A321710 Triangle read by rows: T(n,k) is the number of rooted hypermaps of genus k with n darts.
1, 3, 12, 1, 56, 15, 288, 165, 8, 1584, 1611, 252, 9152, 14805, 4956, 180, 54912, 131307, 77992, 9132, 339456, 1138261, 1074564, 268980, 8064, 2149888, 9713835, 13545216, 6010220, 579744, 13891584, 81968469, 160174960, 112868844, 23235300, 604800, 91287552, 685888171, 1805010948, 1877530740, 684173164, 57170880, 608583680, 5702382933, 19588944336, 28540603884, 16497874380, 2936606400, 68428800, 4107939840, 47168678571, 206254571236, 404562365316, 344901105444, 108502598960, 8099018496
Offset: 1
Examples
Triangle starts: n\k [0] [1] [2] [3] [4] [5] [1] 1; [2] 3; [3] 12, 1; [4] 56, 15; [5] 288, 165, 8; [6] 1584, 1611, 252; [7] 9152, 14805, 4956, 180; [8] 54912, 131307, 77992, 9132; [9] 339456, 1138261, 1074564, 268980, 8064; [10] 2149888, 9713835, 13545216, 6010220, 579744; [11] 13891584, 81968469, 160174960, 112868844, 23235300, 604800; [12] 91287552, 685888171, 1805010948, 1877530740, 684173164, 57170880; [13] ...
Links
- Gheorghe Coserea, Rows n = 1..42, flattened
- Alain Giorgetti and Timothy R. S. Walsh, Enumeration of hypermaps of a given genus, Ars Math. Contemp. 15 (2018) 225-266.
- Timothy R. Walsh, Space-efficient generation of nonisomorphic maps and hypermaps
- T. R. Walsh, Space-Efficient Generation of Nonisomorphic Maps and Hypermaps, J. Int. Seq. 18 (2015) # 15.4.3.
- P. G. Zograf, Enumeration of Grothendieck's Dessins and KP Hierarchy, International Mathematics Research Notices, Volume 2015, Issue 24, 1 January 2015, 13533-13544.
- Peter Zograf, Enumeration of Grothendieck's Dessins and KP Hierarchy, arXiv:1312.2538 [math.CO], 2014.
Crossrefs
Programs
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Mathematica
l1[f_,n_] := Sum[(i-1)t[i]D[f,t[i-1]], {i,2,n}]; m1[f_,n_] := Sum[(i-1)t[j]t[i-j]D[f,t[i-1]] + j(i-j)t[i+1]D[f,t[j],t[i-j]], {i,2,n},{j,i-1}]; ff[1] = x^2 t[1]; ff[n_] := ff[n] = Simplify@(2x l1[ff[n-1],n] + m1[ff[n-1],n] + Sum[t[i+1]j(i-j)D[ff[k],t[j]]D[ff[n-1-k],t[i-j]], {i,2,n-1},{j,i-1},{k,n-2}]) / n; row[n_]:=Reverse[CoefficientList[n ff[n] /. {t[_]->x}, x]][[;;;;2]][[;;Quotient[n+1,2]]]; Table[row[n], {n,14}] (* Andrei Zabolotskii, Jun 27 2025, after the PARI code *) -
PARI
L1(f, N) = sum(i=2, N, (i-1)*t[i]*deriv(f, t[i-1])); M1(f, N) = { sum(i=2, N, sum(j=1, i-1, (i-1)*t[j]*t[i-j]*deriv(f, t[i-1]) + j*(i-j)*t[i+1]*deriv(deriv(f, t[j]), t[i-j]))); }; F(N) = { my(u='x, v='x, f=vector(N)); t=vector(N+1, n, eval(Str("t", n))); f[1] = u*v*t[1]; for (n=2, N, f[n] = (u + v)*L1(f[n-1], n) + M1(f[n-1], n) + sum(i=2, n-1, t[i+1]*sum(j=1, i-1, j*(i-j)*sum(k=1, n-2, deriv(f[k], t[j])*deriv(f[n-1-k], t[i-j])))); f[n] /= n); f; }; seq(N) = { my(f=F(N), v=substvec(f, t, vector(#t, n, 'x)), g=vector(#v, n, Polrev(Vec(n * v[n])))); apply(p->Vecrev(substpol(p, 'x^2, 'x)), g); }; concat(seq(14))
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