A380241 Array read by antidiagonals: T(n,k) is the number of rooted (2k)-regular planar maps with n vertices, n >= 0, k >= 0.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 5, 9, 1, 1, 1, 14, 100, 54, 1, 1, 1, 42, 1225, 3000, 378, 1, 1, 1, 132, 15876, 171500, 110000, 2916, 1, 1, 1, 429, 213444, 10001880, 30012500, 4550000, 24057, 1, 1, 1, 1430, 2944656, 591666768, 7981500240, 5987493750, 204000000, 208494, 1, 1

Offset

0,8

Comments

The zeroth column is included by convention only for consistency with the first row sequences.

The case for regular planar maps of odd valency is more complicated and without simple closed form formulas, so not presented in this sequence. See the references for additional information.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals)

E. A. Bender and E. R. Canfield, The number of degree restricted rooted maps on the sphere, SIAM J. Discrete Math. 7 (1994) 9-15.

Zhicheng Gao and Mizan Rahman, Enumeration of k-poles, Annals of Combinatorics 1 (1997), pp. 55-66.

W. T. Tutte, A Census of Slicings, Canad. J. Math. 14 (1962), 708-722.

W. T. Tutte, A Census of Planar Maps, Canad. J. Math. 15 (1963), 249-271.

Formula

T(n,k) = 2*binomial(2*k-1, k)^n*(n*k)!/(n!*(n*k - n + 2)!) for k > 0.

Example

Array begins:
====================================================================
n\k | 0  1      2          3               4                   5 ...
----+---------------------------------------------------------------
  0 | 1  1      1          1               1                   1 ...
  1 | 1  1      2          5              14                  42 ...
  2 | 1  1      9        100            1225               15876 ...
  3 | 1  1     54       3000          171500            10001880 ...
  4 | 1  1    378     110000        30012500          7981500240 ...
  5 | 1  1   2916    4550000      5987493750       7304332956480 ...
  6 | 1  1  24057  204000000   1302227368750    7310748066293952 ...
  7 | 1  1 208494 9690000000 301107909375000 7794097754539041792 ...
  ...

Prog

(PARI) T(n, k)=if(k==0, 1, 2*binomial(2*k-1, k)^n*(n*k)!/(n!*(n*k - n + 2)!))

Crossrefs

Columns 0..3 are A000012 twice, A000168, A380242.

Rows 0..3 are A000012, A000108, A060150, A380243.

Cf. A269920.

Sequence in context: A292464 A090628 A054387 * A199958 A112734 A351699

Adjacent sequences: A380238 A380239 A380240 * A380242 A380243 A380244

Keyword

nonn,tabl,new

Author

Andrew Howroyd, Jan 22 2025

Status

approved