A365312
Number of strict integer partitions with sum <= n that cannot be linearly combined using nonnegative coefficients to obtain n.
Original entry on oeis.org
0, 0, 0, 1, 1, 3, 2, 6, 4, 8, 7, 16, 6, 24, 17, 24, 20, 46, 22, 62, 31, 63, 57, 106, 35, 122, 90, 137, 88, 212, 74, 262, 134, 267, 206, 345, 121, 476, 294, 484, 232, 698, 242, 837, 389, 763, 571, 1185, 318, 1327, 634, 1392, 727, 1927, 640, 2056, 827, 2233, 1328
Offset: 0
The strict partition (7,3,2) has 19 = 1*7 + 2*3 + 3*2 so is not counted under a(19).
The strict partition (9,6,3) cannot be linearly combined to obtain 19, so is counted under a(19).
The a(0) = 0 through a(11) = 16 strict partitions:
. . . (2) (3) (2) (4) (2) (3) (2) (3) (2)
(3) (5) (3) (5) (4) (4) (3)
(4) (4) (6) (5) (6) (4)
(5) (7) (6) (7) (5)
(6) (7) (8) (6)
(4,2) (8) (9) (7)
(4,2) (6,3) (8)
(6,2) (9)
(10)
(4,2)
(5,4)
(6,2)
(6,3)
(6,4)
(7,3)
(8,2)
The complement for positive coefficients is counted by
A088314.
For positive coefficients we have
A088528.
The complement is counted by
A365311.
A364350 counts combination-free strict partitions, non-strict
A364915.
A364839 counts combination-full strict partitions, non-strict
A364913.
Cf.
A093971,
A237113,
A237668,
A326080,
A363225,
A364272,
A364534,
A364914,
A365043,
A365314,
A365320.
-
combs[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[Select[Join@@Array[IntegerPartitions,n], UnsameQ@@#&],combs[n,#]=={}&]],{n,0,10}]
-
from math import isqrt
from sympy.utilities.iterables import partitions
def A365312(n):
a = {tuple(sorted(set(p))) for p in partitions(n)}
return sum(1 for m in range(1,n+1) for b in partitions(m,m=isqrt(1+(n<<3))>>1) if max(b.values()) == 1 and not any(set(d).issubset(set(b)) for d in a)) # Chai Wah Wu, Sep 13 2023
A365320
Number of pairs of distinct positive integers <= n that cannot be linearly combined with nonnegative coefficients to obtain n.
Original entry on oeis.org
0, 0, 0, 0, 0, 2, 1, 7, 5, 12, 12, 27, 14, 42, 36, 47, 47, 83, 58, 109, 80, 116, 126, 172, 111, 195, 192, 219, 202, 294, 210, 342, 286, 354, 369, 409, 324, 509, 480, 523, 452, 640, 507, 711, 622, 675, 747, 865, 654, 916, 842, 964, 922, 1124, 940, 1147, 1029
Offset: 0
The pair p = (3,6) cannot be linearly combined to obtain 8 or 10, so p is counted under a(8) and a(10), but we have 9 = 1*3 + 1*6 or 9 = 3*3 + 0*6, so p not counted under a(9).
The a(5) = 2 through a(10) = 12 pairs:
(2,4) (4,5) (2,4) (3,6) (2,4) (3,6)
(3,4) (2,6) (3,7) (2,6) (3,8)
(3,5) (5,6) (2,8) (3,9)
(3,6) (5,7) (4,6) (4,7)
(4,5) (6,7) (4,7) (4,8)
(4,6) (4,8) (4,9)
(5,6) (5,6) (6,7)
(5,7) (6,8)
(5,8) (6,9)
(6,7) (7,8)
(6,8) (7,9)
(7,8) (8,9)
The case of positive coefficients is
A365321, for all subsets
A365322.
For all subsets instead of just pairs we have
A365380, complement
A365073.
A004526 counts partitions of length 2, shift right for strict.
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],{2}],combs[n,#]=={}&]],{n,0,30}]
-
from itertools import count
from sympy import divisors
def A365320(n):
a = set()
for i in range(1,n+1):
if not n%i:
a.update(tuple(sorted((i,j))) for j in range(1,n+1) if j!=i)
else:
for j in count(0,i):
if j > n:
break
k = n-j
for d in divisors(k):
if d>=i:
break
a.add((d,i))
return (n*(n-1)>>1)-len(a) # Chai Wah Wu, Sep 13 2023
A088528
Let m = number of ways of partitioning n into parts using all the parts of a subset of {1, 2, ..., n-1} whose sum of all parts of a subset is less than n; a(n) gives number of different subsets of {1, 2, ..., n-1} whose m is 0.
