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A335474 Number of nonempty normal patterns contiguously matched by the n-th composition in standard order. 4
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, 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, 4, 2, 3, 4, 6, 2, 4, 4, 6, 4, 6, 7, 8, 2, 4, 4, 7, 4, 6 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
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.
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).
LINKS
FORMULA
a(n) = A335458(n) - 1.
EXAMPLE
The a(n) patterns for n = 32, 80, 133, 290, 305, 329, 436 are:
(1) (1) (1) (1) (1) (1) (1)
(12) (21) (12) (12) (11) (12)
(321) (21) (21) (12) (21)
(231) (121) (21) (121)
(213) (122) (123)
(2131) (221) (212)
(2331) (1212)
(2123)
(12123)
MATHEMATICA
stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]];
mstype[q_]:=q/.Table[Union[q][[i]]->i, {i, Length[Union[q]]}];
Table[Length[Union[mstype/@ReplaceList[stc[n], {___, s__, ___}:>{s}]]], {n, 0, 100}]
CROSSREFS
The version for Heinz numbers of partitions is A335516(n) - 1.
The non-contiguous version is A335454(n) - 1.
The version allowing empty patterns is A335458.
Patterns are counted by A000670 and ranked by A333217.
The n-th composition has A124771(n) distinct consecutive subsequences.
Knapsack compositions are counted by A325676 and ranked by A333223.
The n-th composition has A334299(n) distinct subsequences.
Minimal avoided patterns are counted by A335465.
Patterns matched by prime indices are counted by A335549.
Sequence in context: A178677 A366888 A243924 * A333939 A272759 A272760
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 21 2020
STATUS
approved

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Last modified May 21 16:37 EDT 2024. Contains 372738 sequences. (Running on oeis4.)