A345166 -id:A345166 - cp's OEIS frontend

A345171 Numbers whose multiset of prime factors has no alternating permutation.

4, 8, 9, 16, 24, 25, 27, 32, 40, 48, 49, 54, 56, 64, 80, 81, 88, 96, 104, 112, 121, 125, 128, 135, 136, 144, 152, 160, 162, 169, 176, 184, 189, 192, 208, 224, 232, 240, 243, 248, 250, 256, 270, 272, 288, 289, 296, 297, 304, 320, 324, 328, 336, 343, 344, 351

Offset: 1

Gus Wiseman, Jun 13 2021

nonn

First differs from A335448 in having 270.
A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,2,2,2,1) has no alternating permutations, even though it has the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2).
Also Heinz numbers of integer partitions without a wiggly permutation, where the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The sequence of terms together with their prime indices begins:
    4: {1,1}
    8: {1,1,1}
    9: {2,2}
   16: {1,1,1,1}
   24: {1,1,1,2}
   25: {3,3}
   27: {2,2,2}
   32: {1,1,1,1,1}
   40: {1,1,1,3}
   48: {1,1,1,1,2}
   49: {4,4}
   54: {1,2,2,2}
   56: {1,1,1,4}
   64: {1,1,1,1,1,1}
   80: {1,1,1,1,3}
   81: {2,2,2,2}
   88: {1,1,1,5}
   96: {1,1,1,1,1,2}

Removing squares of primes A001248 gives A344653, counted by A344654.
A superset of A335448, which is counted by A325535.
Positions of 0's in A345164.
The partitions with these Heinz numbers are counted by A345165.
The complement is A345172, counted by A345170.
The separable case is A345173, counted by A345166.
A001250 counts alternating permutations, complement A348615.
A003242 counts anti-run compositions, complement A261983.
A025047 counts alternating or wiggly compositions, directed A025048, A025049.
A325534 counts separable partitions, ranked by A335433.
A344606 counts alternating permutations of prime indices with twins.
A344742 ranks twins and partitions with an alternating permutation.
A345192 counts non-alternating compositions.
Cf. A001222, A071321, A071322, A316523, A316524, A335126, A344604, A344616, A344652, A344740, A345163, A345168, A345193, A345195, A348380, A348609.

wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]== Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
Select[Range[100],Select[Permutations[Flatten[ ConstantArray@@@FactorInteger[#]]],wigQ]=={}&]

A345163 Number of integer partitions of n with an alternating permutation covering an initial interval of positive integers.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 2, 2, 3, 3, 4, 6, 7, 8, 11, 12, 16, 20, 23, 27, 34, 41, 48, 57, 68, 80, 94, 110, 130, 153, 175, 203, 239, 275, 317, 365, 420, 483, 553, 632, 720, 825, 938, 1064, 1211, 1370, 1550, 1755, 1982, 2235, 2517, 2830, 3182, 3576, 4006, 4487, 5027, 5619, 6275, 7007, 7812

Offset: 0

Gus Wiseman, Jun 12 2021

nonn

A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,3,2,2,2,2,1) has no alternating permutations, even though it has the anti-run permutations (2,3,2,3,2,1,2), (2,3,2,1,2,3,2), and (2,1,2,3,2,3,2).

A partition with k parts is alternating if and only every part has a multiplicity no greater than k/2, except either the smallest or largest part may have a multiplicity of (k+1)/2 when k is odd. - Andrew Howroyd, Jan 31 2024

			The a(3) = 1 through a(12) = 7 partitions:
  21  211  221  321   3211   3221   3321    4321     33221    33321
                2211  22111  22211  32211   33211    43211    43221
                             32111  222111  322111   322211   332211
                                            2221111  332111   432111
                                                     2222111  3222111
                                                     3221111  3321111
                                                              22221111
For example, the partition (3,3,2,1,1,1,1) has the alternating permutations (1,3,1,3,1,2,1), (1,3,1,2,1,3,1), and (1,2,1,3,1,3,1), so is counted under a(12).

