A382302 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,11

LINKS

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

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

EXAMPLE

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

MATHEMATICA

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

PROG

(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

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.

Sequence in context: A331851 A335607 A366192 * A347628 A338796 A085454

Adjacent sequences: A382299 A382300 A382301 * A382303 A382304 A382305

KEYWORD

nonn

AUTHOR

Gus Wiseman, Mar 24 2025

STATUS

approved