cp's OEIS Frontend

A200144 The number of multinomial coefficients, based on a set of partitions of n into m positions, divisible by m entirely.

%I A200144 #18 May 02 2012 13:54:46
%S A200144 1,1,2,3,6,7,14,17,27,34,55,64,100,121,167,213,296,354,489,594,776,
%T A200144 964,1254,1511,1951,2378,2986,3643,4564,5483,6841,8245,10099,12190,
%U A200144 14862,17783,21636,25849,31184
%N A200144 The number of multinomial coefficients, based on a set of partitions of n into m positions, divisible by m entirely.
%C A200144 If n is prime, then the number of multinomial coefficients, based on a set of partitions of n at position m, divided by m entirely, less 1 than the number of partitions of numbers for all m.
%H A200144 Dmitry Kruchinin  <a href="http://arxiv.org/abs/1204.6554">The number of multinomial coefficients based on a set of partitions of n into k parts and divided by k evenly</a>
%e A200144 n=7;
%e A200144   Set of partitions of n into m=4 parts
%e A200144 [1,1,1,4]
%e A200144 [1,1,2,3]
%e A200144 [1,2,2,2]
%e A200144 number of different parts
%e A200144 [3,1]
%e A200144 [2,1,1]
%e A200144 [1,3]
%e A200144 Multinomial coefficient,  divisible by m
%e A200144 4!/(4*(1!*3!))=1
%e A200144 4!/(4*(2!*1!*1!))=2
%e A200144 4!/(4*(1!*3!))=1
%e A200144 Set of partitions of n into m=7 parts
%e A200144 [1,1,1,1,1,1,1]
%e A200144 number of different parts
%e A200144 [7]
%e A200144 Multinomial coefficient,  divisible by m
%e A200144 7!/(7*(7!))=1/7
%o A200144 (Maxima)
%o A200144 /* count number of partitions of n into m parts */
%o A200144 b(n, m):=if n<m then 0 else if m=1 then 1 else b(n-1, m-1)+b(n-m, m);
%o A200144 /* unranking partitions(n, m) , num - numbers partitions of lexicographic order */
%o A200144 array(pa, 100);
%o A200144 gen_partitions(n, m, num, pos):= if n<m then return else
%o A200144                if m=1 then pa[pos]:n else
%o A200144                if num<b(n-1, m-1) then (pa[pos]:1, gen_partitions(n-1, m-1, num, pos+1)) else
%o A200144                if num<b(n-m, m)+b(n-1, m-1) then
%o A200144                 (gen_partitions(n-m, m, num-b(n-1, m-1), pos),
%o A200144                   for i:0 thru m-1 do pa[i+pos]:pa[i+pos]+1);
%o A200144 FindPo(pa,n,po):=block([k,s] ,k:0,po[k]:1,s:pa[0], for i:1 thru n-1 do (if pa[i]=s then po[k]:po[k]+1 else (k:k+1,s:pa[i],po[k]:1)),return (k));
%o A200144 Tep(n,m):=block([d],d:0,for i:0 thru b(n,m)-1 do (gen_partitions(n, m, i, 0), k:FindPo(pa,m,po),
%o A200144         if(denom((m-1)!/prod(po[j]!,j,0,k))=1) then d:d+1), return(d));
%o A200144 makelist(sum(Tep(n,m),m,1,n),n,1,20);
%K A200144 nonn
%O A200144 1,3
%A A200144 _Dmitry Kruchinin_, Nov 11 2011