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 A008307 Table T(n,k) giving number of permutations of [1..n] with order dividing k, read by antidiagonals. 18
 1, 1, 1, 1, 2, 1, 1, 4, 1, 1, 1, 10, 3, 2, 1, 1, 26, 9, 4, 1, 1, 1, 76, 21, 16, 1, 2, 1, 1, 232, 81, 56, 1, 6, 1, 1, 1, 764, 351, 256, 25, 18, 1, 2, 1, 1, 2620, 1233, 1072, 145, 66, 1, 4, 1, 1, 1, 9496, 5769, 6224, 505, 396, 1, 16, 3, 2, 1, 1, 35696, 31041, 33616, 1345, 2052, 1, 56, 9, 4, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS Solutions to x^k = 1 in Symm_n (the symmetric group of degree n). REFERENCES L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 257. J. D. Dixon, B. Mortimer, Permutation Groups, Springer (1996), Exercise 1.2.13. LINKS Alois P. Heinz, Antidiagonals n = 1..141, flattened M. B. Kutler, C. R. Vinroot, On q-Analogs of Recursions for the Number of Involutions and Prime Order Elements in Symmetric Groups, JIS 13 (2010) #10.3.6, eq (5) for primes k. FORMULA T(n+1,k) = Sum_{d|k} (n)_(d-1)*T(n-d+1,k), where (n)_i = n!/(n - i)! = n*(n - 1)*(n - 2)*...*(n - i + 1) is the falling factorial. E.g.f. for n-th row: Sum_{n>=0} T(n,k)*t^n/n! = exp(Sum_{d|k} t^d/d). EXAMPLE Array begins:   1,   1,    1,    1,    1,     1,    1,     1, ...   1,   2,    1,    2,    1,     2,    1,     2, ...   1,   4,    3,    4,    1,     6,    1,     4, ...   1,  10,    9,   16,    1,    18,    1,    16, ...   1,  26,   21,   56,   25,    66,    1,    56, ...   1,  76,   81,  256,  145,   396,    1,   256, ...   1, 232,  351, 1072,  505,  2052,  721,  1072, ...   1, 764, 1233, 6224, 1345, 12636, 5761, 11264, ... MAPLE A:= proc(n, k) option remember; `if`(n<0, 0, `if`(n=0, 1,        add(mul(n-i, i=1..j-1)*A(n-j, k), j=numtheory[divisors](k))))     end: seq(seq(A(1+d-k, k), k=1..d), d=1..12); # Alois P. Heinz, Feb 14 2013 # alternative A008307 := proc(n, m)     local x, d ;     add(x^d/d, d=numtheory[divisors](m)) ;     exp(%) ;     coeftayl(%, x=0, n) ;     %*n! ; end proc: seq(seq(A008307(1+d-k, k), k=1..d), d=1..12) ; # R. J. Mathar, Apr 30 2017 MATHEMATICA t[n_ /; n >= 0, k_ /; k >= 0] := t[n, k] = Sum[(n!/(n - d + 1)!)*t[n - d, k], {d, Divisors[k]}]; t[_, _] = 1; Flatten[ Table[ t[n - k, k], {n, 0, 12}, {k, 1, n}]] (* Jean-François Alcover, Dec 12 2011, after given formula *) CROSSREFS Rows give A000034, A284517, A284518. Columns give A000012, A000085, A001470, A001472, A052501, A053496, A053497, A053498, A053499, A053500, A053501, A053502, A053503, A053504, A053505. Main diagonal gives A074759. - Alois P. Heinz, Feb 14 2013 Sequence in context: A294587 A064645 A285425 * A249694 A192005 A205592 Adjacent sequences:  A008304 A008305 A008306 * A008308 A008309 A008310 KEYWORD nonn,tabl,easy,look,nice AUTHOR EXTENSIONS More terms from Vladeta Jovovic, Apr 13 2001 STATUS approved

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Last modified January 22 07:39 EST 2020. Contains 331139 sequences. (Running on oeis4.)