login
Number L([n],m) of ways the labeled parts of each integer partition of n can be distributed into m nonempty labeled boxes.
1

%I #14 Mar 19 2018 05:41:29

%S 1,2,2,3,6,6,5,16,36,24,7,46,150,240,120,11,114,546,1560,1800,720,15,

%T 614,2058,8400,16800,15120,5040,22,1366,6984,40848,126000,191520,

%U 141120,40320,30,12516,73488,192816,834120,1905120,2328480,1451520,362880

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

%C 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 The first column is equal to A000041 = number of partitions of n (the partition numbers).

%C The main diagonal is equal to the A000142 = Factorial numbers: n!

%C The second diagonal is equal to A001286 = Lah numbers: (n-1)*n!/2.

%C 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 If we normalize the m-th column by m! we get the triangle

%C 1

%C 2 1

%C 3 3 1

%C 5 8 6 1

%C 7 23 25 10 1

%C 11 57 91 65 15 1

%C 15 307 343 350 140 21 1

%C 22 683 1164 1702 1050 266 28 1

%C 30 6258 12248 8034 6951 2646 462 36 1

%C In this triangle we observe:

%C The second diagonal is equal to A000217 = Triangular numbers: a(n) = C(n+1,2) = n(n+1)/2 = 0+1+2+...+n.

%C The third diagonal is composed of numbers belonging to A095660 = Pascal (1,3) triangle.

%D John Riordan: Introduction to Combinatorics, John Wiley & Sons, New York, 1958, ISBN 0-486-42536-3.

%H Thomas Wieder, <a href="/A139359/a139359.txt">Further comments on this sequence</a>

%e Triangle begins:

%e 1

%e 2 2

%e 3 6 6

%e 5 16 36 24

%e 7 46 150 240 120

%e 11 114 546 1560 1800 720

%e 15 614 2058 8400 16800 15120 5040

%e 22 1366 6984 40848 126000 191520 141120 40320

%e 30 12516 73488 192816 834120 1905120 2328480 1451520 362880

%e ...

%Y Cf. A019538, A137383.

%K nonn

%O 1,2

%A _Thomas Wieder_, Apr 14 2008