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

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 41, 54, 72, 91, 115, 145, 177, 215, 258, 308, 364, 424, 491, 568, 651, 742, 838, 940, 1065, 1181, 1320, 1454, 1619, 1757, 1957, 2124, 2329, 2510, 2763, 2934, 3244, 3432, 3752, 3964, 4329, 4531, 4965, 5179, 5627, 5872, 6391, 6577, 7178, 7405

Offset

0,3

Comments

Partitions whose Ferrers diagrams do not contain the pattern 4321 under removal of rows and columns (as defined by Bloom and Saracino).

Links

Alois P. Heinz, Table of n, a(n) for n = 0..5000

J. Bloom and D. Saracino, Rook and Wilf equivalence of integer partitions, arXiv:1808.04238 [math.CO], 2018.

J. Bloom and D. Saracino, Rook and Wilf equivalence of integer partitions, European J. Combin., 71 (2018), 246-267.

Formula

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)).

a(n) = Sum_{k=0..3} A116608(n,k). - Alois P. Heinz, Jul 11 2019

Example

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.

Maple

b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(i<1, 0,
      `if`(t=1, `if`(irem(n, i)=0, 1, 0)+b(n, i-1, t),
       add(b(n-i*j, i-1, t-`if`(j=0, 0, 1)), j=0..n/i))))
    end:
a:= n-> b(n$2, 3):
seq(a(n), n=0..100);  # Alois P. Heinz, Jul 11 2019

Mathematica

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}]]]];
a[n_] := b[n, n, 3];
a /@ Range[0, 100] (* Jean-François Alcover, Feb 27 2020, after Alois P. Heinz *)

Crossrefs

Cf. A265250 (partitions of n with parts having at most 2 distinct magnitudes). Sum of A002134, A002133 and A000005.

Cf. A116608.

Sequence in context: A184644 A209039 A182805 * A218509 A026815 A341913

Adjacent sequences: A309055 A309056 A309057 * A309059 A309060 A309061

Keyword

nonn

Author

Nathan McNew, Jul 09 2019

Status

approved