1 + 2
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [(1,12),(2,3),(4,5),(6,9),(7,8),(10,11)]
 => ? = 1 + 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [(1,12),(2,3),(4,5),(6,11),(7,10),(8,9)]
 => ? = 2 + 2
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [(1,12),(2,3),(4,5),(6,11),(7,10),(8,9)]
 => ? = 2 + 2
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [(1,12),(2,3),(4,7),(5,6),(8,9),(10,11)]
 => ? = 1 + 2
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [(1,12),(2,3),(4,7),(5,6),(8,11),(9,10)]
 => ? = 1 + 2
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [(1,12),(2,3),(4,9),(5,8),(6,7),(10,11)]
 => ? = 2 + 2
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [(1,12),(2,3),(4,9),(5,8),(6,7),(10,11)]
 => ? = 2 + 2
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [(1,12),(2,3),(4,11),(5,10),(6,7),(8,9)]
 => ? = 2 + 2
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [(1,12),(2,3),(4,11),(5,10),(6,7),(8,9)]
 => ? = 2 + 2
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [(1,12),(2,3),(4,11),(5,10),(6,9),(7,8)]
 => ? = 3 + 2
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [(1,12),(2,3),(4,11),(5,10),(6,9),(7,8)]
 => ? = 3 + 2
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [(1,12),(2,5),(3,4),(6,7),(8,9),(10,11)]
 => ? = 1 + 2
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [(1,12),(2,5),(3,4),(6,7),(8,11),(9,10)]
 => ? = 1 + 2
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [(1,12),(2,5),(3,4),(6,9),(7,8),(10,11)]
 => ? = 1 + 2
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [(1,12),(2,5),(3,4),(6,11),(7,10),(8,9)]
 => ? = 2 + 2
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [(1,12),(2,5),(3,4),(6,11),(7,10),(8,9)]
 => ? = 2 + 2
[2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [(1,12),(2,11),(3,4),(5,6),(7,8),(9,10)]
 => ? = 2 + 2
[3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [(1,12),(2,7),(3,6),(4,5),(8,9),(10,11)]
 => ? = 2 + 2
[3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [(1,12),(2,7),(3,6),(4,5),(8,11),(9,10)]
 => ? = 2 + 2
[3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [(1,12),(2,7),(3,6),(4,5),(8,9),(10,11)]
 => ? = 2 + 2
[3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [(1,12),(2,7),(3,6),(4,5),(8,11),(9,10)]
 => ? = 2 + 2
[3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [(1,12),(2,9),(3,8),(4,5),(6,7),(10,11)]
 => ? = 2 + 2
[3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [(1,12),(2,9),(3,8),(4,5),(6,7),(10,11)]
 => ? = 2 + 2
[3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [(1,12),(2,11),(3,10),(4,5),(6,7),(8,9)]
 => ? = 3 + 2
[3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [(1,12),(2,11),(3,10),(4,5),(6,7),(8,9)]
 => ? = 3 + 2
[4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [(1,12),(2,9),(3,8),(4,7),(5,6),(10,11)]
 => ? = 3 + 2
[4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [(1,12),(2,9),(3,8),(4,7),(5,6),(10,11)]
 => ? = 3 + 2
[5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7)]
 => ? = 3 + 2
[5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7)]
 => ? = 3 + 2
[5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7)]
 => ? = 3 + 2
[5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7)]
 => ? = 3 + 2
[5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7)]
 => ? = 3 + 2
[5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7)]
 => ? = 3 + 2
[5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7)]
 => ? = 3 + 2
[5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7)]
 => ? = 3 + 2
[5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7)]
 => ? = 4 + 2
[5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7)]
 => ? = 4 + 2
[5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7)]
 => ? = 4 + 2
[5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7)]
 => ? = 4 + 2
[5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7)]
 => ? = 4 + 2
[5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7)]
 => ? = 4 + 2
[1,2,3,4,5,6] => [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]
 => [(1,14),(2,3),(4,5),(6,7),(8,9),(10,11),(12,13)]
 => ? = 0 + 2
[1,2,3,4,6,5] => [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]
 => [(1,14),(2,3),(4,5),(6,7),(8,9),(10,13),(11,12)]
 => ? = 1 + 2
[1,2,3,5,4,6] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
 => [(1,14),(2,3),(4,5),(6,7),(8,11),(9,10),(12,13)]
 => ? = 1 + 2
[1,2,3,6,4,5] => [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]
 => [(1,14),(2,3),(4,5),(6,7),(8,13),(9,12),(10,11)]
 => ? = 2 + 2
[1,2,3,6,5,4] => [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]
 => [(1,14),(2,3),(4,5),(6,7),(8,13),(9,12),(10,11)]
 => ? = 2 + 2
[1,2,4,3,5,6] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
 => [(1,14),(2,3),(4,5),(6,9),(7,8),(10,11),(12,13)]
 => ? = 1 + 2
[1,2,4,3,6,5] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
 => [(1,14),(2,3),(4,5),(6,9),(7,8),(10,13),(11,12)]
 => ? = 1 + 2
Description
The nesting number of a perfect matching. This is the maximal number of chords in the standard representation of a perfect matching that mutually nest.
