cp's OEIS Frontend

A309058 Partitions of n with parts having at most 3 distinct magnitudes.

%I A309058 #23 Dec 13 2023 20:35:35
%S A309058 1,1,2,3,5,7,11,15,22,30,41,54,72,91,115,145,177,215,258,308,364,424,
%T A309058 491,568,651,742,838,940,1065,1181,1320,1454,1619,1757,1957,2124,2329,
%U A309058 2510,2763,2934,3244,3432,3752,3964,4329,4531,4965,5179,5627,5872,6391,6577,7178,7405
%N A309058 Partitions of n with parts having at most 3 distinct magnitudes.
%C A309058 Partitions whose Ferrers diagrams do not contain the pattern 4321 under removal of rows and columns (as defined by Bloom and Saracino).
%H A309058 Alois P. Heinz, <a href="/A309058/b309058.txt">Table of n, a(n) for n = 0..5000</a>
%H A309058 J. Bloom and D. Saracino, <a href="https://arxiv.org/abs/1808.04238">Rook and Wilf equivalence of integer partitions</a>, arXiv:1808.04238 [math.CO], 2018.
%H A309058 J. Bloom and D. Saracino, <a href="https://doi.org/10.1016/j.ejc.2018.04.002">Rook and Wilf equivalence of integer partitions</a>, European J. Combin., 71 (2018), 246-267.
%F A309058 G.f.: Sum_{i>=1} x^i/(1-x^i) + Sum_{j=1..i-1} x^(i+j)/((1-x^i)*(1-x^j)) + Sum_{k=1..j-1} x^(i+j+k)/((1-x^i)*(1-x^j)*(1-x^k)).
%F A309058 a(n) = Sum_{k=0..3} A116608(n,k). - _Alois P. Heinz_, Jul 11 2019
%e A309058 a(10) = 41 because all of the 42 integer partitions of 10 count (i.e., 10 = 10, 10 = 9+1 = 8+1+1, etc.), except the partition 10 = 4+3+2+1.
%p A309058 b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A309058       `if`(t=1, `if`(irem(n, i)=0, 1, 0)+b(n, i-1, t),
%p A309058        add(b(n-i*j, i-1, t-`if`(j=0, 0, 1)), j=0..n/i))))
%p A309058     end:
%p A309058 a:= n-> b(n$2, 3):
%p A309058 seq(a(n), n=0..100);  # _Alois P. Heinz_, Jul 11 2019
%t A309058 b[n_, i_, t_] := b[n, i, t] = If[n == 0, 1, If[i < 1, 0, If[t == 1, If[Mod[n, i] == 0, 1, 0] + b[n, i - 1, t], Sum[b[n - i*j, i - 1, t - If[j == 0, 0, 1]], {j, 0, n/i}]]]];
%t A309058 a[n_] := b[n, n, 3];
%t A309058 a /@ Range[0, 100] (* _Jean-François Alcover_, Feb 27 2020, after _Alois P. Heinz_ *)
%Y A309058 Cf. A265250 (partitions of n with parts having at most 2 distinct magnitudes). Sum of A002134, A002133 and A000005.
%Y A309058 Cf. A116608.
%K A309058 nonn
%O A309058 0,3
%A A309058 _Nathan McNew_, Jul 09 2019