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A362924
Triangle read by rows: T(n,m), n >= 1, 1 <= m <= n, is number of partitions of the set {1,2,...,n} that have at most one block contained in {1,...,m}.
5
1, 2, 1, 5, 4, 1, 15, 13, 8, 1, 52, 47, 35, 16, 1, 203, 188, 153, 97, 32, 1, 877, 825, 706, 515, 275, 64, 1, 4140, 3937, 3479, 2744, 1785, 793, 128, 1, 21147, 20270, 18313, 15177, 11002, 6347, 2315, 256, 1, 115975, 111835, 102678, 88033, 68303, 45368, 23073, 6817, 512, 1, 678570, 657423, 610989, 536882, 436297, 316305, 191866, 85475, 20195, 1024, 1
OFFSET
1,2
COMMENTS
Also, the maximum number of solutions to an exact cover problem with n items, of which m are secondary.
REFERENCES
D. E. Knuth, The Art of Computer Programming, Volume 4B, exercise 7.2.2.1--185, answer on page 468.
LINKS
Paolo Xausa, Table of n, a(n) for n = 1..11325 (rows 1..150 of the triangle, flattened).
FORMULA
T(n, 1) = Bell number (all set partitions) A000110(n);
T(n, n) = 1 when m=n (the only possibility is a single block);
T(n, n-1) = 2^{n-1} when m=n-1 (a single block or two blocks);
T(n, 2) = A078468(2).
In general, T(n, m) = Sum_{k=0..n-m} Stirling_2(n-m,k)*(k+1)^m.
EXAMPLE
Triangle begins:
[1],
[2, 1],
[5, 4, 1],
[15, 13, 8, 1],
[52, 47, 35, 16, 1],
[203, 188, 153, 97, 32, 1],
[877, 825, 706, 515, 275, 64, 1],
[4140, 3937, 3479, 2744, 1785, 793, 128, 1],
[21147, 20270, 18313, 15177, 11002, 6347, 2315, 256, 1],
[115975, 111835, 102678, 88033, 68303, 45368, 23073, 6817, 512, 1],
[678570, 657423, 610989, 536882, 436297, 316305, 191866, 85475, 20195, 1024, 1],
...
For example, if n=4, m=3, then T(4,3) = 8, because out of the A000110(4) = 15 set partitions of {1,2,3,4}, those that have 2 or more blocks contained in {1,2,3} are
{12,3,4},
{13,2,4},
{14,2,3},
{23,1,4},
{24,1,3},
{34,1,2},
{1,2,3,4},
while
{1234},
{123,4},
{124,3}
{134,2}
{234,1},
{12,34}
{13. 24}.
{14, 23}
do not.
MAPLE
with(combinat);
T:=proc(n, m) local k;
add(stirling2(n-m, k)*(k+1)^m, k=0..n-m);
end;
MATHEMATICA
A362924[n_, m_]:=Sum[StirlingS2[n-m, k](k+1)^m, {k, 0, n-m}];
Table[A362924[n, m], {n, 15}, {m, n}] (* Paolo Xausa, Dec 02 2023 *)
CROSSREFS
See A113547 and A362925 for other versions of this triangle.
Row sums give A005493.
Sequence in context: A128738 A193673 A126181 * A154930 A104259 A137650
KEYWORD
nonn,tabl
AUTHOR
N. J. A. Sloane, Aug 10 2023, based on an email from Don Knuth.
STATUS
approved