A325688 -id:A325688 - cp's OEIS frontend

A325676 Number of compositions of n such that every distinct consecutive subsequence has a different sum.

1, 1, 2, 4, 5, 10, 12, 24, 26, 47, 50, 96, 104, 172, 188, 322, 335, 552, 590, 938, 1002, 1612, 1648, 2586, 2862, 4131, 4418, 6718, 7122, 10332, 11166, 15930, 17446, 24834, 26166, 37146, 41087, 55732, 59592, 84068, 89740, 122106, 133070, 177876, 194024, 262840, 278626

Offset: 0

Gus Wiseman, May 13 2019

nonn

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

Compare to the definition of knapsack partitions (A108917).

			The distinct consecutive subsequences of (1,4,4,3) together with their sums are:
   1: {1}
   3: {3}
   4: {4}
   5: {1,4}
   7: {4,3}
   8: {4,4}
   9: {1,4,4}
  11: {4,4,3}
  12: {1,4,4,3}
Because the sums are all different, (1,4,4,3) is counted under a(12).
The a(1) = 1 through a(6) = 12 compositions:
  (1)  (2)   (3)    (4)     (5)      (6)
       (11)  (12)   (13)    (14)     (15)
             (21)   (22)    (23)     (24)
             (111)  (31)    (32)     (33)
                    (1111)  (41)     (42)
                            (113)    (51)
                            (122)    (114)
                            (221)    (132)
                            (311)    (222)
                            (11111)  (231)
                                     (411)
                                     (111111)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..100

Cf. A000079, A103295, A108917, A169942, A235998, A321143.

Cf. A325466, A325545, A325680, A325682, A325685, A325687, A325688.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@Total/@Union[ReplaceList[#,{_,s__,_}:>{s}]]&]],{n,0,15}]

a(21)-a(22) from Jinyuan Wang, Jun 20 2020
a(23)-a(25) from Robert Price, Jun 19 2021
a(26)-a(46) from Fausto A. C. Cariboni, Feb 10 2022

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}]

A325680 Number of compositions of n such that every distinct circular subsequence has a different sum.

Original entry on oeis.org

1, 1, 2, 4, 5, 6, 8, 14, 16, 29, 24, 42, 46, 78, 66, 146, 133, 242, 208, 386, 352, 620, 494, 948, 842, 1447

Offset: 0

Gus Wiseman, May 13 2019

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

A circular subsequence is a sequence of consecutive terms where the first and last parts are also considered consecutive.

			The a(1) = 1 through a(8) = 16 compositions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (12)   (13)    (14)     (15)      (16)       (17)
             (21)   (22)    (23)     (24)      (25)       (26)
             (111)  (31)    (32)     (33)      (34)       (35)
                    (1111)  (41)     (42)      (43)       (44)
                            (11111)  (51)      (52)       (53)
                                     (222)     (61)       (62)
                                     (111111)  (124)      (71)
                                               (142)      (125)
                                               (214)      (152)
                                               (241)      (215)
                                               (412)      (251)
                                               (421)      (512)
                                               (1111111)  (521)
                                                          (2222)
                                                          (11111111)

Cf. A000079, A008965, A108917, A143823, A169942, A276024.

Cf. A325545, A325676, A325682, A325685, A325687, A325688.

subalt[q_]:=Union[ReplaceList[q,{_,s__,_}:>{s}],DeleteCases[ReplaceList[q,{t___,,u___}:>{u,t}],{}]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@Total/@subalt[#]&]],{n,0,15}]

a(18)-a(25) from Robert Price, Jun 19 2021

A325691 Number of length-3 integer partitions of n whose largest part is not greater than the sum of the other two.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 2, 2, 3, 3, 4, 4, 6, 5, 7, 7, 9, 8, 11, 10, 13, 12, 15, 14, 18, 16, 20, 19, 23, 21, 26, 24, 29, 27, 32, 30, 36, 33, 39, 37, 43, 40, 47, 44, 51, 48, 55, 52, 60, 56, 64, 61, 69, 65, 74, 70, 79, 75, 84, 80, 90, 85, 95, 91, 101, 96, 107, 102, 113

Offset: 0

Gus Wiseman, May 15 2019

Also the number of possible triples of edge-lengths of a triangle with perimeter n, where degenerate (self-intersecting) triangles are allowed.

