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A350252
Number of non-alternating patterns of length n.
13
0, 0, 1, 7, 53, 439, 4121, 43675, 519249, 6867463, 100228877, 1602238783, 27866817297, 524175098299, 10606844137009, 229807953097903, 5308671596791901, 130261745042452855, 3383732450013895721, 92770140175473602755, 2677110186541556215233
OFFSET
0,4
COMMENTS
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)
LINKS
FORMULA
a(n) = A000670(n) - A345194(n).
EXAMPLE
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
MATHEMATICA
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}]
CROSSREFS
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.
Sequence in context: A081338 A367937 A142981 * A120272 A057180 A137612
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 13 2022
EXTENSIONS
Terms a(9) and beyond from Andrew Howroyd, Feb 04 2022
STATUS
approved