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A363945 Triangle read by rows where T(n,k) is the number of integer partitions of n with low mean k.

%I A363945 #8 Jun 30 2023 18:32:50
%S A363945 1,0,1,0,1,1,0,2,0,1,0,2,2,0,1,0,4,2,0,0,1,0,4,3,3,0,0,1,0,7,4,3,0,0,
%T A363945 0,1,0,7,10,0,4,0,0,0,1,0,12,6,7,4,0,0,0,0,1,0,12,16,8,0,5,0,0,0,0,1,
%U A363945 0,19,21,10,0,5,0,0,0,0
%N A363945 Triangle read by rows where T(n,k) is the number of integer partitions of n with low mean k.
%C A363945 Extending the terminology of A124943, the "low mean" of a multiset is its mean rounded down.
%e A363945 Triangle begins:
%e A363945   1
%e A363945   0  1
%e A363945   0  1  1
%e A363945   0  2  0  1
%e A363945   0  2  2  0  1
%e A363945   0  4  2  0  0  1
%e A363945   0  4  3  3  0  0  1
%e A363945   0  7  4  3  0  0  0  1
%e A363945   0  7 10  0  4  0  0  0  1
%e A363945   0 12  6  7  4  0  0  0  0  1
%e A363945   0 12 16  8  0  5  0  0  0  0  1
%e A363945   0 19 21 10  0  5  0  0  0  0  0  1
%e A363945   0 19 24 15 12  0  6  0  0  0  0  0  1
%e A363945   0 30 32 18 14  0  6  0  0  0  0  0  0  1
%e A363945   0 30 58 23 16  0  0  7  0  0  0  0  0  0  1
%e A363945   0 45 47 57  0 19  0  7  0  0  0  0  0  0  0  1
%e A363945 Row k = 8 counts the following partitions:
%e A363945   .  (41111)     (611)   .  (71)  .  .  .  (8)
%e A363945      (32111)     (521)      (62)
%e A363945      (311111)    (5111)     (53)
%e A363945      (22211)     (431)      (44)
%e A363945      (221111)    (422)
%e A363945      (2111111)   (4211)
%e A363945      (11111111)  (332)
%e A363945                  (3311)
%e A363945                  (3221)
%e A363945                  (2222)
%t A363945 meandown[y_]:=If[Length[y]==0,0,Floor[Mean[y]]];
%t A363945 Table[Length[Select[IntegerPartitions[n],meandown[#]==k&]],{n,0,15},{k,0,n}]
%Y A363945 Row sums are A000041.
%Y A363945 Column k = 1 is A025065, ranks A363949.
%Y A363945 For median instead of mean we have triangle A124943, high A124944.
%Y A363945 Column k = 2 is A363745.
%Y A363945 For median instead of mean we have rank statistic A363941, high A363942.
%Y A363945 The rank statistic for this triangle is A363943.
%Y A363945 The high version is A363946, rank statistic A363944.
%Y A363945 For mode instead of mean we have A363952, rank statistic A363486.
%Y A363945 For high mode instead of mean we have A363953, rank statistic A363487.
%Y A363945 A008284 counts partitions by length, A058398 by mean.
%Y A363945 A051293 counts subsets with integer mean, median A000975.
%Y A363945 A067538 counts partitions with integer mean, strict A102627, ranks A316413.
%Y A363945 A349156 counts partitions with non-integer mean, ranks A348551.
%Y A363945 Cf. A002865, A026905, A237984, A327472, A327482, A344296, A362612, A363723, A363724, A363731, A363951.
%K A363945 nonn,tabl
%O A363945 0,8
%A A363945 _Gus Wiseman_, Jun 30 2023