A180682 a(n) is the largest path count within the (right-aligned Ferrers plots of the) partitions of n.

1, 1, 1, 2, 2, 3, 3, 5, 6, 7, 9, 10, 14, 16, 19, 23, 28, 32, 42, 47, 56, 66, 76, 90, 107, 132, 146, 174, 202, 230, 268, 314, 359, 429, 471, 561, 645, 735, 847, 979, 1094, 1247, 1430, 1593, 1859, 2123, 2420, 2768, 3172, 3503, 4019, 4481, 5096, 5691, 6384, 7168, 8151

Offset: 1

Author

Wouter Meeussen, Sep 16 2010

Keywords

• nonn

Examples

			a(13)=14 because the partition of 13 with largest path count is 4432, producing {1,1,1,1}, {1,2,3,4}, {2,5,9}, {5,14} with closing value 14.
		

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..87

Programs

Mathematica

pathcount[p_] := Block[{ferr = (0*Range[#1] &) /@ p}, Last[Fold[Rest[FoldList[Plus, 0, Drop[#1, Length[#1] - Length[#2]] + #2]] &, 1 + First[ferr], Rest[ferr]]]]; f[n_] := Block[{k = mx = 1}, While[k < n + 1, m = Max[ pathcount /@ IntegerPartitions[n, {k}]]; If[m > mx, mx = m]; k++]; mx]; k = 1; lst = {}; While[k < 58, a = f@ k; Print[{k, a}]; AppendTo[lst, a]; k++ ]; lst

Extensions

a(37) onwards from Robert G. Wilson v, Sep 19 2010