A162663 - OEIS

%I #31 Nov 03 2023 16:16:56

%S 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,

%T 203,1,2,28,70,931,1994,6841,877,1,4,12,258,602,9066,16852,51790,4140,

%U 1,3,31,106,2892,6078,99925,158778,428131,21147,1,4,22,329,1144,37778,70402,1224579,1644732,3827967,115975

%N 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.

%C The upper left corner of the array is T(0,1).

%C Without loss of generality, the permutation can be taken to be (1 2 ... k) (k+1 k+2 ... 2k) ... ((n-1)k+1 (n-1)k+2 ... nk).

%C 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.

%H Alois P. Heinz, <a href="/A162663/b162663.txt">Antidiagonals n = 0..140, flattened</a> (first 20 antidiagonals from Franklin T. Adams-Watters)

%H T. S. Motzkin, <a href="/A000262/a000262.pdf">Sorting numbers for cylinders and other classification numbers</a>, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176. [Annotated, scanned copy]

%H OEIS Wiki, <a href="http://oeis.org/wiki/Sorting_numbers">Sorting numbers</a>

%F E.g.f. for column k: exp(Sum_{d|k} (exp(d*x) - 1) / d).

%F 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.

%F 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.

%F T(n,k) = Sum_{j=0..n-1} A036073(n,j)*k^(n-1-j). - _Andrey Zabolotskiy_, Oct 22 2017

%e The table starts:

%e    1,   1,   1,   1,   1

%e    1,   2,   2,   3,   2

%e    2,   7,   8,  16,  10

%e    5,  31,  42, 111,  70

%e   15, 164, 268, 931, 602

%p with(numtheory):

%p A:= proc(n, k) option remember; `if`(n=0, 1, add(binomial(n-1, j-1)

%p        *add(d^(j-1), d=divisors(k))*A(n-j, k), j=1..n))

%p     end:

%p seq(seq(A(n, 1+d-n), n=0..d), d=0..12);  # _Alois P. Heinz_, Oct 29 2015

%t 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. *)

%o (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

%o 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)!)

%Y Columns: A000110, A002872, A002874, A141003, A036075, A141004, A036077, A141005, A141006, A141007, A036081, A141008, A141009, A141010, A141011.

%Y Rows: A000012, A000005, A162664, A162665.

%Y Cf. A084423, A036040, A036073, A080575.

%Y Main diagonal gives A293850.

%K nice,nonn,tabl

%O 0,5

%A _Franklin T. Adams-Watters_, Jul 09 2009

%E Offset set to 0 by _Alois P. Heinz_, Oct 29 2015