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A006827 Number of partitions of 2n with all subsums different from n.

%I A006827 M1351 #52 Apr 20 2024 00:00:04
%S A006827 1,2,5,8,17,24,46,64,107,147,242,302,488,629,922,1172,1745,2108,3104,
%T A006827 3737,5232,6419,8988,10390,14552,17292,23160,27206,36975,41945,57058,
%U A006827 65291,85895,99384,130443,145283,193554,218947,281860,316326,413322,454229,594048
%N A006827 Number of partitions of 2n with all subsums different from n.
%C A006827 Partitions of this type are also called non-biquanimous partitions. - _Gus Wiseman_, Apr 19 2024
%D A006827 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006827 Fausto A. C. Cariboni, <a href="/A006827/b006827.txt">Table of n, a(n) for n = 1..140</a> (terms 1..89 from Alois P. Heinz)
%H A006827 P. Erdős, J. L. Nicolas and A. Sárközy, <a href="http://dx.doi.org/10.1016/0012-365X(89)90086-1">On the number of partitions of n without a given subsum (I)</a>, Discrete Math., 75 (1989), 155-166 = Annals Discrete Math. Vol. 43, Graph Theory and Combinatorics 1988, ed. B. Bollobas.
%F A006827 a(n) = A000041(2*n) - A002219(n).
%F A006827 a(n) = A046663(2*n,n).
%e A006827 From _Gus Wiseman_, Apr 19 2024: (Start)
%e A006827 The a(1) = 1 through a(5) = 17 partitions (A = 10):
%e A006827   (2)  (4)   (6)    (8)     (A)
%e A006827        (31)  (42)   (53)    (64)
%e A006827              (51)   (62)    (73)
%e A006827              (222)  (71)    (82)
%e A006827              (411)  (332)   (91)
%e A006827                     (521)   (433)
%e A006827                     (611)   (442)
%e A006827                     (5111)  (622)
%e A006827                             (631)
%e A006827                             (721)
%e A006827                             (811)
%e A006827                             (3331)
%e A006827                             (4222)
%e A006827                             (6211)
%e A006827                             (7111)
%e A006827                             (22222)
%e A006827                             (61111)
%e A006827 (End)
%p A006827 b:= proc(n, i, s) option remember;
%p A006827       `if`(0 in s or n in s, 0, `if`(n=0, 1, `if`(i<1, 0, b(n, i-1, s)+
%p A006827       `if`(i<=n, b(n-i, i, select(y-> 0<=y and y<=n-i,
%p A006827                  map(x-> [x, x-i][], s))), 0))))
%p A006827     end:
%p A006827 a:= n-> b(2*n, 2*n, {n}):
%p A006827 seq(a(n), n=1..25);  # _Alois P. Heinz_, Jul 10 2012
%t A006827 b[n_, i_, s_] := b[n, i, s] = If[MemberQ[s, 0 | n], 0, If[n == 0, 1, If[i < 1, 0, b[n, i-1, s] + If[i <= n, b[n-i, i, Select[Flatten[Transpose[{s, s-i}]], 0 <= # <= n-i &]], 0]]]]; a[n_] := b[2*n, 2*n, {n}]; Table[Print[an = a[n]]; an, {n, 1, 25}] (* _Jean-François Alcover_, Nov 12 2013, after _Alois P. Heinz_ *)
%o A006827 (Python)
%o A006827 from itertools import combinations_with_replacement
%o A006827 from collections import Counter
%o A006827 from sympy import npartitions
%o A006827 from sympy.utilities.iterables import partitions
%o A006827 def A006827(n): return npartitions(n<<1)-len({tuple(sorted((p+q).items())) for p, q in combinations_with_replacement(tuple(Counter(p) for p in partitions(n)),2)}) # _Chai Wah Wu_, Sep 20 2023
%Y A006827 The complement is counted by A002219, ranks A357976.
%Y A006827 Central diagonal of A046663.
%Y A006827 The strict case is A321142, even bisection of A371794 (odd A078408).
%Y A006827 This is the "bi-" version of A321451, ranks A321453.
%Y A006827 Column k = 0 of A367094.
%Y A006827 These partitions have Heinz numbers A371731.
%Y A006827 Even bisection of A371795 (odd A058695).
%Y A006827 A371783 counts k-quanimous partitions.
%Y A006827 Cf. A035470, A064914, A237258, A305551, A321452, A365543, A365663, A366320, A371736, A371782, A371792.
%K A006827 nonn,nice
%O A006827 1,2
%A A006827 _N. J. A. Sloane_
%E A006827 More terms from _Don Reble_, Nov 03 2001
%E A006827 More terms from _Alois P. Heinz_, Jul 10 2012