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A359893 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.

%I A359893 #7 Jan 22 2023 09:16:56
%S A359893 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 A359893 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 A359893 3,0,1,4,0,0,0,0,0,0,0,0,1
%N A359893 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 A359893 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 A359893 Triangle begins:
%e A359893   1
%e A359893   1  0  1
%e A359893   1  1  0  0  1
%e A359893   2  0  2  0  0  0  1
%e A359893   3  0  1  2  0  0  0  0  1
%e A359893   4  1  2  0  3  0  0  0  0  0  1
%e A359893   6  1  3  0  1  3  0  0  0  0  0  0  1
%e A359893   8  1  6  0  2  0  4  0  0  0  0  0  0  0  1
%e A359893  11  2  7  1  3  0  1  4  0  0  0  0  0  0  0  0  1
%e A359893  15  2 10  3  4  0  2  0  5  0  0  0  0  0  0  0  0  0  1
%e A359893  20  3 13  3  7  0  3  0  1  5  0  0  0  0  0  0  0  0  0  0  1
%e A359893  26  4 19  3 11  1  4  0  2  0  6  0  0  0  0  0  0  0  0  0  0  0  1
%e A359893 For example, row n = 8 counts the following partitions:
%e A359893   611       4211  422    .  332  .  44  .  .  .  .  .  .  .  8
%e A359893   5111            521       431     53
%e A359893   32111           2222              62
%e A359893   41111           3221              71
%e A359893   221111          3311
%e A359893   311111          22211
%e A359893   2111111
%e A359893   11111111
%t A359893 Table[Length[Select[IntegerPartitions[n], Median[#]==k&]],{n,1,10},{k,1,n,1/2}]
%Y A359893 Row sums are A000041.
%Y A359893 Row lengths are 2n-1 = A005408(n-1).
%Y A359893 Column k=1 is A027336(n+1).
%Y A359893 For mean instead of median we have A058398, see also A008284, A327482.
%Y A359893 The mean statistic is ranked by A326567/A326568.
%Y A359893 Omitting half-steps gives A359901.
%Y A359893 The odd-length case is A359902.
%Y A359893 The median statistic is ranked by A360005(n)/2.
%Y A359893 First appearances of medians are ranked by A360006, A360007.
%Y A359893 A027193 counts odd-length partitions, strict A067659, ranked by A026424.
%Y A359893 A067538 counts partitions w/ integer mean, strict A102627, ranked by A316413.
%Y A359893 A240219 counts partitions w/ the same mean as median, complement A359894.
%Y A359893 Cf. A325347, A349156, A359889, A359895, A359906, A359907, A360008.
%K A359893 nonn,tabf
%O A359893 1,10
%A A359893 _Gus Wiseman_, Jan 21 2023