

A335474


Number of nonempty normal patterns contiguously matched by the nth 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
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OFFSET

0,4


COMMENTS

The kth composition in standard order (graded reverselexicographic, 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



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 noncontiguous version is A335454(n)  1.
The version allowing empty patterns is A335458.
The nth composition has A124771(n) distinct consecutive subsequences.
The nth composition has A334299(n) distinct subsequences.
Minimal avoided patterns are counted by A335465.
Patterns matched by prime indices are counted by A335549.
Cf. A034691, A056986, A108917, A124767, A124770, A181796, A269134, A333222, A333224, A335456, A335457.


KEYWORD

nonn


AUTHOR



STATUS

approved



