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Number of integer partitions of n of even rank.
27

%I #29 May 04 2024 05:09:29

%S 1,1,0,3,1,5,3,11,8,18,16,34,33,57,59,98,105,159,179,262,297,414,478,

%T 653,761,1008,1184,1544,1818,2327,2750,3480,4113,5137,6078,7527,8899,

%U 10917,12897,15715,18538,22431,26430,31805,37403,44766,52556,62620,73379

%N Number of integer partitions of n of even rank.

%C The Dyson rank of a nonempty partition is its maximum part minus its number of parts. For this sequence, the rank of an empty partition is 0.

%H Alois P. Heinz, <a href="/A340601/b340601.txt">Table of n, a(n) for n = 0..10000</a>

%H Freeman J. Dyson, <a href="https://doi.org/10.1016/S0021-9800(69)80006-2">A new symmetry of partitions</a>, Journal of Combinatorial Theory 7.1 (1969): 56-61.

%H FindStat, <a href="http://www.findstat.org/StatisticsDatabase/St000145">St000145: The Dyson rank of a partition</a>

%F G.f.: 1 + Sum_{i, j>0} q^(i*j) * ( (1+(-1)^(i+j))/2 + Sum_{k>0} q^k * q_binomial(k,i-2) * (1+(-1)^(i+j+k))/2 ). - _John Tyler Rascoe_, Apr 15 2024

%F a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*n*sqrt(3)). - _Vaclav Kotesovec_, Apr 17 2024

%e The a(1) = 1 through a(9) = 18 partitions (empty column indicated by dot):

%e (1) . (3) (22) (5) (42) (7) (44) (9)

%e (21) (41) (321) (43) (62) (63)

%e (111) (311) (2211) (61) (332) (81)

%e (2111) (322) (521) (333)

%e (11111) (331) (2222) (522)

%e (511) (4211) (531)

%e (2221) (32111) (711)

%e (4111) (221111) (4221)

%e (31111) (4311)

%e (211111) (6111)

%e (1111111) (32211)

%e (33111)

%e (51111)

%e (222111)

%e (411111)

%e (3111111)

%e (21111111)

%e (111111111)

%p b:= proc(n, i, r) option remember; `if`(n=0, 1-max(0, r),

%p `if`(i<1, 0, b(n, i-1, r) +b(n-i, min(n-i, i), 1-

%p `if`(r<0, irem(i, 2), r))))

%p end:

%p a:= n-> b(n$2, -1):

%p seq(a(n), n=0..55); # _Alois P. Heinz_, Jan 22 2021

%t Table[If[n==0,1,Length[Select[IntegerPartitions[n],EvenQ[Max[#]-Length[#]]&]]],{n,0,30}]

%t (* Second program: *)

%t b[n_, i_, r_] := b[n, i, r] = If[n == 0, 1 - Max[0, r], If[i < 1, 0, b[n, i - 1, r] + b[n - i, Min[n - i, i], 1 - If[r < 0, Mod[i, 2], r]]]];

%t a[n_] := b[n, n, -1];

%t a /@ Range[0, 55] (* _Jean-François Alcover_, May 10 2021, after _Alois P. Heinz_ *)

%o (PARI)

%o p_q(k) = {prod(j=1, k, 1-q^j); }

%o GB_q(N, M)= {if(N>=0 && M>=0, p_q(N+M)/(p_q(M)*p_q(N)), 0 ); }

%o A_q(N) = {my(q='q+O('q^N), g=1+sum(i=1,N, sum(j=1,N/i, q^(i*j) * ( ((1/2)*(1+(-1)^(i+j))) + sum(k=1,N-(i*j), ((q^k)*GB_q(k,i-2)) * ((1/2)*(1+(-1)^(i+j+k)))))))); Vec(g)}

%o A_q(50) \\ _John Tyler Rascoe_, Apr 15 2024

%Y Note: Heinz numbers are given in parentheses below.

%Y The positive case is A101708 (A340605).

%Y The Heinz numbers of these partitions are A340602.

%Y The odd version is A340692 (A340603).

%Y - Rank -

%Y A047993 counts partitions of rank 0 (A106529).

%Y A072233 counts partitions by sum and length.

%Y A101198 counts partitions of rank 1 (A325233).

%Y A101707 counts partitions of odd positive rank (A340604).

%Y A101708 counts partitions of even positive rank (A340605).

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

%Y A340653 counts factorizations of rank 0.

%Y - Even -

%Y A024430 counts set partitions of even length.

%Y A027187 counts partitions of even length (A028260).

%Y A027187 (also) counts partitions of even maximum (A244990).

%Y A034008 counts compositions of even length.

%Y A035363 counts partitions into even parts (A066207).

%Y A052841 counts ordered set partitions of even length.

%Y A058696 counts partitions of even numbers (A300061).

%Y A067661 counts strict partitions of even length (A030229).

%Y A236913 counts even-length partitions of even numbers (A340784).

%Y A339846 counts factorizations of even length.

%Y Cf. A000041, A006141, A039900, A064174, A067538, A117409, A200750, A324516.

%K nonn

%O 0,4

%A _Gus Wiseman_, Jan 21 2021