A325676 -id:A325676 - cp's OEIS frontend

A124770 Number of distinct nonempty subsequences for compositions in standard order.

0, 1, 1, 2, 1, 3, 3, 3, 1, 3, 2, 5, 3, 5, 5, 4, 1, 3, 3, 5, 3, 5, 5, 7, 3, 5, 5, 8, 5, 8, 7, 5, 1, 3, 3, 5, 2, 6, 6, 7, 3, 6, 3, 8, 6, 7, 8, 9, 3, 5, 6, 8, 6, 8, 7, 11, 5, 8, 8, 11, 7, 11, 9, 6, 1, 3, 3, 5, 3, 6, 6, 7, 3, 5, 5, 9, 5, 9, 9, 9, 3, 6, 5, 9, 5, 7, 8, 11, 6, 9, 8, 11, 9, 11, 11, 11, 3, 5, 6, 8, 5, 9

Offset: 0

Franklin T. Adams-Watters, Nov 06 2006

The standard order of compositions is given by A066099.

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. This gives a bijective correspondence between nonnegative integers and integer compositions. - Gus Wiseman, Apr 03 2020

			Composition number 11 is 2,1,1; the nonempty subsequences are 1; 2; 1,1; 2,1; 2,1,1; so a(11) = 5.
The table starts:
  0
  1
  1 2
  1 3 3 3
  1 3 2 5 3 5 5 4
  1 3 3 5 3 5 5 7 3 5 5 8 5 8 7 5
From _Gus Wiseman_, Apr 03 2020: (Start)
If the k-th composition in standard order is c, then we say that the STC-number of c is k. The STC-numbers of the distinct subsequences of the composition with STC-number k are given in column k below:
  1  2  1  4  1  1  1  8  1  2   1   1   1   1   1   16  1   2   1   2
        3     2  2  3     4  10  2   4   2   2   3       8   4   4   4
              5  6  7     9      3   12  6   3   7       17  18  3   20
                                 5       5   6   15              9
                                 11      13  14                  19
(End)

Alois P. Heinz, Rows n = 0..14, flattened

Row lengths are A011782.
Allowing empty subsequences gives A124771.
Dominates A333224, the version counting subsequence-sums instead of subsequences.
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 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. A000120, A003022, A029931, A048793, A066099, A070939, A103295, A108917, A143823, A325680.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Length[Union[ReplaceList[stc[n],{_,s__,_}:>{s}]]],{n,0,100}] (* Gus Wiseman, Apr 03 2020 *)

a(n) = A124771(n) - 1. - Gus Wiseman, Apr 03 2020

A032153 Number of ways to partition n elements into pie slices of different sizes.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 5, 6, 8, 11, 19, 22, 32, 41, 57, 92, 114, 155, 209, 280, 364, 587, 707, 984, 1280, 1737, 2213, 2990, 4390, 5491, 7361, 9650, 12708, 16451, 21567, 27506, 40100, 49201, 65701, 84128, 111278, 140595, 184661, 232356, 300680

Offset: 0

Christian G. Bower

Number of strict necklace compositions of n. A strict necklace composition of n is a finite sequence of distinct positive integers summing to n that is lexicographically minimal among all of its cyclic rotations. In other words, it is a strict composition of n starting with its least part. - Gus Wiseman, May 31 2019

			From _Gus Wiseman_, May 31 2019: (Start)
Inequivalent representatives of the a(1) = 1 through a(9) = 11 ways to slice a pie:
  (1)  (2)  (3)   (4)   (5)   (6)    (7)    (8)    (9)
            (12)  (13)  (14)  (15)   (16)   (17)   (18)
                        (23)  (24)   (25)   (26)   (27)
                              (123)  (34)   (35)   (36)
                              (132)  (124)  (125)  (45)
                                     (142)  (134)  (126)
                                            (143)  (135)
                                            (152)  (153)
                                                   (162)
                                                   (234)
                                                   (243)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..5000 (first 2001 terms from Robert Israel)
C. G. Bower, Transforms (2)
Index entries for sequences related to Lyndon words

Cf. A000079, A008289, A008965, A032020, A275972, A325676, A325788.

