A006827 Number of partitions of 2n with all subsums different from n. (Formerly M1351)

1, 2, 5, 8, 17, 24, 46, 64, 107, 147, 242, 302, 488, 629, 922, 1172, 1745, 2108, 3104, 3737, 5232, 6419, 8988, 10390, 14552, 17292, 23160, 27206, 36975, 41945, 57058, 65291, 85895, 99384, 130443, 145283, 193554, 218947, 281860, 316326, 413322, 454229, 594048

Offset

1,2

Comments

Partitions of this type are also called non-biquanimous partitions. - Gus Wiseman, Apr 19 2024

References

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

Links

Fausto A. C. Cariboni, Table of n, a(n) for n = 1..140 (terms 1..89 from Alois P. Heinz)

P. Erdős, J. L. Nicolas and A. Sárközy, On the number of partitions of n without a given subsum (I), Discrete Math., 75 (1989), 155-166 = Annals Discrete Math. Vol. 43, Graph Theory and Combinatorics 1988, ed. B. Bollobas.

Formula

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

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

Example

From Gus Wiseman, Apr 19 2024: (Start)
The a(1) = 1 through a(5) = 17 partitions (A = 10):
  (2)  (4)   (6)    (8)     (A)
       (31)  (42)   (53)    (64)
             (51)   (62)    (73)
             (222)  (71)    (82)
             (411)  (332)   (91)
                    (521)   (433)
                    (611)   (442)
                    (5111)  (622)
                            (631)
                            (721)
                            (811)
                            (3331)
                            (4222)
                            (6211)
                            (7111)
                            (22222)
                            (61111)
(End)

Maple

b:= proc(n, i, s) option remember;
      `if`(0 in s or n in s, 0, `if`(n=0, 1, `if`(i<1, 0, b(n, i-1, s)+
      `if`(i<=n, b(n-i, i, select(y-> 0<=y and y<=n-i,
                 map(x-> [x, x-i][], s))), 0))))
    end:
a:= n-> b(2*n, 2*n, {n}):
seq(a(n), n=1..25);  # Alois P. Heinz, Jul 10 2012

Mathematica

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 *)

Prog

(Python)
from itertools import combinations_with_replacement
from collections import Counter
from sympy import npartitions
from sympy.utilities.iterables import partitions
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

Crossrefs

The complement is counted by A002219, ranks A357976.

Central diagonal of A046663.

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

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

Column k = 0 of A367094.

These partitions have Heinz numbers A371731.

Even bisection of A371795 (odd A058695).

A371783 counts k-quanimous partitions.

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

Sequence in context: A103041 A216307 A176223 * A193992 A112346 A062318

Adjacent sequences: A006824 A006825 A006826 * A006828 A006829 A006830

Keyword

nonn,nice,changed

Author

N. J. A. Sloane

Extensions

More terms from Don Reble, Nov 03 2001

More terms from Alois P. Heinz, Jul 10 2012

Status

approved