

A064173


Number of partitions of n with positive rank.


17



0, 1, 1, 2, 3, 5, 6, 10, 13, 19, 25, 35, 45, 62, 80, 106, 136, 178, 225, 291, 366, 466, 583, 735, 912, 1140, 1407, 1743, 2140, 2634, 3214, 3932, 4776, 5807, 7022, 8495, 10225, 12313, 14762, 17696, 21136, 25236, 30030, 35722, 42367, 50216, 59368, 70138, 82665
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OFFSET

1,4


COMMENTS

The rank of a partition is the largest summand minus the number of summands.
Also number of partitions of n with negative rank.  Omar E. Pol, Mar 05 2012
Column 1 of A208478.  Omar E. Pol, Mar 11 2012
Number of partitions p of n such that max(max(p), number of parts of p) is not a part of p.  Clark Kimberling, Feb 28 2014
The sequence enumerates the semigroup of partitions of positive rank for each number n. The semigroup is a subsemigroup of the monoid of partitions of nonnegative rank under the binary operation "*": Let A be the positive rank partition (a1,...,ak) where ak > k, and let B=(b1,...bj) with bj > j. Then let A*B be the partition (a1b1,...,a1bj,...,akb1,...,akbj), which has akbj > kj, thus having positive rank. For example, the partition (2,3,4) of 9 has rank 1, and its product with itself is (4,6,6,8,8,9,12,12,16) of 81, which has rank 7. A similar situation holds for partitions of negative rankthey are a subsemigroup of the monoid of nonpositive rank partitions.  Richard Locke Peterson, Jul 15 2018


REFERENCES

F. J. Dyson, Some guesses in the theory of partitions, Eureka (Cambridge) 8 (1944), 1015.


LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Mircea Merca, Rank partition functions and truncated theta identities, arXiv:2006.07705 [math.CO], 2020.


FORMULA

a(n) = (A000041(n)  A047993(n))/2.
a(n) = p(n2)  p(n7) + p(n15)  ...  (1)^k*p(n(3*k^2+k)/2) + ..., where p() is A000041().  Vladeta Jovovic, Aug 04 2004
G.f.: Product_{k>=1} (1/(1q^k)) * Sum_{k>=1} ( (1)^k * (q^(3*k^2/2+k/2))) (conjectured).  Thomas Baruchel, May 12 2018


EXAMPLE

a(20) = p(18)  p(13) + p(5) = 385  101 + 7 = 291.


MAPLE

with(combinat): for n from 1 to 30 do P:=partition(n): c:=0: for j from 1 to nops(P) do if P[j][nops(P[j])]>nops(P[j]) then c:=c+1 else c:=c fi od: a[n]:=c: od: seq(a[n], n=1..30); # Emeric Deutsch, Dec 11 2004


MATHEMATICA

Table[Count[IntegerPartitions[n], q_ /; First[q] > Length[q]], {n, 24}] (* Clark Kimberling, Feb 12 2014 *)
Table[Count[IntegerPartitions[n], p_ /; ! MemberQ[p, Max[Max[p], Length[p]]]], {n, 20}] (* Clark Kimberling, Feb 28 2014 *)
P = PartitionsP;
a[n_] := (P[n]  Sum[(1)^k (P[n  (3k^2  k)/2]  P[n  (3k^2 + k)/2]), {k, 1, Floor[(1 + Sqrt[1 + 24n])/6]}])/2;
a /@ Range[48] (* JeanFrançois Alcover, Jan 11 2020, after Wouter Meeussen in A047993 *)


CROSSREFS

Cf. A063995, A064174.
Sequence in context: A027339 A039837 A039838 * A145724 A039843 A305937
Adjacent sequences: A064170 A064171 A064172 * A064174 A064175 A064176


KEYWORD

nonn,changed


AUTHOR

Vladeta Jovovic, Sep 19 2001


STATUS

approved



