A335466 - OEIS

%I #13 Jun 30 2020 09:54:17

%S 13,25,27,29,45,49,51,53,54,55,57,59,61,77,82,89,91,93,97,99,101,102,

%T 103,105,107,108,109,110,111,113,115,117,118,119,121,123,125,141,153,

%U 155,157,162,165,166,173,177,178,179,181,182,183,185,187,189,193,195

%N Numbers k such that the k-th composition in standard order (A066099) matches (1,2,1).

%C A composition of n is a finite sequence of positive integers summing to n. 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.

%C 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 S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation_pattern">Permutation pattern</a>

%H Gus Wiseman, <a href="https://oeis.org/A102726/a102726.txt">Sequences counting and ranking compositions by the patterns they match or avoid.</a>

%H Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vTCPiJVFUXN8IqfLlCXkgP15yrGWeRhFS4ozST5oA4Bl2PYS-XTA3sGsAEXvwW-B0ealpD8qnoxFqN3/pub">Statistics, classes, and transformations of standard compositions</a>

%e The sequence of terms together with the corresponding compositions begins:

%e   13: (1,2,1)

%e   25: (1,3,1)

%e   27: (1,2,1,1)

%e   29: (1,1,2,1)

%e   45: (2,1,2,1)

%e   49: (1,4,1)

%e   51: (1,3,1,1)

%e   53: (1,2,2,1)

%e   54: (1,2,1,2)

%e   55: (1,2,1,1,1)

%e   57: (1,1,3,1)

%e   59: (1,1,2,1,1)

%e   61: (1,1,1,2,1)

%e   77: (3,1,2,1)

%e   82: (2,3,2)

%t stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]];

%t Select[Range[0,100],MatchQ[stc[#],{___,x_,___,y_,___,x_,___}/;x<y]&]

%Y The complement A335467 is the avoiding version.

%Y The (2,1,2)-matching version is A335468.

%Y These compositions are counted by A335470.

%Y Constant patterns are counted by A000005 and ranked by A272919.

%Y Permutations are counted by A000142 and ranked by A333218.

%Y Patterns are counted by A000670 and ranked by A333217.

%Y Non-unimodal compositions are counted by A115981 and ranked by A335373.

%Y Combinatory separations are counted by A269134 and ranked by A334030.

%Y Patterns matched by standard compositions are counted by A335454.

%Y Minimal patterns avoided by a standard composition are counted by A335465.

%Y Cf. A034691, A056986, A108917, A114994, A238279, A333224, A333257, A335446, A335456, A335458, A335509.

%K nonn

%O 1,1

%A _Gus Wiseman_, Jun 15 2020