A325851 -id:A325851

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A238424

Number of partitions of n without three consecutive parts in arithmetic progression.

+10
31

1, 1, 2, 2, 4, 5, 6, 8, 13, 13, 19, 24, 30, 36, 47, 54, 72, 85, 106, 123, 151, 178, 220, 256, 314, 362, 432, 505, 605, 692, 827, 953, 1121, 1303, 1522, 1729, 2037, 2321, 2691, 3095, 3577, 4061, 4699, 5334, 6126, 6959, 7966, 9005, 10317, 11638, 13252, 14977

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

OFFSET

0,3

COMMENTS

Also the number of partitions of n whose first differences are an anti-run, meaning there are no adjacent equal differences. - Gus Wiseman, Mar 31 2020

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000 (terms 0..300 from Joerg Arndt and Alois P. Heinz, terms 301..350 from Fausto A. C. Cariboni)

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts

EXAMPLE

The a(8) = 13 such partitions are:

01: [ 3 2 2 1 ]

02: [ 3 3 1 1 ]

03: [ 3 3 2 ]

04: [ 4 2 1 1 ]

05: [ 4 2 2 ]

06: [ 4 3 1 ]

07: [ 4 4 ]

08: [ 5 2 1 ]

09: [ 5 3 ]

10: [ 6 1 1 ]

11: [ 6 2 ]

12: [ 7 1 ]

13: [ 8 ]

MATHEMATICA

a[n_, r_, d_] := a[n, r, d] = Block[{j}, If[n == 0, 1, Sum[If[j == r+d, 0, a[n-j, j, j - r]], {j, Min[n, r]}]]]; a[n_] := a[n, 2*n+1, 0]; a /@ Range[0, 100] (* Giovanni Resta, Mar 02 2014 *)

Table[Length[Select[IntegerPartitions[n], !MemberQ[Differences[#, 2], 0]&]], {n, 0, 30}] (* Gus Wiseman, Mar 31 2020 *)

CROSSREFS

Cf. A238433 (partitions avoiding equidistant arithmetic progressions).

Cf. A238571 (partitions avoiding any arithmetic progression).

Cf. A238687.

The version for compositions is A238423, with strict case A325849.

The version for permutations is A295370.

The strict case is A332668.

The Heinz numbers of these partitions are the complement of A333195.

Partitions with equal differences are A049988.

Cf. A007862, A175342, A325850, A325851, A325852, A325874, A333631.

KEYWORD

nonn

AUTHOR

Joerg Arndt and Alois P. Heinz, Feb 26 2014

STATUS

approved

A238423

Number of compositions of n avoiding three consecutive parts in arithmetic progression.

+10
16

1, 1, 2, 3, 7, 13, 22, 42, 81, 149, 278, 516, 971, 1812, 3374, 6297, 11770, 21970, 41002, 76523, 142901, 266779, 497957, 929563, 1735418, 3239698, 6047738, 11289791, 21076118, 39344992, 73448769, 137113953, 255965109, 477835991, 892023121, 1665227859

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

OFFSET

0,3

COMMENTS

These are compositions of n whose second-differences are nonzero. - Gus Wiseman, Jun 03 2019

LINKS

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 0..400

Wikipedia, Arithmetic progression

FORMULA

a(n) ~ c * d^n, where d = 1.866800016014240677813344121155900699..., c = 0.540817940878009616510727217687704495... - Vaclav Kotesovec, May 01 2014

EXAMPLE

The a(5) = 13 such compositions are:

01: [ 1 1 2 1 ]

02: [ 1 1 3 ]

03: [ 1 2 1 1 ]

04: [ 1 2 2 ]

05: [ 1 3 1 ]

06: [ 1 4 ]

07: [ 2 1 2 ]

08: [ 2 2 1 ]

09: [ 2 3 ]

10: [ 3 1 1 ]

11: [ 3 2 ]

12: [ 4 1 ]

13: [ 5 ]

MAPLE

# b(n, r, d): number of compositions of n where the leftmost part j

# does not have distance d to the recent part r

b:= proc(n, r, d) option remember; `if`(n=0, 1,

add(`if`(j=r+d, 0, b(n-j, j, j-r)), j=1..n))

end:

a:= n-> b(n, infinity, 0):

seq(a(n), n=0..45);

MATHEMATICA

b[n_, r_, d_] := b[n, r, d] = If[n == 0, 1, Sum[If[j == r + d, 0, b[n - j, j, j - r]], {j, 1, n}]]; a[n_] := b[n, Infinity, 0]; Table[a[n], {n, 0, 45}] (* Jean-François Alcover, Nov 06 2014, after Maple *)

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], !MemberQ[Differences[#, 2], 0]&]], {n, 0, 10}] (* Gus Wiseman, Jun 03 2019 *)

CROSSREFS

Cf. A238424 (equivalent for partitions).

