cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A325535 Number of inseparable partitions of n; see Comments.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 5, 5, 8, 11, 16, 19, 28, 35, 48, 60, 79, 99, 131, 161, 205, 256, 324, 397, 498, 609, 755, 921, 1131, 1372, 1677, 2022, 2452, 2952, 3561, 4260, 5116, 6102, 7291, 8667, 10309, 12210, 14477, 17087, 20177, 23752, 27957, 32804, 38496, 45049, 52704
Offset: 0

Views

Author

Clark Kimberling, May 08 2019

Keywords

Comments

Definition: a partition is separable if there is an ordering of its parts in which no consecutive parts are identical; otherwise the partition is inseparable.
A partition with k parts is inseparable if and only if there is a part whose multiplicity is greater than ceiling(k/2). - Andrew Howroyd, Jan 17 2024

Examples

			For n=5, the partition 1+2+2 is separable as 2+1+2, and 2+1+1+1 is inseparable.
From _Gus Wiseman_, Jun 27 2020: (Start)
The a(2) = 2 through a(9) = 11 inseparable partitions:
  11   111   22     2111    33       2221      44         333
             1111   11111   222      4111      2222       3222
                            3111     31111     5111       6111
                            21111    211111    41111      22221
                            111111   1111111   221111     51111
                                               311111     321111
                                               2111111    411111
                                               11111111   2211111
                                                          3111111
                                                          21111111
                                                          111111111
(End)

Links

Crossrefs

The Heinz numbers of these partitions are given by A335448.
Strict partitions are counted by A000009 and are all separable.
Anti-run compositions are counted by A003242.
Anti-run patterns are counted by A005649.
Partitions whose differences are an anti-run are A238424.
Separable partitions are counted by A325534.
Anti-run compositions are ranked by A333489.
Anti-run permutations of prime indices are counted by A335452.
Cf. A000041, A106356, A238594, A261962, A292884, A332668, A333175.

Programs

  • Mathematica
    u=Table[Length[Select[Map[Quotient[(1 + Length[#]), Max[Map[Length, Split[#]]]] &,
IntegerPartitions[nn]], # > 1 &]], {nn, 50}]
Table[PartitionsP[n] - u[[n]], {n, 1, Length[u]}]
(* Peter J. C. Moses, May 07 2019 *)
Table[Length[Select[IntegerPartitions[n],Select[Permutations[#],!MatchQ[#,{_,x_,x_,_}]&]=={}&]],{n,10}] (* Gus Wiseman, Jun 27 2020 *)
  • PARI
    seq(n) = {Vec(sum(k=1, (n+1)\2, x^(2*k-1)*(1 + x - x^(k-1))/((1-x^(k+1))*prod(j=1, k-1, 1 - x^j, 1 + O(x^(n-2*k+2)))), O(x*x^n)), -(n+1))} \\ Andrew Howroyd, Jan 17 2024

Formula

a(n) = A000041(n) - A325534(n).
a(n) = Sum_{k>=1} x^(2*k-1)*(1 + x - x^(k-1))/((1-x^(k+1))*Product_{j=1..k-1} (1 - x^j)). - Andrew Howroyd, Jan 17 2024

Extensions

a(0)=0 prepended by Andrew Howroyd, Jan 31 2024

A238424 Number of partitions of n without three consecutive parts in arithmetic progression.

Original entry on oeis.org

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
Offset: 0

Views

Author

Joerg Arndt and Alois P. Heinz, Feb 26 2014

Keywords

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

Examples

			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 ]

Links

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.

Programs

  • 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 *)

A333195 Numbers with three consecutive prime indices in arithmetic progression.