Andrew Howroyd, Table of n, a(n) for n = 0..500

Not requiring an alternating permutation gives A000670, ranked by A333217.
The complement in covering partitions is counted by A345162.
Not requiring normality gives A345170, ranked by A345172.
A000041 counts integer partitions.
A001250 counts alternating permutations.
A003242 counts anti-run compositions.
A005649 counts anti-run patterns.
A025047 counts alternating or wiggly compositions, also A025048, A025049.
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A344605 counts alternating patterns with twins.
A345164 counts alternating permutations of prime indices.
A345165 counts partitions without a alternating permutation, ranked by A345171.
A349051 ranks alternating compositions.
Cf. A000070, A103919, A335126, A344604, A344607, A344615, A344653, A344654, A344740, A345166, A345167, A345168.

normQ[m_]:=m=={}||Union[m]==Range[Max[m]];
wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]==Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
Table[Length[Select[IntegerPartitions[n],normQ[#]&&Select[Permutations[#],wigQ]!={}&]],{n,0,15}]

\\ See also A345162 for a faster program.
ok(k,p)={my(S=Set(p)); foreach(S, t, my(c=k+#p-2*(1+#select(x->x==t, p))); if(c<0, return(c==-1 && (t==1||t==k)))); 1}
a(n)={sum(k=1, (sqrtint(8*n+1)-1)\2, s=0; forpart(p=n-binomial(k+1,2), s+=ok(k,Vec(p)), k); s)} \\ Andrew Howroyd, Jan 31 2024

The Heinz numbers of these partitions are A333217 /\ A345172.

a(n) = A000009(n) - A345162(n). - Andrew Howroyd, Jan 31 2024

a(26) onwards from Andrew Howroyd, Jan 31 2024

A344742 Numbers whose prime factors have a permutation with no consecutive monotone triple, i.e., no triple (..., x, y, z, ...) such that either x <= y <= z or x >= y >= z.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77

Offset: 1

Gus Wiseman, Jun 12 2021

nonn

Differs from A335433 in having all squares of primes (A001248) and lacking 270 etc.
Also Heinz numbers of integer partitions that are either a twin (x,x) or have a wiggly permutation.
(1) The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
(2) A sequence is wiggly if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,2,2,2,1) has no wiggly permutations, even though it has anti-run permutations (2,3,2,1,2) and (2,1,2,3,2).

			The sequence of terms together with their prime indices begins:
      1: {}          18: {1,2,2}     36: {1,1,2,2}
      2: {1}         19: {8}         37: {12}
      3: {2}         20: {1,1,3}     38: {1,8}
      4: {1,1}       21: {2,4}       39: {2,6}
      5: {3}         22: {1,5}       41: {13}
      6: {1,2}       23: {9}         42: {1,2,4}
      7: {4}         25: {3,3}       43: {14}
      9: {2,2}       26: {1,6}       44: {1,1,5}
     10: {1,3}       28: {1,1,4}     45: {2,2,3}
     11: {5}         29: {10}        46: {1,9}
     12: {1,1,2}     30: {1,2,3}     47: {15}
     13: {6}         31: {11}        49: {4,4}
     14: {1,4}       33: {2,5}       50: {1,3,3}
     15: {2,3}       34: {1,7}       51: {2,7}
     17: {7}         35: {3,4}       52: {1,1,6}
For example, the prime factors of 120 are (2,2,2,3,5), with the two wiggly permutations (2,3,2,5,2) and (2,5,2,3,2), so 120 is in the sequence.

Positions of nonzero terms in A344606.
The complement is A344653, counted by A344654.
These partitions are counted by A344740.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A001248 lists squares of primes.
A001250 counts wiggly permutations.
A003242 counts anti-run compositions.
A011782 counts compositions.
A025047 counts wiggly compositions (ascend: A025048, descend: A025049).
A056239 adds up prime indices,  row sums of A112798.
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A344604 counts wiggly compositions with twins.
A345164 counts wiggly permutations of prime indices.
A345165 counts partitions without a wiggly permutation, ranked by A345171.
A345170 counts partitions with a wiggly permutation, ranked by A345172.
A345192 counts non-wiggly compositions.
Cf. A000070, A001222, A071321, A071322, A316523, A316524, A344605, A344614, A344616, A344652, A345163, A345166, A345167, A345173.

Select[Range[100],Select[Permutations[Flatten[ConstantArray@@@FactorInteger[#]]],!MatchQ[#,{_,x_,y_,z_,_}/;x<=y<=z||x>=y>=z]&]!={}&]

Union of A345172 (wiggly) and A001248 (squares of primes).

A345173 Numbers whose multiset of prime factors is separable but has no alternating permutation.

