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).

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

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: A352772 A108998 A356265 * A248673 A278881 A335095

Adjacent sequences: A309990 A309991 A309992 * A309994 A309995 A309996

Keyword

nonn,tabl

Author

Bjarki Ágúst Guðmundsson, Aug 26 2019

Status

approved