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A325685 revision #11

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, 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

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OFFSET

0,4

COMMENTS

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

Compare to the definition of perfect partitions (A002033).

LINKS

Table of n, a(n) for n=0..85.

EXAMPLE

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

MATHEMATICA

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

CROSSREFS

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

Sequence in context: A143865 A071168 A091926 * * A109606 A318727

Adjacent sequences: A325682 A325683 A325684 * A325686 A325687 A325688

KEYWORD

nonn,more,changed

AUTHOR

Gus Wiseman, May 13 2019

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

STATUS

editing