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 A162663 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. 21
 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, 70402, 1224579, 1644732, 3827967, 115975 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS 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. LINKS Franklin T. Adams-Watters and Alois P. Heinz, Antidiagonals n = 0..140, flattened (first 20 antidiagonals from Franklin T. Adams-Watters) T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176. [Annotated, scanned copy] OEIS Wiki, Sorting numbers FORMULA E.g.f. for column k: 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. T(n,k) = Sum_{0<=j<=n-1} A036073(n,j)*k^(n-1-j). - Andrey Zabolotskiy, Oct 22 2017 EXAMPLE 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 MAPLE with(numtheory): A:= proc(n, k) option remember; `if`(n=0, 1, add(binomial(n-1, j-1)        *add(d^(j-1), d=divisors(k))*A(n-j, k), j=1..n))     end: seq(seq(A(n, 1+d-n), n=0..d), d=0..12);  # Alois P. Heinz, Oct 29 2015 MATHEMATICA 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. *) PROG (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)!) CROSSREFS Columns: A000110, A002872, A002874, A141003, A036075, A141004, A036077, A141005, A141006, A141007, A036081, A141008, A141009, A141010, A141011. Rows: A000012, A000005, A162664, A162665. Cf. A084423, A036040, A036073, A080575. Main diagonal gives A293850. Sequence in context: A236144 A226328 A307599 * A005007 A188792 A192395 Adjacent sequences:  A162660 A162661 A162662 * A162664 A162665 A162666 KEYWORD nice,nonn,tabl AUTHOR Franklin T. Adams-Watters, Jul 09 2009 EXTENSIONS Offset set to 0 by Alois P. Heinz, Oct 29 2015 STATUS approved

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Last modified November 30 05:31 EST 2020. Contains 338781 sequences. (Running on oeis4.)