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Table, number of partitions of n with k as high median.
23

%I #18 Jul 13 2023 16:28:48

%S 1,1,1,1,1,1,2,1,1,1,3,1,1,1,1,4,3,1,1,1,1,6,4,1,1,1,1,1,8,6,3,1,1,1,

%T 1,1,11,8,5,1,1,1,1,1,1,15,11,7,3,1,1,1,1,1,1,20,15,9,5,1,1,1,1,1,1,1,

%U 26,21,12,8,3,1,1,1,1,1,1,1,35,27,16,10,5,1,1,1,1,1,1,1,1,45,37,21,13,8,3

%N Table, number of partitions of n with k as high median.

%C For a multiset with an odd number of elements, the high median is the same as the median. For a multiset with an even number of elements, the high median is the larger of the two central elements.

%C This table may be read as an upper right triangle with n >= 1 as column index and k >= 1 as row index. - _Peter Munn_, Jul 16 2017

%C Arrange the parts of a partition nonincreasing order. Remove the last part, then the first, then the last remaining part, then the first remaining part, and continue until only a single number, the high median, remains. - _Clark Kimberling_, May 14 2019

%e For the partition [2,1^2], the sole middle element is 1, so that is the high median. For [3,2,1^2], the two middle elements are 1 and 2; the high median is the larger, 2.

%e From _Gus Wiseman_, Jul 12 2023: (Start)

%e Triangle begins:

%e 1

%e 1 1

%e 1 1 1

%e 2 1 1 1

%e 3 1 1 1 1

%e 4 3 1 1 1 1

%e 6 4 1 1 1 1 1

%e 8 6 3 1 1 1 1 1

%e 11 8 5 1 1 1 1 1 1

%e 15 11 7 3 1 1 1 1 1 1

%e 20 15 9 5 1 1 1 1 1 1 1

%e 26 21 12 8 3 1 1 1 1 1 1 1

%e 35 27 16 10 5 1 1 1 1 1 1 1 1

%e 45 37 21 13 8 3 1 1 1 1 1 1 1 1

%e 58 48 29 16 11 5 1 1 1 1 1 1 1 1 1

%e Row n = 8 counts the following partitions:

%e (611) (521) (431) (44) (53) (62) (71) (8)

%e (5111) (422) (332)

%e (41111) (4211) (3311)

%e (32111) (3221)

%e (311111) (2222)

%e (221111) (22211)

%e (2111111)

%e (11111111)

%e (End)

%t Map[BinCounts[#, {1, #[[1]] + 1, 1}] &[Map[#[[Floor[(Length[#] + 1)/2]]] &, IntegerPartitions[#]]] &, Range[13]] (* _Peter J. C. Moses_, May 14 2019 *)

%Y Row sums are A000041.

%Y Column k = 1 is A027336(n-1), ranks A364056.

%Y Column k = 1 in the low version is A027336, ranks A363488.

%Y The low version of this triangle is A124943.

%Y The rank statistic for this triangle is A363942, low version A363941.

%Y A version for mean instead of median is A363946, low A363945.

%Y A version for mode instead of median is A363953, low A363952.

%Y A008284 counts partitions by length, maximum, or decreasing mean.

%Y A026794 counts partitions by minimum, strict A026821.

%Y A325347 counts partitions with integer median, ranks A359908.

%Y A359893 and A359901 count partitions by median.

%Y A360005(n)/2 returns median of prime indices.

%Y Cf. A008289, A025065, A027193, A067538, A237984, A240219, A362608, A363740, A363943, A363944.

%K nonn,tabl

%O 1,7

%A _Franklin T. Adams-Watters_, Nov 13 2006