Original entry on oeis.org

270, 378, 594, 702, 918, 1026, 1242, 1566, 1620, 1674, 1750, 1998, 2214, 2268, 2322, 2538, 2625, 2750, 2862, 3186, 3250, 3294, 3564, 3618, 3834, 3942, 4050, 4125, 4212, 4250, 4266, 4482, 4750, 4806, 4875, 5238, 5454, 5508, 5562, 5670, 5750, 5778, 5886, 6102

Offset: 1

Gus Wiseman, Jun 13 2021

nonn

A multiset is separable if it has an anti-run permutation (no adjacent parts equal).
A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,2,2,2,1) has no alternating permutations, even though it has the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2).
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The terms together with their prime indices begin:
   270: {1,2,2,2,3}
   378: {1,2,2,2,4}
   594: {1,2,2,2,5}
   702: {1,2,2,2,6}
   918: {1,2,2,2,7}
  1026: {1,2,2,2,8}
  1242: {1,2,2,2,9}
  1566: {1,2,2,2,10}
  1620: {1,1,2,2,2,2,3}
  1674: {1,2,2,2,11}
  1750: {1,3,3,3,4}
  1998: {1,2,2,2,12}
  2214: {1,2,2,2,13}
  2268: {1,1,2,2,2,2,4}
  2322: {1,2,2,2,14}

The partitions with these Heinz numbers are counted by A345166.
Permutations of this type are ranked by A345169.
Numbers with a factorization of this type are counted by A348609.
A000041 counts integer partitions.
A001250 counts alternating permutations, complement A348615.
A003242 counts anti-run compositions.
A025047 counts alternating compositions, ascend A025048, descend A025049.
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A344606 counts alternating permutations of prime indices with twins.
A344740 counts twins and partitions with an alternating permutation.
A345164 counts alternating permutations of prime factors.
A345165 counts partitions without an alternating permutation.
A345170 counts partitions with an alternating permutation.
A345192 counts non-alternating compositions, without twins A348377.
A348379 counts factorizations with an alternating permutation.
Cf. A001222, A071321, A316524, A335126, A344614, A344616, A344652, A344653, A345163, A345168, A345193, A347706, A348380, A348613.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]== Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
sepQ[y_]:=!MatchQ[y,{_,x_,x_,_}];
Select[Range[1000],Select[Permutations[primeMS[#]],wigQ]=={}&&!Select[Permutations[primeMS[#]],sepQ]=={}&]

Equals A345171 /\ A335433.

A345169 Numbers k such that the k-th composition in standard order is a non-alternating anti-run.

Original entry on oeis.org

37, 52, 69, 101, 104, 105, 133, 137, 150, 165, 180, 197, 200, 208, 209, 210, 261, 265, 274, 278, 300, 301, 308, 325, 328, 357, 360, 361, 389, 393, 400, 401, 406, 416, 417, 418, 421, 422, 436, 517, 521, 529, 530, 534, 549, 550, 556, 557, 564, 581, 600, 601, 613

Offset: 1

Gus Wiseman, Jun 15 2021

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.
A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,2,2,2,1) has no alternating permutations, even though it does have the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2).
An anti-run (separation or Carlitz composition) is a sequence with no adjacent equal parts.

			The sequence of terms together with their binary indices begins:
     37: (3,2,1)      210: (1,2,3,2)      400: (1,3,5)
     52: (1,2,3)      261: (6,2,1)        401: (1,3,4,1)
     69: (4,2,1)      265: (5,3,1)        406: (1,3,2,1,2)
    101: (1,3,2,1)    274: (4,3,2)        416: (1,2,6)
    104: (1,2,4)      278: (4,2,1,2)      417: (1,2,5,1)
    105: (1,2,3,1)    300: (3,2,1,3)      418: (1,2,4,2)
    133: (5,2,1)      301: (3,2,1,2,1)    421: (1,2,3,2,1)
    137: (4,3,1)      308: (3,1,2,3)      422: (1,2,3,1,2)
    150: (3,2,1,2)    325: (2,4,2,1)      436: (1,2,1,2,3)
    165: (2,3,2,1)    328: (2,3,4)        517: (7,2,1)
    180: (2,1,2,3)    357: (2,1,3,2,1)    521: (6,3,1)
    197: (1,4,2,1)    360: (2,1,2,4)      529: (5,4,1)
    200: (1,3,4)      361: (2,1,2,3,1)    530: (5,3,2)
    208: (1,2,5)      389: (1,5,2,1)      534: (5,2,1,2)
    209: (1,2,4,1)    393: (1,4,3,1)      549: (4,3,2,1)

