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%I #38 May 26 2016 08:20:44
%S 1,2,2,3,4,5,5,7,11,15,7,12,21,36,52,11,19,38,74,135,203,15,30,64,141,
%T 296,566,877,22,45,105,250,592,1315,2610,4140,30,67,165,426,1098,2752,
%U 6393,13082,21147,42,97,254,696,1940,5317,13960,33645,70631,115975
%N Triangular array t read by rows: t(0,k) is p(k), the number of partitions of the k-multiset {0,0,...,0} with k zeros. For 0 <= n < k, t(n, k) is the number of partitions of the k-multiset {0, 0, ..., 0, 1, 2, 3, ..., k-n} with n zeros.
%C First in a series of triangular arrays which comprise subsequences of A096443(n).
%C The second array begins 9 16 26 29 52 92 47 98 198 371 and when the arrays are aligned as illustrated in triangle A126441 with p(n) values they sum to A035310 which counts unordered multisets.
%C Let t(n, k) be the number of ways to partition the k-multiset {0,0,...,0,1,2,3,4,...,k-n} with n zeros, 0 <= n < k. Then t(n, k) = sum_i = 0..k j = 0..n S(n, j) C(i, j) p(k - n - i), where S(n, j) are Stirling numbers of the second kind, C(i, j) are the number of compositions of i distinct objects into j parts, and p is the integer partition function.
%C To see this, partition [n] into j blocks; there are S(n, j) partitions. For such a partition x and for each i, there are C(i, j) ways to distribute i zeros into x, because the blocks of x are all distinct. There are p(k-n-i) ways to partition the remaining k-n-i zeros. Multiplying and summing gives the result. - _George Beck_, Jan 10 2011
%C Values are also part of A096443, A129306 and A249620. Columns are also columns of the last one of these irregular triangles. See "Partitions_of_multisets" link. - _Tilman Piesk_, Nov 09 2014
%H Tilman Piesk, <a href="https://en.wikiversity.org/wiki/Partitions_of_multisets#Right_columns">Partitions of multisets</a> (Wikiversity)
%e This first array includes only the hook cases. A096443(9,14,16) correspond to partitions [2,2], [3,2] and [2,2,1] so these values do not appear in A126442.
%e The array begins:
%e 1
%e 2 2
%e 3 4 5
%e 5 7 11 15
%e 7 12 21 36 52
%t (* The triangle is flattened to a sequence. *)
%t t[n_, k_] := Sum[StirlingS2[n, j] * Binomial[-1 + i + j, i] * PartitionsP[k - n - i], {j, 0, n}, {i, 0, k - n}]; Table[ t[n, k], {k, 10}, {n, 0, k - 1}] // Flatten (* _George Beck_, Jan 10 2011 *)
%Y Cf. A000041, A000070, A082775, A093802, A000291, A002763, A000412, A054225, A035310, A000110, A035098.
%K nonn,tabl
%O 1,2
%A _Alford Arnold_, Jan 28 2007
%E Definition clarified by _George Beck_, Jan 11 2011
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