A365382
Number of relatively prime integer partitions with sum < n that cannot be linearly combined using nonnegative coefficients to obtain n.
Original entry on oeis.org
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 4, 4, 2, 4, 12, 8, 20, 11, 14, 26, 43, 19, 38, 53, 51, 48, 101, 48, 124, 96, 121, 159, 134, 103, 241, 261, 244, 175, 401, 229, 488, 358, 328
Offset: 0
The a(11) = 2 through a(18) = 8 partitions:
(5,4) . (6,5) (6,5) (7,6) (7,5) (7,4) (7,5)
(7,3) (7,4) (8,5) (9,4) (7,6) (7,6) (8,7)
(7,5) (9,4) (9,5) (8,5) (10,7)
(8,3) (10,3) (11,3) (8,7) (11,4)
(9,5) (11,5)
(9,7) (12,5)
(10,3) (13,4)
(11,4) (7,5,5)
(11,5)
(13,3)
(7,4,4)
(10,3,3)
This is the relatively prime case of
A365378.
A364350 counts combination-free strict partitions, non-strict
A364915.
A364839 counts combination-full strict partitions, non-strict
A364913.
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combsu[n_,y_]:=With[{s=Table[{k,i},{k,Union[y]},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[Join@@IntegerPartitions/@Range[n-1],GCD@@#==1&&combsu[n,#]=={}&]],{n,0,20}]
-
from math import gcd
from sympy.utilities.iterables import partitions
def A365382(n):
a = {tuple(sorted(set(p))) for p in partitions(n)}
return sum(1 for m in range(1,n) for b in partitions(m) if gcd(*b.keys()) == 1 and not any(set(d).issubset(set(b)) for d in a)) # Chai Wah Wu, Sep 13 2023
A365542
Number of subsets of {1..n-1} that can be linearly combined using nonnegative coefficients to obtain n.
Original entry on oeis.org
0, 1, 2, 6, 10, 28, 48, 116, 224, 480, 920, 2000, 3840, 7984, 15936, 32320, 63968, 130176, 258304, 521920, 1041664, 2089472, 4171392, 8377856, 16726528, 33509632, 67004416, 134129664, 268111360, 536705024, 1072961536, 2146941952, 4293509120, 8588414976
Offset: 1
The a(2) = 1 through a(5) = 10 partitions:
{1} {1} {1} {1}
{1,2} {2} {1,2}
{1,2} {1,3}
{1,3} {1,4}
{2,3} {2,3}
{1,2,3} {1,2,3}
{1,2,4}
{1,3,4}
{2,3,4}
{1,2,3,4}
For subsets of {1..n} instead of {1..n-1} we have
A365073.
The complement is counted by
A365380.
A364350 counts combination-free strict partitions.
-
combs[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[Subsets[Range[n-1]],combs[n,#]!={}&]],{n,5}]
-
from itertools import combinations
from sympy.utilities.iterables import partitions
def A365542(n):
a = {tuple(sorted(set(p))) for p in partitions(n)}
return sum(1 for m in range(1,n) for b in combinations(range(1,n),m) if any(set(d).issubset(set(b)) for d in a)) # Chai Wah Wu, Sep 12 2023
A365323
Number of integer partitions with sum < n whose distinct parts cannot be linearly combined using all positive coefficients to obtain n.
Original entry on oeis.org
0, 0, 1, 1, 4, 3, 9, 7, 15, 16, 29, 23, 47, 43, 74, 65, 114, 100, 174, 153, 257, 228, 368, 312, 530, 454, 736, 645, 1025, 902, 1402, 1184, 1909, 1626, 2618, 2184, 3412, 2895, 4551, 3887, 5966, 5055, 7796, 6509, 10244, 8462, 13060, 10881, 16834, 14021, 21471
Offset: 1
The partition y = (3,3,2) has distinct parts {2,3}, and we have 9 = 3*2 + 1*3, so y is not counted under a(9).
The a(3) = 1 through a(10) = 16 partitions:
(2) (3) (2) (4) (2) (3) (2) (3)
(3) (5) (3) (5) (4) (4)
(4) (3,2) (4) (6) (5) (6)
(2,2) (5) (7) (6) (7)
(6) (3,3) (7) (8)
(2,2) (4,3) (8) (9)
(3,3) (5,2) (2,2) (3,3)
(4,2) (4,2) (4,4)
(2,2,2) (4,3) (5,2)
(4,4) (5,3)
(5,3) (5,4)
(6,2) (6,3)
(2,2,2) (7,2)
(4,2,2) (3,3,3)
(2,2,2,2) (4,3,2)
(5,2,2)
For strict partitions we have
A088528, nonnegative coefficients
A365312.
For length-2 subsets we have
A365321 (we use n instead of n-1).
A364350 counts combination-free strict partitions, non-strict
A364915.
A364839 counts combination-full strict partitions, non-strict
A364913.
-
combp[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,1,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[Join@@IntegerPartitions/@Range[n-1],combp[n,Union[#]]=={}&]],{n,10}]
-
from sympy.utilities.iterables import partitions
def A365323(n):
a = {tuple(sorted(set(p))) for p in partitions(n)}
return sum(1 for k in range(1,n) for d in partitions(k) if tuple(sorted(set(d))) not in a) # Chai Wah Wu, Sep 12 2023