A344615 -id:A344615 - cp's OEIS frontend

A350138 Number of non-weakly alternating patterns of length n.

0, 0, 0, 2, 32, 338, 3560, 40058, 492664, 6647666, 98210192, 1581844994, 27642067000, 521491848218, 10572345303576, 229332715217954, 5301688511602448, 130152723055769810, 3381930236770946120, 92738693031618794378, 2676532576838728227352

Offset: 0

Gus Wiseman, Dec 24 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.
Conjecture: The directed cases, which count non-weakly up/down or non-weakly down/up patterns, are both equal to the strong case: A350252.

			The a(4) = 32 patterns:
  (1,1,2,3)  (2,1,1,2)  (3,1,1,2)  (4,1,2,3)
  (1,2,2,1)  (2,1,1,3)  (3,1,2,3)  (4,2,1,3)
  (1,2,3,1)  (2,1,2,3)  (3,1,2,4)  (4,3,1,2)
  (1,2,3,2)  (2,1,3,4)  (3,2,1,1)  (4,3,2,1)
  (1,2,3,3)  (2,3,2,1)  (3,2,1,2)
  (1,2,3,4)  (2,3,3,1)  (3,2,1,3)
  (1,2,4,3)  (2,3,4,1)  (3,2,1,4)
  (1,3,2,1)  (2,4,3,1)  (3,3,2,1)
  (1,3,3,2)             (3,4,2,1)
  (1,3,4,2)
  (1,4,3,2)

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

The unordered version is A274230, complement A052955.
The strong case of compositions is A345192, ranked by A345168.
The strict case is A348615, complement A001250.
For compositions we have A349053, complement A349052, ranked by A349057.
The complement is counted by A349058.
The version for partitions is A349061, complement A349060.
The version for permutations of prime indices: A349797, complement A349056.
The version for ordered factorizations is A350139, complement A349059.
The strong case is A350252, complement A345194. Also the directed case?
A003242 = Carlitz compositions, complement A261983, ranked by A333489.
A005649 = anti-run patterns, complement A069321.
A025047/A129852/A129853 = alternating compositions, ranked by A345167.
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, A096441, A336103, A344605, A344614, A344615, A344740, A348610, 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([0], vector(n,i,1) + sum(k=1, n, (vector(n,i,k^i) - 2*R(n, k))*sum(r=k, n, binomial(r, k)*(-1)^(r-k)) ) )} \\ Andrew Howroyd, Jan 13 2024

a(n) = A000670(n) - A349058(n).

a(9) onwards from Andrew Howroyd, Jan 13 2024

A350354 Number of up/down (or down/up) patterns of length n.

Original entry on oeis.org

1, 1, 1, 3, 11, 51, 281, 1809, 13293, 109899, 1009343, 10196895, 112375149, 1341625041, 17249416717, 237618939975, 3491542594727, 54510993341523, 901106621474801, 15723571927404189, 288804851413993941, 5569918636750820751, 112537773142244706427

Offset: 0

Gus Wiseman, Jan 16 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 patten is up/down if it is alternately strictly increasing and strictly decreasing, starting with an increase.
A pattern is up/down if it is alternately strictly increasing and strictly decreasing, starting with an increase. For example, the partition (3,2,2,2,1) has no up/down permutations, even though it does have the anti-run permutation (2,3,2,1,2).
Conjecture: Also the half the number of weakly up/down patterns of length n.
These are the values of the Euler zig-zag polynomials A205497 evaluated at x = 1/2 and normalized by 2^n. - Peter Luschny, Jun 03 2024

			The a(0) = 1 through a(4) = 11 patterns:
  ()  (1)  (1,2)  (1,2,1)  (1,2,1,2)
                  (1,3,2)  (1,2,1,3)
                  (2,3,1)  (1,3,1,2)
                           (1,3,2,3)
                           (1,3,2,4)
                           (1,4,2,3)
                           (2,3,1,2)
                           (2,3,1,3)
                           (2,3,1,4)
                           (2,4,1,3)
                           (3,4,1,2)

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

The version for permutations is A000111, undirected A001250.
For compositions we have A025048, down/up A025049, undirected A025047.
This is the up/down (or down/up) case of A345194.
A205497 are the Euler zig-zag polynomials.
A000670 counts patterns, ranked by A333217.
A005649 counts anti-run patterns.
A019536 counts necklace patterns.
A226316 counts patterns avoiding (1,2,3), weakly A052709.
A335515 counts patterns matching (1,2,3).
A349058 counts weakly alternating patterns.
A350252 counts non-alternating patterns.
Row sums of A079502.
Cf. A000041, A003242, A049774, A124754, A128761, A335457, A344604, A344605, A344615, A348610.