N:= 100: # to get a(0)..a(N)
K:= floor(isqrt(1+8*N)/2):
S:= series(1+add((k-1)!*x^((k^2+k)/2)/mul(1-x^j,j=1..k),k=1..K),x,N+1):
seq(coeff(S,x,j),j=0..N); # Robert Israel, Jul 15 2016
# second Maple program:
b:= proc(n, i, p) option remember; `if`(i*(i+1)/2 `if`(n=0, 1, b(n$2, -1)):
seq(a(n), n=1..50);  # Alois P. Heinz, Aug 12 2020

max=50; s=Sum[(x^(k(k+1)/2-1)*(k-1)!)/QPochhammer[x, x, k], {k, 1, max}] + O[x]^max; CoefficientList[s, x] (* Jean-François Alcover, Jan 19 2016 *)
neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
Table[Length[Select[Join@@Permutations/@Select[IntegerPartitions[n],UnsameQ@@#&],neckQ]],{n,30}] (* Gus Wiseman, May 31 2019 *)

N=66;  q='q+O('q^N);
gf=sum(n=1,N, (n-1)!*q^(n*(n+1)/2) / prod(k=1,n, 1-q^k ) );
Vec(gf)
/* Joerg Arndt, Oct 20 2012 */

seq(n)=[subst(serlaplace(p/y),y,1) | p <- Vec(y-1+prod(k=1, n, 1 + x^k*y + O(x*x^n)))] \\ Andrew Howroyd, Sep 13 2018

"CGK" (necklace, element, unlabeled) transform of 1, 1, 1, 1, ...
G.f.: Sum_{k >= 1} (k-1)! * x^((k^2+k)/2) / (Product_{j=1..k} 1-x^j). - Vladeta Jovovic, Sep 21 2004
a(n) = Sum_{k=1..floor((sqrt(8*n+1)-1)/2)} (k-1)! * A008289(n,k) for n > 0. - Alois P. Heinz, Aug 07 2020

a(0)=1 prepended by Andrew Howroyd, Sep 13 2018

A325769 Number of integer partitions of n whose distinct consecutive subsequences have different sums.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 7, 11, 12, 17, 19, 29, 28, 41, 42, 62, 61, 88, 87, 123, 121, 168, 164, 234, 225, 306, 306, 411, 401, 527, 533, 700, 689, 894, 885, 1163, 1150, 1452, 1469, 1866, 1835, 2333, 2346, 2913, 2913, 3638, 3619, 4511, 4537, 5497, 5576, 6859, 6827, 8263

Offset: 0

Gus Wiseman, May 21 2019

nonn

For example (3,3,1,1) is counted under a(8) because it has distinct consecutive subsequences (), (1), (1,1), (3), (3,1), (3,1,1), (3,3), (3,3,1), (3,3,1,1), all of which have different sums.

The Heinz numbers of these partitions are given by A325778.

			The a(1) = 1 through a(8) = 12 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (1111)  (221)    (51)      (61)       (62)
                            (311)    (222)     (322)      (71)
                            (11111)  (411)     (331)      (332)
                                     (111111)  (421)      (521)
                                               (511)      (611)
                                               (2221)     (2222)
                                               (4111)     (3311)
                                               (1111111)  (5111)
                                                          (11111111)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..300

Cf. A000041, A002033, A143823, A169942, A325676, A325677, A325683, A325768, A325770, A325778.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@Total/@Union[ReplaceList[#,{_,s__,_}:>{s}]]&]],{n,0,30}]

a(41)-a(53) from Fausto A. C. Cariboni, Feb 24 2021

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

A353390 Number of compositions of n whose own run-lengths are a subsequence (not necessarily consecutive).

Original entry on oeis.org

1, 1, 0, 0, 1, 2, 3, 2, 2, 8, 17, 26, 43, 77, 129, 210, 351, 569

Offset: 0

Gus Wiseman, May 15 2022

			The a(0) = 1 through a(9) = 8 compositions (empty columns indicated by dots):
  ()  (1)  .  .  (22)  (122)  (1122)  (11221)  (21122)  (333)
                       (221)  (1221)  (12211)  (22112)  (22113)
                              (2211)                    (22122)
                                                        (31122)
                                                        (121122)
                                                        (122112)
                                                        (211221)
                                                        (221121)
For example, the composition y = (2,2,3,3,1) has run-lengths (2,2,1), which form a (non-consecutive) subsequence, so y is counted under a(11).

