cp's OEIS Frontend

A350250 Numbers k such that the k-th composition in standard order is a non-alternating permutation of an initial interval of positive integers.

%I A350250 #9 Jan 18 2022 05:56:01
%S A350250 37,52,549,550,556,564,581,600,616,649,657,712,786,802,836,840,16933,
%T A350250 16934,16937,16940,16946,16948,16965,16977,16984,16994,17000,17033,
%U A350250 17041,17092,17096,17170,17186,17220,17224,17445,17446,17452,17460,17541,17569,17584
%N A350250 Numbers k such that the k-th composition in standard order is a non-alternating permutation of an initial interval of positive integers.
%C A350250 A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either.
%C A350250 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.
%e A350250 The terms and corresponding permutations begin:
%e A350250      37: (3,2,1)
%e A350250      52: (1,2,3)
%e A350250     549: (4,3,2,1)
%e A350250     550: (4,3,1,2)
%e A350250     556: (4,2,1,3)
%e A350250     564: (4,1,2,3)
%e A350250     581: (3,4,2,1)
%e A350250     600: (3,2,1,4)
%e A350250     616: (3,1,2,4)
%e A350250     649: (2,4,3,1)
%e A350250     657: (2,3,4,1)
%e A350250     712: (2,1,3,4)
%e A350250     786: (1,4,3,2)
%e A350250     802: (1,3,4,2)
%e A350250     836: (1,2,4,3)
%e A350250     840: (1,2,3,4)
%e A350250   16933: (5,4,3,2,1)
%t A350250 stc[n_]:=Differences[Prepend[Join@@Position[ Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A350250 wigQ[y_]:=Or[Length[y]==0, Length[Split[y]]==Length[y] &&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
%t A350250 Select[Range[0,1000],(Sort[stc[#]]==Range[Length[stc[#]]]&&!wigQ[stc[#]])&]
%Y A350250 This is the non-alternating case of A333218.
%Y A350250 This is the restriction of A345168 to permutations, complement A345167.
%Y A350250 These partitions are counted by A348615, complement A001250.
%Y A350250 A003242 counts anti-run compositions, patterns A005649.
%Y A350250 A025047 counts alternating compositions, directed A025048/A025049.
%Y A350250 A345192 counts non-alternating compositions.
%Y A350250 A345194 counts alternating patterns, complement A350252.
%Y A350250 Statistics of standard compositions:
%Y A350250 - Length is A000120.
%Y A350250 - Sum is A070939.
%Y A350250 - Heinz number is A333219.
%Y A350250 - Number of maximal anti-runs is A333381.
%Y A350250 - Number of distinct parts is A334028.
%Y A350250 Classes of standard compositions:
%Y A350250 - Weakly decreasing compositions (partitions) are A114994, strict A333256.
%Y A350250 - Weakly increasing compositions (multisets) are A225620, strict A333255.
%Y A350250 - Strict compositions are A233564.
%Y A350250 - Constant compositions are A272919.
%Y A350250 - Anti-run compositions are A333489, complement A348612.
%Y A350250 Cf. A008965, A059893, A164894, A246534, A333217, A344605, A345162, A350251, A345163, A345171, A345172, A348613.
%K A350250 nonn
%O A350250 1,1
%A A350250 _Gus Wiseman_, Jan 13 2022