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
Keywords
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).
Programs
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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
Comments