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

 

Logo

Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing.
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A309993 Triangle read by rows: T(n,k) is the number of permutations of length n composed of exactly k overlapping adjacent runs (for n >= 1 and 1 <= k <= n). 2
1, 1, 0, 1, 2, 0, 1, 8, 2, 0, 1, 22, 26, 0, 0, 1, 52, 168, 42, 0, 0, 1, 114, 804, 692, 42, 0, 0, 1, 240, 3270, 6500, 1866, 0, 0, 0, 1, 494, 12054, 46304, 34078, 3060, 0, 0, 0, 1, 1004, 41708, 279566, 413878, 122830, 3060, 0, 0, 0, 1, 2026, 138320, 1514324 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

Permutations of A307030 grouped by number of runs. Thus row sums give A307030.

Each column admits a rational generating function (Asinowski et al.).

LINKS

Bjarki Ágúst Guðmundsson, Table of n, a(n) for n = 1..5050

Andrei Asinowski, Cyril Banderier, Sara Billey, Benjamin Hackl, Svante Linusson, Pop-stack sorting and its image: Permutations with overlapping runs (2019), preprint.

Anders Claesson, Bjarki Ágúst Guðmundsson, Jay Pantone, Counting pop-stacked permutations in polynomial time, arXiv:1908.08910 [math.CO], 2019.

FORMULA

G.f. for column k=1: x/(1-x).

G.f. for column k=2: 2*x^3/((1-x)^2*(1-2*x)).

G.f. for column k=3: -2*x^4*(6*x^2 - 3*x - 1)/((1-x)^3*(1-2*x)^2*(1-3*x)).

G.f. for column k=4: -2*x^6*(144*x^4 - 180*x^3 - 5*x^2 + 74*x - 21)/((1-x)^4*(1-2*x)^3*(1-3*x)^2*(1-4*x)).

G.f. for column k=5: 2*x^7*(17280*x^8 - 37600*x^7 + 12784*x^6 + 33060*x^5 - 40581*x^4 + 18982*x^3 - 3856*x^2 + 198*x + 21)/((1-x)^5*(1-2*x)^4*(1-3*x)^3*(1-4*x)^2*(1-5*x)).

EXAMPLE

For n = 3 the permutations with overlapping runs are 123, 132, 213. The first has k = 1 runs, the other two have k = 2 runs. Thus T(3,1) = 1, T(3,2) = 2, T(3,3) = 0.

Triangle begins:

  1;

  1,    0;

  1,    2,     0;

  1,    8,     2,      0;

  1,   22,    26,      0,      0;

  1,   52,   168,     42,      0,      0;

  1,  114,   804,    692,     42,      0,    0;

  1,  240,  3270,   6500,   1866,      0,    0, 0;

  1,  494, 12054,  46304,  34078,   3060,    0, 0, 0;

  1, 1004, 41708, 279566, 413878, 122830, 3060, 0, 0, 0;

  ...

CROSSREFS

Cf. A307030.

Sequence in context: A316649 A065329 A108998 * A248673 A278881 A201637

Adjacent sequences:  A309990 A309991 A309992 * A309994 A309995 A309996

KEYWORD

nonn,tabl

AUTHOR

Bjarki Ágúst Guðmundsson, Aug 26 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 December 8 04:31 EST 2019. Contains 329850 sequences. (Running on oeis4.)