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A171979 Number of partitions of n such that smaller parts do not occur more frequently than greater parts.

%I A171979 #37 Mar 16 2024 13:28:05
%S A171979 1,1,2,3,4,5,8,8,12,14,19,21,30,31,42,50,62,69,91,99,126,144,175,198,
%T A171979 246,275,331,379,452,509,612,686,811,922,1076,1219,1428,1604,1863,
%U A171979 2108,2434,2739,3162,3551,4075,4593,5240,5885,6721,7527,8556,9597,10870
%N A171979 Number of partitions of n such that smaller parts do not occur more frequently than greater parts.
%C A171979 A000009(n) <= a(n) <= A000041(n).
%C A171979 Equivalently, the number of partitions of n such that (maximal multiplicity of parts) = (multiplicity of the maximal part), as in the Mathematica program. - _Clark Kimberling_, Apr 04 2014
%C A171979 Also the number of integer partitions of n whose greatest part is a mode, meaning it appears at least as many times as each of the others. The name "Number of partitions of n such that smaller parts do not occur more frequently than greater parts" seems to describe A100882 = "Number of partitions of n in which the sequence of frequencies of the summands is nonincreasing," which first differs from this at n = 10 due to the partition (3,3,2,1,1). - _Gus Wiseman_, May 07 2023
%H A171979 Alois P. Heinz, <a href="/A171979/b171979.txt">Table of n, a(n) for n = 0..1000</a>
%H A171979 <a href="/index/Par#partN">Index entries for sequences related to partitions</a>
%F A171979 a(n) = p(n,0,1,1) with p(n,i,j,k) = if k<=n then p(n-k,i,j+1,k) +p(n,max(i,j),1,k+1) else (if j<i or n>0 then 0 else 1).
%F A171979 a(n) + A240302(n) = A000041(n).  - _Clark Kimberling_, Apr 04 2014.
%F A171979 G.f.: 1 + Sum_{i, j>0} x^(i*j) * Product_{k=1..i-1} ((1 - x^(k*(j+1)))/(1 - x^k)). - _John Tyler Rascoe_, Mar 09 2024
%e A171979 a(5) = #{5, 4+1, 3+2, 2+2+1, 5x1} = 5;
%e A171979 a(6) = #{6, 5+1, 4+2, 3+3, 3+2+1, 2+2+2, 2+2+1+1, 6x1} = 8;
%e A171979 a(7) = #{7, 6+1, 5+2, 4+3, 4+2+1, 3+3+1, 2+2+2+1, 7x1} = 8;
%e A171979 a(8) = #{8, 7+1, 6+2, 5+3, 5+2+1, 4+4, 4+3+1, 3+3+2, 3+3+1+1, 2+2+2+2, 2+2+2+1+1, 8x1} = 12.
%t A171979 z = 60; f[n_] := f[n] = IntegerPartitions[n]; m[p_] := Max[Map[Length, Split[p]]]  (* maximal multiplicity *)
%t A171979 Table[Count[f[n], p_ /; m[p] == Count[p, Max[p]]], {n, 0, z}] (* this sequence *)
%t A171979 Table[Count[f[n], p_ /; m[p] > Count[p, Max[p]]], {n, 0, z}]  (* A240302 *)
%t A171979 (* _Clark Kimberling_, Apr 04 2014 *)
%t A171979 (* Second program: *)
%t A171979 b[n_, i_, k_] := b[n, i, k] = If[n == 0, If[k == 0, 1, 0],
%t A171979      If[i < 1, 0, b[n, i - 1, k] + Sum[b[n - i*j, i - 1,
%t A171979      If[k == -1, j, If[k == 0, 0, If[j > k, 0, k]]]], {j, 1, n/i}]]];
%t A171979 a[n_] := PartitionsP[n] - b[n, n, -1];
%t A171979 a /@ Range[0, 70] (* _Jean-François Alcover_, Jun 05 2021, after _Alois P. Heinz_ in A240302 *)
%t A171979 Table[Length[Select[IntegerPartitions[n],MemberQ[Commonest[#],Max[#]]&]],{n,0,30}] (* _Gus Wiseman_, May 07 2023 *)
%o A171979 (PARI)
%o A171979 { my(N=53, x='x+O('x^N));
%o A171979 my(gf=1+sum(i=1,N,sum(j=1,floor(N/i),x^(i*j)*prod(k=1,i-1,(1-x^(k*(j+1)))/(1-x^k)))));
%o A171979 Vec(gf) } \\ _John Tyler Rascoe_, Mar 09 2024
%Y A171979 For median instead of mode we have A053263.
%Y A171979 The complement is counted by A240302.
%Y A171979 The case where the maximum is the only mode is A362612.
%Y A171979 A000041 counts integer partitions, strict A000009.
%Y A171979 A362608 counts partitions with a unique mode, complement A362607.
%Y A171979 A362611 counts modes in prime factorization.
%Y A171979 A362614 counts partitions by number of modes.
%Y A171979 Cf. A002865, A008284, A098859, A237984, A275870, A325347, A362610.
%K A171979 nonn
%O A171979 0,3
%A A171979 _Reinhard Zumkeller_, Jan 20 2010