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A277741 Array read by antidiagonals: A(n,k) is the number of unsensed planar maps with n vertices and k faces, n >= 1, k >= 1.

%I A277741 #34 Jun 15 2025 16:55:21
%S A277741 1,1,1,1,2,1,2,5,5,2,3,13,20,13,3,6,35,83,83,35,6,12,104,340,504,340,
%T A277741 104,12,27,315,1401,2843,2843,1401,315,27,65,1021,5809,15578,21420,
%U A277741 15578,5809,1021,65,175,3407,24299,82546,149007,149007,82546,24299,3407,175
%N A277741 Array read by antidiagonals: A(n,k) is the number of unsensed planar maps with n vertices and k faces, n >= 1, k >= 1.
%C A277741 A(n,k) is also the number of multiquadrangulations of the sphere with n stable equilibria and k unstable equilibria.
%C A277741 From _Andrew Howroyd_, Jan 13 2025: (Start)
%C A277741 The planar maps considered are connected and may contain loops and parallel edges.
%C A277741 The number of edges is n + k - 2. (End)
%D A277741 Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, chapter 5.
%H A277741 Richard Kapolnai, Gabor Domokos, and Timea Szabo, <a href="http://dx.doi.org/10.3311/PPee.7074">Generating spherical multiquadrangulations by restricted vertex splittings and the reducibility of equilibrium classes</a>, Periodica Polytechnica Electrical Engineering, 56(1):11-10, 2012. Also <a href="https://arxiv.org/abs/1206.1698">arXiv:1206.1698 [cs.DM]</a>, 2012. See Table 1.
%H A277741 Timothy R. Walsh, <a href="/A007401/a007401.pdf">Number of sensed planar maps with n edges and m vertices</a>, pp. 11-20.
%H A277741 Nicholas C. Wormald, <a href="http://dx.doi.org/10.1016/0012-365X(81)90238-7">Counting unrooted planar maps</a>, Discrete Math. 36 (1981), no. 2, 205-225.
%F A277741 A(n,k) = A(k,n).
%F A277741 A(n,k) = (A379430(n,k) + A379431(n,k))/2. - _Andrew Howroyd_, Jan 14 2025
%e A277741 The array begins:
%e A277741    1,    1,    1,     2,     3,     6,   12,   27, 65, ...
%e A277741    1,    2,    5,    13,    35,   104,  315, 1021, ...
%e A277741    1,    5,   20,    83,   340,  1401, 5809, ...
%e A277741    2,   13,   83,   504,  2843, 15578, ...
%e A277741    3,   35,  340,  2843, 21420, ...
%e A277741    6,  104, 1401, 15578, ...
%e A277741   12,  315, 5809, ...
%e A277741   27, 1021, ...
%e A277741   65, ...
%e A277741   ...
%e A277741 As a triangle, rows give the number of edges (first row is 0 edges):
%e A277741    1;
%e A277741    1,    1;
%e A277741    1,    2,    1;
%e A277741    2,    5,    5,     2;
%e A277741    3,   13,   20,    13,     3;
%e A277741    6,   35,   83,    83,    35,    6;
%e A277741   12,  104,  340,   504,   340,   104,   12;
%e A277741   27,  315, 1401,  2843,  2843,  1401,  315,   27;
%e A277741   65, 1021, 5809, 15578, 21420, 15578, 5809, 1021, 65;
%e A277741   ...
%Y A277741 Antidiagonal sums are A006385.
%Y A277741 Rows 1..2 (equally, columns 1..2) are A006082, A380239.
%Y A277741 Cf. A269920 (rooted), A379430 (sensed), A379431 (achiral), A379432 (2-connected), A384963 (simple).
%K A277741 nonn,tabl
%O A277741 1,5
%A A277741 _N. J. A. Sloane_, Nov 07 2016
%E A277741 Missing terms inserted and definition edited by _Andrew Howroyd_, Jan 13 2025