A364272
Number of strict integer partitions of n containing the sum of some subset of the parts. A variation of sum-full strict partitions.
Original entry on oeis.org
0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 3, 1, 4, 3, 8, 6, 11, 10, 17, 16, 26, 25, 39, 39, 54, 60, 82, 84, 116, 126, 160, 177, 222, 242, 302, 337, 402, 453, 542, 601, 722, 803, 936, 1057, 1234, 1373, 1601, 1793, 2056, 2312, 2658, 2950, 3395, 3789, 4281, 4814, 5452, 6048
Offset: 0
The a(6) = 1 through a(16) = 11 partitions (A=10):
(321) . (431) . (532) (5321) (642) (5431) (743) (6432) (853)
(541) (651) (6421) (752) (6531) (862)
(4321) (5421) (7321) (761) (7431) (871)
(6321) (5432) (7521) (6532)
(6431) (9321) (6541)
(6521) (54321) (7432)
(7421) (7621)
(8321) (8431)
(8521)
(A321)
(64321)
The linear combination-free version is
A364350.
-
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&Intersection[#, Total/@Subsets[#,{2,Length[#]}]]!={}&]],{n,0,30}]
A320348
Number of partition into distinct parts (a_1, a_2, ... , a_m) (a_1 > a_2 > ... > a_m and Sum_{k=1..m} a_k = n) such that a1 - a2, a2 - a_3, ... , a_{m-1} - a_m, a_m are different.
Original entry on oeis.org
1, 1, 1, 2, 3, 2, 4, 4, 4, 6, 9, 7, 13, 12, 13, 16, 22, 17, 28, 28, 31, 36, 50, 45, 63, 62, 74, 78, 102, 92, 123, 123, 146, 148, 191, 181, 228, 233, 280, 283, 348, 350, 420, 437, 518, 523, 616, 641, 727, 774, 884, 911, 1038, 1102, 1240, 1292, 1463, 1530, 1715, 1861, 2002
Offset: 1
n = 9
[9] ********* a_1 = 9.
ooooooooo
------------------------------------
[8, 1] * a_2 = 1.
*******o a_1 - a_2 = 7.
oooooooo
------------------------------------
[7, 2] ** a_2 = 2.
*****oo a_1 - a_2 = 5.
ooooooo
------------------------------------
[5, 4] **** a_2 = 4.
*oooo a_1 - a_2 = 1.
ooooo
------------------------------------
a(9) = 4.
From _Gus Wiseman_, May 04 2019: (Start)
The a(1) = 1 through a(11) = 9 strict partitions with distinct differences (where the last part is taken to be 0) are the following (A = 10, B = 11). The Heinz numbers of these partitions are given by A325388.
(1) (2) (3) (4) (5) (6) (7) (8) (9) (A) (B)
(31) (32) (51) (43) (53) (54) (64) (65)
(41) (52) (62) (72) (73) (74)
(61) (71) (81) (82) (83)
(91) (92)
(631) (A1)
(632)
(641)
(731)
The a(1) = 1 through a(10) = 6 partitions covering an initial interval of positive integers with distinct multiplicities are the following. The Heinz numbers of these partitions are given by A325326.
1 11 111 211 221 21111 2221 22211 22221 222211
1111 2111 111111 22111 221111 2211111 322111
11111 211111 2111111 21111111 2221111
1111111 11111111 111111111 22111111
211111111
1111111111
The a(1) = 1 through a(10) = 6 partitions whose multiplicities cover an initial interval of positive integers and are distinct are the following (A = 10). The Heinz numbers of these partitions are given by A325337.
(1) (2) (3) (4) (5) (6) (7) (8) (9) (A)
(211) (221) (411) (322) (332) (441) (433)
(311) (331) (422) (522) (442)
(511) (611) (711) (622)
(811)
(322111)
(End)
Cf.
A007294,
A007862,
A048767,
A098859,
A179269,
A320509,
A320510,
A325324,
A325325,
A325349,
A325367,
A325404,
A325468.
-
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&UnsameQ@@Differences[Append[#,0]]&]],{n,30}] (* Gus Wiseman, May 04 2019 *)
A364345
Number of integer partitions of n without any three parts (a,b,c) (repeats allowed) satisfying a + b = c. A variation of sum-free partitions.
