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