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A319375 Number T(n,k) of entries in the k-th blocks of all set partitions of [n] when blocks are ordered by decreasing lengths (and increasing smallest elements); triangle T(n,k), n>=1, 1<=k<=n, read by rows.

%I A319375 #49 May 08 2024 17:59:38
%S A319375 1,3,1,10,4,1,35,17,7,1,136,76,36,11,1,577,357,186,81,16,1,2682,1737,
%T A319375 1023,512,162,22,1,13435,8997,5867,3151,1345,295,29,1,72310,49420,
%U A319375 34744,20071,10096,3145,499,37,1,414761,289253,211888,133853,72973,29503,6676,796,46,1
%N A319375 Number T(n,k) of entries in the k-th blocks of all set partitions of [n] when blocks are ordered by decreasing lengths (and increasing smallest elements); triangle T(n,k), n>=1, 1<=k<=n, read by rows.
%H A319375 Alois P. Heinz, <a href="/A319375/b319375.txt">Rows n = 1..141, flattened</a>
%H A319375 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%e A319375 The 5 set partitions of {1,2,3} are:
%e A319375   1   |2  |3
%e A319375   12  |3
%e A319375   13  |2
%e A319375   23  |1
%e A319375   123
%e A319375 so there are 10 elements in the first (largest) blocks, 4 in the second blocks and only 1 in the third blocks.
%e A319375 Triangle T(n,k) begins:
%e A319375       1;
%e A319375       3,     1;
%e A319375      10,     4,     1;
%e A319375      35,    17,     7,     1;
%e A319375     136,    76,    36,    11,     1;
%e A319375     577,   357,   186,    81,    16,    1;
%e A319375    2682,  1737,  1023,   512,   162,   22,   1;
%e A319375   13435,  8997,  5867,  3151,  1345,  295,  29,  1;
%e A319375   72310, 49420, 34744, 20071, 10096, 3145, 499, 37, 1;
%e A319375   ...
%p A319375 b:= proc(n, l) option remember; `if`(n=0, add(l[-i]*
%p A319375       x^i, i=1..nops(l)), add(binomial(n-1, j-1)*
%p A319375       b(n-j, sort([l[], j])), j=1..n))
%p A319375     end:
%p A319375 T:= n-> (p-> (seq(coeff(p, x, i), i=1..n)))(b(n, [])):
%p A319375 seq(T(n), n=1..12);
%p A319375 # second Maple program:
%p A319375 b:= proc(n, i, t) option remember; `if`(n=0, [1, 0], `if`(i<1, 0,
%p A319375       add((p-> p+`if`(t>0 and t-j<1, [0, p[1]*i], 0))(
%p A319375        combinat[multinomial](n, i$j, n-i*j)/j!*
%p A319375       b(n-i*j, min(n-i*j, i-1), max(0, t-j))), j=0..n/i)))
%p A319375     end:
%p A319375 T:= (n, k)-> b(n$2, k)[2]:
%p A319375 seq(seq(T(n, k), k=1..n), n=1..12);  # _Alois P. Heinz_, Mar 02 2020
%t A319375 b[n_, l_] := b[n, l] = If[n == 0, Sum[l[[-i]] x^i, {i, 1, Length[l]}], Sum[ Binomial[n-1, j-1] b[n-j, Sort[Append[l, j]]], {j, 1, n}]];
%t A319375 T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 1, n}]][b[n, {}]];
%t A319375 Array[T, 12] // Flatten (* _Jean-François Alcover_, Dec 28 2018, after _Alois P. Heinz_ *)
%Y A319375 Columns k=1-10 give: A097148, A333059, A333060, A333061, A333062, A333063, A333064, A333065, A333066, A333067.
%Y A319375 Row sums give A070071.
%Y A319375 Cf. A319298, A322384.
%K A319375 nonn,tabl
%O A319375 1,2
%A A319375 _Alois P. Heinz_, Dec 07 2018