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.

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

Author

Gus Wiseman, May 13 2019

Keywords

• nonn

Comments

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

Compare to the definition of perfect partitions (A002033).

Examples

			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
		

Links

• 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.

Crossrefs

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

Programs

Mathematica

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

Extensions

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