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Number of partitions of n with nonnegative rank.
46

%I #80 May 26 2023 08:54:24

%S 1,1,2,3,4,6,9,12,17,23,31,42,56,73,96,125,161,207,265,336,426,536,

%T 672,840,1046,1296,1603,1975,2425,2970,3628,4417,5367,6503,7861,9482,

%U 11412,13702,16423,19642,23447,27938,33231,39453,46767,55342,65386,77135

%N Number of partitions of n with nonnegative rank.

%C The rank of a partition is the largest summand minus the number of summands.

%C This sequence (up to proof) equals "partitions of 2n with even number of parts, ending in 1, with max descent of 1, where the number of odd parts in odd places equals the number of odd parts in even places. (See link and 2nd Mathematica line.) - _Wouter Meeussen_, Mar 29 2013

%C Number of partitions p of n such that max(max(p), number of parts of p) is a part of p. - _Clark Kimberling_, Feb 28 2014

%C From _Gus Wiseman_, Mar 09 2019: (Start)

%C Also the number of integer partitions of n with maximum part greater than or equal to the number of parts. The Heinz numbers of these integer partitions are given by A324521. For example, the a(1) = 1 through a(8) = 12 partitions are:

%C (1) (2) (3) (4) (5) (6) (7) (8)

%C (21) (22) (32) (33) (43) (44)

%C (31) (41) (42) (52) (53)

%C (311) (51) (61) (62)

%C (321) (322) (71)

%C (411) (331) (332)

%C (421) (422)

%C (511) (431)

%C (4111) (521)

%C (611)

%C (4211)

%C (5111)

%C Also the number of integer partitions of n with maximum part less than or equal to the number of parts. The Heinz numbers of these integer partitions are given by A324562. For example, the a(1) = 1 through a(8) = 12 partitions are:

%C (1) (11) (21) (22) (221) (222) (322) (332)

%C (111) (211) (311) (321) (331) (2222)

%C (1111) (2111) (2211) (2221) (3221)

%C (11111) (3111) (3211) (3311)

%C (21111) (4111) (4211)

%C (111111) (22111) (22211)

%C (31111) (32111)

%C (211111) (41111)

%C (1111111) (221111)

%C (311111)

%C (2111111)

%C (11111111)

%C (End)

%H Seiichi Manyama, <a href="/A064174/b064174.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Alois P. Heinz)

%H Cristina Ballantine and Mircea Merca, <a href="https://arxiv.org/abs/1710.05960">Bisected theta series, least r-gaps in partitions, and polygonal numbers</a>, arXiv:1710.05960 [math.CO], 2017.

%H Rekha Biswal, <a href="http://mathoverflow.net/questions/125709">bijection between number of partitions of 2n satisfying certain conditions with number of partitions of n</a>, Mathoverflow.

%H Mircea Merca, <a href="https://arxiv.org/abs/2006.07705">Rank partition functions and truncated theta identities</a>, arXiv:2006.07705 [math.CO], 2020.

%F a(n) = (A000041(n) + A047993(n))/2.

%F a(n) = p(n-1) - p(n-5) + p(n-12) - ... -(-1)^k*p(n-(3*k^2-k)/2) + ..., where p() is A000041(). - _Vladeta Jovovic_, Aug 04 2004

%F G.f.: Sum_{n>=1} x^n * Product_{k=1..n} (1 - x^(n+k-1))/(1 - x^k). - _Paul D. Hanna_, Aug 03 2015

%F A064173(n) + a(n) = A000041(n). - _R. J. Mathar_, Feb 22 2023

%F G.f.: (1/Product_{k>=1} (1-x^k)) * Sum_{k>=1} (-1)^(k-1) * x^(k*(3*k-1)/2). - _Seiichi Manyama_, May 21 2023

%e a(20) = p(19) - p(15) + p(8) = 490 - 176 + 22 = 336.

%p f:= n -> add((-1)^(k+1)*combinat:-numbpart(n-(3*k^2-k)/2),k=1..floor((1+sqrt(24*n+1))/6)):

%p map(f, [$1..100]); # _Robert Israel_, Aug 03 2015

%t Table[Count[IntegerPartitions[n], q_ /; First[q] >= Length[q]], {n, 16}]

%t (* also *)

%t Table[Count[IntegerPartitions[2n],q_/;Last[q]===1 && Max[q-PadRight[Rest[q],Length[q]]]<=1 && Count[First/@Partition[q,2],_?OddQ]==Count[Last/@Partition[q,2],_?OddQ]],{n,16}]

%t (* also *)

%t Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, Max[Max[p], Length[p]]]], {n, 50}] (* _Clark Kimberling_, Feb 28 2014 *)

%o (PARI) {a(n) = my(A=1); A = sum(m=0,n,x^m*prod(k=1,m,(1-x^(m+k-1))/(1-x^k +x*O(x^n)))); polcoeff(A,n)}

%o for(n=1,60,print1(a(n),", ")) \\ _Paul D. Hanna_, Aug 03 2015

%o (PARI) my(N=50, x='x+O('x^N)); Vec(1/prod(k=1, N, 1-x^k)*sum(k=1, N, (-1)^(k-1)*x^(k*(3*k-1)/2))) \\ _Seiichi Manyama_, May 21 2023

%Y Cf. A063995, A064173 (complement).

%Y Row sums of triangle A105806.

%Y Cf. A003114, A006141, A039900, A047993, A090858, A106529, A133121.

%Y Cf. A324516, A324518, A324520, A324521, A324522, A324560, A324562, A324572.

%K nonn

%O 1,3

%A _Vladeta Jovovic_, Sep 20 2001

%E Mathematica programs modified by _Clark Kimberling_, Feb 12 2014