# Using the recurrence by Kyle Petersen from A205497.
G := proc(n) option remember; local F;
if n = 0 then 1/(1 - q*x) else F := G(n - 1);
simplify((p/(p - q))*(subs({p = q, q = p}, F) - subs(p = q, F))) fi end:
A350354 := n -> 2^n*subs({p = 1, q = 1, x = 1/2}, G(n)*(1 - x)^(n + 1)):
seq(A350354(n), n = 0..22);  # Peter Luschny, Jun 03 2024

allnorm[n_]:=If[n<=0,{{}},Function[s, Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
updoQ[y_]:=And@@Table[If[EvenQ[m],y[[m]]>y[[m+1]],y[[m]]

F(p,x) = {sum(k=0, p, (-1)^((k+1)\2)*binomial((p+k)\2, k)*x^k)}
R(n,k) = {Vec(if(k==1, 0, F(k-2,-x)/F(k-1,x)-1) + x + O(x*x^n))}
seq(n)= {concat([1], sum(k=1, n, R(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)) ))} \\ Andrew Howroyd, Feb 04 2022

a(n > 2) = A344605(n)/2.

a(n > 1) = A345194(n)/2.

Terms a(10) and beyond from Andrew Howroyd, Feb 04 2022

A350355 Numbers k such that the k-th composition in standard order is up/down.

Original entry on oeis.org

0, 1, 2, 4, 6, 8, 12, 13, 16, 20, 24, 25, 32, 40, 41, 48, 49, 50, 54, 64, 72, 80, 81, 82, 96, 97, 98, 102, 108, 109, 128, 144, 145, 160, 161, 162, 166, 192, 193, 194, 196, 198, 204, 205, 216, 217, 256, 272, 288, 289, 290, 320, 321, 322, 324, 326, 332, 333, 384

Offset: 1

Gus Wiseman, Jan 15 2022

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 composition is up/down if it is alternately strictly increasing and strictly decreasing, starting with an increase. For example, the partition (3,2,2,2,1) has no up/down permutations, even though it does have the anti-run permutation (2,3,2,1,2).

			The terms together with the corresponding compositions begin:
   0: ()
   1: (1)
   2: (2)
   4: (3)
   6: (1,2)
   8: (4)
  12: (1,3)
  13: (1,2,1)
  16: (5)
  20: (2,3)
  24: (1,4)
  25: (1,3,1)
  32: (6)
  40: (2,4)
  41: (2,3,1)
  48: (1,5)
  49: (1,4,1)
  50: (1,3,2)
  54: (1,2,1,2)

The case of permutations is counted by A000111.
These compositions are counted by A025048, down/up A025049.
The strict case is counted by A129838, undirected A349054.
The weak version is counted by A129852, down/up A129853.
The version for anti-runs is A333489, a superset, complement A348612.
This is the up/down case of A345167, counted by A025047.
Counting patterns of this type gives A350354.
The down/up version is A350356.
A001250 counts alternating permutations, complement A348615.
A003242 counts anti-run compositions.
A011782 counts compositions, unordered A000041.
A345192 counts non-alternating compositions, ranked by A345168.
A349052 counts weakly alternating compositions, complement A349053.
A349057 ranks non-weakly alternating compositions.
Statistics of standard compositions:
- Length is A000120.
- Sum is A070939.
- Heinz number is A333219.
- Number of maximal anti-runs is A333381.
- Number of distinct parts is A334028.
Classes of standard compositions:
- Partitions are A114994, strict A333256.
- Multisets are A225620, strict A333255.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Patterns are A333217.
Cf. A008965, A049774, A095684, A106356, A238279, A344604, A344614, A344615, A345169, A345170, A345172, A349051, A349799.

updoQ[y_]:=And@@Table[If[EvenQ[m],y[[m]]>y[[m+1]],y[[m]]

A345167 = A350355 \/ A350356.

A350356 Numbers k such that the k-th composition in standard order is down/up.

Original entry on oeis.org

0, 1, 2, 4, 5, 8, 9, 16, 17, 18, 22, 32, 33, 34, 38, 44, 45, 64, 65, 66, 68, 70, 76, 77, 88, 89, 128, 129, 130, 132, 134, 140, 141, 148, 152, 153, 176, 177, 178, 182, 256, 257, 258, 260, 262, 264, 268, 269, 276, 280, 281, 296, 297, 304, 305, 306, 310, 352, 353

Offset: 1

Gus Wiseman, Jan 15 2022

nonn

A composition is down/up if it is alternately strictly increasing and strictly decreasing, starting with a decrease. For example, the partition (3,2,2,2,1) has no down/up permutations, even though it does have the anti-run permutation (2,1,2,3,2).

