A001250 -id:A001250 - cp's OEIS frontend

A010026 Triangle read by rows: number of permutations of 1..n by length of longest run.

2, 2, 4, 2, 12, 10, 2, 16, 70, 32, 2, 20, 134, 442, 122, 2, 24, 198, 1164, 3108, 544, 2, 28, 274, 2048, 10982, 24216, 2770, 2, 32, 362, 3204, 22468, 112354, 208586, 15872, 2, 36, 462, 4720, 39420, 264538, 1245676, 1972904, 101042

Offset: 2

N. J. A. Sloane

			Triangle begins:
  2,
  2,  4,
  2, 12,  10,
  2, 16,  70,   32,
  2, 20, 134,  442,   122,
  2, 24, 198, 1164,  3108,    544,
  2, 28, 274, 2048, 10982,  24216,   2770,
  2, 32, 362, 3204, 22468, 112354, 208586, 15872, ...
The row "2, 12, 10" for example means that there are two permutations of [1..4] in which the longest run up or down has length 4, 12 in which the longest run has length 3, and 10 in which the longest run has length 2.
The following table, computed by _Sean A. Irvine_, May 02 2012, gives an extended version of the triangle, oriented the right way round (cf. A211318), and corrects errors in David Kendall and Barton:
n l=0, l=1, l=2, l=3, etc.
----------------------------
1 [0, 1]
2 [0, 0, 2]
3 [0, 0, 4, 2]
4 [0, 0, 10, 12, 2]
5 [0, 0, 32, 70, 16, 2]
6 [0, 0, 122, 442, 134, 20, 2]
7 [0, 0, 544, 3108, 1164, 198, 24, 2]
8 [0, 0, 2770, 24216, 10982, 2048, 274, 28, 2]A049293
9 [0, 0, 15872, 208586, 112354, 22468, 3204, 362, 32, 2]
10 [0, 0, 101042, 1972904, 1245676, 264538, 39420, 4720, 462, 36, 2]
11 [0, 0, 707584, 20373338, 14909340, 3340962, 514296, 64020, 6644, 574, 40, 2]
12 [0, 0, 5405530, 228346522, 191916532, 45173518, 7137818, 913440, 98472, 9024, 698, 44, 2]
13 [0, 0, 44736512, 2763212980, 2646100822, 652209564, 105318770, 13760472, 1523808, 145080, 11908, 834, 48, 2]
14 [0, 0, 398721962, 35926266244, 38932850396, 10024669626, 1649355338, 219040274, 24744720, 2419872, 206388, 15344, 982, 52, 2]
15 [0, 0, 3807514624, 499676669254, 609137502242, 163546399460, 27356466626, 3681354658, 422335056, 42129360, 3690960, 285180, 19380, 1142, 56, 2]

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 262. (Probably contains errors for n >= 13.)

Alois P. Heinz, Rows n = 2..70, flattened

Cf. A211318. Diagonals give: A001250, A001251, A001252, A001253, A230129, A230130, A230131, A230132, A230133.

(* This program is unsuited for a large number of terms *) f[p_List] := Max[Length /@ Split[Differences[p], #1*#2 > 0 &]] + 1; row[n_] := Sort[Tally[f /@ Permutations[Range[n]]], First[#1] > First[#2] &][[All, 2]]; Table[rn = row[n]; Print["n = ", n, " ", rn]; rn, {n, 2, 10}] // Flatten (* Jean-François Alcover, Mar 12 2014 *)
T[n_, length_] := Module[{g, b},
g[u_, o_, t_] := g[u, o, t] = If[u+o == 0, 1, Sum[g[o + j - 1, u - j, 2], {j, 1, u}] + If[tJean-François Alcover, Aug 18 2018, after Alois P. Heinz *)

Edited by N. J. A. Sloane, May 02 2012

A211318 Triangle read by rows: number of permutations of 1..n by length l of longest run (n >= 1, 1 <= l <= n).