Wikipedia, Alternating permutation

A version counting partitions is A345166, ranked by A345173.
These compositions are counted by A345195.
A001250 counts alternating permutations, complement A348615.
A003242 counts anti-run compositions.
A005649 counts anti-run patterns.
A025047 counts alternating or wiggly compositions, also A025048, A025049.
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A345164 counts alternating permutations of prime indices.
A345165 counts partitions w/o an alternating permutation, ranked by A345171.
A345170 counts partitions w/ an alternating permutation, ranked by A345172.
A345192 counts non-alternating compositions.
A345194 counts alternating patterns (with twins: A344605).
Statistics of standard compositions:
- Length is A000120.
- Constant runs are A124767.
- Heinz number is A333219.
- Anti-runs are A333381.
- Runs-resistance is A333628.
- Number of distinct parts is A334028.
- Non-anti-runs are A348612.
Classes of standard compositions:
- Weakly decreasing compositions (partitions) are A114994.
- Weakly increasing compositions (multisets) are A225620.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Strictly increasing compositions (sets) are A333255.
- Strictly decreasing compositions (strict partitions) are A333256.
- Anti-runs are A333489.
- Alternating compositions are A345167.
- Non-Alternating compositions are A345168.
Cf. A001222, A008965, A238279, A344614, A344615, A344652, A344653, A344654, A345162, A345163, A345193, A348609, A348613.

stc[n_]:=Differences[Prepend[Join@@Position[ Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]== Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
sepQ[y_]:=!MatchQ[y,{_,x_,x_,_}];
Select[Range[0,1000],sepQ[stc[#]]&&!wigQ[stc[#]]&]

Intersection of A345168 (non-alternating) and A333489 (anti-run).

A345195 Number of non-alternating anti-run compositions of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 2, 4, 10, 23, 49, 96, 192, 368, 692, 1299, 2403, 4400, 8029, 14556, 26253, 47206, 84574, 151066, 269244, 478826, 849921, 1506309, 2665829, 4711971, 8319763, 14675786, 25865400, 45552678, 80171353, 141015313, 247905305, 435614270, 765132824

Offset: 0

Gus Wiseman, Jun 17 2021

nonn

A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,2,2,2,1) has no alternating permutations, even though it does have the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2).

An anti-run (separation or Carlitz composition) is a sequence with no adjacent equal parts.

			The a(9) = 23 anti-runs:
  (1,2,6)  (1,2,4,2)  (1,2,1,2,3)
  (1,3,5)  (1,2,5,1)  (1,2,3,1,2)
  (2,3,4)  (1,3,4,1)  (1,2,3,2,1)
  (4,3,2)  (1,4,3,1)  (1,3,2,1,2)
  (5,3,1)  (1,5,2,1)  (2,1,2,3,1)
  (6,2,1)  (2,1,2,4)  (2,1,3,2,1)
           (2,4,2,1)  (3,2,1,2,1)
           (3,1,2,3)
           (3,2,1,3)
           (4,2,1,2)

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Non-anti-run compositions are counted by A261983.
A version counting partitions is A345166, ranked by A345173.
These compositions are ranked by A345169.
Non-alternating compositions are counted by A345192, ranked by A345168.
A001250 counts alternating permutations, complement A348615.
A003242 counts anti-run compositions, ranked by A333489.
A011782 counts compositions.
A025047 counts alternating or wiggly compositions, ranked by A345167.
A032020 counts strict compositions.
A106356 counts compositions by number of maximal anti-runs.
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A345164 counts alternating permutations of prime indices, w/ twins A344606.
A345165 counts partitions w/o an alternating permutation, ranked by A345171.
A345170 counts partitions w/ an alternating permutation, ranked by A345172.
A345194 counts alternating patterns (with twins: A344605).
Cf. A005649, A008965, A114901, A178470, A333755, A344604, A344614, A344654, A344740, A345162, A345163, A348380, A348612, A348613.

wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]== Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
sepQ[y_]:=!MatchQ[y,{_,x_,x_,_}];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], sepQ[#]&&!wigQ[#]&]],{n,0,15}]

a(n) = A003242(n) - A025047(n).

a(21) onwards from Andrew Howroyd, Jan 31 2024

A345162 Number of integer partitions of n with no alternating permutation covering an initial interval of positive integers.

