login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A113547 Triangle read by rows: number of labeled partitions of n with maximin m. 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, 1, 128, 793, 1785, 2744, 3479, 3937, 4140, 4140, 1, 256, 2315, 6347, 11002, 15177, 18313, 20270, 21147, 21147, 1, 512, 6817, 23073, 45368, 68303, 88033, 102678, 111835, 115975, 115975 (list; table; graph; refs; listen; history; text; internal format)
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

Cf. A008277, A000110, A271466.

Sequence in context: A063415 A098977 A247311 * A218580 A259697 A330664

Adjacent sequences:  A113544 A113545 A113546 * A113548 A113549 A113550

KEYWORD

nonn,tabl

AUTHOR

Franklin T. Adams-Watters, Jan 13 2006

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified February 24 04:36 EST 2020. Contains 332197 sequences. (Running on oeis4.)