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A359902 Triangle read by rows where T(n,k) is the number of odd-length integer partitions of n with median k.

%I A359902 #6 Jan 22 2023 09:16:44
%S A359902 1,0,1,1,0,1,1,0,0,1,2,1,0,0,1,2,2,0,0,0,1,4,2,1,0,0,0,1,4,3,2,0,0,0,
%T A359902 0,1,7,4,3,1,0,0,0,0,1,8,6,3,2,0,0,0,0,0,1,12,8,4,3,1,0,0,0,0,0,1,14,
%U A359902 11,5,4,2,0,0,0,0,0,0,1
%N A359902 Triangle read by rows where T(n,k) is the number of odd-length integer partitions of n with median k.
%C A359902 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 A359902 Triangle begins:
%e A359902   1
%e A359902   0  1
%e A359902   1  0  1
%e A359902   1  0  0  1
%e A359902   2  1  0  0  1
%e A359902   2  2  0  0  0  1
%e A359902   4  2  1  0  0  0  1
%e A359902   4  3  2  0  0  0  0  1
%e A359902   7  4  3  1  0  0  0  0  1
%e A359902   8  6  3  2  0  0  0  0  0  1
%e A359902  12  8  4  3  1  0  0  0  0  0  1
%e A359902  14 11  5  4  2  0  0  0  0  0  0  1
%e A359902  21 14  8  4  3  1  0  0  0  0  0  0  1
%e A359902  24 20 10  5  4  2  0  0  0  0  0  0  0  1
%e A359902  34 25 15  6  5  3  1  0  0  0  0  0  0  0  1
%e A359902 For example, row n = 9 counts the following partitions:
%e A359902   (7,1,1)              (5,2,2)      (3,3,3)  (4,4,1)  .  .  .  .  (9)
%e A359902   (3,3,1,1,1)          (6,2,1)      (4,3,2)
%e A359902   (4,2,1,1,1)          (2,2,2,2,1)  (5,3,1)
%e A359902   (5,1,1,1,1)          (3,2,2,1,1)
%e A359902   (2,2,1,1,1,1,1)
%e A359902   (3,1,1,1,1,1,1)
%e A359902   (1,1,1,1,1,1,1,1,1)
%t A359902 Table[Length[Select[IntegerPartitions[n],OddQ[Length[#]]&&Median[#]==k&]],{n,15},{k,n}]
%Y A359902 Column k=1 is A002865(n-1).
%Y A359902 Row sums are A027193 (odd-length ptns), strict A067659.
%Y A359902 This is the odd-length case of A359901, with half-steps A359893.
%Y A359902 The median statistic is ranked by A360005(n)/2.
%Y A359902 First appearances of medians are ranked by A360006, A360007.
%Y A359902 A000041 counts partitions, strict A000009.
%Y A359902 A058398 counts partitions by mean, see also A008284, A327482.
%Y A359902 A067538 counts partitions w/ integer mean, strict A102627, ranked by A316413.
%Y A359902 A240219 counts partitions w/ the same mean as median, complement A359894.
%Y A359902 A325347 counts partitions w/ integer median, complement A307683.
%Y A359902 A326567/A326568 gives mean of prime indices.
%Y A359902 Cf. A008289, A026424, A327472, A359889, A359895, A359906, A359907, A359910.
%K A359902 nonn,tabl
%O A359902 1,11
%A A359902 _Gus Wiseman_, Jan 21 2023