OFFSET
1,5
COMMENTS
The maximin of a partition is the maximum over all parts of the minimum label in each part. If the rows are reversed, the result is the number of partitions of n with minimax m.
The number of restricted growth functions of length n where the maximum appears first at position m. The RGF's are defined here as f(1)=1 and f(i) <=1+max_{1<=j<i} f(j). - R. J. Mathar, Mar 18 2016
LINKS
G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Todd Tichenor, A note on graph compositions and their connection to minimax of set partitions, arXiv:1709.00393 [math.CO], 2017.
FORMULA
T(n, m) = Sum_{k=1..m} S2(m-1, k-1)*k^(n-m), where S2 is the Stirling numbers of the second kind (A008277). T(n, n)=T(n, n-1)=B(n-1), where B is the Bell numbers (A000110). T(n, n-2)=B(n-1)-B(n-3).
Conjectures: T(n,3) = A007689(n-3). T(n,4) = 2^(n-4)+3^(n-3)+4^(n-4).- R. J. Mathar, Mar 13 2016
EXAMPLE
Maximin [123]=max(1)=1, maximin [12|3]=max(1,3)=3, maximin [13|2]=max(1,2)=2, maximin [1|23]=max(1,2)=2 and maximin [1|2|3]=max(1,2,3)=3, so for n=3 the multiset of maximins is {1,2,2,3,3}, making the 3rd line 1,2,2.
1;
1, 1;
1, 2, 2;
1, 4, 5, 5;
1, 8, 13, 15, 15;
1, 16, 35, 47, 52, 52;
1, 32, 97, 153, 188, 203, 203;
1, 64, 275, 515, 706, 825, 877, 877;
MAPLE
A113547 := proc(n, m)
add(combinat[stirling2](m-1, k-1)*k^(n-m), k=1..m) ;
end proc:
seq(seq( A113547(n, m), m=1..n), n=1..10) ; # R. J. Mathar, Mar 13 2016
MATHEMATICA
T[n_, n_] := BellB[n - 1]; T[n_, n_ - 1] := BellB[n - 1]; T[n_, n_ - 2] := BellB[n - 1] - BellB[n - 3]; T[n_, m_] := Sum[StirlingS2[m - 1, k - 1]*k^(n - m), {k, 1, m}]; Table[T[n, m], {n, 1, 5}, {m, 1, n}] (* G. C. Greubel, May 06 2017 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Franklin T. Adams-Watters, Jan 13 2006
STATUS
approved