A390567 Number of integer compositions of n with all distinct first sums.

1, 1, 2, 3, 6, 9, 16, 27, 44, 73, 114, 191, 302, 473, 728, 1183, 1800, 2769, 4198, 6555, 9850, 14941, 22284, 33467, 50276, 74741, 110322, 162019, 238638, 350809, 513380, 746099, 1079044, 1555337, 2255666, 3251579, 4668238, 6660713, 9501776, 13475439, 19227708

Offset

0,3

Comments

The first sums of a nonempty sequence (a, b, c, d, ...) are (a+b, b+c, c+d, ...).

Links

Table of n, a(n) for n=0..40.

Example

The composition (2,2,1,1) has first sums (4,3,2) so is counted under a(6).
The a(1) = 1 through a(6) = 16 compositions:
  (1)  (2)    (3)    (4)      (5)      (6)
       (1,1)  (1,2)  (1,3)    (1,4)    (1,5)
              (2,1)  (2,2)    (2,3)    (2,4)
                     (3,1)    (3,2)    (3,3)
                     (1,1,2)  (4,1)    (4,2)
                     (2,1,1)  (1,1,3)  (5,1)
                              (1,2,2)  (1,1,4)
                              (2,2,1)  (1,2,3)
                              (3,1,1)  (1,3,2)
                                       (2,1,3)
                                       (2,3,1)
                                       (3,1,2)
                                       (3,2,1)
                                       (4,1,1)
                                       (1,1,2,2)
                                       (2,2,1,1)

Mathematica

firsums[c_]:=Table[c[[i]]+c[[i+1]], {i, Length[c]-1}];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], UnsameQ@@firsums[#]&]], {n, 0, 15}]

Crossrefs

For partitions or multisets instead of compositions we have A000726, ranks A004709.

For multiplicities instead of first sums we have A242882.

For first differences instead of first sums we have A325545, ranks A389597.

For run lengths instead of first sums we have A329739, ranks A351596.

For equal instead of distinct first sums we have A342527.

These compositions are ranked by A390673.

A011782 counts compositions.

A175342 counts arithmetic progressions, ranks A389731, subsets A051336.

A390432 lists first sums of standard compositions.

Cf. A000740, A008965, A049988, A325325, A325551, A325554, A361003, A389601, A389741, A389743, A389744.

Sequence in context: A118033 A048810 A331680 * A367205 A327475 A017915

Adjacent sequences: A390564 A390565 A390566 * A390568 A390569 A390570

Keyword

nonn

Author

Gus Wiseman, Nov 15 2025

Extensions

a(26)-a(40) from Christian Sievers, Jan 15 2026

Status

approved