The number of triples (a,b,c) for 1 <= a <= b <= c <= a+b and a+b+c = n. - Yuchun Ji, Oct 15 2020

			The a(3) = 1 through a(12) = 6 partitions:
  (111)  (211)  (221)  (222)  (322)  (332)  (333)  (433)  (443)  (444)
                       (321)  (331)  (422)  (432)  (442)  (533)  (543)
                                     (431)  (441)  (532)  (542)  (552)
                                                   (541)  (551)  (633)
                                                                 (642)
                                                                 (651)

Stefano Spezia, Table of n, a(n) for n = 0..10000
Colin Defant, Michael Joseph, Matthew Macauley, and Alex McDonough, Torsors and tilings from toric toggling, arXiv:2305.07627 [math.CO], 2023. See g.f. at p. 20.
Index entries for linear recurrences with constant coefficients, signature (0,1,1,1,-1,-1,-1,0,1).

Cf. A001399, A005044 (nondegenerate triangles), A008642, A069905, A124278.

Cf. A325688, A325690, A325694.

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

Conjectures from Colin Barker, May 16 2019: (Start)
G.f.: x^3*(1 + x - 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)
a(n) = A005044(n+3) - A000035(n+3). i.e., remove the only one triple (a=0,b,b) if n is even from the A005044 which is the number of triples (a,b,c) for 0 <= a <= b <= c <= a+b and a+b+c = n. - Yuchun Ji, Oct 15 2020
The above conjectured formulas are true. - Stefano Spezia, May 19 2023

A325686 Number of strict length-3 compositions x + y + z = n satisfying x + y != z and x != y + z.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 2, 6, 8, 18, 16, 30, 34, 48, 48, 72, 72, 96, 98, 126, 128, 162, 160, 198, 202, 240, 240, 288, 288, 336, 338, 390, 392, 450, 448, 510, 514, 576, 576, 648, 648, 720, 722, 798, 800, 882, 880, 966, 970, 1056, 1056, 1152, 1152, 1248, 1250, 1350, 1352

Offset: 0

Gus Wiseman, May 13 2019

nonn

A strict composition of n is a finite sequence of distinct positive integers summing to n.
From Kevin O'Bryant, Jun 02 2025: (Start)
Also the number of Sidon sets in {0,1,...,n} with 4 elements that contain both 0 and n.
Also, the number of 3-tuples of positive integers with the 6 numbers x, y, z, x+y, y+z, x+y+z=n all distinct. (End)

			The a(6) = 2 through a(10) = 16 compositions:
  (132)  (124)  (125)  (126)  (127)
  (231)  (142)  (143)  (135)  (136)
         (214)  (152)  (153)  (154)
         (241)  (215)  (162)  (163)
         (412)  (251)  (216)  (172)
         (421)  (341)  (234)  (217)
                (512)  (243)  (253)
                (521)  (261)  (271)
                       (315)  (316)
                       (324)  (352)
                       (342)  (361)
                       (351)  (451)
                       (423)  (613)
                       (432)  (631)
                       (513)  (712)
                       (531)  (721)
                       (612)
                       (621)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..5000

Column k = 3 of A325677.
Cf. A000079, A001399, A005044, A108917, A124278, A143823, A266223, A275972.
Cf. A325676, A325688, A325689, A325690, A325691.

Table[Length[Cases[Join@@Permutations/@IntegerPartitions[n,{3}],{x_,y_,z_}/;x!=y!=z&&x+y!=z &&x!=y+z]],{n,0,30}]

Conjectures from Colin Barker, May 14 2019: (Start)
G.f.: 2*x^6*(1 + 3*x + 3*x^2 + 5*x^3) / ((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>9. (End)
Above conjecture confirmed for n <= 5000. - Fausto A. C. Cariboni, Feb 17 2022

A325689 Number of length-3 compositions of n such that no part is the sum of the other two.

