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

1, 1, 2, 3, 5, 7, 11, 14, 20, 25, 32, 39, 49, 56, 68, 79, 91, 103, 119, 132, 150, 165, 183, 202, 224, 241, 264, 287, 311, 334, 362, 385, 415, 442, 472, 503, 535, 563, 599, 634, 670, 703, 743, 778, 820, 859, 899, 942, 988, 1027, 1074, 1119, 1167, 1214, 1266

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

Table of n, a(n) for n=0..54.

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.

Jianing Song, 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

Wikipedia, Nilpotent matrix

Crossrefs

Cf. A309098, A309099, A309058.

Sequence in context: A252797 A325333 A036608 * A309098 A136185 A319471

Adjacent sequences: A309094 A309095 A309096 * A309098 A309099 A309100

Keyword

nonn

Author

Jonathan S. Bloom, Jul 12 2019

Extensions

More terms from Alois P. Heinz, Jul 12 2019

Status

approved