Original entry on oeis.org
0, 0, 1, 1, 3, 3, 6, 6, 10, 12, 17, 18, 26, 30, 40, 44, 58, 66, 84, 95, 120, 135, 166, 186, 230, 257, 314, 350, 421, 476, 561, 626, 749, 831, 986, 1095, 1276, 1424, 1666, 1849, 2138, 2388, 2741, 3042, 3522, 3879, 4441, 4928, 5617, 6222, 7084, 7802, 8852, 9800
Offset: 1
a(5)=3 because there are three different subsets, {2}, {3} & {4}; a(6)=3 because there are three different subsets, {4}, {5} & {2,3}.
From _Gus Wiseman_, Sep 10 2023: (Start)
The set {3,5} is not counted under a(8) because 1*3 + 1*5 = 8, but it is counted under a(9) and a(10), and it is not counted under a(11) because 2*3 + 1*5 = 11.
The a(3) = 1 through a(11) = 17 subsets:
{2} {3} {2} {4} {2} {3} {2} {3} {2}
{3} {5} {3} {5} {4} {4} {3}
{4} {2,3} {4} {6} {5} {6} {4}
{5} {7} {6} {7} {5}
{6} {2,5} {7} {8} {6}
{2,4} {3,4} {8} {9} {7}
{2,4} {2,5} {8}
{2,6} {2,7} {9}
{3,4} {3,5} {10}
{3,5} {3,6} {2,4}
{4,5} {2,6}
{2,3,4} {2,8}
{3,6}
{3,7}
{4,5}
{4,6}
{2,3,5}
(End)
For sets with max < n instead of sum < n we have
A365045, nonempty
A070880.
For sets with max <= n we have
A365322.
-
combp[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,1,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[Select[Subsets[Range[n]],0Gus Wiseman, Sep 12 2023 *)
A365314
Number of unordered pairs of distinct positive integers <= n that can be linearly combined using nonnegative coefficients to obtain n.
Original entry on oeis.org
0, 0, 1, 3, 6, 8, 14, 14, 23, 24, 33, 28, 52, 36, 55, 58, 73, 53, 95, 62, 110, 94, 105, 81, 165, 105, 133, 132, 176, 112, 225, 123, 210, 174, 192, 186, 306, 157, 223, 218, 328, 180, 354, 192, 324, 315, 288, 216, 474, 260, 383, 311, 404, 254, 491, 338, 511, 360
Offset: 0
We have 19 = 4*3 + 1*7, so the pair (3,7) is counted under a(19).
The a(2) = 1 through a(7) = 14 pairs:
(1,2) (1,2) (1,2) (1,2) (1,2) (1,2)
(1,3) (1,3) (1,3) (1,3) (1,3)
(2,3) (1,4) (1,4) (1,4) (1,4)
(2,3) (1,5) (1,5) (1,5)
(2,4) (2,3) (1,6) (1,6)
(3,4) (2,5) (2,3) (1,7)
(3,5) (2,4) (2,3)
(4,5) (2,5) (2,5)
(2,6) (2,7)
(3,4) (3,4)
(3,5) (3,7)
(3,6) (4,7)
(4,6) (5,7)
(5,6) (6,7)
For all subsets instead of just pairs we have
A365073, complement
A365380.
The case of positive coefficients is
A365315, for all subsets
A088314.
A004526 counts partitions of length 2, shift right for strict.
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],{2}], combs[n,#]!={}&]],{n,0,30}]
-
from itertools import count
from sympy import divisors
def A365314(n):
a = set()
for i in range(1,n+1):
if not n%i:
a.update(tuple(sorted((i,j))) for j in range(1,n+1) if j!=i)
else:
for j in count(0,i):
if j > n:
break
k = n-j
for d in divisors(k):
if d>=i:
break
a.add((d,i))
return len(a) # Chai Wah Wu, Sep 12 2023
A365321
Number of pairs of distinct positive integers <= n that cannot be linearly combined with positive coefficients to obtain n.
