login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A326254 Number of non-capturing set partitions of {1..n}. 9
1, 1, 2, 5, 14, 41, 123, 374, 1147, 3538, 10958, 34042, 105997 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Conjectured to be equal to A054391.

A set partition is capturing if it has two blocks of the form {...x...y...} and {...z...t...} where x < z and y > t or x > z and y < t. Capturing is a weaker condition than nesting, so for example {{1,3,5},{2,4}} is capturing but not nesting.

LINKS

Table of n, a(n) for n=0..12.

Eric Marberg, Crossings and nestings in colored set partitions, arXiv preprint arXiv:1203.5738 [math.CO], 2012.

FORMULA

a(n) = A000110(n) - A326243(n).

EXAMPLE

The a(0) = 1 through a(4) = 14 non-capturing set partitions:

  {}  {{1}}  {{1,2}}    {{1,2,3}}      {{1,2,3,4}}

             {{1},{2}}  {{1},{2,3}}    {{1},{2,3,4}}

                        {{1,2},{3}}    {{1,2},{3,4}}

                        {{1,3},{2}}    {{1,2,3},{4}}

                        {{1},{2},{3}}  {{1,2,4},{3}}

                                       {{1,3},{2,4}}

                                       {{1,3,4},{2}}

                                       {{1},{2},{3,4}}

                                       {{1},{2,3},{4}}

                                       {{1,2},{3},{4}}

                                       {{1},{2,4},{3}}

                                       {{1,3},{2},{4}}

                                       {{1,4},{2},{3}}

                                       {{1},{2},{3},{4}}

MATHEMATICA

sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];

capXQ[stn_]:=MatchQ[stn, {___, {___, x_, ___, y_, ___}, ___, {___, z_, ___, t_, ___}, ___}/; x<z&&y>t||x>z&&y<t];

Table[Length[Select[sps[Range[n]], !capXQ[#]&]], {n, 0, 5}]

CROSSREFS

Capturing set partitions are A326243.

Non-crossing set partitions are A000108.

Cf. A000110, A001519, A016098, A054391, A058681, A099947, A122880.

Cf. A326212, A326237, A326245, A326246, A326249, A326255, A326256, A326260.

Sequence in context: A124302 A088355 A113485 * A054391 A176677 A108626

Adjacent sequences:  A326251 A326252 A326253 * A326255 A326256 A326257

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Jun 20 2019

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified September 22 05:47 EDT 2019. Contains 327287 sequences. (Running on oeis4.)