cp's OEIS Frontend

A362558 Number of integer partitions of n without a nonempty initial consecutive subsequence summing to n/2.

%I A362558 #15 Apr 28 2023 15:46:33
%S A362558 1,1,1,3,2,7,6,15,11,30,27,56,44,101,93,176,149,297,271,490,432,792,
%T A362558 744,1255,1109,1958,1849,3010,2764,4565,4287,6842,6328,10143,9673,
%U A362558 14883,13853,21637,20717,31185,29343,44583,42609,63261,60100,89134,85893,124754
%N A362558 Number of integer partitions of n without a nonempty initial consecutive subsequence summing to n/2.
%C A362558 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 A362558 The a(1) = 1 through a(7) = 15 partitions:
%e A362558   (1)  (2)  (3)    (4)   (5)      (6)     (7)
%e A362558             (21)   (31)  (32)     (42)    (43)
%e A362558             (111)        (41)     (51)    (52)
%e A362558                          (221)    (222)   (61)
%e A362558                          (311)    (411)   (322)
%e A362558                          (2111)   (2211)  (331)
%e A362558                          (11111)          (421)
%e A362558                                           (511)
%e A362558                                           (2221)
%e A362558                                           (3211)
%e A362558                                           (4111)
%e A362558                                           (22111)
%e A362558                                           (31111)
%e A362558                                           (211111)
%e A362558                                           (1111111)
%e A362558 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 A362558 Table[Length[Select[IntegerPartitions[n],!MemberQ[Accumulate[#],n/2]&]],{n,0,15}]
%Y A362558 The odd bisection is A058695.
%Y A362558 The version for compositions is A213173.
%Y A362558 The complement is counted by A322439 aerated.
%Y A362558 The even bisection is A362051.
%Y A362558 For mean instead of median we have A362559.
%Y A362558 A000041 counts integer partitions, strict A000009.
%Y A362558 A325347 counts partitions with integer median, complement A307683.
%Y A362558 A359893/A359901/A359902 count partitions by median.
%Y A362558 Cf. A058398, A108917, A169942, A325676, A353864, A360254, A360672, A360675, A360686, A360687, A362560.
%K A362558 nonn
%O A362558 0,4
%A A362558 _Gus Wiseman_, Apr 24 2023