A391629 Number of multisets summing to n (or reversed partitions of n) that are not the first sums of any integer partition.

1, 1, 2, 3, 6, 9, 14, 19, 29, 40, 55, 73, 100, 133, 175, 227, 296, 382, 489, 623, 791, 1000, 1254, 1569, 1957, 2434, 3009, 3714, 4564, 5600

Offset

1,3

Comments

Here, a multiset is a finite weakly increasing sequence of positive integers.

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

Also the number of integer partitions y of n such that the k-based partial alternating sums are not positive and weakly increasing for any k. Here, the k-based partial alternating sums pas(q,k) of a finite sequence q are given by pas(q,k)_j = (-1)^j * k + Sum_{i=1..j} (-1)^(i+j) * q_i.

Links

Table of n, a(n) for n=1..30.

Formula

Conjecture: a(n) = A000041(n) - A368684(n).

Example

The a(1) = 1 through a(7) = 14 partitions:
  (1)  (11)  (21)   (31)    (32)     (33)      (43)
             (111)  (211)   (41)     (42)      (52)
                    (1111)  (221)    (51)      (61)
                            (311)    (321)     (322)
                            (2111)   (411)     (331)
                            (11111)  (2211)    (421)
                                     (3111)    (511)
                                     (21111)   (2221)
                                     (111111)  (3211)
                                               (4111)
                                               (22111)
                                               (31111)
                                               (211111)
                                               (1111111)

Mathematica

prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
pas[y_, k_]:=Table[(-1)^j*k + Sum[(-1)^(i+j)*y[[i]], {i, j}], {j, 0, Length[y]}];
Table[Length[Select[IntegerPartitions[n], Select[Table[pas[#, b], {b, 1, Max[#]}], LessEqual@@#&]=={}&]], {n, 20}]

Crossrefs

For strict partitions on the right we appear to have A000065.

For strict partitions on the left we appear to have A111133.

The complement appears to be counted by A368684.

For partitions with all parts > 1 we have A390431.

These partitions are ranked by A390445.

The complement is counted by A390446, ranks A390448.

The unreversed version is A390447.

A000041 counts integer partitions, strict A000009.

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

A390429 counts distinct possible first sums of partitions.

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

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

Cf. A008484, A046099, A295341, A325325, A389811, A390678, A391620, A391641, A391645, A391981.

Sequence in context: A061925 A073736 A379736 * A101593 A349502 A226893

Adjacent sequences: A391626 A391627 A391628 * A391630 A391631 A391632

Keyword

nonn,more

Author

Gus Wiseman, Dec 31 2025

Status

approved