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 A064173 Number of partitions of n with positive rank. 17
 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 (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 Alois P. Heinz, Table of n, a(n) for n = 1..10000 Mircea Merca, Rank partition functions and truncated theta identities, arXiv:2006.07705 [math.CO], 2020. 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 *) 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 (* Jean-François Alcover, Jan 11 2020, after Wouter Meeussen in A047993 *) CROSSREFS Cf. A063995, A064174. Sequence in context: A027339 A039837 A039838 * A145724 A039843 A305937 Adjacent sequences:  A064170 A064171 A064172 * A064174 A064175 A064176 KEYWORD nonn,changed AUTHOR Vladeta Jovovic, Sep 19 2001 STATUS approved

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Last modified September 24 17:09 EDT 2020. Contains 337321 sequences. (Running on oeis4.)