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

%I A309099 #54 Jan 29 2025 12:31:28
%S A309099 1,1,2,3,5,7,11,15,21,28,37,46,59,72,87,104,124,144,168,192,220,250,
%T A309099 282,314,352,391,432,475,522,569,622,675,732,791,852,915,985,1055,
%U A309099 1127,1201,1281,1361,1447,1533,1623,1717,1813,1909,2013,2118,2227,2338,2453
%N A309099 Number of partitions of n avoiding the partition (4,3,1).
%C A309099 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.
%H A309099 Alois P. Heinz, <a href="/A309099/b309099.txt">Table of n, a(n) for n = 0..10000</a>
%H A309099 Jonathan Bloom and Nathan McNew, <a href="https://arxiv.org/abs/1908.03953">Counting pattern-avoiding integer partitions</a>, arXiv:1908.03953 [math.CO], 2019.
%H A309099 J. Bloom and D. Saracino, <a href="https://arxiv.org/abs/1808.04221">On Criteria for rook equivalence of Ferrers boards</a>, arXiv:1808.04221 [math.CO], 2018.
%H A309099 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 A309099 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.
%H A309099 J. Bloom and D. Saracino, <a href="https://doi.org/10.1016/j.ejc.2018.08.006">On Criteria for rook equivalence of Ferrers boards</a>, European J. Combin., 76 (2018), 199-207.
%H A309099 Joachim König, <a href="https://mathoverflow.net/a/460384/231922">Closed-form for the number of partitions of n avoiding the partition (4,3,1)</a>, answer to question on MathOverflow (2023).
%F A309099 a(n) = A078567(n+1) - A002378(n-1) for n > 0 with a(0) = 1. - _Mikhail Kurkov_, Dec 20 2023 [verification needed]
%p A309099 b:= proc(n) option remember; `if`(n<1, [0$2],
%p A309099       (p-> p+[numtheory[tau](n), p[1]])(b(n-1)))
%p A309099     end:
%p A309099 a:= n-> b(n+1)[2]+`if`(n=0, 1, n*(1-n)):
%p A309099 seq(a(n), n=0..55);  # _Alois P. Heinz_, Dec 20 2023
%t A309099 b[n_] := b[n] = If[n < 1, {0, 0}, With[{p = b[n-1]},
%t A309099    p + {DivisorSigma[0, n], p[[1]]}]];
%t A309099 a[n_] := b[n+1][[2]] + If[n == 0, 1, n*(1-n)];
%t A309099 Table[a[n], {n, 0, 55}] (* _Jean-François Alcover_, Jan 29 2025, after _Alois P. Heinz_ *)
%o A309099 (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]
%Y A309099 Cf. A002378, A078567, A309097, A309098, A309058.
%K A309099 nonn
%O A309099 0,3
%A A309099 _Jonathan S. Bloom_, Jul 12 2019
%E A309099 More terms from _Alois P. Heinz_, Jul 12 2019