The OEIS is supported by the many generous donors to the OEIS Foundation.
%I #7 Jan 22 2023 09:16:56
%S 1,1,0,1,1,1,0,0,1,2,0,2,0,0,0,1,3,0,1,2,0,0,0,0,1,4,1,2,0,3,0,0,0,0,
%T 0,1,6,1,3,0,1,3,0,0,0,0,0,0,1,8,1,6,0,2,0,4,0,0,0,0,0,0,0,1,11,2,7,1,
%U 3,0,1,4,0,0,0,0,0,0,0,0,1
%N Triangle read by rows where T(n,k) is the number of integer partitions of n with median k, where k ranges from 1 to n in steps of 1/2.
%C The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).
%e Triangle begins:
%e 1
%e 1 0 1
%e 1 1 0 0 1
%e 2 0 2 0 0 0 1
%e 3 0 1 2 0 0 0 0 1
%e 4 1 2 0 3 0 0 0 0 0 1
%e 6 1 3 0 1 3 0 0 0 0 0 0 1
%e 8 1 6 0 2 0 4 0 0 0 0 0 0 0 1
%e 11 2 7 1 3 0 1 4 0 0 0 0 0 0 0 0 1
%e 15 2 10 3 4 0 2 0 5 0 0 0 0 0 0 0 0 0 1
%e 20 3 13 3 7 0 3 0 1 5 0 0 0 0 0 0 0 0 0 0 1
%e 26 4 19 3 11 1 4 0 2 0 6 0 0 0 0 0 0 0 0 0 0 0 1
%e For example, row n = 8 counts the following partitions:
%e 611 4211 422 . 332 . 44 . . . . . . . 8
%e 5111 521 431 53
%e 32111 2222 62
%e 41111 3221 71
%e 221111 3311
%e 311111 22211
%e 2111111
%e 11111111
%t Table[Length[Select[IntegerPartitions[n], Median[#]==k&]],{n,1,10},{k,1,n,1/2}]
%Y Row sums are A000041.
%Y Row lengths are 2n-1 = A005408(n-1).
%Y Column k=1 is A027336(n+1).
%Y For mean instead of median we have A058398, see also A008284, A327482.
%Y The mean statistic is ranked by A326567/A326568.
%Y Omitting half-steps gives A359901.
%Y The odd-length case is A359902.
%Y The median statistic is ranked by A360005(n)/2.
%Y First appearances of medians are ranked by A360006, A360007.
%Y A027193 counts odd-length partitions, strict A067659, ranked by A026424.
%Y A067538 counts partitions w/ integer mean, strict A102627, ranked by A316413.
%Y A240219 counts partitions w/ the same mean as median, complement A359894.
%Y Cf. A325347, A349156, A359889, A359895, A359906, A359907, A360008.
%K nonn,tabf
%O 1,10
%A _Gus Wiseman_, Jan 21 2023