A391620 Number of integer partitions of n > 0 that are not the first sums of any composition with all parts > 1.

1, 2, 3, 4, 6, 10, 14, 20, 28, 39, 53, 72, 96, 128, 168, 220, 285, 369, 472, 603, 765, 968, 1216, 1525, 1901, 2366, 2929, 3618, 4450, 5464, 6681, 8154, 9918, 12041, 14572, 17606, 21210, 25510, 30602, 36650, 43792, 52246, 62194, 73927, 87700

Offset

1,2

Comments

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

Conjecture: These are integer partitions of n whose least part is < 4.

Also the number of integer partitions of n whose b-based partial alternating sums are not all > 1 for any integer b. Here, the b-based partial alternating sums of a sequence q are given by pas(q,b)_j = (-1)^j * b + Sum_{i=1..j} (-1)^(i+j) * q_i.

Links

Table of n, a(n) for n=1..45.

Example

The partition (5,4) is the first sums of (3,2,2) so is not counted under a(9).
The a(1) = 1 through a(6) = 10 partitions:
  (1)  (2)    (3)      (2,2)      (3,2)        (3,3)
       (1,1)  (2,1)    (3,1)      (4,1)        (4,2)
              (1,1,1)  (2,1,1)    (2,2,1)      (5,1)
                       (1,1,1,1)  (3,1,1)      (2,2,2)
                                  (2,1,1,1)    (3,2,1)
                                  (1,1,1,1,1)  (4,1,1)
                                               (2,2,1,1)
                                               (3,1,1,1)
                                               (2,1,1,1,1)
                                               (1,1,1,1,1,1)

Mathematica

pas[y_, k_]:=Table[(-1)^j*k+Sum[(-1)^(i-j)*y[[i]], {i, j}], {j, 0, Length[y]}];
agc[m_]:=Select[Table[pas[m, b], {b, 1, Max[m]}], Min@@#>1&];
Table[Length[Select[IntegerPartitions[n], agc[#]=={}&]], {n, 15}]

Crossrefs

The complement appears to be counted by A008484, ranks A007775.

For a unique choice we appear to have A026797.

These partitions appear to have ranks A080671.

For more than one choice we appear to have A185325.

The complement for compositions is A391235, ranks A391626.

For compositions instead of partitions we have A391641.

A004709 ranks partitions with distinct first sums, count A000726.

A046099 ranks partitions without distinct first sums, count A295341.

A357213 counts compositions by sum of first sums.

A390307 lists first sums of prime indices (reverse A390362), row ranks A390449.

A390673 ranks compositions with distinct first sums, count A390567.

A390674 ranks compositions with equal first sums, count A342527.

A390676 ranks compositions that are first sums, union of A390568.

A390677 ranks compositions that are not first sums, count A391680.

A391629 counts multisets that are not first sums of partitions, see A390446, A390447.

Cf. A000041, A008965, A011782, A070211, A390448, A390675, A390678, A391621, A391642, A391644, A391645.

Sequence in context: A318558 A347867 A394251 * A256248 A089223 A240057

Adjacent sequences: A391617 A391618 A391619 * A391621 A391622 A391623

Keyword

nonn

Author

Gus Wiseman, Dec 30 2025

Status

approved