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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

%I

%S 1,1,2,3,5,6,9,12,13,17,23,25,30,37,38,48,61,61,67,85,89,105,120,122,

%T 138,163,174,193,216,231,252,289,289,324,369,389,428,477,481,519,587,

%U 628,661,729,760,827,902,940,999,1095,1140,1238,1337,1380,1464,1616

%N 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.

%C 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.

%C Can anyone supply a generating function or asymptotic formula?

%H Vaclav Kotesovec, <a href="/A074658/b074658.txt">Table of n, a(n) for n = 0..500</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FerrersDiagram.html">Ferrers Diagram.</a>

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

%t 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 *)

%Y The first and second differences are in A074659 and A074660.

%K nonn

%O 0,3

%A _Dean Hickerson_, Aug 29 2002

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Last modified April 6 08:06 EDT 2020. Contains 333267 sequences. (Running on oeis4.)