This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A309097 #49 Sep 01 2025 20:04:37
%S A309097 1,1,2,3,5,7,11,14,20,25,32,39,49,56,68,79,91,103,119,132,150,165,183,
%T A309097 202,224,241,264,287,311,334,362,385,415,442,472,503,535,563,599,634,
%U A309097 670,703,743,778,820,859,899,942,988,1027,1074,1119,1167,1214,1266
%N A309097 Number of partitions of n avoiding the partition (4,2,1).
%C A309097 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.
%C A309097 Conjecture: for n > 0, a(n) is the number of ordered pairs (r, l) such that there exists a nilpotent matrix of order n whose rank is r and nilpotent index is l. Actually, such a matrix exists if and only if ceiling(n/(n-r)) <= l <= r+1, see my proof below. If this conjecture is true, then a(n) = (n^2 + 3n)/2 - A006590(n) for n > 0. - _Jianing Song_, Nov 04 2019
%H A309097 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 A309097 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 A309097 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 A309097 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., 76 (2018), 199-207.
%H A309097 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., 71 (2018), 246-267.
%H A309097 Jianing Song, <a href="/A309097/a309097.txt">Proof that there exists a nilpotent matrix of order n whose rank is r and nilpotent index is l if and only if ceiling(n/(n-r)) <= l <= r+1</a>
%H A309097 Wikipedia, <a href="https://en.wikipedia.org/wiki/Nilpotent_matrix">Nilpotent matrix</a>
%F A309097 G.f.: 1 + Sum_{i>=1} b(i) * ( 1 + Sum_{j>i} b(j) * ( 1 + b(j+1) ) ) where b(k)=x^k/(1-x^k). - _Christian Sievers_, Sep 01 2025
%o A309097 (PARI) lista(n)=my(b(k)=x^k/(1-x^k)+O(x*x^n));Vec(1+sum(i=1,n,b(i)*(1+sum(j=i+1,n,b(j)*(1+b(j+1)))))) \\ _Christian Sievers_, Sep 01 2025
%Y A309097 Cf. A309098, A309099, A309058.
%K A309097 nonn,changed
%O A309097 0,3
%A A309097 _Jonathan S. Bloom_, Jul 12 2019
%E A309097 More terms from _Alois P. Heinz_, Jul 12 2019