cp's OEIS Frontend

A258152 Partition array in Abramowitz-Stegun order for the number of ways of putting n stones into a rectangular m X n grid of squares such that each of the m rows contains at least one stone.

%I A258152 #13 Jun 28 2015 01:14:42
%S A258152 1,1,4,1,18,27,1,32,36,288,256,1,50,200,750,1500,5000,3125,1,72,450,
%T A258152 400,1620,10800,3375,17280,48600,97200,46656,1,98,882,2450,3087,30870,
%U A258152 25725,46305,48020,432180,259308,420175,1512630,2117682,823543
%N A258152 Partition array in Abramowitz-Stegun order for the number of ways of putting n stones into a rectangular m X n grid of squares such that each of the m rows contains at least one stone.
%C A258152 Motivated by A258371 by _Adam J.T. Partridge_.
%C A258152 The sequence for the row lengths is A000041(n).
%C A258152 The k-th partition of n in Abramowitz-Stegun (A-St) order is denoted by P(n, k) = [1^e(n,k,1), ..., n^e(n,k,n)], with nonnegative exponents summing to m, the number of parts. Here j^0 is not 1, the corresponding P(n, k) list entry is missing, that is, this part j does not appear.
%C A258152 If the k-th partition of n in A-St order has m parts (see A008284) then the irregular triangle entry a(n, k) gives the number of ways of putting n stones into a rectangular m X n grid of squares such that each of the m rows contains at least one stone.
%C A258152 The triangle version of this partition array, with the numbers belonging to the same number of parts m of the partition of n summed, is given in A259051.
%F A258152 a(n, k) = (m!/product_{j=1..n} e(n,k,j)!)* product_{j=1..n} binomial(n, j)^e(n,k,j), for n >= 1 and k = 1, 2, ..., A000041(n). Note that for e(n,k,j) = 0 the binomial does not contribute to the product.
%e A258152 The irregular triangle a(n, k), with entries belong to the same number of parts m = 1, ..., n enclosed in brackets, begins:
%e A258152 n\k 1   2     3     4     5      6      7 ...
%e A258152 1  [1]
%e A258152 2: [1] [4]
%e A258152 3: [1] [18] [27]
%e A258152 4: [1] [32   36] [288] [256]
%e A258152 5: [1] [50  200] [750  1500] [5000] [3125]
%e A258152 ...
%e A258152 n=6: [1] [72  450   400] [1620  10800  3375] [17280, 48600] [97200] [46656],
%e A258152 n=7: [1] [98  882  2450] [3087  30870  25725  46305] [48020 432180  259308] [420175 1512630] [2117682] [823543].
%e A258152 a(4, 3) = 36 because the third partition of 4 is (2^2), and m = 2, hence (2/(2!))*binomial(4,2)^2 = 6^2 = 36. Both of the m=2 rows of n squares are occupied with two stones, hence the binomial(4,2)^2.
%Y A258152 Cf. A258371, A259051.
%K A258152 nonn,easy,tabf
%O A258152 1,3
%A A258152 _Wolfdieter Lang_, Jun 17 2015