A335838 Number of normal patterns contiguously matched by integer partitions of n.

1, 2, 5, 9, 18, 31, 54, 89, 145, 225, 349, 524, 778, 1137, 1645, 2330, 3293, 4586, 6341, 8676, 11794, 15880, 21292, 28298, 37419, 49163, 64301, 83576, 108191, 139326, 178699, 228183, 290286, 367760, 464374, 584146, 732481, 915468, 1140773, 1417115, 1755578

Offset: 0

Author

Gus Wiseman, Jun 27 2020

Keywords

• nonn

Comments

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

Examples

			The patterns contiguously matched by (3,2,2,1) are: (), (1), (1,1), (2,1), (2,1,1), (2,2,1), (3,2,2,1). Note that (3,2,1) is not contiguously matched. See A335837 for a larger example.
		

Links

• Wikipedia, Permutation pattern
• Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

Crossrefs

The version for compositions in standard order is A335474.

The version for compositions is A335457.

The not necessarily contiguous version is A335837.

Patterns are counted by A000670 and ranked by A333217.

Patterns contiguously matched by prime indices are counted by A335516.

Contiguous divisors are counted by A335519.

Minimal patterns avoided by prime indices are counted by A335550.

Cf. A056986, A108917, A333257, A335454, A335456, A335458, A335549.

Programs

Mathematica

mstype[q_]:=q/.Table[Union[q][[i]]->i,{i,Length[Union[q]]}];
Table[Sum[Length[Union[mstype/@ReplaceList[y,{_,s___,_}:>{s}]]],{y,IntegerPartitions[n]}],{n,0,8}]

Extensions

More terms from Jinyuan Wang, Jun 27 2020