Original entry on oeis.org

0, 0, 0, 1, 0, 6, 4, 15, 12, 28, 24, 45, 40, 66, 60, 91, 84, 120, 112, 153, 144, 190, 180, 231, 220, 276, 264, 325, 312, 378, 364, 435, 420, 496, 480, 561, 544, 630, 612, 703, 684, 780, 760, 861, 840, 946, 924, 1035, 1012, 1128, 1104, 1225, 1200, 1326, 1300, 1431

Offset: 0

Gus Wiseman, May 15 2019

nonn

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

Confirmed recurrence relation from Colin Barker for n <= 5000. - Fausto A. C. Cariboni, Feb 15 2022

			The a(3) = 1 through a(8) = 12 compositions (empty columns not shown):
  (111)  (113)  (114)  (115)  (116)
         (122)  (141)  (124)  (125)
         (131)  (222)  (133)  (152)
         (212)  (411)  (142)  (161)
         (221)         (151)  (215)
         (311)         (214)  (233)
                       (223)  (251)
                       (232)  (323)
                       (241)  (332)
                       (313)  (512)
                       (322)  (521)
                       (331)  (611)
                       (412)
                       (421)
                       (511)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..5000

Cf. A000079, A001399, A005044, A008642, A069905, A124278, A266223.

Cf. A325676, A325688, A325690, A325691, A325694.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{3}],And@@Table[#[[i]]!=Total[Delete[#,i]],{i,3}]&]],{n,0,30}]

Conjectures from Colin Barker, May 16 2019: (Start)
G.f.: x^3*(1 - x + 4*x^2) / ((1 - x)^3*(1 + x)^2) for n>5.
a(n) = -(5 + 3*(-1)^n - 2*n) * (n-2) / 4 for n>0.
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
(End)

A325696 Number of length-3 strict compositions of n such that no part is the sum of the other two.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 6, 6, 18, 12, 30, 30, 48, 42, 72, 66, 96, 90, 126, 120, 162, 150, 198, 192, 240, 228, 288, 276, 336, 324, 390, 378, 450, 432, 510, 498, 576, 558, 648, 630, 720, 702, 798, 780, 882, 858, 966, 948, 1056, 1032, 1152, 1128, 1248, 1224, 1350

Offset: 0

Gus Wiseman, May 15 2019

nonn

A composition of n is a finite sequence of positive integers summing to n. It is strict if all parts are distinct.

			The a(6) = 6 through a(10) = 12 compositions:
  (124)  (125)  (126)  (127)
  (142)  (152)  (135)  (136)
  (214)  (215)  (153)  (163)
  (241)  (251)  (162)  (172)
  (412)  (512)  (216)  (217)
  (421)  (521)  (234)  (271)
                (243)  (316)
                (261)  (361)
                (315)  (613)
                (324)  (631)
                (342)  (712)
                (351)  (721)
                (423)
                (432)
                (513)
                (531)
                (612)
                (621)

Cf. A000079, A001399, A005044, A266223.

Cf. A325686, A325688, A325689 (non-strict case), A325695.

Table[Length[Cases[Join@@Permutations/@IntegerPartitions[n,{3}],{x_,y_,z_}/;x!=y!=z&&x+y!=z&&x!=y+z&&y!=x+z]],{n,0,30}]

Conjectures from Colin Barker, May 16 2019: (Start)
G.f.: 6*x^7*(1 + x + 2*x^2) / ((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>9.
(End)
a(n) = 6 * A325695(n). - Alois P. Heinz, Jun 18 2020

cp's OEIS Frontend

A325676 Number of compositions of n such that every distinct consecutive subsequence 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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A325680 Number of compositions of n such that every distinct circular subsequence has a different sum.

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A325691 Number of length-3 integer partitions of n whose largest part is not greater than the sum of the other two.

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A325686 Number of strict length-3 compositions x + y + z = n satisfying x + y != z and x != y + z.

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A325689 Number of length-3 compositions of n such that no part is the sum of the other two.

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A325696 Number of length-3 strict compositions of n such that no part is the sum of the other two.

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