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A322786 Irregular triangle read by rows where T(n,k) is the number of multiset partitions of a multiset with d = A027750(n,k) copies of each integer from 1 to n/d.

%I A322786 #19 Mar 05 2025 22:05:26
%S A322786 1,2,2,5,3,15,9,5,52,7,203,66,31,11,877,15,4140,712,109,22,21147,686,
%T A322786 30,115975,10457,339,42,678570,56,4213597,198091,27036,6721,1043,77,
%U A322786 27644437,101,190899322,4659138,2998,135,1382958545,1688360,58616,176
%N A322786 Irregular triangle read by rows where T(n,k) is the number of multiset partitions of a multiset with d = A027750(n,k) copies of each integer from 1 to n/d.
%H A322786 Andrew Howroyd, <a href="/A322786/b322786.txt">Table of n, a(n) for n = 1..207</a> (first 50 rows)
%F A322786 T(n,k) = A001055(A002110(n/d)^d), where d = A027750(n,k).
%F A322786 T(n,k) = A219727(d, n/d), where d = A027750(n, k). - _Andrew Howroyd_, Jan 11 2020
%e A322786 Triangle begins:
%e A322786         1
%e A322786         2       2
%e A322786         5       3
%e A322786        15       9       5
%e A322786        52       7
%e A322786       203      66      31      11
%e A322786       877      15
%e A322786      4140     712     109      22
%e A322786     21147     686      30
%e A322786    115975   10457     339      42
%e A322786    678570      56
%e A322786   4213597  198091   27036    6721    1043      77
%e A322786 For example, row 4 counts the following multiset partitions.
%e A322786   {{1,2,3,4}}        {{1,1,2,2}}        {{1,1,1,1}}
%e A322786   {{1},{2,3,4}}      {{1},{1,2,2}}      {{1},{1,1,1}}
%e A322786   {{1,2},{3,4}}      {{1,1},{2,2}}      {{1,1},{1,1}}
%e A322786   {{1,3},{2,4}}      {{1,2},{1,2}}      {{1},{1},{1,1}}
%e A322786   {{1,4},{2,3}}      {{2},{1,1,2}}      {{1},{1},{1},{1}}
%e A322786   {{2},{1,3,4}}      {{1},{1},{2,2}}
%e A322786   {{3},{1,2,4}}      {{1},{2},{1,2}}
%e A322786   {{4},{1,2,3}}      {{2},{2},{1,1}}
%e A322786   {{1},{2},{3,4}}    {{1},{1},{2},{2}}
%e A322786   {{1},{3},{2,4}}
%e A322786   {{1},{4},{2,3}}
%e A322786   {{2},{3},{1,4}}
%e A322786   {{2},{4},{1,3}}
%e A322786   {{3},{4},{1,2}}
%e A322786   {{1},{2},{3},{4}}
%t A322786 u[n_,k_]:=u[n,k]=If[n==1,1,Sum[u[n/d,d],{d,Select[Rest[Divisors[n]],#<=k&]}]];
%t A322786 Table[Table[u[Array[Prime,n/d,1,Times]^d,Array[Prime,n/d,1,Times]^d],{d,Divisors[n]}],{n,10}]
%o A322786 (PARI) \\ needs T(n,k) from A219727.
%o A322786 Row(n)={[T(d,n/d) | d<-divisors(n)]}
%o A322786 { for(n=1, 12, print(Row(n))) } \\ _Andrew Howroyd_, Jan 11 2020
%Y A322786 Row sums are A322784. First column is A000110.
%Y A322786 Cf. A001055, A005176, A027750, A056239, A072774, A100778, A219727, A295193, A306017, A319190, A319612, A322784, A322785, A322787, A322788, A322792.
%K A322786 nonn,tabf
%O A322786 1,2
%A A322786 _Gus Wiseman_, Dec 26 2018
%E A322786 Edited by _Peter Munn_, Mar 05 2025