Original entry on oeis.org

1, 0, 2, 0, 4, 2, 0, 10, 12, 2, 0, 32, 70, 16, 2, 0, 122, 442, 134, 20, 2, 0, 544, 3108, 1164, 198, 24, 2, 0, 2770, 24216, 10982, 2048, 274, 28, 2, 0, 15872, 208586, 112354, 22468, 3204, 362, 32, 2, 0, 101042, 1972904, 1245676, 264538, 39420, 4720, 462, 36, 2, 0

Offset: 1

N. J. A. Sloane, May 02 2012, based on computations by Sean A. Irvine

			Triangle begins:
n l=1, l=2, l=3, etc...
1 [1]
2 [0, 2]
3 [0, 4, 2]
4 [0, 10, 12, 2]
5 [0, 32, 70, 16, 2]
6 [0, 122, 442, 134, 20, 2]
7 [0, 544, 3108, 1164, 198, 24, 2]
8 [0, 2770, 24216, 10982, 2048, 274, 28, 2]
9 [0, 15872, 208586, 112354, 22468, 3204, 362, 32, 2]
10 [0, 101042, 1972904, 1245676, 264538, 39420, 4720, 462, 36, 2]
11 [0, 707584, 20373338, 14909340, 3340962, 514296, 64020, 6644, 574, 40, 2]
12 [0, 5405530, 228346522, 191916532, 45173518, 7137818, 913440, 98472, 9024, 698, 44, 2]
13 [0, 44736512, 2763212980, 2646100822, 652209564, 105318770, 13760472, 1523808, 145080, 11908, 834, 48, 2]
14 [0, 398721962, 35926266244, 38932850396, 10024669626, 1649355338, 219040274, 24744720, 2419872, 206388, 15344, 982, 52, 2]
15 [0, 3807514624, 499676669254, 609137502242, 163546399460, 27356466626, 3681354658, 422335056, 42129360, 3690960, 285180, 19380, 1142, 56, 2],
...
More rows than usual are shown, in order to correct errors in David, Kendall and Barton.

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 262. (Contains errors for n >= 13.)
Sean A. Irvine, Posting to Sequence Fans Mailing List, May 02 2012

Alois P. Heinz, Rows n = 1..70, flattened (rows n = 1..16 from Wouter Meeussen)
Max A. Alekseyev, On the number of permutations with bounded run lengths, arXiv:1205.4581 [math.CO], 2012-2013. [From _N. J. A. Sloane_, Oct 23 2012]

Mirror image of triangle in A010026.

Columns l=2-10 give: A001250, A001251, A001252, A001253, A230129, A230130, A230131, A230132, A230133.

<Wouter Meeussen, May 09 2012 *)
T[n_, length_] := Module[{g, b},
g[u_, o_, t_] := g[u, o, t] = If[u+o == 0, 1, Sum[g[o + j - 1, u - j, 2], {j, 1, u}] + If[t1, 1] = 0;
Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Aug 18 2018, after Alois P. Heinz *)

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

A349054 Number of alternating strict compositions of n. Number of alternating (up/down or down/up) permutations of strict integer partitions of n.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 9, 11, 15, 21, 35, 41, 59, 75, 103, 155, 193, 255, 339, 443, 569, 841, 1019, 1365, 1743, 2295, 2879, 3785, 5151, 6417, 8301, 10625, 13567, 17229, 21937, 27509, 37145, 45425, 58345, 73071, 93409, 115797, 147391, 182151, 229553, 297061, 365625

Offset: 0

Gus Wiseman, Dec 21 2021

nonn

A strict composition of n is a finite sequence of distinct positive integers summing to n.
A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either.
The case starting with an increase (or decrease, it doesn't matter in the enumeration) is counted by A129838.

			The a(1) = 1 through a(7) = 11 compositions:
  (1)  (2)  (3)    (4)    (5)    (6)      (7)
            (1,2)  (1,3)  (1,4)  (1,5)    (1,6)
            (2,1)  (3,1)  (2,3)  (2,4)    (2,5)
                          (3,2)  (4,2)    (3,4)
                          (4,1)  (5,1)    (4,3)
                                 (1,3,2)  (5,2)
                                 (2,1,3)  (6,1)
                                 (2,3,1)  (1,4,2)
                                 (3,1,2)  (2,1,4)
                                          (2,4,1)
                                          (4,1,2)

