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A319298
Number T(n,k) of entries in the k-th blocks of all set partitions of [n] when blocks are ordered by increasing lengths (and increasing smallest elements); triangle T(n,k), n>=1, 1<=k<=n, read by rows.
14
1, 3, 1, 7, 7, 1, 21, 25, 13, 1, 66, 101, 71, 21, 1, 258, 366, 396, 166, 31, 1, 1079, 1555, 1877, 1247, 337, 43, 1, 4987, 7099, 9199, 7855, 3305, 617, 57, 1, 25195, 34627, 47371, 47245, 27085, 7681, 1045, 73, 1, 136723, 184033, 253108, 284968, 203278, 79756, 16126, 1666, 91, 1
OFFSET
1,2
LINKS
EXAMPLE
The 5 set partitions of {1,2,3} are:
1 |2 |3
1 |23
2 |13
3 |12
123
so there are 7 elements in the first (smallest) blocks, 7 in the second blocks and only 1 in the third blocks.
Triangle T(n,k) begins:
1;
3, 1;
7, 7, 1;
21, 25, 13, 1;
66, 101, 71, 21, 1;
258, 366, 396, 166, 31, 1;
1079, 1555, 1877, 1247, 337, 43, 1;
4987, 7099, 9199, 7855, 3305, 617, 57, 1;
25195, 34627, 47371, 47245, 27085, 7681, 1045, 73, 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);
# second Maple program:
b:= proc(n, i, t) option remember; `if`(n=0, [1, 0], `if`(i>n, 0,
add((p-> p+`if`(t>0 and t-j<1, [0, p[1]*i], 0))(b(n-i*j, i+1,
max(0, t-j))/j!*combinat[multinomial](n, i$j, n-i*j)), j=0..n/i)))
end:
T:= (n, k)-> b(n, 1, 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, {}]];
Table[T[n], {n, 1, 12}] // Flatten (* Jean-François Alcover, Dec 28 2018, after Alois P. Heinz *)
CROSSREFS
Row sums give A070071.
Sequence in context: A374839 A369892 A359576 * A101748 A058606 A135284
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Dec 07 2018
STATUS
approved