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A171979 Number of partitions of n such that smaller parts do not occur more frequently than greater parts. 4
1, 1, 2, 3, 4, 5, 8, 8, 12, 14, 19, 21, 30, 31, 42, 50, 62, 69, 91, 99, 126, 144, 175, 198, 246, 275, 331, 379, 452, 509, 612, 686, 811, 922, 1076, 1219, 1428, 1604, 1863, 2108, 2434, 2739, 3162, 3551, 4075, 4593, 5240, 5885, 6721, 7527, 8556, 9597, 10870 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

A000009(n) <= a(n) <= A000041(n).

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

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Index entries for sequences related to partitions

FORMULA

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).

a(n) + A240302(n) = A000041(n).  - Clark Kimberling, Apr 04 2014.

EXAMPLE

a(5) = #{5, 4+1, 3+2, 2+2+1, 5x1} = 5;

a(6) = #{6, 5+1, 4+2, 3+3, 3+2+1, 2+2+2, 2+2+1+1, 6x1} = 8;

a(7) = #{7, 6+1, 5+2, 4+3, 4+2+1, 3+3+1, 2+2+2+1, 7x1} = 8;

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.

MATHEMATICA

z = 60; f[n_] := f[n] = IntegerPartitions[n]; m[p_] := Max[Map[Length, Split[p]]]  (* maximal multiplicity *)

Table[Count[f[n], p_ /; m[p] == Count[p, Max[p]]], {n, 0, z}] (* A171979 *)

Table[Count[f[n], p_ /; m[p] > Count[p, Max[p]]], {n, 0, z}]  (* A240302 *)

(* Clark Kimberling, Apr 04 2014 *)

CROSSREFS

Cf. A240302.

Sequence in context: A316496 A332339 A100882 * A181694 A297166 A297163

Adjacent sequences:  A171976 A171977 A171978 * A171980 A171981 A171982

KEYWORD

nonn

AUTHOR

Reinhard Zumkeller, Jan 20 2010

STATUS

approved

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Last modified April 11 14:44 EDT 2021. Contains 342886 sequences. (Running on oeis4.)