The version for partitions is A325702.
The recursive version is A353391, ranked by A353431.
The consecutive case is A353392, ranked by A353432.
These compositions are ranked by A353402.
The reverse version is A353403.
The recursive consecutive version is A353430.
A003242 counts anti-run compositions, ranked by A333489.
A011782 counts compositions.
A047966 counts uniform partitions, compositions A329738.
A169942 counts Golomb rulers, ranked by A333222.
A325676 counts knapsack compositions, ranked by A333223, partitions A108917.
A325705 counts partitions containing all of their distinct multiplicities.
A329739 counts compositions with all distinct run-lengths, for runs A351013.
A353400 counts compositions with all run-lengths > 2.
Cf. A005811, A103295, A114901, A181591, A238279, A242882, A324572, A333755, A351017, A353401, A353426.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], MemberQ[Subsets[#],Length/@Split[#]]&]],{n,0,15}]

A325685 Number of compositions of n whose distinct consecutive subsequences have different sums, and such that these sums cover an initial interval of positive integers.

Original entry on oeis.org

1, 1, 1, 3, 1, 5, 3, 5, 3, 9, 1, 9, 5, 7, 5, 11, 1, 13, 5, 9, 5, 13, 3, 13, 7, 9, 5, 17, 1, 17, 5, 9, 9, 15, 5, 15, 5, 13, 5, 21, 1, 17, 9, 9, 9, 17, 3, 21, 7, 13, 5, 17, 5, 21, 9, 13, 5, 21, 1, 21, 9, 11, 13, 19, 5, 17, 5, 17, 5, 29, 1, 21, 9, 9, 13, 17, 5, 25, 7, 17, 7

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 perfect partitions (A002033).

			The distinct consecutive subsequences of (3,4,1,1) together with their sums are:
   1: {1}
   2: {1,1}
   3: {3}
   4: {4}
   5: {4,1}
   6: {4,1,1}
   7: {3,4}
   8: {3,4,1}
   9: {3,4,1,1}
Because the sums are all different and cover {1...9}, it follows that (3,4,1,1) is counted under a(9).
The a(1) = 1 through a(9) = 9 compositions:
  1   11   12    1111   113     132      1114      1133       1143
           21           122     231      1222      3311       1332
           111          221     111111   2221      11111111   2331
                        311              4111                 3411
                        11111            1111111              11115
                                                              12222
                                                              22221
                                                              51111
                                                              111111111

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..100
Fausto A. C. Cariboni, All compositions that yield a(n) for n = 1..100, Feb 21 2022.

Cf. A000079, A002033, A103295, A126796, A143823, A169942, A325676, A325677, A325683, A325684.

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

a(21)-a(25) from Jinyuan Wang, Jun 26 2020

a(21)-a(25) corrected, a(26)-a(80) from Fausto A. C. Cariboni, Feb 21 2022

A325778 Heinz numbers of integer partitions whose distinct consecutive subsequences have different sums.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 64, 65, 66, 67, 68, 69, 71, 73, 74, 75, 76, 77

Offset: 1

Gus Wiseman, May 20 2019

nonn

First differs from A299702 in having 462.

The enumeration of these partitions by sum is given by A325769.

			Most small numbers are in the sequence. However, the sequence of non-terms together with their prime indices begins:
  12: {1,1,2}
  24: {1,1,1,2}
  30: {1,2,3}
  36: {1,1,2,2}
  40: {1,1,1,3}
  48: {1,1,1,1,2}
  60: {1,1,2,3}
  63: {2,2,4}
  70: {1,3,4}
  72: {1,1,1,2,2}
  80: {1,1,1,1,3}
  84: {1,1,2,4}
  90: {1,2,2,3}
  96: {1,1,1,1,1,2}

Complement of A325777.