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 2, 3, 3, 5, 6, 6, 8, 10, 11, 15, 16, 18, 23, 27, 30, 35, 41, 47, 54, 62, 71, 82, 92, 103, 121, 137, 151, 173, 195, 220, 248, 277, 311, 350, 393, 435, 488, 546, 605, 678, 754, 835, 928, 1029, 1141, 1267, 1400, 1544, 1712, 1891, 2081, 2298, 2533, 2785, 3068

Offset: 0

Gus Wiseman, Jun 12 2021

nonn

A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,3,2,2,2,2,1) has no alternating permutations, even though it has anti-run permutations (2,3,2,3,2,1,2), (2,3,2,1,2,3,2), and (2,1,2,3,2,3,2).

Sequences covering an initial interval (patterns) are counted by A000670 and ranked by A333217.

			The a(2) = 1 through a(10) = 6 partitions:
  11  111  1111  2111   21111   2221     221111    22221      32221
                 11111  111111  211111   2111111   321111     222211
                                1111111  11111111  2211111    3211111
                                                   21111111   22111111
                                                   111111111  211111111
                                                              1111111111

Andrew Howroyd, Table of n, a(n) for n = 0..500

The complement in covering partitions is counted by A345163.
Not requiring normality gives A345165, ranked by A345171.
The separable case is A345166.
A000041 counts integer partitions.
A000670 counts patterns, ranked by A333217.
A001250 counts alternating permutations.
A003242 counts anti-run compositions.
A005649 counts anti-run patterns.
A025047 counts alternating or wiggly compositions, directed A025048/A025049.
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A344604 counts alternating compositions with twins.
A344605 counts alternating patterns with twins.
A345164 counts alternating permutations of prime indices.
A345170 counts partitions with a alternating permutation, ranked by A345172.
Cf. A000070, A103919, A335126, A344614, A344615, A344653, A344654, A344740, A344742, A345167, A345168, A345192, A348609.

normQ[m_]:=m=={}||Union[m]==Range[Max[m]];
wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]==Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
Table[Length[Select[IntegerPartitions[n],normQ[#]&&Select[Permutations[#],wigQ[#]&]=={}&]],{n,0,15}]

P(n,m)={Vec(1/prod(k=1, m, 1-y*x^k, 1+O(x*x^n)))}
a(n) = {(n >= 2) + sum(k=2, (sqrtint(8*n+1)-1)\2, my(r=n-binomial(k+1,2), v=P(r, k)); sum(i=1, min(k,2*r\k), sum(j=k-1, (2*r-(k-1)*(i-1))\(i+1), my(p=(j+k+(i==1||i==k))\2); if(p*i<=r, polcoef(v[r-p*i+1],j-p)) )))} \\ Andrew Howroyd, Jan 31 2024

a(n) = A000009(n) - A345163(n). - Andrew Howroyd, Jan 31 2024

a(26) onwards from Andrew Howroyd, Jan 31 2024

A349800 Number of integer compositions of n that are weakly alternating and have at least two adjacent equal parts.

Original entry on oeis.org

0, 0, 1, 1, 4, 9, 16, 33, 62, 113, 205, 373, 664, 1190, 2113, 3744, 6618, 11683, 20564, 36164, 63489, 111343, 195042, 341357, 596892, 1042976, 1821179, 3178145, 5543173, 9663545, 16839321, 29332231, 51075576, 88908912, 154722756, 269186074, 468221264

Offset: 0

Gus Wiseman, Dec 16 2021

nonn

We define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either.

This sequence counts compositions that are weakly but not strongly alternating; also weakly alternating non-anti-run compositions.

			The a(2) = 1 through a(6) = 16 compositions:
  (1,1)  (1,1,1)  (2,2)      (1,1,3)      (3,3)
                  (1,1,2)    (1,2,2)      (1,1,4)
                  (2,1,1)    (2,2,1)      (2,2,2)
                  (1,1,1,1)  (3,1,1)      (4,1,1)
                             (1,1,1,2)    (1,1,1,3)
                             (1,1,2,1)    (1,1,2,2)
                             (1,2,1,1)    (1,1,3,1)
                             (2,1,1,1)    (1,3,1,1)
                             (1,1,1,1,1)  (2,2,1,1)
                                          (3,1,1,1)
                                          (1,1,1,1,2)
                                          (1,1,1,2,1)
                                          (1,1,2,1,1)
                                          (1,2,1,1,1)
                                          (2,1,1,1,1)
                                          (1,1,1,1,1,1)

