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