OFFSET
0,6
COMMENTS
A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes of {a,a,b,b,b,c,d,d,d} are {b,d}.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..2000 (first 301 terms from John Tyler Rascoe)
FORMULA
G.f.: Sum_{u>0} A(u,x) where A(u,x) = Sum_{i>0} Sum_{j>u} ( x^(i*(u+j))*(1-x^u)*(1-x^j) )/( (1-x^(u*i))*(1-x^(j*i)) ) * Product_{k>0} ( (1-x^(k*(i+[k>j])))/(1-x^k) ) is the g.f. for partitions of this kind with least mode u and [] is the Iverson bracket. - John Tyler Rascoe, Apr 05 2024
EXAMPLE
The partition (3,2,2,1,1) has greatest multiplicity 2, and two parts of multiplicity 2 (namely 1 and 2), so is counted under a(9).
The a(3) = 1 through a(9) = 9 partitions:
(21) (31) (32) (42) (43) (53) (54)
(41) (51) (52) (62) (63)
(321) (61) (71) (72)
(2211) (421) (431) (81)
(521) (432)
(3311) (531)
(621)
(32211)
(222111)
MAPLE
b:= proc(n, i, m, t) option remember; `if`(n=0, `if`(t=2, 1, 0), `if`(i<1, 0,
add(b(n-i*j, i-1, max(j, m), `if`(j>m, 1, `if`(j=m, 2, t))), j=0..n/i)))
end:
a:= n-> b(n$2, 0$2):
seq(a(n), n=0..51); # Alois P. Heinz, May 05 2024
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Length[Commonest[#]]>1&]], {n, 0, 30}]
PROG
(PARI)
G_x(N)={my(x='x+O('x^(N-1)), Ib(k, j) = if(k>j, 1, 0), A_x(u)=sum(i=1, N-u, sum(j=u+1, N-u, (x^(i*(u+j))*(1-x^u)*(1-x^j))/((1-x^(u*i))*(1-x^(j*i))) * prod(k=1, N-i*(u+j), (1-x^(k*(i+Ib(k, j))))/(1-x^k)))));
concat([0, 0, 0], Vec(sum(u=1, N, A_x(u))))}
G_x(51) \\ John Tyler Rascoe, Apr 05 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 30 2023
STATUS
approved