Matching statistic: St001207
(load all 53 compositions to match this statistic)
Mapping
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 57%
Values
[1] => [1] => ? = 0
[1,2] => [1,2] => 0
[2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => 1
[2,1,3] => [2,1,3] => 1
[3,1,2] => [3,1,2] => 2
[3,2,1] => [3,2,1] => 2
[1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => 1
[1,3,2,4] => [1,3,2,4] => 1
[1,4,2,3] => [1,4,2,3] => 2
[1,4,3,2] => [1,4,3,2] => 2
[2,1,3,4] => [2,1,3,4] => 1
[2,1,4,3] => [2,1,4,3] => 1
[3,1,2,4] => [3,1,2,4] => 2
[3,2,1,4] => [3,2,1,4] => 2
[3,4,1,2] => [2,4,1,3] => 2
[3,4,2,1] => [1,4,3,2] => 2
[4,3,1,2] => [4,3,1,2] => 3
[4,3,2,1] => [4,3,2,1] => 3
[1,2,3,4,5] => [1,2,3,4,5] => ? = 0
[1,2,3,5,4] => [1,2,3,5,4] => ? = 1
[1,2,4,3,5] => [1,2,4,3,5] => ? = 1
[1,2,5,3,4] => [1,2,5,3,4] => ? = 2
[1,2,5,4,3] => [1,2,5,4,3] => ? = 2
[1,3,2,4,5] => [1,3,2,4,5] => ? = 1
[1,3,2,5,4] => [1,3,2,5,4] => ? = 1
[1,4,2,3,5] => [1,4,2,3,5] => ? = 2
[1,4,3,2,5] => [1,4,3,2,5] => ? = 2
[1,4,5,2,3] => [1,3,5,2,4] => ? = 2
[1,4,5,3,2] => [1,2,5,4,3] => ? = 2
[1,5,4,2,3] => [1,5,4,2,3] => ? = 3
[1,5,4,3,2] => [1,5,4,3,2] => ? = 3
[2,1,3,4,5] => [2,1,3,4,5] => ? = 1
[2,1,3,5,4] => [2,1,3,5,4] => ? = 1
[2,1,4,3,5] => [2,1,4,3,5] => ? = 1
[2,1,5,3,4] => [2,1,5,3,4] => ? = 2
[2,1,5,4,3] => [2,1,5,4,3] => ? = 2
[2,3,4,5,1] => [1,2,3,5,4] => ? = 2
[3,1,2,4,5] => [3,1,2,4,5] => ? = 2
[3,1,2,5,4] => [3,1,2,5,4] => ? = 2
[3,2,1,4,5] => [3,2,1,4,5] => ? = 2
[3,2,1,5,4] => [3,2,1,5,4] => ? = 2
[3,4,1,2,5] => [2,4,1,3,5] => ? = 2
[3,4,2,1,5] => [1,4,3,2,5] => ? = 2
[3,4,5,1,2] => [1,3,5,2,4] => ? = 3
[3,4,5,2,1] => [1,2,5,4,3] => ? = 3
[4,3,1,2,5] => [4,3,1,2,5] => ? = 3
[4,3,2,1,5] => [4,3,2,1,5] => ? = 3
[5,1,2,3,4] => [5,1,2,3,4] => ? = 3
[5,1,2,4,3] => [5,1,2,4,3] => ? = 3
[5,1,3,2,4] => [5,1,3,2,4] => ? = 3
[5,1,3,4,2] => [5,1,2,4,3] => ? = 3
[5,2,1,3,4] => [5,2,1,3,4] => ? = 3
[5,2,1,4,3] => [5,2,1,4,3] => ? = 3
[5,2,3,1,4] => [5,1,3,2,4] => ? = 3
[5,2,3,4,1] => [5,1,2,4,3] => ? = 3
[5,4,1,2,3] => [5,4,1,2,3] => ? = 4
[5,4,1,3,2] => [5,4,1,3,2] => ? = 4
[5,4,2,1,3] => [5,4,2,1,3] => ? = 4
[5,4,2,3,1] => [5,4,1,3,2] => ? = 4
[5,4,3,1,2] => [5,4,3,1,2] => ? = 4
[5,4,3,2,1] => [5,4,3,2,1] => ? = 4
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 0
[1,2,3,4,6,5] => [1,2,3,4,6,5] => ? = 1
[1,2,3,5,4,6] => [1,2,3,5,4,6] => ? = 1
[1,2,3,6,4,5] => [1,2,3,6,4,5] => ? = 2
[1,2,3,6,5,4] => [1,2,3,6,5,4] => ? = 2
[1,2,4,3,5,6] => [1,2,4,3,5,6] => ? = 1
Description
The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.