cp's OEIS Frontend

A335474 Number of nonempty normal patterns contiguously matched by the n-th composition in standard order.

%I A335474 #10 Jun 30 2020 01:55:23
%S A335474 0,1,1,2,1,2,2,3,1,2,2,4,2,4,4,4,1,2,2,4,2,4,4,6,2,4,4,7,4,7,6,5,1,2,
%T A335474 2,4,2,3,4,6,2,4,3,6,4,6,7,8,2,4,4,7,3,7,6,10,4,7,6,10,6,10,8,6,1,2,2,
%U A335474 4,2,3,4,6,2,4,4,6,4,6,7,8,2,4,4,7,4,6
%N A335474 Number of nonempty normal patterns contiguously matched by the n-th composition in standard order.
%C A335474 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 A335474 We define a (normal) 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 A335474 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation_pattern">Permutation pattern</a>
%H A335474 Gus Wiseman, <a href="https://oeis.org/A102726/a102726.txt">Sequences counting and ranking compositions by the patterns they match or avoid.</a>
%H A335474 Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vTCPiJVFUXN8IqfLlCXkgP15yrGWeRhFS4ozST5oA4Bl2PYS-XTA3sGsAEXvwW-B0ealpD8qnoxFqN3/pub">Statistics, classes, and transformations of standard compositions</a>
%F A335474 a(n) = A335458(n) - 1.
%e A335474 The a(n) patterns for n = 32, 80, 133, 290, 305, 329, 436 are:
%e A335474       (1)  (1)   (1)    (1)    (1)     (1)     (1)
%e A335474            (12)  (21)   (12)   (12)    (11)    (12)
%e A335474                  (321)  (21)   (21)    (12)    (21)
%e A335474                         (231)  (121)   (21)    (121)
%e A335474                                (213)   (122)   (123)
%e A335474                                (2131)  (221)   (212)
%e A335474                                        (2331)  (1212)
%e A335474                                                (2123)
%e A335474                                                (12123)
%t A335474 stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]];
%t A335474 mstype[q_]:=q/.Table[Union[q][[i]]->i,{i,Length[Union[q]]}];
%t A335474 Table[Length[Union[mstype/@ReplaceList[stc[n],{___,s__,___}:>{s}]]],{n,0,100}]
%Y A335474 The version for Heinz numbers of partitions is A335516(n) - 1.
%Y A335474 The non-contiguous version is A335454(n) - 1.
%Y A335474 The version allowing empty patterns is A335458.
%Y A335474 Patterns are counted by A000670 and ranked by A333217.
%Y A335474 The n-th composition has A124771(n) distinct consecutive subsequences.
%Y A335474 Knapsack compositions are counted by A325676 and ranked by A333223.
%Y A335474 The n-th composition has A334299(n) distinct subsequences.
%Y A335474 Minimal avoided patterns are counted by A335465.
%Y A335474 Patterns matched by prime indices are counted by A335549.
%Y A335474 Cf. A034691, A056986, A108917, A124767, A124770, A181796, A269134, A333222, A333224, A335456, A335457.
%K A335474 nonn
%O A335474 0,4
%A A335474 _Gus Wiseman_, Jun 21 2020