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A064173 Number of partitions of n with positive rank. 16
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 (list; graph; refs; listen; history; text; internal format)
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 rank--they 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), 10-15.

LINKS

Table of n, a(n) for n=1..48.

FORMULA

a(n) = (A000041(n) - A047993(n))/2.

a(n) = p(n-2) - p(n-7) + p(n-15) - ... - (-1)^k*p(n-(3*k^2+k)/2) + ..., where p() is A000041(). - Vladeta Jovovic, Aug 04 2004

G.f.: Product_{k>=1} (1/(1-q^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 *)

CROSSREFS

Cf. A063995, A064174.

Sequence in context: A027339 A039837 A039838 * A145724 A039843 A305937

Adjacent sequences:  A064170 A064171 A064172 * A064174 A064175 A064176

KEYWORD

nonn

AUTHOR

Vladeta Jovovic, Sep 19 2001

STATUS

approved

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Last modified October 16 21:10 EDT 2019. Contains 328103 sequences. (Running on oeis4.)