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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. 10
1, 1, 1, 3, 1, 5, 3, 5, 3, 9, 1, 9, 5, 7, 5, 11, 1, 13, 5, 9, 5 (list; graph; refs; listen; history; text; internal format)
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..20.

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 A307806

Adjacent sequences:  A325682 A325683 A325684 * A325686 A325687 A325688

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, May 13 2019

STATUS

approved

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Last modified April 5 12:34 EDT 2020. Contains 333241 sequences. (Running on oeis4.)