Wikipedia, Alternating permutation

Ranking sequences are put in parentheses below.
This is the strict case of A025047/A025048/A025049 (A345167).
This is the alternating case of A032020 (A233564).
The unordered case (partitions) is A065033.
The directed case is A129838.
A001250 = alternating permutations (A349051), complement A348615 (A350250).
A003242 = Carlitz (anti-run) compositions, complement A261983.
A011782 = compositions, unordered A000041.
A345165 = partitions without an alternating permutation (A345171).
A345170 = partitions with an alternating permutation (A345172).
A345192 = non-alternating compositions (A345168).
A345195 = non-alternating anti-run compositions (A345169).
A349800 = weakly but not strongly alternating compositions (A349799).
A349052 = weakly alternating compositions, complement A349053 (A349057).
Cf. A000111, A008965, A015723, A114901, A128761, A129852, A129853, A218074, A333213, A344614, A345164, A345194, A349060, A349795.

g:= proc(u, o) option remember;
      `if`(u+o=0, 1, add(g(o-1+j, u-j), j=1..u))
    end:
b:= proc(n, k) option remember; `if`(k<0 or n<0, 0,
      `if`(k=0, `if`(n=0, 2, 0), b(n-k, k)+b(n-k, k-1)))
    end:
a:= n-> add(b(n, k)*g(k, 0), k=0..floor((sqrt(8*n+1)-1)/2))-1:
seq(a(n), n=0..46);  # Alois P. Heinz, Dec 22 2021

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

a(n) = 2 * A129838(n) - 1.

G.f.: Sum_{n>0} A001250(n)*x^(n*(n+1)/2)/Product_{k=1..n}(1-x^k).

A349058 Number of weakly alternating patterns of length n.

Original entry on oeis.org

1, 1, 3, 11, 43, 203, 1123, 7235, 53171, 439595, 4037371, 40787579, 449500595, 5366500163, 68997666867, 950475759899, 13966170378907, 218043973366091, 3604426485899203, 62894287709616755, 1155219405655975763, 22279674547003283003, 450151092568978825707

Offset: 0

Gus Wiseman, Dec 04 2021

nonn

We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217.

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

			The a(1) = 1 through a(3) = 11 patterns:
  (1)  (1,1)  (1,1,1)
       (1,2)  (1,1,2)
       (2,1)  (1,2,1)
              (1,2,2)
              (1,3,2)
              (2,1,1)
              (2,1,2)
              (2,1,3)
              (2,2,1)
              (2,3,1)
              (3,1,2)

Andrew Howroyd, Table of n, a(n) for n = 0..200

The strict case is A001250, complement A348615.
The strong case of compositions is A025047, ranked by A345167.
The unordered version is A052955.
The strong case is A345194, with twins A344605. Also the directed case.
The version for compositions is A349052, complement A349053.
The version for permutations of prime indices: A349056, complement A349797.
The version for compositions is ranked by A349057.
The version for ordered factorizations is A349059, strong A348610.
The version for partitions is A349060, complement A349061.
A003242 counts Carlitz (anti-run) compositions.
A005649 counts anti-run patterns.
A344604 counts alternating compositions with twins.
A345163 counts normal partitions with an alternating permutation.
A345170 counts partitions w/ an alternating permutation, complement A345165.
A345192 counts non-alternating compositions, ranked by A345168.
A349055 counts multisets w/ an alternating permutation, complement A349050.
Cf. A049774, A096441, A129852, A129853, A336103, A344614, A344615, A344740, A345164, A348613, A349794.

allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s, y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+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/@allnorm[n],whkQ[#]||whkQ[-#]&]],{n,0,6}]

R(n,k)={my(v=vector(k,i,1), u=vector(n)); for(r=1, n, if(r%2==0, my(s=v[k]); forstep(i=k, 2, -1, v[i] = s - v[i-1]); v[1] = s); for(i=2, k, v[i] += v[i-1]); u[r]=v[k]); u}
seq(n)= {concat([1], -vector(n,i,1) + 2*sum(k=1, n, R(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)) ) )} \\ Andrew Howroyd, Jan 13 2024

a(9)-a(18) from Alois P. Heinz, Dec 10 2021

a(19) onwards from Andrew Howroyd, Jan 13 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