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Wikipedia, Alternating permutation

This is the weakly alternating case of A345192, ranked by A345168.
The case of partitions is A349795, ranked by A350137.
The version counting permutations of prime indices is A349798.
These compositions are ranked by A349799.
A001250 = alternating permutations, ranked by A349051, complement A348615.
A003242 = Carlitz (anti-run) compositions, ranked by A333489.
A025047/A025048/A025049 = alternating compositions, ranked by A345167.
A261983 = non-anti-run compositions, ranked by A348612.
A345165 = partitions without an alternating permutation, ranked by A345171.
A345170 = partitions with an alternating permutation, ranked by A345172.
A345173 = non-alternating anti-run partitions, ranked by A345166.
A345195 = non-alternating anti-run compositions, ranked by A345169.
A348377 = non-alternating non-twin compositions.
A349801 = non-alternating partitions, ranked by A289553.
Weakly alternating:
- A349052 = compositions, directed A129852/A129853, complement A349053.
- A349056 = permutations of prime indices, complement A349797.
- A349057 = complement of standard composition numbers (too dense).
- A349058 = patterns, complement A350138.
- A349059 = ordered factorizations, complement A350139.
- A349060 = partitions, complement A349061.
Cf. A008965, A011782, A027383, A096441, A274230, A333213, A344614, A344615, A348382, A348613, A349796, A350140.

wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]==Length[y] &&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
whkQ[y_]:=And@@Table[If[EvenQ[m],y[[m]]<=y[[m+1]],y[[m]]>=y[[m+1]]],{m,1,Length[y]-1}];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],(whkQ[#]||whkQ[-#])&&!wigQ[#]&]],{n,0,10}]

a(n) = A349052(n) - A025047(n). - Andrew Howroyd, Jan 31 2024

a(21) onwards from Andrew Howroyd, Jan 31 2024

A348377 Number of non-alternating compositions of n, excluding twins (x,x).

Original entry on oeis.org

0, 0, 0, 1, 3, 9, 19, 45, 98, 208, 436, 906, 1861, 3803, 7731, 15659, 31628, 63747, 128257, 257722, 517338, 1037652, 2079983, 4167325, 8346203, 16710572, 33449694, 66944254, 133959020, 268028868, 536231902, 1072737537, 2145905284, 4292486690, 8586035992

Offset: 0

Gus Wiseman, Oct 26 2021

nonn

First differs from A348382 at a(6) = 19, A348382(6) = 17. The two non-alternating non-twin compositions of 6 that are not an anti-run are (1,2,3) and (3,2,1).

A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,2,2,2,1) has no alternating permutations, even though it does have the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2). Alternating permutations of multisets are a generalization of alternating or up-down permutations of {1..n}.

			The a(3) = 1 through a(6) = 19 compositions:
  (1,1,1)  (1,1,2)    (1,1,3)      (1,1,4)
           (2,1,1)    (1,2,2)      (1,2,3)
           (1,1,1,1)  (2,2,1)      (2,2,2)
                      (3,1,1)      (3,2,1)
                      (1,1,1,2)    (4,1,1)
                      (1,1,2,1)    (1,1,1,3)
                      (1,2,1,1)    (1,1,2,2)
                      (2,1,1,1)    (1,1,3,1)
                      (1,1,1,1,1)  (1,2,2,1)
                                   (1,3,1,1)
                                   (2,1,1,2)
                                   (2,2,1,1)
                                   (3,1,1,1)
                                   (1,1,1,1,2)
                                   (1,1,1,2,1)
                                   (1,1,2,1,1)
                                   (1,2,1,1,1)
                                   (2,1,1,1,1)
                                   (1,1,1,1,1,1)

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Wikipedia, Alternating permutation

The version for patterns is A000670(n) - A344605(n).
Non-twin compositions are counted by A051049.
The complement is counted by A344604.
An unordered version is A344654.
The complement is ranked by A345167 \/ A007582.
These compositions are ranked by A345168 \ A007582.
Including twins gives A345192, complement A025047.
The version for factorizations is A347706, or A348380 with twins.
The non-anti-run case is A348382.
A001250 counts alternating permutations.
A011782 counts compositions, strict A032020.
A106356 counts compositions by number of maximal anti-runs.
A114901 counts compositions where each part is adjacent to an equal part.
A261983 counts non-anti-run compositions, complement A003242.
A325535 counts inseparable partitions, ranked by A335448.
A344614 counts compositions avoiding (1,2,3) and (3,2,1) adjacent.
A345165 = partitions with no alternating permutations, ranked by A345171.
A345170 = partitions with an alternating permutation, ranked by A345172.
Cf. A005649, A178470, A238279, A333755, A335126, A344653, A344740, A345166, A345169, A345173, A348379, A348381.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],MatchQ[#,{_,x_,y_,z_,_}/;x<=y<=z||x>=y>=z]&]],{n,0,15}]

