A362558 - OEIS

A362558

Number of integer partitions of n without a nonempty initial consecutive subsequence summing to n/2.

%I #15 Apr 28 2023 15:46:33

%S 1,1,1,3,2,7,6,15,11,30,27,56,44,101,93,176,149,297,271,490,432,792,

%T 744,1255,1109,1958,1849,3010,2764,4565,4287,6842,6328,10143,9673,

%U 14883,13853,21637,20717,31185,29343,44583,42609,63261,60100,89134,85893,124754

%N Number of integer partitions of n without a nonempty initial consecutive subsequence summing to n/2.

%C Also the number of n-multisets of positive integers that (1) have integer median, (2) cover an initial interval, and (3) have weakly decreasing multiplicities.

%e The a(1) = 1 through a(7) = 15 partitions:

%e (1) (2) (3) (4) (5) (6) (7)

%e (21) (31) (32) (42) (43)

%e (111) (41) (51) (52)

%e (221) (222) (61)

%e (311) (411) (322)

%e (2111) (2211) (331)

%e (11111) (421)

%e (511)

%e (2221)

%e (3211)

%e (4111)

%e (22111)

%e (31111)

%e (211111)

%e (1111111)

%e The partition y = (3,2,1,1,1) has nonempty initial consecutive subsequences (3,2,1,1,1), (3,2,1,1), (3,2,1), (3,2), (3), with sums 8, 7, 6, 5, 3. Since 4 is missing, y is counted under a(8).

%t Table[Length[Select[IntegerPartitions[n],!MemberQ[Accumulate[#],n/2]&]],{n,0,15}]

%Y The odd bisection is A058695.

%Y The version for compositions is A213173.

%Y The complement is counted by A322439 aerated.

%Y The even bisection is A362051.

%Y For mean instead of median we have A362559.

%Y A000041 counts integer partitions, strict A000009.

%Y A325347 counts partitions with integer median, complement A307683.

%Y A359893/A359901/A359902 count partitions by median.

%Y Cf. A058398, A108917, A169942, A325676, A353864, A360254, A360672, A360675, A360686, A360687, A362560.

%K nonn

%O 0,4

%A _Gus Wiseman_, Apr 24 2023