A380617 -id:A380617 - cp's OEIS frontend

A380615 Triangle read by rows: T(n,k) is the number of sensed combinatorial maps with n edges and k vertices, 1 <= k <= n + 1.

1, 1, 1, 2, 2, 1, 5, 8, 5, 2, 18, 38, 34, 14, 3, 105, 275, 288, 154, 42, 6, 902, 2614, 3102, 1959, 705, 140, 14, 9749, 30346, 39242, 27898, 11956, 3142, 473, 34, 127072, 415360, 573654, 446078, 217000, 68544, 13886, 1670, 95, 1915951, 6513999, 9484003, 7911844, 4230802, 1523176, 373188, 60614, 5969, 280

Offset: 0

Andrew Howroyd, Jan 28 2025

By duality, also the number of sensed combinatorial maps with n edges and k faces.

			Triangle begins:
n\k |      1       2       3       4       5      6      7     8   9
----+----------------------------------------------------------------
  0 |      1
  1 |      1,      1
  2 |      2,      2,      1;
  3 |      5,      8,      5,      2;
  4 |     18,     38,     34,     14,      3;
  5 |    105,    275,    288,    154,     42,     6;
  6 |    902,   2614,   3102,   1959,    705,   140,    14;
  7 |   9749,  30346,  39242,  27898,  11956,  3142,   473,   34;
  8 | 127072, 415360, 573654, 446078, 217000, 68544, 13886, 1670, 95;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Row sums are A170946.
Main diagonal is A002995(n+1).
Second diagonal gives A380237.
Columns 1..3 are A007769, A380618, A380619.
Cf. A053979 (rooted), A379430 (planar), A380616 (unsensed), A380617 (achiral).

InvEulerMTS(p)={my(n=serprec(p, x)-1, q=log(p), vars=variables(p)); sum(i=1, n, moebius(i)*substvec(q + O(x*x^(n\i)), vars, apply(v->v^i, vars))/i)}
b(k,r)={if(k%2, if(r%2, 0, my(j=r/2); k^j*(2*j)!/(j!*2^j)), sum(j=0, r\2, binomial(r, 2*j)*k^j*(2*j)!/(j!*2^j)))}
C(k,r,y)={my(p=sumdiv(k,d,eulerphi(k/d)*y^d)/k); sum(i=0, r, abs(stirling(r,i,1))*p^i)/r!}
S(n,k,y)={sum(r=0, 2*n\k, if(k*r%2==0, x^(k*r/2)*b(k,r)*C(k,r,y)), O(x*x^n))}
G(n,y='y)={prod(k=1, 2*n, S(n,k,y))}
T(n)={[Vecrev(p/y) | p<-Vec(y+InvEulerMTS(G(n)))]}
{ my(A=T(10)); for(i=1, #A, print(A[i])) }

A380616 Triangle read by rows: T(n,k) is the number of unsensed combinatorial maps with n edges and k vertices, 1 <= k <= n + 1.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 5, 8, 5, 2, 17, 33, 30, 13, 3, 79, 198, 208, 118, 35, 6, 554, 1571, 1894, 1232, 472, 104, 12, 5283, 16431, 21440, 15545, 6879, 1914, 315, 27, 65346, 213831, 296952, 233027, 115134, 37311, 7881, 1021, 65, 966156, 3288821, 4799336, 4019360, 2163112, 787065, 196267, 32857, 3407, 175

Offset: 0

Andrew Howroyd, Jan 28 2025

By duality, also the number of unsensed combinatorial maps with n edges and k faces.

			Triangle begins:
n\k |     1       2       3       4       5      6     7     8   9
----+--------------------------------------------------------------
  0 |     1;
  1 |     1,      1;
  2 |     2,      2,      1;
  3 |     5,      8,      5,      2;
  4 |    17,     33,     30,     13,      3;
  5 |    79,    198,    208,    118,     35,     6;
  6 |   554,   1571,   1894,   1232,    472,   104,   12;
  7 |  5283,  16431,  21440,  15545,   6879,  1914,  315,   27;
  8 | 65346, 213831, 296952, 233027, 115134, 37311, 7881, 1021, 65;
  ...

Row sums are A214816.
Main diagonal is A006082(n+1).
Columns 1..3 are A054499, A380620, A380621.
Cf. A053979 (rooted), A277741 (planar), A380615 (sensed), A380617 (achiral).

T(n,k) = (A380615(n,k) + A380617(n,k))/2.

cp's OEIS Frontend

A380615 Triangle read by rows: T(n,k) is the number of sensed combinatorial maps with n edges and k vertices, 1 <= k <= n + 1.

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