cp's OEIS Frontend

A335454 Number of normal patterns matched by the n-th composition in standard order (A066099).

%I A335454 #21 Mar 12 2025 17:14:51
%S A335454 1,2,2,3,2,3,3,4,2,3,3,5,3,6,5,5,2,3,3,5,3,5,6,7,3,6,5,9,5,9,7,6,2,3,
%T A335454 3,5,3,4,5,7,3,5,4,7,5,10,9,9,3,6,5,9,4,9,10,12,5,9,7,13,7,12,9,7,2,3,
%U A335454 3,5,3,4,5,7,3,5,5,7,6,10,9,9,3,5,6,8,5
%N A335454 Number of normal patterns matched by the n-th composition in standard order (A066099).
%C A335454 We define a (normal) pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670. 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).
%C A335454 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.
%H A335454 John Tyler Rascoe, <a href="/A335454/b335454.txt">Table of n, a(n) for n = 0..8192</a>
%H A335454 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation_pattern">Permutation pattern</a>
%H A335454 Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vTCPiJVFUXN8IqfLlCXkgP15yrGWeRhFS4ozST5oA4Bl2PYS-XTA3sGsAEXvwW-B0ealpD8qnoxFqN3/pub">Statistics, classes, and transformations of standard compositions</a>
%H A335454 Gus Wiseman, <a href="https://oeis.org/A102726/a102726.txt">Sequences counting and ranking compositions by the patterns they match or avoid.</a>
%e A335454 The a(n) patterns for n = 0, 1, 3, 7, 11, 13, 23, 83, 27, 45:
%e A335454   0:  1:   11:   111:   211:   121:   2111:   2311:   1211:   2121:
%e A335454 ---------------------------------------------------------------------
%e A335454   ()  ()   ()    ()     ()     ()     ()      ()      ()      ()
%e A335454       (1)  (1)   (1)    (1)    (1)    (1)     (1)     (1)     (1)
%e A335454            (11)  (11)   (11)   (11)   (11)    (11)    (11)    (11)
%e A335454                  (111)  (21)   (12)   (21)    (12)    (12)    (12)
%e A335454                         (211)  (21)   (111)   (21)    (21)    (21)
%e A335454                                (121)  (211)   (211)   (111)   (121)
%e A335454                                       (2111)  (231)   (121)   (211)
%e A335454                                               (2311)  (211)   (212)
%e A335454                                                       (1211)  (221)
%e A335454                                                               (2121)
%t A335454 stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]];
%t A335454 mstype[q_]:=q/.Table[Union[q][[i]]->i,{i,Length[Union[q]]}];
%t A335454 Table[Length[Union[mstype/@Subsets[stc[n]]]],{n,0,30}]
%o A335454 (Python)
%o A335454 from itertools import combinations
%o A335454 def comp(n):
%o A335454     # see A357625
%o A335454     return
%o A335454 def A335465(n):
%o A335454     A,B,C = set(),set(),comp(n)
%o A335454     c = range(len(C))
%o A335454     for j in c:
%o A335454         for k in combinations(c, j):
%o A335454             A.add(tuple(C[i] for i in k))
%o A335454     for i in A:
%o A335454         D = {v: rank + 1 for rank, v in enumerate(sorted(set(i)))}
%o A335454         B.add(tuple(D[v] for v in i))
%o A335454     return len(B)+1 # _John Tyler Rascoe_, Mar 12 2025
%Y A335454 References found in the links are not all included here.
%Y A335454 Summing over indices with binary length n gives A335456(n).
%Y A335454 The contiguous case is A335458.
%Y A335454 The version for Heinz numbers of partitions is A335549.
%Y A335454 Patterns are counted by A000670 and ranked by A333217.
%Y A335454 The n-th composition has A124771(n) distinct consecutive subsequences.
%Y A335454 Knapsack compositions are counted by A325676 and ranked by A333223.
%Y A335454 The n-th composition has A333257(n) distinct subsequence-sums.
%Y A335454 The n-th composition has A334299(n) distinct subsequences.
%Y A335454 Minimal avoided patterns are counted by A335465.
%Y A335454 Cf. A034691, A056986, A108917, A124767, A124770, A158005, A269134, A333218, A333222, A333224, A334030.
%K A335454 nonn,look
%O A335454 0,2
%A A335454 _Gus Wiseman_, Jun 14 2020