A309099 Number of partitions of n avoiding the partition (4,3,1).

1, 1, 2, 3, 5, 7, 11, 15, 21, 28, 37, 46, 59, 72, 87, 104, 124, 144, 168, 192, 220, 250, 282, 314, 352, 391, 432, 475, 522, 569, 622, 675, 732, 791, 852, 915, 985, 1055, 1127, 1201, 1281, 1361, 1447, 1533, 1623, 1717, 1813, 1909, 2013, 2118, 2227, 2338, 2453

Offset: 0

Author

Jonathan S. Bloom, Jul 12 2019

Keywords

• nonn

Comments

We say a partition alpha contains mu provided that one can delete rows and columns from (the Ferrers board of) alpha and then top/right justify to obtain mu. If this is not possible then we say alpha avoids mu. For example, the only partitions avoiding (2,1) are those whose Ferrers boards are rectangles.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000
• Jonathan Bloom and Nathan McNew, Counting pattern-avoiding integer partitions, arXiv:1908.03953 [math.CO], 2019.
• J. Bloom and D. Saracino, On Criteria for rook equivalence of Ferrers boards, arXiv:1808.04221 [math.CO], 2018.
• 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.
• J. Bloom and D. Saracino, On Criteria for rook equivalence of Ferrers boards, European J. Combin., 76 (2018), 199-207.
• Joachim König, Closed-form for the number of partitions of n avoiding the partition (4,3,1), answer to question on MathOverflow (2023).

Crossrefs

Cf. A002378, A078567, A309097, A309098, A309058.

Programs

Maple

b:= proc(n) option remember; `if`(n<1, [0$2],
      (p-> p+[numtheory[tau](n), p[1]])(b(n-1)))
    end:
a:= n-> b(n+1)[2]+`if`(n=0, 1, n*(1-n)):
seq(a(n), n=0..55);  # Alois P. Heinz, Dec 20 2023

Mathematica

b[n_] := b[n] = If[n < 1, {0, 0}, With[{p = b[n-1]},
   p + {DivisorSigma[0, n], p[[1]]}]];
a[n_] := b[n+1][[2]] + If[n == 0, 1, n*(1-n)];
Table[a[n], {n, 0, 55}] (* Jean-François Alcover, Jan 29 2025, after Alois P. Heinz *)

PARI

a(n) = if(n == 0, 1, sum(i = 1, n, (n - i + 1) * numdiv(i)) - n * (n - 1)) \\ Mikhail Kurkov, Dec 20 2023 [verification needed]

Formula

a(n) = A078567(n+1) - A002378(n-1) for n > 0 with a(0) = 1. - Mikhail Kurkov, Dec 20 2023 [verification needed]

Extensions

More terms from Alois P. Heinz, Jul 12 2019