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A074658 Number of 'convex' partitions of n; i.e., partitions of n for which the Ferrers graph is the intersection of a convex set and the integer lattice. 3
1, 1, 2, 3, 5, 6, 9, 12, 13, 17, 23, 25, 30, 37, 38, 48, 61, 61, 67, 85, 89, 105, 120, 122, 138, 163, 174, 193, 216, 231, 252, 289, 289, 324, 369, 389, 428, 477, 481, 519, 587, 628, 661, 729, 760, 827, 902, 940, 999, 1095, 1140, 1238, 1337, 1380, 1464, 1616 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Equivalently, a partition n = a_1 + ... + a_m with a_1 >= ... >= a_m >= 1 is convex if a_j >= floor(a_i + (a_k - a_i)*(j-i)/(k-i)) whenever 1 <= i < j < k <= m.

Can anyone supply a generating function or asymptotic formula?

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..500

Eric Weisstein's World of Mathematics, Ferrers Diagram.

EXAMPLE

The only partition of 5 which is not convex is 5 = 3 + 1 + 1, so a(5) = A000041(5) - 1 = 6.

MATHEMATICA

f[1, 1, 0, 0]=1; f[n_, m_, r_, s_] := Module[{nn, rr, ss}, If[GCD[r, s]!=1||2r*n>(m+1)(2r+s*m)||(nn=n-m*s+(r+1)(s-1)/2)<m-r, 0, f[n, m, r, s]=Sum[If[s*rr>=r*ss, f[nn, m-r, rr, ss], 0], {rr, 0, m-r-1}, {ss, 0, nn-m+r}]]]; a[n_] := Module[{r, s}, If[n<=1, 1, Sum[f[n, m, r, s], {m, 1, n}, {r, 0, m-1}, {s, 0, n-m}]]] (* f[n, m, r, s] = number of convex partitions of n into m parts, with bottom slope of convex hull equal to r/s *)

CROSSREFS

The first and second differences are in A074659 and A074660.

Sequence in context: A227070 A032718 A086191 * A186106 A329161 A124866

Adjacent sequences:  A074655 A074656 A074657 * A074659 A074660 A074661

KEYWORD

nonn

AUTHOR

Dean Hickerson, Aug 29 2002

STATUS

approved

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Last modified February 21 23:21 EST 2020. Contains 332113 sequences. (Running on oeis4.)