A325592 -id:A325592 - cp's OEIS frontend

A333224 Number of distinct positive consecutive subsequence-sums of the k-th composition in standard order.

0, 1, 1, 2, 1, 3, 3, 3, 1, 3, 2, 4, 3, 4, 4, 4, 1, 3, 3, 5, 3, 5, 4, 5, 3, 4, 5, 5, 5, 5, 5, 5, 1, 3, 3, 5, 2, 5, 5, 6, 3, 6, 3, 6, 5, 6, 5, 6, 3, 4, 6, 6, 5, 6, 6, 6, 5, 6, 6, 6, 6, 6, 6, 6, 1, 3, 3, 5, 3, 6, 6, 7, 3, 5, 5, 7, 4, 6, 6, 7, 3, 6, 4, 7, 5, 7, 6

Offset: 0

Gus Wiseman, Mar 18 2020

nonn

The k-th composition in standard order (row k of A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again.

			The composition (4,3,1,2) has positive subsequence-sums 1, 2, 3, 4, 6, 7, 8, 10, so a(550) = 8.

Dominated by A124770.
Compositions where every subinterval has a different sum are counted by A169942 and A325677 and ranked by A333222. The case of partitions is counted by A325768 and ranked by A325779.
Positive subset-sums of partitions are counted by A276024 and A299701.
Knapsack partitions are counted by A108917 and A325592 and ranked by A299702.
Strict knapsack partitions are counted by A275972 and ranked by A059519 and A301899.
Knapsack compositions are counted by A325676 and A325687 and ranked by A333223. The case of partitions is counted by A325769 and ranked by A325778, for which the number of distinct consecutive subsequences is given by A325770.
Allowing empty subsequences gives A333257.
Cf. A000120, A003022, A029931, A048793, A066099, A070939, A103295, A124767, A143823, A295235, A325680, A333217.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Length[Union[ReplaceList[stc[n],{_,s__,_}:>Plus[s]]]],{n,0,100}]

a(n) = A333257(n) - 1.

A325862 Number of integer partitions of n such that every set of distinct parts has a different sum.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 10, 14, 19, 26, 34, 46, 58, 77, 93, 122, 146, 188, 217, 282, 327, 410, 470, 596, 673, 848, 947, 1178, 1325, 1629, 1798, 2213, 2444, 2962, 3247, 3935, 4292, 5149, 5579, 6674, 7247, 8590, 9221, 10964, 11804, 13870, 14843, 17480, 18675, 21866

Offset: 0

Gus Wiseman, May 31 2019

nonn

A knapsack partition (A108917, A299702) is an integer partition such that every submultiset has a different sum. The one non-knapsack partition counted under a(4) is (2,1,1).

			The a(1) = 1 through a(7) = 14 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (21)   (22)    (32)     (33)      (43)
             (111)  (31)    (41)     (42)      (52)
                    (211)   (221)    (51)      (61)
                    (1111)  (311)    (222)     (322)
                            (2111)   (411)     (331)
                            (11111)  (2211)    (421)
                                     (3111)    (511)
                                     (21111)   (2221)
                                     (111111)  (4111)
                                               (22111)
                                               (31111)
                                               (211111)
                                               (1111111)
The three non-knapsack partitions counted under a(6) are:
  (2,2,1,1)
  (3,1,1,1)
  (2,1,1,1,1)

Dominates A108917.

Cf. A002033, A034444, A196723, A275972, A276024, A299702, A325592, A325856, A325863, A325864, A325865, A325877.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@Plus@@@Subsets[Union[#]]&]],{n,0,20}]

A325687 Triangle read by rows where T(n,k) is the number of length-k compositions of n such that every distinct consecutive subsequence has a different sum.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 0, 1, 1, 4, 4, 0, 1, 1, 5, 5, 0, 0, 1, 1, 6, 12, 4, 0, 0, 1, 1, 7, 12, 5, 0, 0, 0, 1, 1, 8, 25, 8, 4, 0, 0, 0, 1, 1, 9, 24, 12, 3, 0, 0, 0, 0, 1, 1, 10, 40, 32, 8, 4, 0, 0, 0, 0, 1, 1, 11, 41, 41, 6, 3, 0, 0, 0, 0, 0, 1

Offset: 1

Gus Wiseman, May 13 2019

A composition of n is a finite sequence of positive integers summing to n.