A349797 Number of non-weakly alternating permutations of the multiset of prime factors of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 4, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 6, 0, 0, 0

Offset: 1

Gus Wiseman, Dec 24 2021

nonn

First differs from 2 * A326291 at a(90) = 4, A326291(90) = 3.
The first odd term is a(144) = 7, whose non-weakly alternating permutations are shown in the example below.
We define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either. Then a sequence is alternating in the sense of A025047 iff it is a weakly alternating anti-run.
For n > 1, the multiset of prime factors of n is row n of A027746. The prime indices A112798 can also be used.

			The following are the weakly alternating permutations for selected n.
n = 30    60     72      120     144      180
   ---------------------------------------------
    235   2235   22332   22235   222332   22353
    532   2352   23223   22352   223223   23235
          2532   23322   22532   223322   23325
          3225   32232   23225   232232   23523
          5223           23522   233222   23532
          5322           25223   322223   25323
                         25322   322322   32235
                         32252            32253
                         52232            32352
                         53222            32532
                                          33225
                                          35223
                                          35322
                                          52233
                                          52332
                                          53223
                                          53232

Counting all permutations of prime factors gives A008480.
Compositions not of this type are counted by A349052/A129852/A129853.
Compositions of this type are counted by A349053, ranked by A349057.
The complement is counted by A349056.
Partitions of this type are counted by A349061, complement A349060.
The version counting patterns is A350138, complement A349058.
The version counting ordered factorizations is A350139, complement A349059.
The strong case is counted by A350251, complement A345164.
Positions of nonzero terms are A350353.
A001250 counts alternating permutations, complement A348615.
A025047 = alternating compositions, ranked by A345167, complement A345192.
A056239 adds up prime indices, row sums of A112798, row lengths A001222.
A071321 gives the alternating sum of prime factors, reverse A071322.
A335452 counts anti-run permutations of prime factors, complement A336107.
A345165 counts partitions w/o an alternating permutation, ranked by A345171.
A345170 counts partitions w/ an alternating permutation, ranked by A345172.
A348379 counts factorizations with an alternating permutation.
Cf. A003242, A335433, A335448, A344614, A344652, A344653, A345173, A348613, A349798, A350252, A349800.

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[Permutations[Flatten[ConstantArray@@@ FactorInteger[n]]], !whkQ[#]&&!whkQ[-#]&]],{n,100}]

a(n) = A008480(n) - A349056(n).

A350252 Number of non-alternating patterns of length n.

Original entry on oeis.org

0, 0, 1, 7, 53, 439, 4121, 43675, 519249, 6867463, 100228877, 1602238783, 27866817297, 524175098299, 10606844137009, 229807953097903, 5308671596791901, 130261745042452855, 3383732450013895721, 92770140175473602755, 2677110186541556215233

Offset: 0

Gus Wiseman, Jan 13 2022

nonn

We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217.
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 alternating pattern is necessarily an anti-run (A005649).
Conjecture: Also the number of non-weakly up/down (or down/up) patterns of length n. For example:
- The a(3) = 7 non-weakly up/down patterns:
  (121), (122), (123), (132), (221), (231), (321)
- The a(3) = 7 non-weakly down/up patterns:
  (112), (123), (211), (212), (213), (312), (321)
- The a(3) = 7 non-alternating patterns (see example for more):
  (111), (112), (122), (123), (211), (221), (321)

			The a(2) = 1 and a(3) = 7 non-alternating patterns:
  (1,1)  (1,1,1)
         (1,1,2)
         (1,2,2)
         (1,2,3)
         (2,1,1)
         (2,2,1)
         (3,2,1)
The a(4) = 53 non-alternating patterns:
  2112   3124   4123   1112   2134   1234   3112   2113   1123
  2211   3214   4213   1211   2314   1243   3123   2123   1213
  2212   3412   4312   1212   2341   1324   3211   2213   1223
         3421   4321   1221   2413   1342   3212   2311   1231
                       1222   2431   1423   3213   2312   1232
                                     1432   3312   2313   1233
                                            3321   2321   1312
                                                   2331   1321
                                                          1322
                                                          1323
                                                          1332