For n > 0, a(n) = A345192(n) - 1 if n is even; otherwise A345192(n).

a(26) onwards from Andrew Howroyd, Jan 31 2024

A348609 Numbers with a separable factorization without an alternating permutation.

Original entry on oeis.org

216, 270, 324, 378, 432, 486, 540, 594, 640, 648, 702, 756, 768, 810, 864, 896, 918, 960, 972, 1024, 1026, 1080, 1134, 1152, 1188, 1242, 1280, 1296, 1344, 1350, 1404, 1408, 1458, 1500, 1512, 1536, 1566, 1620, 1664, 1674, 1728, 1750, 1782, 1792, 1836, 1890

Offset: 1

Gus Wiseman, Oct 30 2021

nonn

A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
A multiset is separable if it has a permutation that is an anti-run, meaning there are no adjacent equal parts. Alternatively, a multiset is separable if its greatest multiplicity is greater than the sum of the remaining multiplicities plus one.
A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,2,2,2,1) has no alternating permutations, even though it does have the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2). Alternating permutations of multisets are a generalization of alternating or up-down permutations of sets.
Note that 216 has separable prime factorization (2*2*2*3*3*3) with an alternating permutation, but the separable factorization (2*3*3*3*4) is has no alternating permutation. See also A345173.

			The terms and their prime factorizations begin:
  216 = 2*2*2*3*3*3
  270 = 2*3*3*3*5
  324 = 2*2*3*3*3*3
  378 = 2*3*3*3*7
  432 = 2*2*2*2*3*3*3
  486 = 2*3*3*3*3*3
  540 = 2*2*3*3*3*5
  594 = 2*3*3*3*11
  640 = 2*2*2*2*2*2*2*5
  648 = 2*2*2*3*3*3*3
  702 = 2*3*3*3*13
  756 = 2*2*3*3*3*7
  768 = 2*2*2*2*2*2*2*2*3
  810 = 2*3*3*3*3*5
  864 = 2*2*2*2*2*3*3*3

Partitions of this type are counted by A345166, ranked by A345173 (a superset).
Compositions of this type are counted by A345195, ranked by A345169.
A001055 counts factorizations, strict A045778, ordered A074206.
A001250 counts alternating permutations, complement A348615.
A025047 counts alternating compositions, complement A345192, ranked by A345167.
A335434 counts separable factorizations, with twins A348383, complement A333487.
A339846 counts even-length factorizations.
A339890 counts odd-length factorizations.
A345165 counts partitions w/o an alternating permutation, complement A345170.
A347438 counts factorizations with alternating product 1, additive A119620.
A348379 counts factorizations w/ an alternating permutation, complement A348380.
A348610 counts alternating ordered factorizations, complement A348613.
Cf. A001248, A003242, A038548, A325534, A336103, A344614, A345162, A347050, A347437, A347706, A348377, A348381, A348382.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
sepQ[m_]:=Select[Permutations[m],!MatchQ[#,{_,x_,x_,_}]&]!={};
wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]==Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
Select[Range[1000],Function[n,Select[facs[n],sepQ[#]&&Select[Permutations[#],wigQ]=={}&]!={}]]

cp's OEIS Frontend

A345171 Numbers whose multiset of prime factors has no alternating permutation.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A345163 Number of integer partitions of n with an alternating permutation covering an initial interval of positive integers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A344742 Numbers whose prime factors have a permutation with no consecutive monotone triple, i.e., no triple (..., x, y, z, ...) such that either x <= y <= z or x >= y >= z.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A345173 Numbers whose multiset of prime factors is separable but has no alternating permutation.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A345169 Numbers k such that the k-th composition in standard order is a non-alternating anti-run.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A345195 Number of non-alternating anti-run compositions of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A345162 Number of integer partitions of n with no alternating permutation covering an initial interval of positive integers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A349800 Number of integer compositions of n that are weakly alternating and have at least two adjacent equal parts.