Cf. A002033, A056239, A112798, A143823, A169942, A299702, A301899, A325676, A325768, A325769, A325770, A325779.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],UnsameQ@@Total/@Union[ReplaceList[primeMS[#],{_,s__,_}:>{s}]]&]

A353391 Number of compositions of n that are empty, a singleton, or whose run-lengths are a subsequence that is already counted.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 1, 1, 4, 5, 7, 9, 11, 15, 22, 38, 45, 87, 93

Offset: 0

Gus Wiseman, May 15 2022

			The a(9) = 4 through a(14) = 15 compositions (A..E = 10..14):
  (9)       (A)       (B)       (C)       (D)       (E)
  (333)     (2233)    (141122)  (2244)    (161122)  (2255)
  (121122)  (3322)    (221123)  (4422)    (221125)  (5522)
  (221121)  (131122)  (221132)  (151122)  (221134)  (171122)
            (221131)  (221141)  (221124)  (221143)  (221126)
                      (231122)  (221142)  (221152)  (221135)
                      (321122)  (221151)  (221161)  (221153)
                                (241122)  (251122)  (221162)
                                (421122)  (341122)  (221171)
                                          (431122)  (261122)
                                          (521122)  (351122)
                                                    (531122)
                                                    (621122)
                                                    (122121122)
                                                    (221121221)

The non-recursive version is A353390, ranked by A353402.
The non-recursive consecutive version is A353392, ranked by A353432.
The non-recursive reverse version is A353403.
The unordered version is A353426, ranked by A353393 (nonprime A353389).
The consecutive version is A353430.
These compositions are ranked by A353431.
A003242 counts anti-run compositions, ranked by A333489.
A011782 counts compositions.
A329738 counts uniform compositions, partitions A047966.
A114901 counts compositions with no runs of length 1.
A169942 counts Golomb rulers, ranked by A333222.
A325676 counts knapsack compositions, ranked by A333223.
A325705 counts partitions containing all of their distinct multiplicities.
A329739 counts compositions with all distinct run-length.
Cf. A005811, A032020, A103295, A114640, A165413, A181591, A242882, A324572, A325702, A333755, A351013, A353401.

yosQ[y_]:=Length[y]<=1||MemberQ[Subsets[y],Length/@Split[y]]&&yosQ[Length/@Split[y]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],yosQ]],{n,0,15}]

A353402 Numbers k such that the k-th composition in standard order has its own run-lengths as a subsequence (not necessarily consecutive).

Original entry on oeis.org

0, 1, 10, 21, 26, 43, 53, 58, 107, 117, 174, 186, 292, 314, 346, 348, 349, 373, 430, 442, 570, 585, 586, 629, 676, 693, 696, 697, 698, 699, 804, 826, 858, 860, 861, 885, 954, 1082, 1141, 1173, 1210, 1338, 1353, 1387, 1392, 1393, 1394, 1396, 1397, 1398, 1466

Offset: 0

Gus Wiseman, May 15 2022

nonn

First differs from A353432 (the consecutive case) in having 0 and 53.

The k-th composition in standard order (graded reverse-lexicographic, 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. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The initial terms, their binary expansions, and the corresponding standard compositions:
    0:          0  ()
    1:          1  (1)
   10:       1010  (2,2)
   21:      10101  (2,2,1)
   26:      11010  (1,2,2)
   43:     101011  (2,2,1,1)
   53:     110101  (1,2,2,1)
   58:     111010  (1,1,2,2)
  107:    1101011  (1,2,2,1,1)
  117:    1110101  (1,1,2,2,1)
  174:   10101110  (2,2,1,1,2)
  186:   10111010  (2,1,1,2,2)
  292:  100100100  (3,3,3)
  314:  100111010  (3,1,1,2,2)
  346:  101011010  (2,2,1,2,2)
  348:  101011100  (2,2,1,1,3)
  349:  101011101  (2,2,1,1,2,1)
  373:  101110101  (2,1,1,2,2,1)
  430:  110101110  (1,2,2,1,1,2)
  442:  110111010  (1,2,1,1,2,2)

