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URL: https://oeis.org/A319375

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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.
13
1, 3, 1, 10, 4, 1, 35, 17, 7, 1, 136, 76, 36, 11, 1, 577, 357, 186, 81, 16, 1, 2682, 1737, 1023, 512, 162, 22, 1, 13435, 8997, 5867, 3151, 1345, 295, 29, 1, 72310, 49420, 34744, 20071, 10096, 3145, 499, 37, 1, 414761, 289253, 211888, 133853, 72973, 29503, 6676, 796, 46, 1
OFFSET
1,2
LINKS
EXAMPLE
The 5 set partitions of {1,2,3} are:
1 |2 |3
12 |3
13 |2
23 |1
123
so there are 10 elements in the first (largest) blocks, 4 in the second blocks and only 1 in the third blocks.
Triangle T(n,k) begins:
1;
3, 1;
10, 4, 1;
35, 17, 7, 1;
136, 76, 36, 11, 1;
577, 357, 186, 81, 16, 1;
2682, 1737, 1023, 512, 162, 22, 1;
13435, 8997, 5867, 3151, 1345, 295, 29, 1;
72310, 49420, 34744, 20071, 10096, 3145, 499, 37, 1;
...
MAPLE
b:= proc(n, l) option remember; `if`(n=0, add(l[-i]*
x^i, i=1..nops(l)), add(binomial(n-1, j-1)*
b(n-j, sort([l[], j])), j=1..n))
end:
T:= n-> (p-> (seq(coeff(p, x, i), i=1..n)))(b(n, [])):
seq(T(n), n=1..12);
# Alternative:
b:= proc(n, i, t) option remember; `if`(n=0, [1, 0], `if`(i<1, 0,
add((p-> p+`if`(t>0 and t-j<1, [0, p[1]*i], 0))(
combinat[multinomial](n, i$j, n-i*j)/j!*
b(n-i*j, min(n-i*j, i-1), max(0, t-j))), j=0..n/i)))
end:
T:= (n, k)-> b(n$2, k)[2]:
seq(seq(T(n, k), k=1..n), n=1..12); # Alois P. Heinz, Mar 02 2020
MATHEMATICA
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[n_] := Function[p, Table[Coefficient[p, x, i], {i, 1, n}]][b[n, {}]];
Array[T, 12] // Flatten (* Jean-François Alcover, Dec 28 2018, after Alois P. Heinz *)
CROSSREFS
Row sums give A070071.
Sequence in context: A280787 A126954 A176992 * A107870 A078817 A316193
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Dec 07 2018
STATUS
approved