A006827 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

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.