cp's OEIS Frontend

A362607 Number of integer partitions of n with more than one mode.

%I A362607 #31 May 05 2024 14:32:18
%S A362607 0,0,0,1,1,2,4,4,6,9,13,13,23,23,33,45,56,64,90,101,137,169,208,246,
%T A362607 320,379,469,567,702,828,1035,1215,1488,1772,2139,2533,3076,3612,4333,
%U A362607 5117,6113,7168,8557,10003,11862,13899,16385,19109,22525,26198,30729,35736
%N A362607 Number of integer partitions of n with more than one mode.
%C A362607 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}.
%H A362607 Alois P. Heinz, <a href="/A362607/b362607.txt">Table of n, a(n) for n = 0..2000</a> (first 301 terms from John Tyler Rascoe)
%F A362607 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
%e A362607 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).
%e A362607 The a(3) = 1 through a(9) = 9 partitions:
%e A362607   (21)  (31)  (32)  (42)    (43)   (53)    (54)
%e A362607               (41)  (51)    (52)   (62)    (63)
%e A362607                     (321)   (61)   (71)    (72)
%e A362607                     (2211)  (421)  (431)   (81)
%e A362607                                    (521)   (432)
%e A362607                                    (3311)  (531)
%e A362607                                            (621)
%e A362607                                            (32211)
%e A362607                                            (222111)
%p A362607 b:= proc(n, i, m, t) option remember; `if`(n=0, `if`(t=2, 1, 0), `if`(i<1, 0,
%p A362607       add(b(n-i*j, i-1, max(j, m), `if`(j>m, 1, `if`(j=m, 2, t))), j=0..n/i)))
%p A362607     end:
%p A362607 a:= n-> b(n$2, 0$2):
%p A362607 seq(a(n), n=0..51);  # _Alois P. Heinz_, May 05 2024
%t A362607 Table[Length[Select[IntegerPartitions[n],Length[Commonest[#]]>1&]],{n,0,30}]
%o A362607 (PARI)
%o A362607 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)))));
%o A362607 concat([0,0,0],Vec(sum(u=1,N, A_x(u))))}
%o A362607 G_x(51) \\ _John Tyler Rascoe_, Apr 05 2024
%Y A362607 For parts instead of multiplicities we have A002865.
%Y A362607 For median instead of mode we have A238479, complement A238478.
%Y A362607 These partitions have ranks A362605.
%Y A362607 The complement is counted by A362608, ranks A356862.
%Y A362607 For co-mode we have A362609, ranks A362606.
%Y A362607 For co-mode complement we have A362610, ranks A359178.
%Y A362607 A000041 counts integer partitions.
%Y A362607 A359893 counts partitions by median.
%Y A362607 A362611 counts modes in prime factorization, co-modes A362613.
%Y A362607 A362614 counts partitions by number of modes, co-modes A362615.
%Y A362607 Cf. A008284, A098859, A237984, A275870, A304442, A327472, A353864, A353865, A360071, A362612.
%K A362607 nonn
%O A362607 0,6
%A A362607 _Gus Wiseman_, Apr 30 2023