A391620 - OEIS

%I #12 Jan 04 2026 23:27:44

%S 1,2,3,4,6,10,14,20,28,39,53,72,96,128,168,220,285,369,472,603,765,

%T 968,1216,1525,1901,2366,2929,3618,4450,5464,6681,8154,9918,12041,

%U 14572,17606,21210,25510,30602,36650,43792,52246,62194,73927,87700

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

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

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

%C 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.

%e The partition (5,4) is the first sums of (3,2,2) so is not counted under a(9).

%e The a(1) = 1 through a(6) = 10 partitions:

%e   (1)  (2)    (3)      (2,2)      (3,2)        (3,3)

%e        (1,1)  (2,1)    (3,1)      (4,1)        (4,2)

%e               (1,1,1)  (2,1,1)    (2,2,1)      (5,1)

%e                        (1,1,1,1)  (3,1,1)      (2,2,2)

%e                                   (2,1,1,1)    (3,2,1)

%e                                   (1,1,1,1,1)  (4,1,1)

%e                                                (2,2,1,1)

%e                                                (3,1,1,1)

%e                                                (2,1,1,1,1)

%e                                                (1,1,1,1,1,1)

%t pas[y_,k_]:=Table[(-1)^j*k+Sum[(-1)^(i-j)*y[[i]],{i,j}],{j,0,Length[y]}];

%t agc[m_]:=Select[Table[pas[m,b],{b,1,Max[m]}],Min@@#>1&];

%t Table[Length[Select[IntegerPartitions[n],agc[#]=={}&]],{n,15}]

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

%Y For a unique choice we appear to have A026797.

%Y These partitions appear to have ranks A080671.

%Y For more than one choice we appear to have A185325.

%Y The complement for compositions is A391235, ranks A391626.

%Y For compositions instead of partitions we have A391641.

%Y A004709 ranks partitions with distinct first sums, count A000726.

%Y A046099 ranks partitions without distinct first sums, count A295341.

%Y A357213 counts compositions by sum of first sums.

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

%Y A390673 ranks compositions with distinct first sums, count A390567.

%Y A390674 ranks compositions with equal first sums, count A342527.

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

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

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

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

%K nonn

%O 1,2

%A _Gus Wiseman_, Dec 30 2025