Original entry on oeis.org
0, 0, 1, 2, 4, 6, 10, 13, 18, 24, 30, 37, 46, 54, 63, 77, 85, 99, 111, 127, 141, 161, 171, 194, 210, 235, 246, 277, 293, 322, 342, 372, 389, 428, 441, 491, 504, 545, 561, 612, 635, 680, 701, 753, 773, 836, 846, 911, 932, 1000, 1017, 1082, 1103, 1176, 1193
Offset: 0
For the pair p = (2,3) we have 4 = 2*2 + 0*3, so p is not counted under A365320(4), but it is not possible to write 4 as a positive linear combination of 2 and 3, so p is counted under a(4).
The a(2) = 1 through a(7) = 13 pairs:
(1,2) (1,3) (1,4) (1,5) (1,6) (1,7)
(2,3) (2,3) (2,4) (2,3) (2,4)
(2,4) (2,5) (2,5) (2,6)
(3,4) (3,4) (2,6) (2,7)
(3,5) (3,4) (3,5)
(4,5) (3,5) (3,6)
(3,6) (3,7)
(4,5) (4,5)
(4,6) (4,6)
(5,6) (4,7)
(5,6)
(5,7)
(6,7)
For all subsets instead of just pairs we have
A365322, complement
A088314.
A004526 counts partitions of length 2, shift right for strict.
A364350 counts combination-free strict partitions.
Cf.
A070880,
A088571,
A088809,
A151897,
A326020,
A365043,
A365073,
A365311,
A365312,
A365378,
A365380.
-
combp[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,1,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[Subsets[Range[n],{2}], combp[n,#]=={}&]],{n,0,30}]
-
from itertools import count
from sympy import divisors
def A365321(n):
a = set()
for i in range(1,n+1):
for j in count(i,i):
if j >= n:
break
for d in divisors(n-j):
if d>=i:
break
a.add((d,i))
return (n*(n-1)>>1)-len(a) # Chai Wah Wu, Sep 12 2023
A365379
Number of integer partitions with sum <= n whose distinct parts can be linearly combined using nonnegative coefficients to obtain n.
Original entry on oeis.org
0, 1, 3, 5, 10, 14, 27, 35, 61, 83, 128, 166, 264, 327, 482, 632, 882, 1110, 1565, 1938, 2663, 3339, 4401, 5471, 7290, 8921, 11555, 14291, 18280, 22303, 28507, 34507, 43534, 52882, 65798, 79621, 98932, 118629, 146072, 175562, 214708, 256351, 312583, 371779
Offset: 0
The partition (4,2,2) cannot be linearly combined to obtain 9, so is not counted under a(9). On the other hand, the same partition (4,2,2) has distinct parts {2,4} and has 10 = 1*2 + 2*4, so is counted under a(10).
The a(1) = 1 through a(5) = 14 partitions:
(1) (1) (1) (1) (1)
(2) (3) (2) (5)
(11) (11) (4) (11)
(21) (11) (21)
(111) (21) (31)
(22) (32)
(31) (41)
(111) (111)
(211) (211)
(1111) (221)
(311)
(1111)
(2111)
(11111)
For subsets with positive coefficients we have
A088314, complement
A088528.
The case of strict partitions with positive coefficients is also
A088314.
The complement is counted by
A365378.
A364350 counts combination-free strict partitions, non-strict
A364915.
A364839 counts combination-full strict partitions, non-strict
A364913.
-
combs[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[Join@@Array[IntegerPartitions,n],combs[n,Union[#]]!={}&]],{n,0,10}]
-
from sympy.utilities.iterables import partitions
def A365379(n):
a = {tuple(sorted(set(p))) for p in partitions(n)}
return sum(1 for m in range(1,n+1) for b in partitions(m) if any(set(d).issubset(set(b)) for d in a)) # Chai Wah Wu, Sep 13 2023
A365315
Number of unordered pairs of distinct positive integers <= n that can be linearly combined using positive coefficients to obtain n.
Original entry on oeis.org
0, 0, 0, 1, 2, 4, 5, 8, 10, 12, 15, 18, 20, 24, 28, 28, 35, 37, 42, 44, 49, 49, 60, 59, 66, 65, 79, 74, 85, 84, 93, 93, 107, 100, 120, 104, 126, 121, 142, 129, 145, 140, 160, 150, 173, 154, 189, 170, 196, 176, 208, 193, 223, 202, 238, 203, 241, 227, 267, 235
Offset: 0
We have 19 = 4*3 + 1*7, so the pair (3,7) is counted under a(19).