			The distinct consecutive subsequences of (1,1,3,3) are (1), (1,1), (3), (1,3), (1,1,3), (3,3), (1,3,3), (1,1,3,3), all of which have different sums, so (1,1,3,3) is counted under a(8).
Triangle begins:
  1
  1  1
  1  2  1
  1  3  0  1
  1  4  4  0  1
  1  5  5  0  0  1
  1  6 12  4  0  0  1
  1  7 12  5  0  0  0  1
  1  8 25  8  4  0  0  0  1
  1  9 24 12  3  0  0  0  0  1
  1 10 40 32  8  4  0  0  0  0  1
  1 11 41 41  6  3  0  0  0  0  0  1
  1 12 60 76 14  4  4  0  0  0  0  0  1
  1 13 60 88 16  6  3  0  0  0  0  0  0  1
Row n = 8 counts the following compositions:
  (8)  (17)  (116)  (1115)  (11111111)
       (26)  (125)  (1133)
       (35)  (143)  (2222)
       (44)  (152)  (3311)
       (53)  (215)  (5111)
       (62)  (233)
       (71)  (251)
             (332)
             (341)
             (512)
             (521)
             (611)

Row sums are A325676.
Column k = 2 is A000027.
Column k = 3 is A325688.
Cf. A000079, A007318, A048004, A108917, A143823, A169942, A266223, A325592, A325680, A325685.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{k}],UnsameQ@@Total/@Union[ReplaceList[#,{_,s__,_}:>{s}]]&]],{n,15},{k,n}]

A326017 Triangle read by rows where T(n,k) is the number of knapsack partitions of n with maximum k.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 2, 1, 1, 0, 1, 1, 1, 2, 1, 1, 0, 1, 1, 2, 3, 2, 1, 1, 0, 1, 1, 2, 1, 3, 2, 1, 1, 0, 1, 1, 2, 2, 4, 3, 2, 1, 1, 0, 1, 1, 2, 3, 1, 4, 3, 2, 1, 1, 0, 1, 1, 3, 3, 4, 6, 4, 3, 2, 1, 1, 0, 1, 1, 1, 1, 3, 1, 6, 4

Offset: 0

Gus Wiseman, Jun 03 2019

An integer partition is knapsack if every distinct submultiset has a different sum.

			Triangle begins:
  1
  0  1
  0  1  1
  0  1  1  1
  0  1  1  1  1
  0  1  1  2  1  1
  0  1  1  1  2  1  1
  0  1  1  2  3  2  1  1
  0  1  1  2  1  3  2  1  1
  0  1  1  2  2  4  3  2  1  1
  0  1  1  2  3  1  4  3  2  1  1
  0  1  1  3  3  4  6  4  3  2  1  1
  0  1  1  1  1  3  1  6  4  3  2  1  1
  0  1  1  3  3  5  4  7  6  4  3  2  1  1
  0  1  1  2  3  5  4  1  7  6  4  3  2  1  1
  0  1  1  2  3  4  6  6 11  7  6  4  3  2  1  1
Row n = 9 counts the following partitions:
  (111111111)  (22221)  (333)   (432)  (54)     (63)    (72)   (81)  (9)
                        (3222)  (441)  (522)    (621)   (711)
                                       (531)    (6111)
                                       (51111)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..10010
Fausto A. C. Cariboni, Conjectures on columns of T(n,k), Jun 05 2021.

Row sums are A108917.
Column k = 3 is A326034.
Cf. A002033, A196723, A275972, A276024.
Cf. A325592, A325857, A325862, A325863, A325864, A325877, A326016, A326018.

ks[n_]:=Select[IntegerPartitions[n],UnsameQ@@Total/@Union[Subsets[#]]&];
Table[Length[Select[ks[n],Length[#]==k==0||Max@@#==k&]],{n,0,15},{k,0,n}]

A325690 Number of length-3 integer partitions of n whose largest part is not the sum of the other two.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 2, 4, 3, 7, 6, 10, 9, 14, 13, 19, 17, 24, 23, 30, 28, 37, 35, 44, 42, 52, 50, 61, 58, 70, 68, 80, 77, 91, 88, 102, 99, 114, 111, 127, 123, 140, 137, 154, 150, 169, 165, 184, 180, 200, 196, 217, 212, 234, 230, 252, 247, 271, 266, 290, 285, 310

Offset: 0

Gus Wiseman, May 15 2019

nonn

Confirmed recurrence relation from Colin Barker for n <= 10000. - Fausto A. C. Cariboni, Feb 19 2022