The version for partitions is A325755, counted by A325702.
These compositions are counted by A353390.
The recursive version is A353431, counted by A353391.
The consecutive case is A353432, counted by A353392.
A005811 counts runs in binary expansion.
A011782 counts compositions.
A066099 lists compositions in standard order, reverse A228351.
A333769 lists run-lengths of compositions in standard order.
Words with all distinct run-lengths: A032020, A044813, A098859, A130091, A329739, A351017.
Statistics of standard compositions:
- Length is A000120, sum A070939.
- Runs are counted by A124767, distinct A351014.
- Subsequences are counted by A334299, consecutive A124770/A124771.
- Runs-resistance is A333628.
Classes of standard compositions:
- Partitions are A114994, strict A333255, rev A225620, strict rev A333256.
- Runs are A272919.
- Golomb rulers are A333222, counted by A169942.
- Knapsack compositions are A333223, counted by A325676.
- Anti-runs are A333489, counted by A003242.
Cf. A114640, A165413, A181819, A318928, A325705, A329738, A333224/A333257, A333755, A353393, A353403, A353430.

stc[n_]:=Differences[Prepend[Join@@Position[ Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
rosQ[y_]:=Length[y]==0||MemberQ[Subsets[y],Length/@Split[y]];
Select[Range[0,100],rosQ[stc[#]]&]

A325592 Triangle read by rows where T(n,k) is the number of length-k knapsack partitions of n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 0, 1, 0, 1, 2, 2, 0, 1, 0, 1, 3, 2, 0, 0, 1, 0, 1, 3, 4, 2, 0, 0, 1, 0, 1, 4, 3, 3, 0, 0, 0, 1, 0, 1, 4, 7, 2, 2, 0, 0, 0, 1, 0, 1, 5, 6, 4, 2, 0, 0, 0, 0, 1, 0, 1, 5, 10, 6, 4, 2, 0, 0, 0, 0, 1, 0, 1, 6, 9, 5, 1, 2, 0, 0, 0, 0, 0, 1

Offset: 0

Gus Wiseman, May 15 2019

A knapsack partition of n is an integer partition of n whose distinct submultisets all have different sums.

			Triangle begins:
  1
  0  1
  0  1  1
  0  1  1  1
  0  1  2  0  1
  0  1  2  2  0  1
  0  1  3  2  0  0  1
  0  1  3  4  2  0  0  1
  0  1  4  3  3  0  0  0  1
  0  1  4  7  2  2  0  0  0  1
  0  1  5  6  4  2  0  0  0  0  1
  0  1  5 10  6  4  2  0  0  0  0  1
  0  1  6  9  5  1  2  0  0  0  0  0  1
  0  1  6 14 10  5  2  2  0  0  0  0  0  1
  0  1  7 13 11  3  3  2  0  0  0  0  0  0  1
  0  1  7 19 16  7  3  2  2  0  0  0  0  0  0  1
Row n = 12 counts the following partitions (A = 10, B = 11, C = 12):
   (C)  (66)   (444)   (3333)  (81111)  (222222)  (111111111111)
        (75)   (543)   (5511)           (711111)
        (84)   (552)   (7221)
        (93)   (732)   (7311)
        (A2)   (741)   (9111)
        (B1)   (822)
               (831)
               (921)
               (A11)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..10010

Row sums are A000041.
Column k = 2 is A004526.
Column k = 3 is A325690.
Cf. A002219, A006827, A108917, A143823, A169942, A275972, A276024, A292886, A321143, A325676, A325687.

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

cp's OEIS Frontend

A124770 Number of distinct nonempty subsequences for compositions in standard order.

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A032153 Number of ways to partition n elements into pie slices of different sizes.

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Maple

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A325769 Number of integer partitions of n whose distinct consecutive subsequences have different sums.

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

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A353390 Number of compositions of n whose own run-lengths are a subsequence (not necessarily consecutive).

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A325685 Number of compositions of n whose distinct consecutive subsequences have different sums, and such that these sums cover an initial interval of positive integers.

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A325778 Heinz numbers of integer partitions whose distinct consecutive subsequences have different sums.

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A353391 Number of compositions of n that are empty, a singleton, or whose run-lengths are a subsequence that is already counted.

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