Andrew Howroyd, Table of n, a(n) for n = 0..200

The unordered version is A122746.
The version for compositions is A345192, ranked by A345168, weak A349053.
The complement is counted by A345194, weak A349058.
The version for factorizations is A348613, complement A348610, weak A350139.
The strict case (permutations) is A348615, complement A001250.
The weak version for partitions is A349061, complement A349060.
The weak version for perms of prime indices is A349797, complement A349056.
The weak version is A350138.
The version for perms of prime indices is A350251, complement A345164.
A000670 = patterns (ranked by A333217).
A003242 = anti-run compositions, complement A261983, ranked by A333489.
A005649 = anti-run patterns, complement A069321.
A019536 = necklace patterns.
A025047/A129852/A129853 = alternating compositions, ranked by A345167.
A226316 = patterns avoiding (1,2,3), weakly A052709, complement A335515.
A345163 = normal partitions w/ alternating permutation, complement A345162.
A345170 = partitions w/ alternating permutation, complement A345165.
A349055 = normal multisets w/ alternating permutation, complement A349050.
Cf. A049774, A052955, A096441, A128761, A274230, A335456, A335457, A336103, A344605, A344614.

allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]==Length[y]&& Length[Split[Sign[Differences[y]]]]==Length[y]-1];
Table[Length[Select[Join@@Permutations/@allnorm[n],!wigQ[#]&]],{n,0,6}]

a(n) = A000670(n) - A345194(n).

Terms a(9) and beyond from Andrew Howroyd, Feb 04 2022

A001251 Number of permutations of order n with the length of longest run equal to 3.

Original entry on oeis.org

0, 0, 2, 12, 70, 442, 3108, 24216, 208586, 1972904, 20373338, 228346522, 2763212980, 35926266244, 499676669254, 7405014187564, 116511984902094, 1940073930857802, 34087525861589564, 630296344519286304, 12235215845125112122, 248789737587365945992

Offset: 1

N. J. A. Sloane

nonn

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 262. (Terms for n>=13 are incorrect.)
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 1..450 (first 100 terms from Max Alekseyev)
Max A. Alekseyev, On the number of permutations with bounded runs length, arXiv preprint arXiv:1205.4581 [math.CO], 2012-2013. - From _N. J. A. Sloane_, Oct 23 2012
Richard Ehrenborg and JiYoon Jung, Descent pattern avoidance, arXiv preprint:1312.2027 [math.CO], 6 Dec 2013.

Cf. A001250, A001252, A001253, A010026, A211318.

length = 3;
g[u_, o_, t_] := g[u, o, t] = If[u+o == 0, 1, Sum[g[o + j - 1, u - j, 2], {j, 1, u}] + If[tJean-François Alcover, Aug 18 2018, after Alois P. Heinz *)

a(n) ~ c * d^n * n!, where d = 0.92403585760753647721113386869798700855648617941... is the root of the equation 8 - 2*sin(sqrt(phi)/d) * (2*sqrt(5*(phi-1)) * cosh(sqrt(phi-1)/d) + 2*sinh(sqrt(phi-1)/d)) + 2*cos(sqrt(phi)/d) * (6*cosh(sqrt(phi-1)/d) + 2*sqrt(5*phi) * sinh(sqrt(phi-1)/d)) = 0, phi = A001622 = (1+sqrt(5))/2 is the golden ratio and c = 1.259371257828351725264434486385284120241474052544197367866029465830756911... - Vaclav Kotesovec, Sep 06 2014, updated Aug 18 2018

Corrected and extended by Max Alekseyev at the suggestion of Sean A. Irvine, May 04 2012

cp's OEIS Frontend

A010026 Triangle read by rows: number of permutations of 1..n by length of longest run.

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A211318 Triangle read by rows: number of permutations of 1..n by length l of longest run (n >= 1, 1 <= l <= n).

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

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A349054 Number of alternating strict compositions of n. Number of alternating (up/down or down/up) permutations of strict integer partitions of n.

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A349058 Number of weakly alternating patterns of length n.

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

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

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