			The a(3) = 1 through a(13) = 14 partitions (A = 10, B = 11):
  (111)  (221)  (222)  (322)  (332)  (333)  (433)  (443)  (444)   (544)
         (311)  (411)  (331)  (521)  (432)  (442)  (533)  (543)   (553)
                       (421)  (611)  (441)  (622)  (542)  (552)   (643)
                       (511)         (522)  (631)  (551)  (732)   (652)
                                     (531)  (721)  (632)  (741)   (661)
                                     (621)  (811)  (641)  (822)   (733)
                                     (711)         (722)  (831)   (742)
                                                   (731)  (921)   (751)
                                                   (821)  (A11)   (832)
                                                   (911)          (841)
                                                                  (922)
                                                                  (931)
                                                                  (A21)
                                                                  (B11)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..10000

Column k = 3 of A325592.
Cf. A000041, A001399, A005044, A008642, A069905, A266223.
Cf. A325689, A325691, A325694.

Table[Length[Select[IntegerPartitions[n,{3}],#[[1]]!=#[[2]]+#[[3]]&]],{n,0,30}]

Conjectures from Colin Barker, May 15 2019: (Start)
G.f.: x^3*(1 + x^2 + x^3 + x^4) / ((1 - x)^3*(1 + x)^2*(1 + x^2)*(1 + x + x^2)).
a(n) = a(n-2) + a(n-3) + a(n-4) - a(n-5) - a(n-6) - a(n-7) + a(n-9) for n>8.
(End)

A326034 Number of knapsack partitions of n with largest part 3.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 1, 2, 2, 2, 2, 3, 1, 3, 2, 2, 2, 3, 1, 3, 2, 2, 2, 3, 1, 3, 2, 2, 2, 3, 1, 3, 2, 2, 2, 3, 1, 3, 2, 2, 2, 3, 1, 3, 2, 2, 2, 3, 1, 3, 2, 2, 2, 3, 1, 3, 2, 2, 2, 3, 1, 3, 2, 2, 2, 3, 1, 3, 2, 2, 2, 3, 1, 3, 2, 2, 2, 3, 1, 3, 2, 2, 2, 3, 1, 3, 2

Offset: 0

Gus Wiseman, Jun 04 2019

nonn

An integer partition is knapsack if every distinct submultiset has a different sum.
Appears to repeat the terms (2,2,2,3,1,3) ad infinitum.
I computed terms a(n) for n = 0..5000 and (2,2,2,3,1,3) is repeated continuously starting at a(8). - Fausto A. C. Cariboni, May 14 2021

			The initial values count the following partitions:
   3: (3)
   4: (3,1)
   5: (3,2)
   5: (3,1,1)
   6: (3,3)
   7: (3,3,1)
   7: (3,2,2)
   8: (3,3,2)
   8: (3,3,1,1)
   9: (3,3,3)
   9: (3,2,2,2)
  10: (3,3,3,1)
  10: (3,3,2,2)
  11: (3,3,3,2)
  11: (3,3,3,1,1)
  11: (3,2,2,2,2)
  12: (3,3,3,3)
  13: (3,3,3,3,1)
  13: (3,3,3,2,2)
  13: (3,2,2,2,2,2)
  14: (3,3,3,3,2)
  14: (3,3,3,3,1,1)
  15: (3,3,3,3,3)
  15: (3,2,2,2,2,2,2)

Cf. A002033, A108917, A275972, A276024, A299702.

Cf. A325592, A326015, A326016, A326017, A326018.

sums[ptn_]:=sums[ptn]=If[Length[ptn]==1,ptn,Union@@(Join[sums[#],sums[#]+Total[ptn]-Total[#]]&/@Union[Table[Delete[ptn,i],{i,Length[ptn]}]])];
kst[n_]:=Select[IntegerPartitions[n,All,{1,2,3}],Length[sums[Sort[#]]]==Times@@(Length/@Split[#]+1)-1&];
Table[Length[Select[kst[n],Max@@#==3&]],{n,0,30}]

A326035 Number of uniform knapsack partitions of n.

Original entry on oeis.org

1, 1, 2, 3, 4, 4, 6, 6, 9, 10, 12, 12, 17, 16, 20, 25, 27, 29, 35, 39, 44, 57, 53, 66, 75, 84, 84, 114, 112, 131, 133, 162, 167, 209, 192, 242, 250, 289, 279, 363, 348, 417, 404, 502, 487, 608, 557, 706, 682, 835, 773, 1004, 922, 1149, 1059, 1344, 1257, 1595

Offset: 0

Gus Wiseman, Jun 04 2019

nonn

An integer partition is uniform if all parts appear with the same multiplicity, and knapsack if every distinct submultiset has a different sum.