For the pair p = (2,3), we have 4 = 2*2 + 0*3, so p is counted under A365314(4), but it is not possible to write 4 as a positive linear combination of 2 and 3, so p is not counted under a(4).
The a(3) = 1 through a(10) = 15 pairs:
(1,2) (1,2) (1,2) (1,2) (1,2) (1,2) (1,2) (1,2)
(1,3) (1,3) (1,3) (1,3) (1,3) (1,3) (1,3)
(1,4) (1,4) (1,4) (1,4) (1,4) (1,4)
(2,3) (1,5) (1,5) (1,5) (1,5) (1,5)
(2,4) (1,6) (1,6) (1,6) (1,6)
(2,3) (1,7) (1,7) (1,7)
(2,5) (2,3) (1,8) (1,8)
(3,4) (2,4) (2,3) (1,9)
(2,6) (2,5) (2,3)
(3,5) (2,7) (2,4)
(3,6) (2,6)
(4,5) (2,8)
(3,4)
(3,7)
(4,6)
For all subsets instead of just pairs we have
A088314, complement
A365322.
The case of nonnegative coefficients is
A365314, for all subsets
A365073.
A004526 counts partitions of length 2, shift right for strict.
A364350 counts combination-free strict partitions.
Cf.
A070880,
A088809,
A326020,
A364534,
A365043,
A365311,
A365312,
A365378,
A365379,
A365380,
A365383.
-
combp[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,1,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[Subsets[Range[n],{2}],combp[n,#]!={}&]],{n,0,30}]
-
from itertools import count
from sympy import divisors
def A365315(n):
a = set()
for i in range(1,n+1):
for j in count(i,i):
if j >= n:
break
for d in divisors(n-j):
if d>=i:
break
a.add((d,i))
return len(a) # Chai Wah Wu, Sep 13 2023
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.
-
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
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
A365383
Triangle read by rows where T(n,k) is the number of integer partitions of n that can be linearly combined with nonnegative coefficients to obtain k.
Original entry on oeis.org
1, 2, 1, 3, 2, 2, 5, 3, 4, 3, 7, 5, 6, 6, 6, 11, 7, 9, 8, 9, 7, 15, 11, 13, 13, 14, 13, 14, 22, 15, 19, 17, 20, 17, 20, 16, 30, 22, 26, 26, 27, 26, 28, 26, 27, 42, 30, 37, 34, 39, 33, 40, 34, 39, 34, 56, 42, 50, 49, 52, 50, 54, 51, 54, 53, 53
Offset: 0
Triangle begins:
1
2 1
3 2 2
5 3 4 3
7 5 6 6 6
11 7 9 8 9 7
15 11 13 13 14 13 14
22 15 19 17 20 17 20 16
30 22 26 26 27 26 28 26 27
42 30 37 34 39 33 40 34 39 34
56 42 50 49 52 50 54 51 54 53 53
77 56 68 64 71 63 73 63 71 65 70 62
101 77 91 89 95 90 97 93 97 97 98 94 99
135 101 122 115 127 115 130 114 131 119 130 117 132 116
176 135 159 156 165 157 170 161 167 168 166 165 172 164 166
Row n = 6 counts the following partitions:
(6) (51) (51) (51) (51) (51)
(51) (411) (42) (411) (42) (411)
(42) (321) (411) (33) (411) (321)
(411) (3111) (321) (321) (321) (3111)
(33) (2211) (3111) (3111) (3111) (2211)
(321) (21111) (222) (2211) (222) (21111)
(3111) (111111) (2211) (21111) (2211) (111111)
(222) (21111) (111111) (21111)
(2211) (111111) (111111)
(21111)
(111111)
A364350 counts combination-free strict partitions, non-strict
A364915.
A364839 counts combination-full strict partitions, non-strict
A364913.
-
combu[n_,y_]:=With[{s=Table[{k,i},{k,Union[y]},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[IntegerPartitions[n],combu[k,#]!={}&]],{n,0,12},{k,0,n-1}]
Showing 1-10 of 10 results.
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