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A270236 Triangle T(n,p) read by rows: the number of occurrences of p in the restricted growth functions of length n. 17
1, 3, 1, 9, 5, 1, 30, 21, 8, 1, 112, 88, 47, 12, 1, 463, 387, 253, 97, 17, 1, 2095, 1816, 1345, 675, 184, 23, 1, 10279, 9123, 7304, 4418, 1641, 324, 30, 1, 54267, 48971, 41193, 28396, 13276, 3645, 536, 38, 1, 306298, 279855, 243152, 183615, 102244, 36223, 7473, 842, 47, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
The RG functions used here are defined by f(1)=1, f(j) <= 1+max_{i<j} f(i).
T(n,p) is the number of elements in the p-th subset in all set partitions of [n]. - Joerg Arndt, Mar 14 2016
LINKS
FORMULA
T(n,n) = 1.
Conjecture: T(n,n-1) = 2+n*(n-1)/2 for n>1.
Conjecture: T(n+1,n-1) = 2+n*(n+1)*(3*n^2-5*n+26)/24 for n>1.
Sum_{k=1..n} k * T(n,k) = A346772(n). - Alois P. Heinz, Aug 03 2021
EXAMPLE
The two restricted growth functions of length 2 are (1,1) and (1,2). The 1 appears 3 times and the 2 once, so T(2,1)=3 and T(2,2)=1.
1;
3,1;
9,5,1;
30,21,8,1;
112,88,47,12,1;
463,387,253,97,17,1;
2095,1816,1345,675,184,23,1;
10279,9123,7304,4418,1641,324,30,1;
54267,48971,41193,28396,13276,3645,536,38,1;
306298,279855,243152,183615,102244,36223,7473,842,47,1;
1838320,1695902,1506521,1211936,770989,334751,90223,14303,1267,57,1;
11677867,10856879,9799547,8237223,5795889,2965654,995191,207186,25820, 1839,68,1;
MAPLE
b:= proc(n, m) option remember; `if`(n=0, [1, 0], add((p->
[p[1], p[2]+p[1]*x^j])(b(n-1, max(m, j))), j=1..m+1))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n, 0)[2]):
seq(T(n), n=0..12); # Alois P. Heinz, Mar 14 2016
MATHEMATICA
b[n_, m_] := b[n, m] = If[n == 0, {1, 0}, Sum[Function[p, {p[[1]], p[[2]] + p[[1]]*x^j}][b[n-1, Max[m, j]]], {j, 1, m+1}]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 1, n}]][b[n, 0][[2]] ]; Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Apr 07 2016, after Alois P. Heinz *)
CROSSREFS
Cf. A070071 (row sums).
T(2n+1,n+1) gives A270529.
Sequence in context: A331257 A112626 A050155 * A140714 A112932 A337680
KEYWORD
nonn,tabl
AUTHOR
R. J. Mathar, Mar 13 2016
STATUS
approved

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Last modified July 23 21:46 EDT 2024. Contains 374575 sequences. (Running on oeis4.)