Original entry on oeis.org
1, 1, 2, 2, 4, 5, 7, 10, 13, 16, 21, 27, 34, 43, 54, 67, 83, 102, 122, 151, 182, 218, 258, 313, 366, 443, 513, 611, 713, 844, 975, 1149, 1325, 1554, 1780, 2079, 2381, 2761, 3145, 3647, 4134, 4767, 5408, 6200, 7014, 8035, 9048, 10320, 11639, 13207, 14836, 16850
Offset: 0
The a(1) = 1 through a(8) = 13 partitions:
(1) (2) (3) (4) (5) (6) (7) (8)
(11) (111) (22) (32) (33) (43) (44)
(31) (41) (51) (52) (53)
(1111) (311) (222) (61) (62)
(11111) (411) (322) (71)
(3111) (331) (332)
(111111) (511) (611)
(4111) (2222)
(31111) (3311)
(1111111) (5111)
(41111)
(311111)
(11111111)
For subsets of {1..n} instead of partitions we have
A007865 (sum-free sets), differences
A288728.
-
Table[Length[Select[IntegerPartitions[n],Select[Tuples[Union[#],3],#[[1]]+#[[2]]==#[[3]]&]=={}&]],{n,0,30}]
A364349
Number of strict integer partitions of n containing the sum of no subset of the parts.
Original entry on oeis.org
1, 1, 1, 2, 2, 3, 3, 5, 5, 8, 7, 11, 11, 15, 14, 21, 21, 28, 29, 38, 38, 51, 50, 65, 68, 82, 83, 108, 106, 130, 136, 163, 168, 206, 210, 248, 266, 307, 322, 381, 391, 457, 490, 553, 582, 675, 703, 797, 854, 952, 1000, 1147, 1187, 1331, 1437, 1564, 1656, 1869
Offset: 0
The partition y = (7,5,3,1) has no subset with sum in y, so is counted under a(16).
The partition y = (15,8,4,2,1) has subset {1,2,4,8} with sum in y, so is not counted under a(31).
The a(1) = 1 through a(9) = 8 partitions:
(1) (2) (3) (4) (5) (6) (7) (8) (9)
(2,1) (3,1) (3,2) (4,2) (4,3) (5,3) (5,4)
(4,1) (5,1) (5,2) (6,2) (6,3)
(6,1) (7,1) (7,2)
(4,2,1) (5,2,1) (8,1)
(4,3,2)
(5,3,1)
(6,2,1)
The complement in strict partitions is counted by
A364272.
The linear combination-free version is
A364350.
A236912 counts sum-free partitions (not re-using parts), complement
A237113.
-
Table[Length[Select[IntegerPartitions[n],Function[ptn,UnsameQ@@ptn&&Select[Subsets[ptn,{2,Length[ptn]}],MemberQ[ptn,Total[#]]&]=={}]]],{n,0,30}]
A364346
Number of strict integer partitions of n such that there is no ordered triple of parts (a,b,c) (repeats allowed) satisfying a + b = c. A variation of sum-free strict partitions.
Original entry on oeis.org
1, 1, 1, 1, 2, 3, 2, 4, 4, 5, 5, 8, 9, 11, 11, 16, 16, 20, 20, 25, 30, 34, 38, 42, 50, 58, 64, 73, 80, 90, 105, 114, 128, 148, 158, 180, 201, 220, 241, 277, 306, 333, 366, 404, 447, 497, 544, 592, 662, 708, 797, 861, 954, 1020, 1131, 1226, 1352, 1456, 1600
Offset: 0
The a(1) = 1 through a(14) = 11 partitions (A..E = 10..14):
1 2 3 4 5 6 7 8 9 A B C D E
31 32 51 43 53 54 64 65 75 76 86
41 52 62 72 73 74 93 85 95
61 71 81 82 83 A2 94 A4
531 91 92 B1 A3 B3
A1 543 B2 C2
641 732 C1 D1
731 741 652 851
831 751 932
832 941
931 A31
For subsets of {1..n} we have
A007865 (sum-free sets), differences
A288728.
A236912 counts sum-free partitions not re-using parts, complement
A237113.
Cf.
A002865,
A025065,
A085489,
A093971,
A108917,
A111133,
A240861,
A275972,
A320347,
A325862,
A326083,
A363260.
