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

1, 1, 2, 3, 6, 7, 14, 17, 27, 34, 55, 64, 100, 121, 167, 213, 296, 354, 489, 594, 776, 964, 1254, 1511, 1951, 2378, 2986, 3643, 4564, 5483, 6841, 8245, 10099, 12190, 14862, 17783, 21636, 25849, 31184

Offset

1,3

Comments

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.

Links

Table of n, a(n) for n=1..39.

Dmitry Kruchinin The number of multinomial coefficients based on a set of partitions of n into k parts and divided by k evenly

Example

n=7;
  Set of partitions of n into m=4 parts
[1,1,1,4]
[1,1,2,3]
[1,2,2,2]
number of different parts
[3,1]
[2,1,1]
[1,3]
Multinomial coefficient,  divisible by m
4!/(4*(1!*3!))=1
4!/(4*(2!*1!*1!))=2
4!/(4*(1!*3!))=1
Set of partitions of n into m=7 parts
[1,1,1,1,1,1,1]
number of different parts
[7]
Multinomial coefficient,  divisible by m
7!/(7*(7!))=1/7

Prog

(Maxima)
/* count number of partitions of n into m parts */
b(n, m):=if n<m then 0 else if m=1 then 1 else b(n-1, m-1)+b(n-m, m);
/* unranking partitions(n, m) , num - numbers partitions of lexicographic order */
array(pa, 100);
gen_partitions(n, m, num, pos):= if n<m then return else
               if m=1 then pa[pos]:n else
               if num<b(n-1, m-1) then (pa[pos]:1, gen_partitions(n-1, m-1, num, pos+1)) else
               if num<b(n-m, m)+b(n-1, m-1) then
                (gen_partitions(n-m, m, num-b(n-1, m-1), pos),
                  for i:0 thru m-1 do pa[i+pos]:pa[i+pos]+1);
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));
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),
        if(denom((m-1)!/prod(po[j]!, j, 0, k))=1) then d:d+1), return(d));
makelist(sum(Tep(n, m), m, 1, n), n, 1, 20);

Crossrefs

Sequence in context: A322367 A319811 A000837 * A056498 A325093 A018652

Adjacent sequences: A200141 A200142 A200143 * A200145 A200146 A200147

Keyword

nonn

Author

Dmitry Kruchinin, Nov 11 2011

Status

approved