This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A064173 #88 Jan 07 2025 19:33:57
%S A064173 0,1,1,2,3,5,6,10,13,19,25,35,45,62,80,106,136,178,225,291,366,466,
%T A064173 583,735,912,1140,1407,1743,2140,2634,3214,3932,4776,5807,7022,8495,
%U A064173 10225,12313,14762,17696,21136,25236,30030,35722,42367,50216,59368,70138,82665
%N A064173 Number of partitions of n with positive rank.
%C A064173 The rank of a partition is the largest summand minus the number of summands.
%C A064173 Also number of partitions of n with negative rank. - _Omar E. Pol_, Mar 05 2012
%C A064173 Column 1 of A208478. - _Omar E. Pol_, Mar 11 2012
%C A064173 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
%C A064173 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
%H A064173 Alois P. Heinz, <a href="/A064173/b064173.txt">Table of n, a(n) for n = 1..10000</a>
%H A064173 F. J. Dyson, <a href="https://archim.org.uk/eureka/archive/Eureka-8.pdf">Some guesses in the theory of partitions</a>, Eureka (Cambridge) 8 (1944), 10-15.
%H A064173 FindStat, <a href="http://www.findstat.org/StatisticsDatabase/St000145">St000145: The Dyson rank of a partition</a>
%H A064173 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 A064173 a(n) = (A000041(n) - A047993(n))/2.
%F A064173 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
%F A064173 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
%F A064173 G.f.: Sum_{k>=1} x^k * Product_{j=1..k} (1-x^(k+j-2))/(1-x^j). - _Seiichi Manyama_, Jan 25 2022
%F A064173 a(n)+A064174(n) = A000041(n). - _R. J. Mathar_, Feb 22 2023
%e A064173 a(20) = p(18) - p(13) + p(5) = 385 - 101 + 7 = 291.
%e A064173 From _Gus Wiseman_, Feb 09 2021: (Start)
%e A064173 The a(2) = 1 through a(9) = 13 partitions of positive rank:
%e A064173 (2) (3) (4) (5) (6) (7) (8) (9)
%e A064173 (31) (32) (33) (43) (44) (54)
%e A064173 (41) (42) (52) (53) (63)
%e A064173 (51) (61) (62) (72)
%e A064173 (411) (421) (71) (81)
%e A064173 (511) (422) (432)
%e A064173 (431) (441)
%e A064173 (521) (522)
%e A064173 (611) (531)
%e A064173 (5111) (621)
%e A064173 (711)
%e A064173 (5211)
%e A064173 (6111)
%e A064173 (End)
%p A064173 A064173 := proc(n)
%p A064173 a := 0 ;
%p A064173 for p in combinat[partition](n) do
%p A064173 r := max(op(p))-nops(p) ;
%p A064173 if r > 0 then
%p A064173 a := a+1 ;
%p A064173 end if;
%p A064173 end do:
%p A064173 a ;
%p A064173 end proc:
%p A064173 seq(A064173(n),n=0..40) ;# _Emeric Deutsch_, Dec 11 2004
%t A064173 Table[Count[IntegerPartitions[n], q_ /; First[q] > Length[q]], {n, 24}] (* _Clark Kimberling_, Feb 12 2014 *)
%t A064173 Table[Count[IntegerPartitions[n], p_ /; ! MemberQ[p, Max[Max[p], Length[p]]]], {n, 20}] (* _Clark Kimberling_, Feb 28 2014 *)
%t A064173 P = PartitionsP;
%t A064173 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;
%t A064173 a /@ Range[48] (* _Jean-François Alcover_, Jan 11 2020, after _Wouter Meeussen_ in A047993 *)
%o A064173 (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
%Y A064173 Note: A-numbers of ranking sequences are in parentheses below.
%Y A064173 The negative-rank version is also A064173 (A340788).
%Y A064173 The case of odd positive rank is A101707 (A340604).
%Y A064173 The case of even positive rank is A101708 (A340605).
%Y A064173 These partitions are ranked by (A340787).
%Y A064173 A063995/A105806 count partitions by rank.
%Y A064173 A072233 counts partitions by sum and length.
%Y A064173 A168659 counts partitions whose length is a multiple of the greatest part.
%Y A064173 A200750 counts partitions whose length and greatest part are coprime.
%Y A064173 - Rank -
%Y A064173 A064174 counts partitions of nonnegative/nonpositive rank (A324562/A324521).
%Y A064173 A101198 counts partitions of rank 1 (A325233).
%Y A064173 A257541 gives the rank of the partition with Heinz number n.
%Y A064173 A340601 counts partitions of even rank (A340602).
%Y A064173 A340692 counts partitions of odd rank (A340603).
%Y A064173 - Balance -
%Y A064173 A047993 counts balanced partitions (A106529).
%Y A064173 A340599 counts alt-balanced factorizations.
%Y A064173 A340653 counts balanced factorizations.
%Y A064173 Cf. A003114, A006141, A039900, A096401, A117193, A117409, A143773, A324516, A324518, A324520.
%K A064173 nonn
%O A064173 1,4
%A A064173 _Vladeta Jovovic_, Sep 19 2001