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A364060 Triangle read by rows where T(n,k) is the number of integer partitions of n with rounded mean k.

%I A364060 #7 Jul 08 2023 23:06:24
%S A364060 1,0,1,0,1,1,0,1,1,1,0,2,2,0,1,0,2,4,0,0,1,0,2,5,3,0,0,1,0,4,7,0,3,0,
%T A364060 0,1,0,4,8,5,4,0,0,0,1,0,4,14,7,4,0,0,0,0,1,0,7,21,8,0,5,0,0,0,0,1,0,
%U A364060 7,22,11,10,0,5,0,0,0,0,1
%N A364060 Triangle read by rows where T(n,k) is the number of integer partitions of n with rounded mean k.
%C A364060 We use the "rounding half to even" rule, see link.
%H A364060 Wikipedia, <a href="https://en.wikipedia.org/wiki/Rounding">Rounding</a>.
%e A364060 Triangle begins:
%e A364060   1
%e A364060   0  1
%e A364060   0  1  1
%e A364060   0  1  1  1
%e A364060   0  2  2  0  1
%e A364060   0  2  4  0  0  1
%e A364060   0  2  5  3  0  0  1
%e A364060   0  4  7  0  3  0  0  1
%e A364060   0  4  8  5  4  0  0  0  1
%e A364060   0  4 14  7  4  0  0  0  0  1
%e A364060   0  7 21  8  0  5  0  0  0  0  1
%e A364060   0  7 22 11 10  0  5  0  0  0  0  1
%e A364060   0  7 36 15 12  0  6  0  0  0  0  0  1
%e A364060   0 12 32 36 14  0  6  0  0  0  0  0  0  1
%e A364060   0 12 53 23 23 16  0  7  0  0  0  0  0  0  1
%e A364060   0 12 80 30 27 19  0  0  7  0  0  0  0  0  0  1
%e A364060 Row n = 7 counts the following partitions:
%e A364060   .  (31111)    (511)   .  (61)  .  .  (7)
%e A364060      (22111)    (421)      (52)
%e A364060      (211111)   (4111)     (43)
%e A364060      (1111111)  (331)
%e A364060                 (322)
%e A364060                 (3211)
%e A364060                 (2221)
%t A364060 Table[If[n==k==0,1,Length[Select[IntegerPartitions[n], Round[Mean[#]]==k&]]],{n,0,15},{k,0,n}]
%Y A364060 Row sums are A000041.
%Y A364060 The rank statistic for this triangle is A363489.
%Y A364060 The version for low mean is A363945, rank statistic A363943.
%Y A364060 The version for high mean is A363946, rank statistic A363944.
%Y A364060 Column k = 1 is A363947 (A026905 tripled).
%Y A364060 A008284 counts partitions by length, A058398 by mean.
%Y A364060 A026905 redoubled counts partitions with high mean 2, ranks A363950.
%Y A364060 A051293 counts subsets with integer mean, median A000975.
%Y A364060 A067538 counts partitions with integer mean, strict A102627, ranks A316413.
%Y A364060 More triangles: A124943, A124944, A363952, A363953.
%Y A364060 Cf. A002865, A025065, A237984, A327472, A327482, A363723, A363724, A363731.
%K A364060 nonn,tabl
%O A364060 0,12
%A A364060 _Gus Wiseman_, Jul 07 2023