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Revision History for A240312

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Number of partitions p of n such that (maximal multiplicity of the parts of p) = (maximal part of p).
(history; published version)

#6 by Andrey Zabolotskiy at Mon Dec 25 17:38:13 EST 2023

STATUS

editing

approved

#5 by Andrey Zabolotskiy at Mon Dec 25 17:38:11 EST 2023

NAME

Number of partitions p of n such that (maximal multiplicity of the parts of p) = (maximal part of p).

STATUS

approved

editing

#4 by N. J. A. Sloane at Mon Apr 14 11:10:34 EDT 2014

STATUS

proposed

approved

#3 by Clark Kimberling at Sun Apr 13 09:19:21 EDT 2014

STATUS

editing

proposed

#2 by Clark Kimberling at Sat Apr 05 15:46:33 EDT 2014

NAME

allocated for Clark KimberlingNumber of partitions p of n such that (maximal multiplicity of the parts of p) = maximal part of p).

DATA

1, 1, 0, 0, 2, 1, 2, 0, 2, 3, 5, 5, 9, 7, 11, 11, 18, 15, 28, 27, 41, 43, 62, 64, 91, 96, 127, 140, 184, 200, 260, 287, 365, 410, 511, 573, 717, 803, 985, 1120, 1359, 1538, 1859, 2106, 2522, 2870, 3407, 3872, 4586, 5207, 6128, 6976, 8167, 9284, 10844, 12321

OFFSET

0,5

FORMULA

a(n) = A240311(n) - A240310(n) for n >= 0.

a(n) + A240310(n) + A240314(n) = A000041(n) for n >= 0.

EXAMPLE

a(6) counts these 2 partitions: 3111, 2211.

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] < Max[p]], {n, 0, z}] (* A240310 *)

Table[Count[f[n], p_ /; m[p] <= Max[p]], {n, 0, z}] (* A240311 *)

Table[Count[f[n], p_ /; m[p] == Max[p]], {n, 0, z}] (* A240312 *)

Table[Count[f[n], p_ /; m[p] >= Max[p]], {n, 0, z}] (* A240313 *)

Table[Count[f[n], p_ /; m[p] > Max[p]], {n, 0, z}] (* A240314 *)

CROSSREFS

Cf. A240310, A240311, A240313, A240314, A000041.

KEYWORD

allocated

nonn,easy

AUTHOR

Clark Kimberling, Apr 05 2014

STATUS

approved

editing

Discussion

Sat Apr 12

18:06

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#1 by Clark Kimberling at Thu Apr 03 18:13:19 EDT 2014

NAME

allocated for Clark Kimberling

KEYWORD

allocated

STATUS

approved