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A200144 The number of multinomial coefficients, based on a set of partitions of n into m positions, divisible by m entirely. 0
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified October 18 02:23 EDT 2019. Contains 328135 sequences. (Running on oeis4.)