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A053391
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Number of cycle types of direct products of two degree-n permutations.
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0
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1, 1, 2, 5, 10, 27, 43, 118, 183, 414, 700, 1554, 2229, 5002, 7591, 14267, 22378, 42866, 62093, 116639, 170909, 297002, 447730, 765973, 1096141, 1861030, 2707679, 4356383, 6351345, 10173716, 14425510, 22886088
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OFFSET
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0,3
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COMMENTS
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If f is a permutation of A and g is a permutation of B, the direct product of f and g is the permutation of AXB that maps (a, b) to (f(a), g(b)). The cycle type of the direct product is determined by the cycle types of f and g. - David Wasserman, Mar 01 2002
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LINKS
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EXAMPLE
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I will use the notation (i, j, k, ...) for a permutation with i 1-cycles, j 2-cycles, k 3-cycles, etc. and * for direct product. There are 3 cycle types of 3-element permutations: (3), (1, 1) and (0, 0, 1). (3)*(3) has cycle type (9); (3)*(1, 1) has cycle type (3, 3); (3)*(0, 0, 1) has cycle type (0, 0, 3); (1, 1)*(1, 1) has cycle type (4, 1); (1, 1)*(0, 0, 1) has cycle type (0, 0, 1, 0, 0, 1); (0, 0, 1)*(0, 0, 1) has cycle type (0, 0, 3). Since there are 5 distinct answers, a(3) = 5. - David Wasserman, Mar 01 2002
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PROG
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(PARI)
PartProdPoly(p, q)={sum(i=1, #p, sum(j=1, #q, my(g=gcd(p[i], q[j])); g*'x^(p[i]*q[j]/g)))}
a(n)={my(M=Map()); forpart(p=n, forpart(q=n, mapput(M, PartProdPoly(p, q), 1) )); #M} \\ Andrew Howroyd, Mar 20 2018
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CROSSREFS
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A000041 gives the number of cycle types of an n-element permutation.
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KEYWORD
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more,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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