			The a(1) = 1 through a(8) = 9 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (1111)  (11111)  (51)      (61)       (62)
                                     (222)     (421)      (71)
                                     (111111)  (1111111)  (521)
                                                          (2222)
                                                          (3311)
                                                          (11111111)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..650

Cf. A002033, A047966, A072774, A108917, A275972, A276024, A299702.

Cf. A325592, A325858, A326015, A326016, A326017, A326036, A326037.

sums[ptn_]:=sums[ptn]=If[Length[ptn]==1,ptn,Union@@(Join[sums[#],sums[#]+Total[ptn]-Total[#]]&/@Union[Table[Delete[ptn,i],{i,Length[ptn]}]])];
ks[n_]:=Select[IntegerPartitions[n],Length[sums[Sort[#]]]==Times@@(Length/@Split[#]+1)-1&];
Table[Length[Select[ks[n],SameQ@@Length/@Split[#]&]],{n,30}]

A334268 Number of compositions of n where every distinct subsequence (not necessarily contiguous) has a different sum.

Original entry on oeis.org

1, 1, 2, 4, 5, 10, 10, 24, 24, 43, 42, 88, 72, 136, 122, 242, 213, 392, 320, 630, 490, 916, 742, 1432, 1160, 1955, 1604, 2826, 2310, 3850, 2888, 5416, 4426, 7332, 5814, 10046, 7983, 12946, 10236, 17780, 14100, 22674, 17582, 30232, 23674, 37522, 29426, 49832

Offset: 0

Gus Wiseman, Jun 02 2020

nonn

A composition of n is a finite sequence of positive integers summing to n.

The contiguous case is A325676.

			The a(1) = 1 through a(6) = 19 compositions:
  (1)  (2)    (3)      (4)        (5)          (6)
       (1,1)  (1,2)    (1,3)      (1,4)        (1,5)
              (2,1)    (2,2)      (2,3)        (2,4)
              (1,1,1)  (3,1)      (3,2)        (3,3)
                       (1,1,1,1)  (4,1)        (4,2)
                                  (1,1,3)      (5,1)
                                  (1,2,2)      (1,1,4)
                                  (2,2,1)      (2,2,2)
                                  (3,1,1)      (4,1,1)
                                  (1,1,1,1,1)  (1,1,1,1,1,1)

These compositions are ranked by A334967.
Compositions where every restriction to a subinterval has a different sum are counted by A169942 and A325677 and ranked by A333222. The case of partitions is counted by A325768 and ranked by A325779.
Positive subset-sums of partitions are counted by A276024 and A299701.
Knapsack partitions are counted by A108917 and A325592 and ranked by A299702, while the strict case is counted by A275972 and ranked by A059519 and A301899.
Knapsack compositions are counted by A325676 and A325687 and ranked by A333223. The case of partitions is counted by A325769 and ranked by A325778, for which the number of distinct consecutive subsequences is given by A325770.
Cf. A003022, A029931, A103295, A143823, A325680, A333224.

b:= proc(n, s) option remember; `if`(n=0, 1, add((h->
      `if`(nops(h)=nops(map(l-> add(i, i=l), h)),
       b(n-j, h), 0))({s[], map(l-> [l[], j], s)[]}), j=1..n))
    end:
a:= n-> b(n, {[]}):
seq(a(n), n=0..23);  # Alois P. Heinz, Jun 03 2020

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@Total/@Union[Subsets[#]]&]],{n,0,15}]

a(18)-a(47) from Alois P. Heinz, Jun 03 2020

cp's OEIS Frontend

A333224 Number of distinct positive consecutive subsequence-sums of the k-th composition in standard order.

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A325862 Number of integer partitions of n such that every set of distinct parts has a different sum.

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A325687 Triangle read by rows where T(n,k) is the number of length-k compositions of n such that every distinct consecutive subsequence has a different sum.

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A326017 Triangle read by rows where T(n,k) is the number of knapsack partitions of n with maximum k.

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A325690 Number of length-3 integer partitions of n whose largest part is not the sum of the other two.

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A326034 Number of knapsack partitions of n with largest part 3.

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A326035 Number of uniform knapsack partitions of n.

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A334268 Number of compositions of n where every distinct subsequence (not necessarily contiguous) has a different sum.

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