A359899 - OEIS

%I #12 Jan 21 2023 14:28:21

%S 0,1,1,1,1,1,2,1,1,3,1,1,4,1,1,6,1,1,6,1,5,7,1,1,8,12,1,9,2,1,33,1,1,

%T 11,1,50,12,1,1,13,70,1,46,1,1,122,1,1,16,102,155,17,1,1,30,216,258,

%U 19,1,1,310,1,1,666,1,382,23,1,1,23,1596,1,393,1,1

%N Number of strict odd-length integer partitions of n whose parts have the same mean as median.

%H Andrew Howroyd, <a href="/A359899/b359899.txt">Table of n, a(n) for n = 0..1000</a>

%F a(p) = 1 for prime p. - _Andrew Howroyd_, Jan 21 2023

%e The a(30) = 33 partitions:

%e   (30)  (11,10,9)  (8,7,6,5,4)

%e         (12,10,8)  (9,7,6,5,3)

%e         (13,10,7)  (9,8,6,4,3)

%e         (14,10,6)  (9,8,6,5,2)

%e         (15,10,5)  (10,7,6,4,3)

%e         (16,10,4)  (10,7,6,5,2)

%e         (17,10,3)  (10,8,6,4,2)

%e         (18,10,2)  (10,8,6,5,1)

%e         (19,10,1)  (10,9,6,3,2)

%e                    (10,9,6,4,1)

%e                    (11,7,6,4,2)

%e                    (11,7,6,5,1)

%e                    (11,8,6,3,2)

%e                    (11,8,6,4,1)

%e                    (11,9,6,3,1)

%e                    (12,7,6,3,2)

%e                    (12,7,6,4,1)

%e                    (12,8,6,3,1)

%e                    (12,9,6,2,1)

%e                    (13,7,6,3,1)

%e                    (13,8,6,2,1)

%e                    (14,7,6,2,1)

%e                    (11,10,6,2,1)

%t Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&OddQ[Length[#]]&&Mean[#]==Median[#]&]],{n,0,30}]

%o (PARI) \\ Q(n,k,m) is g.f. for k strict parts of max size m.

%o Q(n,k,m)={polcoef(prod(i=1, m, 1 + y*x^i + O(x*x^n)), k, y)}

%o a(n)={if(n==0, 0, sumdiv(n, d, if(d%2, my(m=n/d, h=d\2, r=n-m*(h+1)); if(r>=h*(h+1), polcoef(Q(r, h, m-1)*Q(r, h, r), r)))))} \\ _Andrew Howroyd_, Jan 21 2023

%Y Strict odd-length case of A240219, complement A359894, ranked by A359889.

%Y Strict case of A359895, complement A359896, ranked by A359891.

%Y Odd-length case of A359897, complement A359898.

%Y The complement is counted by A359900.

%Y A000041 counts partitions, strict A000009.

%Y A008284/A058398/A327482 count partitions by mean, ranked by A326567/A326568.

%Y A008289 counts strict partitions by mean.

%Y A027193 counts odd-length partitions, strict A067659, ranked by A026424.

%Y A067538 counts ptns with integer mean, strict A102627, ranked by A316413.

%Y A237984 counts ptns containing their mean, strict A240850, ranked by A327473.

%Y A325347 counts ptns with integer median, strict A359907, ranked by A359908.

%Y A359893 and A359901 count partitions by median, odd-length A359902.

%Y Cf. A000016, A065795, A066571, A240851, A359903, A359906, A359910.

%K nonn

%O 0,7

%A _Gus Wiseman_, Jan 20 2023