OFFSET

0,3

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.

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

LINKS

Jonathan Bloom, 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., 76 (2018), 199-207.

J. Bloom and D. Saracino On Criteria for rook equivalence of Ferrers boards, European J. Combin., 71 (2018), 246-267.

Wikipedia, Nilpotent matrix

CROSSREFS

KEYWORD

nonn

AUTHOR

Jonathan S. Bloom, Jul 12 2019

EXTENSIONS

More terms from Alois P. Heinz, Jul 12 2019

STATUS

approved