W. Lang, Oct 19 2007 A035206 tabf, a partition number array a(n,k): a(n,k) enumerates the number of distributions of n identical objects into n distinguishable boxes with m boxes occupied. The occupation of the m=m(n,k):=A036043(n,k) boxes is specified by the k-th partition of n in the Abramowitz-Stegun order. m=m(n,k) is the number of parts of the k-th partition of n. For the Abramowitz-Stegun (A-St) order of partitions see pp. 831-2 of the reference given in A117506. n\k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 ... 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 3 6 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 4 12 6 12 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 5 20 20 30 30 20 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 6 30 30 15 60 120 20 60 90 30 1 0 0 0 0 0 0 0 0 0 0 0 7 7 42 42 42 105 210 105 105 140 420 140 105 210 42 1 0 0 0 0 0 0 0 8 8 56 56 56 28 168 336 336 168 168 280 840 420 840 70 280 1120 560 168 420 56 1 . . . . n\k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 ... The next two rows, for n=9 and n=10, are: n=9: [9, 72, 72, 72, 72, 252, 504, 504, 252, 252, 504, 84, 504, 1512, 1512, 1512, 1512, 504, 630, 2520, 1260, 3780, 630, 504, 2520, 1680, 252, 756, 72, 1]. n=10: [10, 90, 90, 90, 90, 45, 360, 720, 720, 720, 360, 720, 360, 360, 840, 2520, 2520, 1260, 2520, 5040, 840, 840, 1260, 1260, 5040, 5040, 7560, 7560, 5040, 252, 1260, 6300, 3150, 12600, 3150, 840, 5040, 4200, 360, 1260, 90, 1]. The row sums give, for n>=1: A001700(n-1) = [1,3,10,35,126,462,1716,6435,24310,92378,...] = binomial(2*n-1,n-1). They coincide with the row sums of triangle A103371. Example: a(5,5) relates to the partition (1,2^2) of n=5. Here m=3, and 5 indistinguishable (identical) balls are put into boxes b1,...,b5 with m=3 boxes occupied, one with one ball and two with two balls. Therefore, a(5,5) = binomial(5,3)*3!/(1!*2!) = 10*3 = 30. ########################################### e.o.f. #####################################################################################