cp's OEIS Frontend

A325987 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with k submultisets, k > 0.

%I A325987 #16 Aug 22 2019 09:58:43
%S A325987 1,0,1,0,1,1,0,1,0,2,0,1,1,1,1,1,0,1,0,2,0,3,0,1,0,1,1,3,0,1,1,2,1,1,
%T A325987 0,1,0,3,0,3,0,4,0,1,0,3,0,1,1,3,1,3,0,3,2,1,0,4,0,1,1,1,0,1,0,5,0,3,
%U A325987 0,5,0,3,0,6,0,1,0,3,0,2,0,1,0,1,1,4,0
%N A325987 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with k submultisets, k > 0.
%C A325987 The number of submultisets of a partition is the product of its multiplicities, each plus one.
%H A325987 Alois P. Heinz, <a href="/A325987/b325987.txt">Rows n = 0..60, flattened</a>
%F A325987 Sum_{k=1..A088881(n)} k * T(n,k) = A000712(n). - _Alois P. Heinz_, Aug 17 2019
%e A325987 Triangle begins:
%e A325987   1
%e A325987   0 1
%e A325987   0 1 1
%e A325987   0 1 0 2
%e A325987   0 1 1 1 1 1
%e A325987   0 1 0 2 0 3 0 1
%e A325987   0 1 1 3 0 1 1 2 1 1
%e A325987   0 1 0 3 0 3 0 4 0 1 0 3
%e A325987   0 1 1 3 1 3 0 3 2 1 0 4 0 1 1 1
%e A325987   0 1 0 5 0 3 0 5 0 3 0 6 0 1 0 3 0 2 0 1
%e A325987   0 1 1 4 0 5 0 7 2 1 1 4 0 1 2 5 0 3 0 2 1 0 0 2
%e A325987 Row n = 7 counts the following partitions (empty columns not shown):
%e A325987   (7)  (43)  (322)  (421)      (31111)  (3211)
%e A325987        (52)  (331)  (2221)              (22111)
%e A325987        (61)  (511)  (4111)              (211111)
%e A325987                     (1111111)
%t A325987 Table[Length[Select[IntegerPartitions[n],Times@@(1+Length/@Split[#])==k&]],{n,0,10},{k,1,Max@@(Times@@(1+Length/@Split[#])&)/@IntegerPartitions[n]}]
%Y A325987 Row lengths are A088881.
%Y A325987 Row sums are A000041.
%Y A325987 Diagonal n = k is A325830 interspersed with zeros.
%Y A325987 Diagonal n + 1 = k is A325828.
%Y A325987 Diagonal n - 1 = k is A325836.
%Y A325987 Column k = 3 appears to be A137719.
%Y A325987 Cf. A000005, A000712, A002033, A005179, A088880, A108917, A126796.
%Y A325987 Cf. A325694, A325792, A325793, A325831, A325832, A325833, A325834.
%K A325987 nonn,look,tabf
%O A325987 0,10
%A A325987 _Gus Wiseman_, May 30 2019