A064173 Number of partitions of n with positive rank.

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

Offset: 1

Author

Vladeta Jovovic, Sep 19 2001

Keywords

• nonn

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

Examples

			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)
		

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.

Crossrefs

Note: A-numbers of ranking sequences are in parentheses below.

The negative-rank version is also A064173 (A340788).

The case of odd positive rank is A101707 (A340604).

The case of even positive rank is A101708 (A340605).

These partitions are ranked by (A340787).

A063995/A105806 count partitions by rank.

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 -

A064174 counts partitions of nonnegative/nonpositive rank (A324562/A324521).

A101198 counts partitions of rank 1 (A325233).

A257541 gives the rank of the partition with Heinz number n.

A340601 counts partitions of even rank (A340602).

A340692 counts partitions of odd rank (A340603).

- Balance -

A047993 counts balanced partitions (A106529).

A340599 counts alt-balanced factorizations.

A340653 counts balanced factorizations.

Cf. A003114, A006141, A039900, A096401, A117193, A117409, A143773, A324516, A324518, A324520.

Programs

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;
a /@ Range[48] (* Jean-François Alcover, Jan 11 2020, after Wouter Meeussen in A047993 *)

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

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

G.f.: Sum_{k>=1} x^k * Product_{j=1..k} (1-x^(k+j-2))/(1-x^j). - Seiichi Manyama, Jan 25 2022

a(n)+A064174(n) = A000041(n). - R. J. Mathar, Feb 22 2023