Cf. A238569 (equivalent for any 3-term arithmetic progression).

Cf. A238686, A295370.

Cf. A007862, A049988, A175342, A325545, A325849, A325851, A325874, A325875.

KEYWORD

nonn

AUTHOR

Joerg Arndt and Alois P. Heinz, Feb 26 2014

STATUS

approved

A295370

Number of permutations of [n] avoiding three consecutive terms in arithmetic progression.

+10
16

1, 1, 2, 4, 18, 80, 482, 3280, 26244, 231148, 2320130, 25238348, 302834694, 3909539452, 54761642704, 816758411516, 13076340876500, 221396129723368, 3985720881222850, 75503196628737920, 1510373288335622576, 31634502738658957588, 696162960370556156224, 15978760340940405262668

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

OFFSET

0,3

COMMENTS

These are permutations of n whose second-differences are nonzero. - Gus Wiseman, Jun 03 2019

LINKS

Table of n, a(n) for n=0..23.

Wikipedia, Arithmetic progression

Index entries related to non-averaging sequences

EXAMPLE

a(3) = 4: 132, 213, 231, 312.

a(4) = 18: 1243, 1324, 1342, 1423, 2134, 2143, 2314, 2413, 2431, 3124, 3142, 3241, 3412, 3421, 4132, 4213, 4231, 4312.

MAPLE

b:= proc(s, j, k) option remember; `if`(s={}, 1,

add(`if`(k=0 or 2*j<>i+k, b(s minus {i}, i,

`if`(2*i-j in s, j, 0)), 0), i=s))

end:

a:= n-> b({$1..n}, 0$2):

seq(a(n), n=0..12);

MATHEMATICA

Table[Length[Select[Permutations[Range[n]], !MemberQ[Differences[#, 2], 0]&]], {n, 0, 5}] (* Gus Wiseman, Jun 03 2019 *)

b[s_, j_, k_] := b[s, j, k] = If[s == {}, 1, Sum[If[k == 0 || 2*j != i + k, b[s~Complement~{i}, i, If[MemberQ[s, 2*i - j ], j, 0]], 0], {i, s}]];

a[n_] := a[n] = b[Range[n], 0, 0];

Table[Print[n, " ", a[n]]; a[n], {n, 0, 16}] (* Jean-François Alcover, Nov 20 2023, after Alois P. Heinz *)

CROSSREFS

Column k=0 of A295390.

Cf. A003407, A174080, A238423, A238424, A338765.

Cf. A049988, A175342, A279945, A325545, A325849, A325850, A325851, A325874, A325875.

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Nov 20 2017

EXTENSIONS

a(22)-a(23) from Vaclav Kotesovec, Mar 22 2022

STATUS

approved

A325874

Number of integer partitions of n whose differences of all degrees > 1 are nonzero.

+10
10

1, 1, 2, 2, 4, 5, 6, 8, 12, 13, 19, 24, 26, 33, 45, 52, 66, 78, 92, 113, 129, 160, 192, 231, 268, 305, 361, 436, 501, 591, 665, 783, 897, 1071, 1228, 1361, 1593, 1834, 2101, 2452, 2685, 3129, 3526, 4067, 4568, 5189, 5868, 6655, 7565, 8468, 9400

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

OFFSET

0,3

COMMENTS

The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) are (-3,-2). The zeroth differences are the sequence itself, while k-th differences for k > 0 are the differences of the (k-1)-th differences. If m is the length of the sequence, its differences of all degrees are the union of the zeroth through m-th differences.

The case for all degrees including 1 is A325852.

LINKS

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..220

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts

EXAMPLE

The a(1) = 1 through a(9) = 13 partitions:

(1) (2) (3) (4) (5) (6) (7) (8) (9)

(11) (21) (22) (32) (33) (43) (44) (54)

(31) (41) (42) (52) (53) (63)

(211) (221) (51) (61) (62) (72)

(311) (411) (322) (71) (81)

(2211) (331) (332) (441)

(421) (422) (522)

(511) (431) (621)

(521) (711)

(611) (4221)

(3221) (4311)

(3311) (5211)

(32211)

MATHEMATICA

Table[Length[Select[IntegerPartitions[n], !MemberQ[Union@@Table[Differences[#, i], {i, 2, Length[#]}], 0]&]], {n, 0, 30}]

CROSSREFS

Cf. A049988, A238423, A325325, A325468, A325545, A325849, A325850, A325851, A325852, A325875, A325876.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jun 02 2019

STATUS

approved

A325852

Number of (strict) integer partitions of n whose differences of all degrees are nonzero.