Original entry on oeis.org

8, 16, 24, 27, 30, 32, 40, 48, 54, 56, 60, 64, 72, 80, 81, 88, 96, 104, 105, 108, 110, 112, 120, 125, 128, 135, 136, 144, 150, 152, 160, 162, 168, 176, 184, 189, 192, 200, 208, 210, 216, 220, 224, 232, 238, 240, 243, 248, 250, 256, 264, 270, 272, 273, 280, 288
Offset: 1

Views

Author

Gus Wiseman, Mar 29 2020

Keywords

Comments

Also numbers whose first differences of prime indices do not form an anti-run, meaning there are adjacent equal differences.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Examples

			The sequence of terms together with their prime indices begins:
    8: {1,1,1}          105: {2,3,4}
   16: {1,1,1,1}        108: {1,1,2,2,2}
   24: {1,1,1,2}        110: {1,3,5}
   27: {2,2,2}          112: {1,1,1,1,4}
   30: {1,2,3}          120: {1,1,1,2,3}
   32: {1,1,1,1,1}      125: {3,3,3}
   40: {1,1,1,3}        128: {1,1,1,1,1,1,1}
   48: {1,1,1,1,2}      135: {2,2,2,3}
   54: {1,2,2,2}        136: {1,1,1,7}
   56: {1,1,1,4}        144: {1,1,1,1,2,2}
   60: {1,1,2,3}        150: {1,2,3,3}
   64: {1,1,1,1,1,1}    152: {1,1,1,8}
   72: {1,1,1,2,2}      160: {1,1,1,1,1,3}
   80: {1,1,1,1,3}      162: {1,2,2,2,2}
   81: {2,2,2,2}        168: {1,1,1,2,4}
   88: {1,1,1,5}        176: {1,1,1,1,5}
   96: {1,1,1,1,1,2}    184: {1,1,1,9}
  104: {1,1,1,6}        189: {2,2,2,4}

Links

Crossrefs

Anti-run compositions are counted by A003242.
Normal anti-runs of length n + 1 are counted by A005649.
Strict partitions with equal differences are A049980.
Partitions with equal differences are A049988.
These are the Heinz numbers of the partitions *not* counted by A238424.
Permutations avoiding triples in arithmetic progression are A295370.
Strict partitions avoiding triples in arithmetic progression are A332668.
Anti-run compositions are ranked by A333489.
Cf. A006560, A007862, A238423, A307824, A325328, A325849, A325852.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],MatchQ[Differences[primeMS[#]],{_,x_,x_,_}]&]

A333631 Number of permutations of {1..n} with three consecutive terms in arithmetic progression.

Original entry on oeis.org

0, 0, 0, 2, 6, 40, 238, 1760, 14076, 131732, 1308670, 14678452, 176166906, 2317481348, 32416648496, 490915956484, 7846449011500, 134291298372632, 2416652824505150, 46141903780094080, 922528719841017424, 19456439433050482412, 427837767407051523776, 9873256397944571377332
Offset: 0

Views

Author

Gus Wiseman, Mar 31 2020

Keywords

Comments

Also permutations whose second differences have at least one zero.

Examples

			The a(3) = 2 and a(4) = 6 permutations:
  (1,2,3)  (1,2,3,4)
  (3,2,1)  (1,4,3,2)
           (2,3,4,1)
           (3,2,1,4)
           (4,1,2,3)
           (4,3,2,1)

Links

Crossrefs

The complement is counted by A295370.
The version for prime indices is A333195.
Strict partitions with equal differences are A049980.
Partitions with equal differences are A049988.
Compositions without triples in arithmetic progression are A238423.
Partitions without triples in arithmetic progression are A238424.
Strict partitions without triples in arithmetic progression are A332668.
Cf. A000142, A007862, A175342, A325849, A325850.

Programs

  • Mathematica
    Table[Select[Permutations[Range[n]],MatchQ[Differences[#],{_,x_,x_,_}]&]//Length,{n,0,8}]

Formula

a(n) = n! - A295370(n).

Extensions

a(11)-a(21) (using A295370) from Giovanni Resta, Apr 07 2020
a(22)-a(23) (using A295370) from Alois P. Heinz, Jan 27 2024
Showing 1-4 of 4 results.

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