A006827 - OEIS

%I M1351 #52 Apr 20 2024 00:00:04

%S 1,2,5,8,17,24,46,64,107,147,242,302,488,629,922,1172,1745,2108,3104,

%T 3737,5232,6419,8988,10390,14552,17292,23160,27206,36975,41945,57058,

%U 65291,85895,99384,130443,145283,193554,218947,281860,316326,413322,454229,594048

%N Number of partitions of 2n with all subsums different from n.

%C Partitions of this type are also called non-biquanimous partitions. - _Gus Wiseman_, Apr 19 2024

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

%H 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 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 a(n) = A000041(2*n) - A002219(n).

%F a(n) = A046663(2*n,n).

%e From _Gus Wiseman_, Apr 19 2024: (Start)

%e The a(1) = 1 through a(5) = 17 partitions (A = 10):

%e   (2)  (4)   (6)    (8)     (A)

%e        (31)  (42)   (53)    (64)

%e              (51)   (62)    (73)

%e              (222)  (71)    (82)

%e              (411)  (332)   (91)

%e                     (521)   (433)

%e                     (611)   (442)

%e                     (5111)  (622)

%e                             (631)

%e                             (721)

%e                             (811)

%e                             (3331)

%e                             (4222)

%e                             (6211)

%e                             (7111)

%e                             (22222)

%e                             (61111)

%e (End)

%p b:= proc(n, i, s) option remember;

%p       `if`(0 in s or n in s, 0, `if`(n=0, 1, `if`(i<1, 0, b(n, i-1, s)+

%p       `if`(i<=n, b(n-i, i, select(y-> 0<=y and y<=n-i,

%p                  map(x-> [x, x-i][], s))), 0))))

%p     end:

%p a:= n-> b(2*n, 2*n, {n}):

%p seq(a(n), n=1..25);  # _Alois P. Heinz_, Jul 10 2012

%t 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 (Python)

%o from itertools import combinations_with_replacement

%o from collections import Counter

%o from sympy import npartitions

%o from sympy.utilities.iterables import partitions

%o 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 The complement is counted by A002219, ranks A357976.

%Y Central diagonal of A046663.

%Y The strict case is A321142, even bisection of A371794 (odd A078408).

%Y This is the "bi-" version of A321451, ranks A321453.

%Y Column k = 0 of A367094.

%Y These partitions have Heinz numbers A371731.

%Y Even bisection of A371795 (odd A058695).

%Y A371783 counts k-quanimous partitions.

%Y Cf. A035470, A064914, A237258, A305551, A321452, A365543, A365663, A366320, A371736, A371782, A371792.

%K nonn,nice

%O 1,2

%A _N. J. A. Sloane_

%E More terms from _Don Reble_, Nov 03 2001

%E More terms from _Alois P. Heinz_, Jul 10 2012