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A139359 Number L([n],m) of ways the labeled parts of each integer partition of n can be distributed into m nonempty labeled boxes.

%I A139359 #14 Mar 19 2018 05:41:29
%S A139359 1,2,2,3,6,6,5,16,36,24,7,46,150,240,120,11,114,546,1560,1800,720,15,
%T A139359 614,2058,8400,16800,15120,5040,22,1366,6984,40848,126000,191520,
%U A139359 141120,40320,30,12516,73488,192816,834120,1905120,2328480,1451520,362880
%N A139359 Number L([n],m) of ways the labeled parts of each integer partition of n can be distributed into m nonempty labeled boxes.
%C A139359 This formula is related to a formula given by Riordan, see Riordan, 1958, page 94. Furthermore, this formula is related to the distribution of labeled elements into labeled boxes, as described by A019538.
%C A139359 The first column is equal to A000041 = number of partitions of n (the partition numbers).
%C A139359 The main diagonal is equal to the A000142 = Factorial numbers: n!
%C A139359 The second diagonal is equal to A001286 = Lah numbers: (n-1)*n!/2.
%C A139359 The third diagonal is equal to A019538 = Triangle of numbers T(n,k) = k!*Stirling2(n,k) read by rows (n >= 1, 1 <= k <= n).
%C A139359 If we normalize the m-th column by m! we get the triangle
%C A139359   1
%C A139359   2 1
%C A139359   3 3 1
%C A139359   5 8 6 1
%C A139359   7 23 25 10 1
%C A139359   11 57 91 65 15 1
%C A139359   15 307 343 350 140 21 1
%C A139359   22 683 1164 1702 1050 266 28 1
%C A139359   30 6258 12248 8034 6951 2646 462 36 1
%C A139359 In this triangle we observe:
%C A139359 The second diagonal is equal to A000217 = Triangular numbers: a(n) = C(n+1,2) = n(n+1)/2 = 0+1+2+...+n.
%C A139359 The third diagonal is composed of numbers belonging to A095660 = Pascal (1,3) triangle.
%D A139359 John Riordan: Introduction to Combinatorics, John Wiley & Sons, New York, 1958, ISBN 0-486-42536-3.
%H A139359 Thomas Wieder, <a href="/A139359/a139359.txt">Further comments on this sequence</a>
%e A139359 Triangle begins:
%e A139359   1
%e A139359   2 2
%e A139359   3 6 6
%e A139359   5 16 36 24
%e A139359   7 46 150 240 120
%e A139359   11 114 546 1560 1800 720
%e A139359   15 614 2058 8400 16800 15120 5040
%e A139359   22 1366 6984 40848 126000 191520 141120 40320
%e A139359   30 12516 73488 192816 834120 1905120 2328480 1451520 362880
%e A139359   ...
%Y A139359 Cf. A019538, A137383.
%K A139359 nonn
%O A139359 1,2
%A A139359 _Thomas Wieder_, Apr 14 2008