cp's OEIS Frontend

A124496 Triangle read by rows: T(n,k) is the number of set partitions of {1,2,...,n} in which the size of the last block is k, 1<=k<=n; the blocks are ordered with increasing least elements.

%I A124496 #35 Jan 10 2022 08:11:33
%S A124496 1,1,1,3,1,1,9,4,1,1,31,14,5,1,1,121,54,20,6,1,1,523,233,85,27,7,1,1,
%T A124496 2469,1101,400,125,35,8,1,1,12611,5625,2046,635,175,44,9,1,1,69161,
%U A124496 30846,11226,3488,952,236,54,10,1,1,404663,180474,65676,20425,5579,1366,309,65,11,1,1
%N A124496 Triangle read by rows: T(n,k) is the number of set partitions of {1,2,...,n} in which the size of the last block is k, 1<=k<=n; the blocks are ordered with increasing least elements.
%C A124496 Number of restricted growth functions of length n with a multiplicity k of the maximum value. RGF's are here defined as f(1)=1, f(i) <= 1+max_{1<=j<i} f(j). - _R. J. Mathar_, Mar 18 2016
%C A124496 This is table 9.2 in the Gould-Quaintance reference. - _Peter Luschny_, Apr 25 2016
%H A124496 Alois P. Heinz, <a href="/A124496/b124496.txt">Rows n = 1..141, flattened</a>
%H A124496 H. W. Gould and Jocelyn Quaintance, <a href="http://www.doiserbia.nb.rs/img/doi/1452-8630/2007/1452-86300702371G.pdf">A linear binomial recurrence and the Bell numbers and polynomials</a>. Applicable Analysis and Discrete Mathematics, 1 (2007), 371-385.
%F A124496 The row enumerating polynomial P[n](t)=Q[n](t,1), where Q[1](t,s)=ts and Q[n](t,s)=s*dQ[n-1](t,s)/ds +(t-1)Q[n-1](t,s)+tsQ[n-1](1,s) for n>=2.
%F A124496 A008275^-1*ONES*A008275 or A008277*ONES*A008277^-1 where ONES is a triangle with all entries = 1. [From _Gerald McGarvey_, Aug 20 2009]
%F A124496 Conjectures: T(n,n-3) = A000096(n). T(n,n-4)= A055831(n+1). - _R. J. Mathar_, Mar 13 2016
%e A124496 T(4,2) = 4 because we have 13|24, 14|23, 12|34 and 1|2|34.
%e A124496 Triangle starts:
%e A124496   1;
%e A124496   1,1;
%e A124496   3,1,1;
%e A124496   9,4,1,1;
%e A124496   31,14,5,1,1;
%e A124496   121,54,20,6,1,1;
%e A124496   523,233,85,27,7,1,1;
%e A124496   2469,1101,400,125,35,8,1,1;
%e A124496   12611,5625,2046,635,175,44,9,1,1;
%e A124496   69161,30846,11226,3488,952,236,54,10,1,1;
%e A124496   404663,180474,65676,20425,5579,1366,309,65,11,1,1;
%e A124496   2512769,1120666,407787,126817,34685,8494,1893,395,77,12,1,1;
%e A124496   ...
%p A124496 Q[1]:=t*s: for n from 2 to 12 do Q[n]:=expand(t*s*subs(t=1,Q[n-1])+s*diff(Q[n-1],s)+t*Q[n-1]-Q[n-1]) od:for n from 1 to 12 do P[n]:=sort(subs(s=1,Q[n])) od: for n from 1 to 12 do seq(coeff(P[n],t,j),j=1..n) od;
%p A124496 # second Maple program:
%p A124496 T:= proc(n, k) option remember; `if`(n=k, 1,
%p A124496       add(T(n-j, k)*binomial(n-1, j-1), j=1..n-k))
%p A124496     end:
%p A124496 seq(seq(T(n, k), k=1..n), n=1..12);  # _Alois P. Heinz_, Jul 05 2016
%t A124496 T[n_, k_] := T[n, k] = If[n == k, 1, Sum[T[n-j, k]*Binomial[n-1, j-1], {j, 1, n-k}]];
%t A124496 Table[Table[T[n, k], {k, 1, n}], {n, 1, 12}] // Flatten; (* _Jean-François Alcover_, Jul 21 2016, after _Alois P. Heinz_ *)
%Y A124496 Row sums are the Bell numbers (A000110). It seems that T(n, 1), T(n, 2), T(n, 3) and T(n, 4) are given by A040027, A045501, A045499 and A045500, respectively. A121207 gives a very similar triangle.
%Y A124496 T(2n,n) gives A297924.
%Y A124496 Cf. A000110, A040027, A045501, A045499, A045500, A186020, A160185.
%K A124496 nonn,tabl
%O A124496 1,4
%A A124496 _Emeric Deutsch_, Nov 14 2006