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

1, 2, 2, 3, 6, 6, 5, 16, 36, 24, 7, 46, 150, 240, 120, 11, 114, 546, 1560, 1800, 720, 15, 614, 2058, 8400, 16800, 15120, 5040, 22, 1366, 6984, 40848, 126000, 191520, 141120, 40320, 30, 12516, 73488, 192816, 834120, 1905120, 2328480, 1451520, 362880

Offset: 1

Author

Thomas Wieder, Apr 14 2008

Keywords

• nonn

Comments

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.

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

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

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

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).

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

1

2 1

3 3 1

5 8 6 1

7 23 25 10 1

11 57 91 65 15 1

15 307 343 350 140 21 1

22 683 1164 1702 1050 266 28 1

30 6258 12248 8034 6951 2646 462 36 1

In this triangle we observe:

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

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

Examples

			Triangle begins:
  1
  2 2
  3 6 6
  5 16 36 24
  7 46 150 240 120
  11 114 546 1560 1800 720
  15 614 2058 8400 16800 15120 5040
  22 1366 6984 40848 126000 191520 141120 40320
  30 12516 73488 192816 834120 1905120 2328480 1451520 362880
  ...
		

References

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

Links

• Thomas Wieder, Further comments on this sequence

Crossrefs

Cf. A019538, A137383.