-
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Select[Tuples[#,3],#[[1]]+#[[2]]==#[[3]]&]=={}&]],{n,0,15}]
-
from collections import Counter
from itertools import combinations_with_replacement
from sympy.utilities.iterables import partitions
def A364346(n): return sum(1 for p in partitions(n) if max(p.values(),default=1)==1 and not any(q[0]+q[1]==q[2] for q in combinations_with_replacement(sorted(Counter(p).elements()),3))) # Chai Wah Wu, Sep 20 2023
A363226
Number of strict integer partitions of n containing some three possibly equal parts (a,b,c) such that a + b = c. A variation of sum-full strict partitions.
Original entry on oeis.org
0, 0, 0, 1, 0, 0, 2, 1, 2, 3, 5, 4, 6, 7, 11, 11, 16, 18, 26, 29, 34, 42, 51, 62, 72, 84, 101, 119, 142, 166, 191, 226, 262, 300, 354, 405, 467, 540, 623, 705, 807, 927, 1060, 1206, 1369, 1551, 1760, 1998, 2248, 2556, 2861, 3236, 3628, 4100, 4587, 5152, 5756
Offset: 0
The a(3) = 1 through a(15) = 11 partitions (A=10, B=11, C=12):
21 . . 42 421 431 63 532 542 84 643 653 A5
321 521 432 541 632 642 742 743 843
621 631 821 651 841 752 942
721 5321 921 A21 761 C21
4321 5421 5431 842 6432
6321 6421 B21 6531
7321 5432 7431
6431 7521
6521 8421
7421 9321
8321 54321
For subsets of {1..n} we have
A093971 (sum-full sets), complement
A007865.
A236912 counts sum-free partitions not re-using parts, complement
A237113.
-
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Select[Tuples[#,3],#[[1]]+#[[2]]==#[[3]]&]!={}&]],{n,0,30}]
-
from itertools import combinations_with_replacement
from collections import Counter
from sympy.utilities.iterables import partitions
def A363226(n): return sum(1 for p in partitions(n) if max(p.values(),default=0)==1 and any(q[0]+q[1]==q[2] for q in combinations_with_replacement(sorted(Counter(p).elements()),3))) # Chai Wah Wu, Sep 20 2023
A363260
Number of integer partitions of n with parts disjoint from first differences of parts, meaning no part is the difference of two consecutive parts.
Original entry on oeis.org
1, 1, 2, 2, 4, 5, 7, 10, 13, 17, 21, 28, 35, 46, 57, 70, 87, 110, 130, 165, 198, 238, 285, 349, 410, 498, 583, 702, 819, 983, 1136, 1353, 1570, 1852, 2137, 2520, 2898, 3390, 3891, 4540, 5191, 6028, 6889, 7951, 9082, 10450, 11884, 13650, 15508, 17728, 20113
Offset: 0
The a(1) = 1 through a(8) = 13 partitions:
(1) (2) (3) (4) (5) (6) (7) (8)
(11) (111) (22) (32) (33) (43) (44)
(31) (41) (51) (52) (53)
(1111) (311) (222) (61) (62)
(11111) (411) (322) (71)
(3111) (331) (332)
(111111) (511) (611)
(4111) (2222)
(31111) (3311)
(1111111) (5111)
(41111)
(311111)
(11111111)
For all differences of pairs parts we have
A364345.
For subsets of {1..n} instead of partitions we have
A364463.
A325325 counts partitions with distinct first-differences.
-
Table[Length[Select[IntegerPartitions[n],Intersection[#,-Differences[#]]=={}&]],{n,0,30}]
-
from collections import Counter
from sympy.utilities.iterables import partitions
def A363260(n): return sum(1 for s,p in map(lambda x: (x[0],tuple(sorted(Counter(x[1]).elements()))), partitions(n,size=True)) if set(p).isdisjoint({p[i+1]-p[i] for i in range(s-1)})) # Chai Wah Wu, Sep 26 2023
A364463
Number of subsets of {1..n} with elements disjoint from first differences of elements.
Original entry on oeis.org
1, 2, 3, 6, 10, 18, 30, 54, 92, 167, 290, 525, 935, 1704, 3082, 5664, 10386, 19249, 35701, 66702, 124855, 234969, 443174, 839254, 1592925, 3032757, 5786153, 11066413, 21204855, 40712426, 78294085, 150815154, 290922900, 561968268, 1086879052, 2104570243
Offset: 0
The a(0) = 1 through a(5) = 18 subsets:
{} {} {} {} {} {}
{1} {1} {1} {1} {1}
{2} {2} {2} {2}
{3} {3} {3}
{1,3} {4} {4}
{2,3} {1,3} {5}
{1,4} {1,3}
{2,3} {1,4}
{3,4} {1,5}
{2,3,4} {2,3}
{2,5}
{3,4}
{3,5}
{4,5}
{1,3,5}
{2,3,4}
{3,4,5}
{2,3,4,5}
For all differences of pairs of elements we have
A007865.
