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
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..10000
F. J. Dyson, Some guesses in the theory of partitions, Eureka (Cambridge) 8 (1944), 10-15.
FindStat, St000145: The Dyson rank of a partition
Mircea Merca, Rank partition functions and truncated theta identities, arXiv:2006.07705 [math.CO], 2020.
FORMULA
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
G.f.: Sum_{k>=1} x^k * Product_{j=1..k} (1-x^(k+j-2)/(1-x^j). - Seiichi Manyama, Jan 25 2022
EXAMPLE
a(20) = p(18) - p(13) + p(5) = 385 - 101 + 7 = 291.
From Gus Wiseman, Feb 09 2021: (Start)
The a(2) = 1 through a(9) = 13 partitions of positive rank:
(2) (3) (4) (5) (6) (7) (8) (9)
(31) (32) (33) (43) (44) (54)
(41) (42) (52) (53) (63)
(51) (61) (62) (72)
(411) (421) (71) (81)
(511) (422) (432)
(431) (441)
(521) (522)
(611) (531)
(5111) (621)
(711)
(5211)
(6111)
(End)
MAPLE
A064173 := proc(n)
a := 0 ;
for p in combinat[partition](n) do
r := max(op(p))-nops(p) ;
if r > 0 then
a := a+1 ;
end if;
end do:
a ;
end proc:
seq(A064173(n), n=0..40) ; # 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;
PROG
(PARI) my(N=66, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, x^k*prod(j=1, k, (1-x^(k+j-2))/(1-x^j))))) \\ Seiichi Manyama, Jan 25 2022
CROSSREFS
Note: A-numbers of ranking sequences are in parentheses below.
These partitions are ranked by (A340787).
A072233 counts partitions by sum and length.
A168659 counts partitions whose length is a multiple of the greatest part.
A200750 counts partitions whose length and greatest part are coprime.
- Rank -
A257541 gives the rank of the partition with Heinz number n.
- Balance -
A340599 counts alt-balanced factorizations.
A340653 counts balanced factorizations.
KEYWORD
nonn
AUTHOR
Vladeta Jovovic, Sep 19 2001
STATUS
approved