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A322788 Irregular triangle read by rows where T(n,k) is the number of uniform multiset partitions of a multiset with d = A027750(n,k) copies of each integer from 1 to n/d.

%I A322788 #17 Mar 05 2025 22:04:55
%S A322788 1,2,2,2,2,5,4,3,2,2,27,11,6,4,2,2,142,29,8,4,282,12,3,1073,101,8,4,2,
%T A322788 2,32034,1581,234,75,20,6,2,2,136853,2660,10,4,1527528,1985,91,4,
%U A322788 4661087,64596,648,20,5,2,2,227932993,1280333,41945,231,28,6
%N A322788 Irregular triangle read by rows where T(n,k) is the number of uniform multiset partitions of a multiset with d = A027750(n,k) copies of each integer from 1 to n/d.
%C A322788 A multiset partition is uniform if all parts have the same size.
%H A322788 Andrew Howroyd, <a href="/A322788/b322788.txt">Table of n, a(n) for n = 1..482</a> (rows 1..100)
%F A322788 T(n,k) = A322794(A002110(n/d)^d), where d = A027750(n,k).
%e A322788 Triangle begins:
%e A322788      1
%e A322788      2    2
%e A322788      2    2
%e A322788      5    4    3
%e A322788      2    2
%e A322788     27   11    6    4
%e A322788      2    2
%e A322788    142   29    8    4
%e A322788    282   12    3
%e A322788   1073  101    8    4
%e A322788 The multiset partitions counted under row 6:
%e A322788   {123456}          {112233}          {111222}          {111111}
%e A322788   {123}{456}        {112}{233}        {111}{222}        {111}{111}
%e A322788   {124}{356}        {113}{223}        {112}{122}        {11}{11}{11}
%e A322788   {125}{346}        {122}{133}        {11}{12}{22}      {1}{1}{1}{1}{1}{1}
%e A322788   {126}{345}        {123}{123}        {12}{12}{12}
%e A322788   {134}{256}        {11}{22}{33}      {1}{1}{1}{2}{2}{2}
%e A322788   {135}{246}        {11}{23}{23}
%e A322788   {136}{245}        {12}{12}{33}
%e A322788   {145}{236}        {12}{13}{23}
%e A322788   {146}{235}        {13}{13}{22}
%e A322788   {156}{234}        {1}{1}{2}{2}{3}{3}
%e A322788   {12}{34}{56}
%e A322788   {12}{35}{46}
%e A322788   {12}{36}{45}
%e A322788   {13}{24}{56}
%e A322788   {13}{25}{46}
%e A322788   {13}{26}{45}
%e A322788   {14}{23}{56}
%e A322788   {14}{25}{36}
%e A322788   {14}{26}{35}
%e A322788   {15}{23}{46}
%e A322788   {15}{24}{36}
%e A322788   {15}{26}{34}
%e A322788   {16}{23}{45}
%e A322788   {16}{24}{35}
%e A322788   {16}{25}{34}
%e A322788   {1}{2}{3}{4}{5}{6}
%t A322788 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A322788 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A322788 Table[Length[Select[mps[Join@@Table[Range[n/d],{d}]],SameQ@@Length/@#&]],{n,10},{d,Divisors[n]}]
%Y A322788 Row sums are A322785. First column is A038041.
%Y A322788 Cf. A001055, A005176, A027750, A056239, A072774, A100778, A295193, A306017, A319190, A319612, A322784, A322785, A322786, A322789, A322792.
%K A322788 nonn,look,tabf
%O A322788 1,2
%A A322788 _Gus Wiseman_, Dec 26 2018
%E A322788 More terms from _Alois P. Heinz_, Jan 30 2019
%E A322788 Terms a(38) and beyond from _Andrew Howroyd_, Feb 03 2022
%E A322788 Edited by _Peter Munn_, Mar 05 2025