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A362607
Number of integer partitions of n with more than one mode.
36
0, 0, 0, 1, 1, 2, 4, 4, 6, 9, 13, 13, 23, 23, 33, 45, 56, 64, 90, 101, 137, 169, 208, 246, 320, 379, 469, 567, 702, 828, 1035, 1215, 1488, 1772, 2139, 2533, 3076, 3612, 4333, 5117, 6113, 7168, 8557, 10003, 11862, 13899, 16385, 19109, 22525, 26198, 30729, 35736
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
For parts instead of multiplicities we have A002865.
For median instead of mode we have A238479, complement A238478.
These partitions have ranks A362605.
The complement is counted by A362608, ranks A356862.
For co-mode we have A362609, ranks A362606.
For co-mode complement we have A362610, ranks A359178.
A000041 counts integer partitions.
A359893 counts partitions by median.
A362611 counts modes in prime factorization, co-modes A362613.
A362614 counts partitions by number of modes, co-modes A362615.
Sequence in context: A359678 A008133 A237828 * A340626 A351489 A022471
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 30 2023
STATUS
approved