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.

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

Offset: 0

Author

Franklin T. Adams-Watters, Jul 09 2009

Keywords

• nice
• nonn
• tabl

Comments

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

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).

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.

Examples

			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
		

Links

• 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

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.

Programs

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. *)

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)!)

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_{j=0..n-1} A036073(n,j)*k^(n-1-j). - Andrey Zabolotskiy, Oct 22 2017

Extensions

Offset set to 0 by Alois P. Heinz, Oct 29 2015