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

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, 26, 32, 34, 37, 44, 47, 55, 62, 72, 78, 94, 103, 118, 132, 151, 163, 189, 205, 230, 251, 284, 307, 346, 377, 420, 462, 515, 562, 629, 690, 763

Offset: 0

Author

Gus Wiseman, Mar 24 2025

Keywords

• nonn

Examples

			The a(n) partitions for n = 1, 2, 10, 13, 14, 19, 20, 21:
  1  .  32221   332221   333221   4333321     43333211    43333221
        322111  333211   3322211  43322221    44322221    433332111
                3322111  3332111  433321111   433222211   443222211
                4321111           443221111   443321111   444321111
                                  543211111   4332221111  4332222111
                                  4322221111              4333221111
                                                          4432221111
                                                          5432211111
		

Links

• John Tyler Rascoe, Table of n, a(n) for n = 0..500

Crossrefs

Without the middle statistic we have A000009, ranked by A055932.

Counting partitions by the LHS gives A008284 (strict A008289), rank statistic A061395.

Counting partitions by the middle statistic gives A091602, rank statistic A051903.

Counting partitions by the RHS gives A116608/A365676, rank statistic A001221.

Without the LHS we have A239964, ranked by A212166.

Without the RHS we have A240312, ranked by A381542.

The Heinz numbers of these partitions are listed by A381543.

A000041 counts integer partitions.

A047993 counts partitions with max part = length, ranks A106529.

A116598 counts ones in partitions, rank statistic A007814.

A381438 counts partitions by last part part of section-sum partition.

Cf. A047966, A130091, A237984, A239455, A241131, A351293, A362608, A363719, A381079, A381544, A382303.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],Max@@#==Max@@Length/@Split[#]==Length[Union[#]]&]],{n,0,30}]

PARI

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))))))}
A_x(60) \\ John Tyler Rascoe, Mar 25 2025

Formula

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