A241761 -id:A241761

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A241735

Number of partitions p = [x(1), ..., x(k)], where x(1) >= x(2) >= ... >= x(k), of n such that max(x(i) - x(i-1)) is a part of p.

+10
4

0, 0, 0, 1, 1, 2, 4, 5, 7, 10, 15, 19, 28, 34, 46, 61, 81, 101, 137, 168, 218, 273, 349, 431, 550, 676, 849, 1043, 1298, 1579, 1959, 2373, 2913, 3526, 4301, 5178, 6293, 7544, 9109, 10895, 13091, 15591, 18666, 22158, 26402, 31269, 37120, 43813, 51853, 61027

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

OFFSET

0,6

LINKS

Table of n, a(n) for n=0..49.

FORMULA

a(n) + A241736(n) = A000041(n) for n >= 0.

EXAMPLE

a(6) counts these 4 partitions: 42, 321, 2211, 21111.

MATHEMATICA

z = 55; f[n_] := f[n] = IntegerPartitions[n]; g[p_] := Max[-Differences[p]]; g1[p_] := Min[-Differences[p]];

Table[Count[f[n], p_ /; MemberQ[p, g[p]]], {n, 0, z}] (* A241735 *)

Table[Count[f[n], p_ /; ! MemberQ[p, g[p]]], {n, 0, z}] (* A241736 *)

Table[Count[f[n], p_ /; MemberQ[p, g1[p]]], {n, 0, z}] (* A241760 *)

Table[Count[f[n], p_ /; ! MemberQ[p, g1[p]]], {n, 0, z}](* A241761 *)

CROSSREFS

Cf. A241736, A241760, A241761, A000041.

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Apr 28 2014

STATUS

approved

A241736

Number of partitions p = [x(1), ..., x(k)], where x(1) >= x(2) >= ... >= x(k), of n such that max(x(i) - x(i-1)) is not a part of p.

+10
4

1, 1, 2, 2, 4, 5, 7, 10, 15, 20, 27, 37, 49, 67, 89, 115, 150, 196, 248, 322, 409, 519, 653, 824, 1025, 1282, 1587, 1967, 2420, 2986, 3645, 4469, 5436, 6617, 8009, 9705, 11684, 14093, 16906, 20290, 24247, 28992, 34508, 41103, 48773, 57865, 68438, 80941

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

OFFSET

0,3

COMMENTS

The partition {n} is included in the count.

LINKS

Table of n, a(n) for n=0..47.

FORMULA

a(n) + A241735(n) = A000041(n) for n >= 0.

EXAMPLE

a(6) counts these 7 partitions: 6, 51, 411, 33, 3111, 222, 11111.

MATHEMATICA

z = 55; f[n_] := f[n] = IntegerPartitions[n]; g[p_] := Max[-Differences[p]]; g1[p_] := Min[-Differences[p]];

Table[Count[f[n], p_ /; MemberQ[p, g[p]]], {n, 0, z}] (* A241735 *)

Table[Count[f[n], p_ /; ! MemberQ[p, g[p]]], {n, 0, z}] (* A241736 *)

Table[Count[f[n], p_ /; MemberQ[p, g1[p]]], {n, 0, z}] (* A241760 *)

Table[Count[f[n], p_ /; ! MemberQ[p, g1[p]]], {n, 0, z}](* A241761 *)

CROSSREFS

Cf. A241736, A241760, A241761, A000041.

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Apr 28 2014

STATUS

approved

A241760

Number of partitions p = [x(1), ..., x(k)], where x(1) >= x(2) >= ... >= x(k), of n such that min(x(i) - x(i-1)) is a part of p.

+10
4

0, 0, 0, 1, 0, 0, 2, 1, 2, 2, 3, 2, 6, 5, 8, 9, 11, 11, 17, 17, 23, 26, 32, 36, 47, 51, 63, 71, 84, 93, 115, 127, 151, 171, 201, 225, 266, 297, 346, 389, 448, 501, 580, 648, 740, 831, 945, 1056, 1201, 1340, 1517, 1695, 1910, 2130, 2399, 2670

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

OFFSET

0,7

LINKS

Table of n, a(n) for n=0..55.

FORMULA

a(n) + A241761(n) = A000041(n) for n >= 0.

EXAMPLE

a(6) counts these 2 partitions: 42, 321.

MATHEMATICA

z = 55; f[n_] := f[n] = IntegerPartitions[n]; g[p_] := Max[-Differences[p]]; g1[p_] := Min[-Differences[p]];

Table[Count[f[n], p_ /; MemberQ[p, g[p]]], {n, 0, z}] (* A241735 *)

Table[Count[f[n], p_ /; ! MemberQ[p, g[p]]], {n, 0, z}] (* A241736 *)

Table[Count[f[n], p_ /; MemberQ[p, g1[p]]], {n, 0, z}] (* A241760 *)

Table[Count[f[n], p_ /; ! MemberQ[p, g1[p]]], {n, 0, z}](* A241761 *)

CROSSREFS

Cf. A241735, A241736, A241761, A000041.

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Apr 28 2014

STATUS

approved

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