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A290095
a(n) = A275725(A060126(n)); prime factorization encodings of cycle-polynomials computed for finite permutations listed in reversed colexicographic ordering.
8
2, 4, 18, 8, 8, 12, 150, 100, 54, 16, 16, 24, 54, 16, 90, 40, 36, 16, 16, 24, 40, 60, 16, 36, 1470, 980, 882, 392, 392, 588, 750, 500, 162, 32, 32, 48, 162, 32, 270, 80, 108, 32, 32, 48, 80, 120, 32, 72, 750, 500, 162, 32, 32, 48, 1050, 700, 378, 112, 112, 168, 450, 200, 162, 32, 32, 72, 200, 300, 32, 48, 108, 32, 162, 32, 270, 80, 108, 32, 378, 112, 630, 280
OFFSET
0,1
COMMENTS
In this context "cycle-polynomials" are single-variable polynomials where the coefficients (encoded with the exponents of prime factorization of n) are equal to the lengths of cycles in the permutation listed with index n in table A055089 (A195663). See the examples.
FORMULA
a(n) = A275725(A060126(n)).
Other identities:
A046523(a(n)) = A290096(n).
A056170(a(n)) = A055090(n).
A046660(a(n)) = A055091(n).
A072411(a(n)) = A055092(n).
A275812(a(n)) = A055093(n).
EXAMPLE
Consider the first eight permutations (indices 0-7) listed in A055089:
1 [Only the first 1-cycle explicitly listed thus a(0) = 2^1 = 2]
2,1 [One transposition (2-cycle) in beginning, thus a(1) = 2^2 = 4]
1,3,2 [One fixed element in beginning, then transposition, thus a(2) = 2^1 * 3^2 = 18]
3,1,2 [One 3-cycle, thus a(3) = 2^3 = 8]
2,3,1 [One 3-cycle, thus a(4) = 2^3 = 8]
3,2,1 [One transposition jumping over a fixed element, a(5) = 2^2 * 3^1 = 12]
1,2,4,3 [Two 1-cycles, then a 2-cycle, thus a(6) = 2^1 * 3^1 * 5^2 = 150].
2,1,4,3 [Two 2-cycles, not crossed, thus a(7) = 2^2 * 5^2 = 100].
CROSSREFS
KEYWORD
nonn
AUTHOR
Antti Karttunen, Aug 17 2017
STATUS
approved