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
KEYWORD
nonn
AUTHOR
Nathan McNew, Jul 09 2019
STATUS
approved