cp's OEIS Frontend

A350252 Number of non-alternating patterns of length n.

%I A350252 #10 Feb 04 2022 18:10:52
%S A350252 0,0,1,7,53,439,4121,43675,519249,6867463,100228877,1602238783,
%T A350252 27866817297,524175098299,10606844137009,229807953097903,
%U A350252 5308671596791901,130261745042452855,3383732450013895721,92770140175473602755,2677110186541556215233
%N A350252 Number of non-alternating patterns of length n.
%C A350252 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.
%C A350252 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).
%C A350252 Conjecture: Also the number of non-weakly up/down (or down/up) patterns of length n. For example:
%C A350252 - The a(3) = 7 non-weakly up/down patterns:
%C A350252   (121), (122), (123), (132), (221), (231), (321)
%C A350252 - The a(3) = 7 non-weakly down/up patterns:
%C A350252   (112), (123), (211), (212), (213), (312), (321)
%C A350252 - The a(3) = 7 non-alternating patterns (see example for more):
%C A350252   (111), (112), (122), (123), (211), (221), (321)
%H A350252 Andrew Howroyd, <a href="/A350252/b350252.txt">Table of n, a(n) for n = 0..200</a>
%F A350252 a(n) = A000670(n) - A345194(n).
%e A350252 The a(2) = 1 and a(3) = 7 non-alternating patterns:
%e A350252   (1,1)  (1,1,1)
%e A350252          (1,1,2)
%e A350252          (1,2,2)
%e A350252          (1,2,3)
%e A350252          (2,1,1)
%e A350252          (2,2,1)
%e A350252          (3,2,1)
%e A350252 The a(4) = 53 non-alternating patterns:
%e A350252   2112   3124   4123   1112   2134   1234   3112   2113   1123
%e A350252   2211   3214   4213   1211   2314   1243   3123   2123   1213
%e A350252   2212   3412   4312   1212   2341   1324   3211   2213   1223
%e A350252          3421   4321   1221   2413   1342   3212   2311   1231
%e A350252                        1222   2431   1423   3213   2312   1232
%e A350252                                      1432   3312   2313   1233
%e A350252                                             3321   2321   1312
%e A350252                                                    2331   1321
%e A350252                                                           1322
%e A350252                                                           1323
%e A350252                                                           1332
%t A350252 allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
%t A350252 wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]==Length[y]&& Length[Split[Sign[Differences[y]]]]==Length[y]-1];
%t A350252 Table[Length[Select[Join@@Permutations/@allnorm[n],!wigQ[#]&]],{n,0,6}]
%Y A350252 The unordered version is A122746.
%Y A350252 The version for compositions is A345192, ranked by A345168, weak A349053.
%Y A350252 The complement is counted by A345194, weak A349058.
%Y A350252 The version for factorizations is A348613, complement A348610, weak A350139.
%Y A350252 The strict case (permutations) is A348615, complement A001250.
%Y A350252 The weak version for partitions is A349061, complement A349060.
%Y A350252 The weak version for perms of prime indices is A349797, complement A349056.
%Y A350252 The weak version is A350138.
%Y A350252 The version for perms of prime indices is A350251, complement A345164.
%Y A350252 A000670 = patterns (ranked by A333217).
%Y A350252 A003242 = anti-run compositions, complement A261983, ranked by A333489.
%Y A350252 A005649 = anti-run patterns, complement A069321.
%Y A350252 A019536 = necklace patterns.
%Y A350252 A025047/A129852/A129853 = alternating compositions, ranked by A345167.
%Y A350252 A226316 = patterns avoiding (1,2,3), weakly A052709, complement A335515.
%Y A350252 A345163 = normal partitions w/ alternating permutation, complement A345162.
%Y A350252 A345170 = partitions w/ alternating permutation, complement A345165.
%Y A350252 A349055 = normal multisets w/ alternating permutation, complement A349050.
%Y A350252 Cf. A049774, A052955, A096441, A128761, A274230, A335456, A335457, A336103, A344605, A344614.
%K A350252 nonn
%O A350252 0,4
%A A350252 _Gus Wiseman_, Jan 13 2022
%E A350252 Terms a(9) and beyond from _Andrew Howroyd_, Feb 04 2022