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%I #31 Aug 11 2023 10:02:51
%S 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,
%T 203,203,1,64,275,515,706,825,877,877,1,128,793,1785,2744,3479,3937,
%U 4140,4140,1,256,2315,6347,11002,15177,18313,20270,21147,21147,1,512,6817,23073,45368,68303,88033,102678,111835,115975,115975
%N Triangle read by rows: number of labeled partitions of n with maximin m.
%C 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.
%C 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
%H G. C. Greubel, <a href="/A113547/b113547.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%H Todd Tichenor, <a href="https://arxiv.org/abs/1709.00393">A note on graph compositions and their connection to minimax of set partitions</a>, arXiv:1709.00393 [math.CO], 2017.
%F 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).
%F 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
%e 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.
%e 1;
%e 1, 1;
%e 1, 2, 2;
%e 1, 4, 5, 5;
%e 1, 8, 13, 15, 15;
%e 1, 16, 35, 47, 52, 52;
%e 1, 32, 97, 153, 188, 203, 203;
%e 1, 64, 275, 515, 706, 825, 877, 877;
%p A113547 := proc(n,m)
%p add(combinat[stirling2](m-1,k-1)*k^(n-m),k=1..m) ;
%p end proc:
%p seq(seq( A113547(n,m),m=1..n),n=1..10) ; # _R. J. Mathar_, Mar 13 2016
%t 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 *)
%Y Cf. A008277, A000110, A271466.
%Y A362924 and A362925 are other versions of this triangle. - _N. J. A. Sloane_, Aug 10 2023
%K nonn,tabl
%O 1,5
%A _Franklin T. Adams-Watters_, Jan 13 2006
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