A000701 One half of number of non-self-conjugate partitions; also half of number of asymmetric Ferrers graphs with n nodes.

0, 0, 1, 1, 2, 3, 5, 7, 10, 14, 20, 27, 37, 49, 66, 86, 113, 146, 190, 242, 310, 392, 497, 623, 782, 973, 1212, 1498, 1851, 2274, 2793, 3411, 4163, 5059, 6142, 7427, 8972, 10801, 12989, 15572, 18646, 22267, 26561, 31602, 37556, 44533, 52743, 62338, 73593

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

Also number of cycle types of odd permutations.

Also number of partitions of n with an odd number of even parts. There is no restriction on the odd parts. - N. Sato, Jul 20 2005. E.g., a(6)=5 because we have [6],[4,1,1],[3,2,1],[2,2,2] and [2,1,1,1,1]. - Emeric Deutsch, Mar 02 2006

Also number of partitions of n with largest part not congruent to n modulo 2: a(2*n)=A027193(2*n), a(2*n+1)=A027187(2*n+1); a(n)=A000041(n)-A046682(n). - Reinhard Zumkeller, Apr 22 2006

From Gus Wiseman, Mar 31 2022: (Start)

Also the number of integer partitions of n with Heinz number greater than that of their conjugate, where the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). These partitions are ranked by A352490. The complement is counted by A046682. For example, the a(n) partitions for n = 2...8 are:

(11) (111) (211) (221) (222) (331) (2222)

(1111) (2111) (2211) (2221) (3221)

(11111) (3111) (3211) (3311)

(21111) (22111) (22211)

(111111) (31111) (32111)

(211111) (41111)

(1111111) (221111)

(311111)

(2111111)

(11111111)

Also the number of integer partitions of n with Heinz number less than that of their conjugate, ranked by A352487. For example, the a(n) partitions for n = 2...8 are:

(2) (3) (4) (5) (6) (7) (8)

(31) (32) (33) (43) (44)

(41) (42) (52) (53)

(51) (61) (62)

(411) (322) (71)

(421) (422)

(511) (431)

(521)

(611)

(5111)

(End)

Examples

			G.f. = x^2 + x^3 + 2*x^4 + 3*x^5 + 5*x^6 + 7*x^7 + 10*x^8 + 14*x^9 + ...
		

References

• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
• Brian Hopkins and Michael A. Jones, Shift-Induced Dynamical Systems on Partitions and Compositions, Electron. J. Combin. 13 (2006), Research Paper 80, see p. 10.
• M. Osima, On the irreducible representations of the symmetric group, Canad. J. Math., 4 (1952), 381-384.

Crossrefs

Cf. A000041, A027187, A027193, A046682, A118302, A236559, A236914.

A000700 counts self-conjugate partitions, ranked by A088902.

A330644 counts non-self-conjugate partitions, ranked by A352486.

Heinz number (rank) and partition:

- A122111 = rank of conjugate.

- A296150 = parts of partition, conjugate A321649.

- A352487 = rank less than conjugate, counted by A000701.

- A352488 = rank greater than or equal to conjugate, counted by A046682.

- A352489 = rank less than or equal to conjugate, counted by A046682.

- A352490 = rank greater than conjugate, counted by A000701.

- A352491 = rank minus conjugate.

Cf. A114088, A115994, A171966, A238352, A258116, A325039.

Programs

Maple

with(combinat); A000701 := n->(numbpart(n)-A000700(n))/2;

Mathematica

a41 = PartitionsP; a700[n_] := SeriesCoefficient[ Product[1 + x^k, {k, 1, n, 2}], {x, 0, n}]; a[0] = 0; a[n_] := (a41[n] - a700[n])/2; Table[a[n], {n, 0, 48}] (* Jean-François Alcover, Feb 21 2012, after first formula *)
a[ n_] := SeriesCoefficient[ (1 / QPochhammer[ x] - 1 / QPochhammer[ x, -x]) / 2, {x, 0, n}]; (* Michael Somos, Aug 25 2015 *)
a[ n_] := SeriesCoefficient[ (1 - EllipticTheta[ 4, 0, x^2]) / (2 QPochhammer[ x]), {x, 0, n}]; (* Michael Somos, Aug 25 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x] Sum[ x^(2 k) / QPochhammer[ x^2, x^2, k], {k, 1, n/2, 2}], {x, 0, n}] (* Michael Somos, Aug 25 2015 *)
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ (1 / QPochhammer[ x, x, k]^2 - 1 / QPochhammer[ x^2, x^2, k]) x^k^2, {k, Sqrt@n}] / 2, {x, 0, n}]]; (* Michael Somos, Aug 25 2015 *)
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[Length[Select[IntegerPartitions[n],Times@@Prime/@#>Times@@Prime/@conj[#]&]],{n,0,15}] (* Gus Wiseman, Mar 31 2022 *)

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (1 - eta(x^2 + A)^2 / eta(x^4 + A) ) / (2 * eta(x + A)), n))}; /* Michael Somos, Aug 25 2015 */

PARI

q='q+O('q^60); concat([0, 0], Vec((1-eta(q^2)^2/eta(q^4))/(2*eta(q)))) \\ Altug Alkan, Sep 26 2018

Formula

a(n) = (A000041(n) - A000700(n))/2.

From Bill Gosper, Aug 08 2005: (Start)

Sum a(n) q^n = q^2 + q^3 + 2 q^4 + 3 q^5 + 5 q^6 + 7 q^7 + ...

= -( Sum_{n>=1} (-q^2)^(n^2) ) / ( Sum_{ n = -oo..oo } (-1)^n q^(n(3n-1)/2) )

= (- q; q){oo} Sum{n>=1} q^(2(2n-1))/(q^2;q^2)_{2n-1}

= (1/(q;q)_oo - 1/(q;-q)_oo)/2

= (1/(q;q)_oo - (-q;q^2)_oo)/2

= Sum{k>=0} ( 1/((q;q)_k)^2 - 1/(q^2;q^2)_k ) q^(k^2)/2

using the "q-Pochhammer" notation (a;q)n := Product{k=0..n-1} (1 - a*q^k).

(End)

a(n) = p(n-2) - p(n-8) + p(n-18) - p(n-32) + ... + (-1)^(k+1)*p(n-2*k^2) + ..., where p() is A000041(). E.g., a(20) = p(18) - p(12) + p(2) = 385 - 77 + 2 = 310. - Vladeta Jovovic, Aug 08 2004

G.f.: (1/2)*(1 - Product_{j>=1} (1-x^(2j))/(1+x^(2j)))/Product_{j>=1} (1 - x^j). - Emeric Deutsch, Mar 02 2006

a(2*n) = A236559(n). a(2*n + 1) = A236914(n). - Michael Somos, Aug 25 2015

a(n) = A330644(n)/2. - Omar E. Pol, Jan 10 2020

a(n) = A000041(n) - A046682(n) = A046682(n) - A000700(n). - Gus Wiseman, Mar 31 2022

Extensions

Better description and more terms from Christian G. Bower, Apr 27 2000