A101199 Number of partitions of n with rank 2 (the rank of a partition is the largest part minus the number of parts).
0, 0, 1, 0, 1, 1, 2, 2, 3, 3, 6, 6, 9, 10, 15, 16, 23, 27, 36, 42, 55, 64, 84, 98, 124, 147, 185, 217, 270, 318, 391, 461, 562, 661, 802, 942, 1132, 1331, 1592, 1864, 2220, 2597, 3077, 3593, 4240, 4940, 5811, 6758, 7916, 9192, 10737, 12438, 14488, 16755, 19459, 22465, 26024, 29987
Offset: 1
Keywords
Examples
a(6)=1 because the 11 partitions 6,51,42,411,33,321,3111,222,2211,21111,111111 have ranks 5,3,2,1,1,0,-1,-1,-2,-3,-5, respectively.
References
- George E. Andrews, The Theory of Partitions, Addison-Wesley, Reading, Mass., 1976.
Links
- Seiichi Manyama, Table of n, a(n) for n = 1..10000
Programs
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Maple
with(combinat): for n from 1 to 45 do P:=partition(n): c:=0: for j from 1 to nops(P) do if P[j][nops(P[j])]-nops(P[j])=2 then c:=c+1 else c:=c fi od: a[n]:=c: od: seq(a[n],n=1..45);
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Mathematica
Table[Count[Max[#]-Length[#]&/@IntegerPartitions[n],2],{n,60}] (* Harvey P. Dale, Dec 22 2018 *)
Formula
a(n) ~ Pi * exp(Pi*sqrt(2*n/3)) / (3 * 2^(9/2) * n^(3/2)). - Vaclav Kotesovec, May 26 2023
Extensions
More terms from Joerg Arndt, Oct 07 2012
Comments