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A162663
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Table by antidiagonals, T(n,k) is the number of partitions of 1..(nk) that are invariant under a permutation consisting of n k-cycles.
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6
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1, 1, 1, 1, 2, 2, 1, 2, 7, 5, 1, 3, 8, 31, 15, 1, 2, 16, 42, 164, 52, 1, 4, 10, 111, 268, 999, 203, 1, 2, 28, 70, 931, 1994, 6841, 877, 1, 4, 12, 258, 602, 9066, 16852, 51790, 4140, 1, 3, 31, 106, 2892, 6078, 99925, 158778, 428131, 21147, 1, 4, 22, 329, 1144, 37778
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OFFSET
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1,5
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COMMENTS
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The upper left corner of the array is T(0,1).
Wlog, the permutation can be taken to be (1 2 ... k) (k+1 k+2 ... 2k) ... ((n-1)k+1 (n-1)k+2 ... nk).
Note that it is the partition that is invariant, not the individual parts. Thus for n=k=2 with permutation (1 2)(3 4), the partition 1,3|2,4 is counted; it maps to 2,4|1,3, which is the same partition.
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LINKS
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Franklin T. Adams-Watters, Table of n, a(n) for n=0..210
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FORMULA
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E.g.f for column k is exp(sum( d|k, (exp(d*x) - 1) / d).
Equivalently, column k is the exponential transform of a(n) = sum( d|k, d^(n-1)); this represents a set of n k-cycles, each repeating the same d elements (parts), but starting in different places.
T(n,k) = sum( P a partition of n, SP(P) * product( (sigma_{i-1}(k))^(P(i)-1) )), where SP is A036040 or A080575, and P(i) is the number of parts in P of size i.
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EXAMPLE
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The table starts:
1,1,1,1,1
1,2,2,3,2
2,7,8,16,10
5,31,42,111,70
15,164,268,931,602
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MATHEMATICA
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max = 11; ClearAll[col]; col[k_] := col[k] = CoefficientList[ Series[ Exp[ Sum[ (Exp[d*x] - 1)/d, {d, Divisors[k]}]], {x, 0, max}], x]*Range[0, max]!; t[n_, k_] := col[k][[n]]; Flatten[ Table[ t[n-k+1, k], {n, 1, max}, {k, n, 1, -1}] ] (* Jean-François Alcover, Aug 08 2012, after e.g.f. *)
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PROG
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(PARI) amat(n, m)=local(r); r=matrix(n, m, i, j, 1); for(k=1, n-1, for(j=1, m, r[k+1, j]=sum (i=1, k, binomial(k-1, i-1)*sumdiv(j, d, r[k-i+1, j]*d^(i-1))))); r
acol(n, k)=local(fn); fn=exp(sumdiv(k, d, (exp(d*x+x*O(x^n))-1)/d)); vector(n+ 1, i, polcoeff(fn, i-1)*(i-1)!)
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CROSSREFS
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Columns: A000110, A002872, A002874, A141003, A036075, A141004, A036077, A141005, A141006, A141007, A036081, A141008, A141009, A141010, A141011.
Rows: A000012, A000005, A162664, A162665.
Cf. A084423, A036040, A080575.
Sequence in context: A091562 A106585 A057227 * A005007 A188792 A192395
Adjacent sequences: A162660 A162661 A162662 * A162664 A162665 A162666
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KEYWORD
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nice,nonn,tabl
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AUTHOR
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Franklin T. Adams-Watters, Jul 09 2009
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STATUS
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approved
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