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A309097 Number of partitions of n avoiding the partition (4,2,1). 4

%I

%S 1,1,2,3,5,7,11,14,20,25,32,39,49,56,68,79,91,103,119,132,150,165,183,

%T 202,224,241,264,287,311,334,362,385,415,442,472,503,535,563,599,634,

%U 670,703,743,778,820,859,899,942,988,1027,1074,1119,1167,1214,1266

%N Number of partitions of n avoiding the partition (4,2,1).

%C 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 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 Jonathan Bloom, Nathan McNew, <a href="https://arxiv.org/abs/1908.03953">Counting pattern-avoiding integer partitions</a>, arXiv:1908.03953 [math.CO], 2019.

%H 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 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 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 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 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 Wikipedia, <a href="https://en.wikipedia.org/wiki/Nilpotent_matrix">Nilpotent matrix</a>

%Y Cf. A309098, A309099, A309058.

%K nonn

%O 0,3

%A _Jonathan S. Bloom_, Jul 12 2019

%E More terms from _Alois P. Heinz_, Jul 12 2019

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Last modified September 29 10:06 EDT 2020. Contains 337428 sequences. (Running on oeis4.)