The complement is counted by
A364466.
A364465 counts subsets with distinct first differences, partitions
A325325.
Cf.
A011782,
A025065,
A229816,
A236912,
A237113,
A237667,
A240861,
A320347,
A323092,
A326083,
A364347.
-
Table[Length[Select[Subsets[Range[n]],Intersection[#,Differences[#]]=={}&]],{n,0,10}]
-
from itertools import combinations
def A364463(n): return sum(1 for l in range(n+1) for c in combinations(range(1,n+1),l) if set(c).isdisjoint({c[i+1]-c[i] for i in range(l-1)})) # Chai Wah Wu, Sep 26 2023
A364464
Number of strict integer partitions of n where no part is the difference of two consecutive parts.
Original entry on oeis.org
1, 1, 1, 1, 2, 3, 2, 4, 4, 6, 5, 8, 9, 12, 13, 16, 16, 21, 23, 29, 34, 38, 41, 49, 57, 64, 73, 86, 95, 110, 120, 135, 160, 171, 197, 219, 247, 277, 312, 342, 386, 431, 476, 527, 598, 640, 727, 796, 893, 966, 1097, 1178, 1327, 1435, 1602, 1740, 1945, 2084, 2337
Offset: 0
The strict partition y = (9,5,3,1) has differences (4,2,2), and these are disjoint from the parts, so y is counted under a(18).
The a(1) = 1 through a(9) = 6 strict partitions:
(1) (2) (3) (4) (5) (6) (7) (8) (9)
(3,1) (3,2) (5,1) (4,3) (5,3) (5,4)
(4,1) (5,2) (6,2) (7,2)
(6,1) (7,1) (8,1)
(4,3,2)
(5,3,1)
For length instead of differences we have
A240861, non-strict
A229816.
For all differences of pairs of elements we have
A364346, for subsets
A007865.
For subsets instead of strict partitions we have
A364463, complement
A364466.
A320347 counts strict partitions w/ distinct differences, non-strict
A325325.
-
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Intersection[#,-Differences[#]]=={}&]],{n,0,15}]
-
from collections import Counter
from sympy.utilities.iterables import partitions
def A364464(n): return sum(1 for s,p in map(lambda x: (x[0],tuple(sorted(Counter(x[1]).elements()))), filter(lambda p:max(p[1].values(),default=1)==1,partitions(n,size=True))) if set(p).isdisjoint({p[i+1]-p[i] for i in range(s-1)})) # Chai Wah Wu, Sep 26 2023
A364466
Number of subsets of {1..n} where some element is a difference of two consecutive elements.
Original entry on oeis.org
0, 0, 1, 2, 6, 14, 34, 74, 164, 345, 734, 1523, 3161, 6488, 13302, 27104, 55150, 111823, 226443, 457586, 923721, 1862183, 3751130, 7549354, 15184291, 30521675, 61322711, 123151315, 247230601, 496158486, 995447739, 1996668494, 4004044396, 8027966324, 16092990132, 32255168125
Offset: 0
The a(0) = 0 through a(5) = 14 subsets:
. . {1,2} {1,2} {1,2} {1,2}
{1,2,3} {2,4} {2,4}
{1,2,3} {1,2,3}
{1,2,4} {1,2,4}
{1,3,4} {1,2,5}
{1,2,3,4} {1,3,4}
{1,4,5}
{2,3,5}
{2,4,5}
{1,2,3,4}
{1,2,3,5}
{1,2,4,5}
{1,3,4,5}
{1,2,3,4,5}
The complement is counted by
A364463.
For subset-sums instead of differences we have
A364534, complement
A325864.
A325325 counts partitions with all distinct differences, strict
A320347.
-
Table[Length[Select[Subsets[Range[n]],Intersection[#,Differences[#]]!={}&]],{n,0,10}]
-
from itertools import combinations
def A364466(n): return sum(1 for l in range(n+1) for c in combinations(range(1,n+1),l) if not set(c).isdisjoint({c[i+1]-c[i] for i in range(l-1)})) # Chai Wah Wu, Sep 26 2023
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