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A382302 Number of integer partitions of n with greatest part, greatest multiplicity, and number of distinct parts all equal.

%I A382302 #14 Mar 26 2025 08:30:52
%S A382302 0,1,0,0,1,1,1,0,1,0,2,2,2,4,3,3,4,4,3,6,5,8,8,13,13,16,17,21,22,25,
%T A382302 26,32,34,37,44,47,55,62,72,78,94,103,118,132,151,163,189,205,230,251,
%U A382302 284,307,346,377,420,462,515,562,629,690,763
%N A382302 Number of integer partitions of n with greatest part, greatest multiplicity, and number of distinct parts all equal.
%H A382302 John Tyler Rascoe, <a href="/A382302/b382302.txt">Table of n, a(n) for n = 0..500</a>
%F A382302 G.f.: Sum_{i>0} (B(i+1,i,x) - B(i,i,x)) where B(a,c,x) = Product_{j=1..c} (x^j - x^(a*j))/(1 - x^j). - _John Tyler Rascoe_, Mar 25 2025
%e A382302 The a(n) partitions for n = 1, 2, 10, 13, 14, 19, 20, 21:
%e A382302   1  .  32221   332221   333221   4333321     43333211    43333221
%e A382302         322111  333211   3322211  43322221    44322221    433332111
%e A382302                 3322111  3332111  433321111   433222211   443222211
%e A382302                 4321111           443221111   443321111   444321111
%e A382302                                   543211111   4332221111  4332222111
%e A382302                                   4322221111              4333221111
%e A382302                                                           4432221111
%e A382302                                                           5432211111
%t A382302 Table[Length[Select[IntegerPartitions[n],Max@@#==Max@@Length/@Split[#]==Length[Union[#]]&]],{n,0,30}]
%o A382302 (PARI)
%o A382302 A_x(N) = {if(N<1,[0],my(x='x+O('x^(N+1))); concat([0],Vec(sum(i=1,N, prod(j=1,i, (x^j-x^((i+1)*j))/(1-x^j)) - prod(j=1,i, (x^j-x^(i*j))/(1-x^j))))))}
%o A382302 A_x(60) \\ _John Tyler Rascoe_, Mar 25 2025
%Y A382302 Without the middle statistic we have A000009, ranked by A055932.
%Y A382302 Counting partitions by the LHS gives A008284 (strict A008289), rank statistic A061395.
%Y A382302 Counting partitions by the middle statistic gives A091602, rank statistic A051903.
%Y A382302 Counting partitions by the RHS gives A116608/A365676, rank statistic A001221.
%Y A382302 Without the LHS we have A239964, ranked by A212166.
%Y A382302 Without the RHS we have A240312, ranked by A381542.
%Y A382302 The Heinz numbers of these partitions are listed by A381543.
%Y A382302 A000041 counts integer partitions.
%Y A382302 A047993 counts partitions with max part = length, ranks A106529.
%Y A382302 A116598 counts ones in partitions, rank statistic A007814.
%Y A382302 A381438 counts partitions by last part part of section-sum partition.
%Y A382302 Cf. A047966, A130091, A237984, A239455, A241131, A351293, A362608, A363719, A381079, A381544, A382303.
%K A382302 nonn
%O A382302 0,11
%A A382302 _Gus Wiseman_, Mar 24 2025