			The terms together with the corresponding compositions begin:
   0: ()
   1: (1)
   2: (2)
   4: (3)
   5: (2,1)
   8: (4)
   9: (3,1)
  16: (5)
  17: (4,1)
  18: (3,2)
  22: (2,1,2)
  32: (6)
  33: (5,1)
  34: (4,2)
  38: (3,1,2)
  44: (2,1,3)
  45: (2,1,2,1)

Gus Wiseman, Statistics, classes, and transformations of standard compositions
Wikipedia, Alternating permutation

The case of permutations is counted by A000111.
These compositions are counted by A025049, up/down A025048.
The strict case is counted by A129838, undirected A349054.
The weak version is counted by A129853, up/down A129852.
The version for anti-runs is A333489, a superset, complement A348612.
This is the down/up case of A345167, counted by A025047.
Counting patterns of this type gives A350354.
The up/down version is A350355.
A001250 counts alternating permutations, complement A348615.
A003242 counts anti-run compositions.
A011782 counts compositions, unordered A000041.
A345192 counts non-alternating compositions, ranked by A345168.
A349052 counts weakly alternating compositions, complement A349053.
A349057 ranks non-weakly alternating compositions.
Statistics of standard compositions:
- Length is A000120.
- Sum is A070939.
- Heinz number is A333219.
- Number of maximal anti-runs is A333381.
- Number of distinct parts is A334028.
Classes of standard compositions:
- Partitions are A114994, strict A333256.
- Multisets are A225620, strict A333255.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Patterns are A333217.
Cf. A008965, A049774, A095684, A106356, A238279, A344604, A344614, A344615, A345169, A345170, A345172, A349051, A349799.

doupQ[y_]:=And@@Table[If[EvenQ[m],y[[m]]y[[m+1]]],{m,1,Length[y]-1}];
stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],doupQ[stc[#]]&]

A345167 = A350355 \/ A350356.

A350140 Nonsquarefree numbers whose prime signature has at least one odd part other the first or last.

Original entry on oeis.org

60, 84, 120, 132, 140, 150, 156, 168, 204, 220, 228, 240, 260, 264, 270, 276, 280, 294, 300, 308, 312, 315, 336, 340, 348, 364, 372, 378, 380, 408, 420, 440, 444, 456, 460, 476, 480, 490, 492, 495, 516, 520, 528, 532, 540, 552, 560, 564, 572, 580, 585, 588

Offset: 1

Gus Wiseman, Dec 25 2021

nonn

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization.

Also Heinz numbers of non-weakly alternating non-strict integer partitions, where we define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either. These partitions are counted by A349796. This sequence involves the somewhat degenerate case where no strict increases are allowed.

			The terms together with their Heinz partitions begin (A-E = 10-14):
     60: (3211)      276: (9211)      420: (43211)
     84: (4211)      280: (43111)     440: (53111)
    120: (32111)     294: (4421)      444: (C211)
    132: (5211)      300: (33211)     456: (82111)
    140: (4311)      308: (5411)      460: (9311)
    150: (3321)      312: (62111)     476: (7411)
    156: (6211)      315: (4322)      480: (3211111)
    168: (42111)     336: (421111)    490: (4431)
    204: (7211)      340: (7311)      492: (D211)
    220: (5311)      348: (A211)      495: (5322)
    228: (8211)      364: (6411)      516: (E211)
    240: (321111)    372: (B211)      520: (63111)
    260: (6311)      378: (42221)     528: (521111)
    264: (52111)     380: (8311)      532: (8411)
    270: (32221)     408: (72111)     540: (322211)

Including all nonsquarefree numbers gives A013929, complement A005117.
Subsets include A088860 and A110286.
Signatures of this type are counted by A274230, complement A027383.
The strict instead of non-strict version is A336568, counted by A347548.
A version for compositions allowing strict is A349057, counted by A349053.
Allowing strict partitions gives A349794, counted by A349061.
These partitions are counted by A349796.
The complement in nonsquarefree partitions is A350137, counted by A349795.
A000041 = integer partitions, strict A000009.
A001250 = alternating permutations, ranked by A349051, complement A348615.
A003242 = Carlitz (anti-run) compositions.
A025047/A025048/A025049 = alternating compositions, ranked by A345167.
A056239 adds up prime indices, row sums of A112798, row lengths A001222.
A096441 = weakly alternating 0-appended partitions.
A124010 = prime signature, sorted A118914.
A345164 = alternating permutations of prime indices, complement A350251.
A345170 = partitions w/ an alternating permutation, ranked by A345172.
A349052/A129852/A129853 = weakly alternating compositions.
A349056 = weakly alternating permutations of prime indices.
A349058 = weakly alternating patterns, complement A350138.
A349060 = weakly alternating partitions, strong A349801.
A349798 = weakly but not strongly alternating perms of prime indices.
Cf. A000111, A047967, A333213, A335448, A344615, A344653, A345173, A349054, A349059, A349797, A349799.

Select[Range[300],!SquareFreeQ[#]&&PrimeNu[#]>1&& !And@@EvenQ/@Take[Last/@FactorInteger[#],{2,-2}]&]

Complement of A005117 in A349794.

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A350138 Number of non-weakly alternating patterns of length n.

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A350354 Number of up/down (or down/up) patterns of length n.

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A350355 Numbers k such that the k-th composition in standard order is up/down.

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A350356 Numbers k such that the k-th composition in standard order is down/up.

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A350140 Nonsquarefree numbers whose prime signature has at least one odd part other the first or last.

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