+10
8

1, 1, 1, 2, 2, 3, 3, 5, 6, 6, 9, 11, 11, 15, 19, 19, 26, 31, 31, 41, 49, 53, 62, 75, 81, 97, 112, 124, 145, 171, 175, 215, 244, 274, 307, 344, 388, 446, 497, 561, 599, 700, 779, 881, 981, 1054, 1184, 1340, 1500, 1669, 1767, 2031, 2237, 2486, 2765, 2946, 3300

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

OFFSET

0,4

COMMENTS

The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) are (-3,-2). The zeroth differences are the sequence itself, while k-th differences for k > 0 are the differences of the (k-1)-th differences. The differences of all degrees of a sequence are the union of its zeroth through m-th differences, where m is the length of the sequence.

LINKS

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..250

EXAMPLE

The a(1) = 1 through a(11) = 11 partitions (A = 10, B = 11):

(1) (2) (3) (4) (5) (6) (7) (8) (9) (A) (B)

(21) (31) (32) (42) (43) (53) (54) (64) (65)

(41) (51) (52) (62) (63) (73) (74)

(61) (71) (72) (82) (83)

(421) (431) (81) (91) (92)

(521) (621) (532) (A1)

(541) (542)

(631) (632)

(721) (641)

(731)

(821)

MATHEMATICA

Table[Length[Select[IntegerPartitions[n], !MemberQ[Union@@Table[Differences[#, i], {i, Length[#]}], 0]&]], {n, 0, 30}]

CROSSREFS

The case for only degrees > 1 is A325874.

Cf. A049988, A175342, A238423, A279945, A295370, A325328, A325468, A325545, A325850, A325851, A325875.

KEYWORD

nonn

AUTHOR

Gus Wiseman, May 31 2019

STATUS

approved

A325850

Number of permutations of {1..n} whose differences of all degrees are nonzero.

+10
7

1, 1, 2, 4, 18, 72, 446, 2804, 21560, 184364, 1788514

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

OFFSET

0,3

COMMENTS

The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) are (-3,-2). The zeroth differences are the sequence itself, while k-th differences for k > 0 are the differences of the (k-1)-th differences. The differences of all degrees of a sequence are the union of its zeroth through m-th differences, where m is the length of the sequence.

LINKS

Table of n, a(n) for n=0..10.

EXAMPLE

The a(1) = 1 through a(4) = 18 permutations:

(1) (12) (132) (1243)

(21) (213) (1324)

(231) (1342)

(312) (1423)

(2134)

(2143)

(2314)

(2413)

(2431)

(3124)

(3142)

(3241)

(3412)

(3421)

(4132)

(4213)

(4231)

(4312)

MATHEMATICA

Table[Length[Select[Permutations[Range[n]], !MemberQ[Union@@Table[Differences[#, i], {i, Length[#]}], 0]&]], {n, 0, 5}]

CROSSREFS

Dominated by A295370, the case for only differences of degree 2.

Cf. A049988, A175342, A238423, A279945, A325545, A325851, A325852, A325874, A325875.

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, May 31 2019

STATUS

approved

A325875

Number of compositions of n whose differences of all degrees > 1 are nonzero.

+10
7

1, 1, 2, 3, 7, 13, 20, 38, 69, 129, 222, 407, 726, 1313, 2318, 4146, 7432, 13296, 23759, 42458, 75714

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

OFFSET

0,3

COMMENTS

A composition of n is a finite sequence of positive integers with sum n.

The case for all degrees including 1 is A325851.

LINKS

Table of n, a(n) for n=0..20.

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts

EXAMPLE

The a(1) = 1 through a(6) = 20 compositions:

(1) (2) (3) (4) (5) (6)

(11) (12) (13) (14) (15)

(21) (22) (23) (24)

(31) (32) (33)

(112) (41) (42)

(121) (113) (51)

(211) (122) (114)

(131) (132)

(212) (141)

(221) (213)

(311) (231)

(1121) (312)

(1211) (411)

(1122)

(1131)

(1212)

(1311)

(2121)

(2211)

(11211)

MATHEMATICA

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], !MemberQ[Union@@Table[Differences[#, i], {i, 2, Length[#]}], 0]&]], {n, 0, 10}]

CROSSREFS

Cf. A049988, A238423, A325325, A325468, A325545, A325849, A325850, A325851, A325852, A325874, A325876.

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Jun 02 2019

STATUS

approved

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