A350746 Triangle read by rows: T(n,k) is the number of labeled quasi-loop-threshold graphs on vertex set [n] with k components, for n >= 1 and 1 <= k <= n.

2, 3, 4, 16, 18, 8, 133, 155, 72, 16, 1521, 1810, 910, 240, 32, 22184, 26797, 14145, 4180, 720, 64, 393681, 480879, 262514, 83230, 16520, 2016, 128, 8233803, 10144283, 5675866, 1888873, 409360, 58912, 5376, 256

Offset: 1

Author

David Galvin, Jan 13 2022

Keywords

• nonn
• tabl

Comments

The family of quasi-loop-threshold graphs is the smallest family of looped graphs that contains K_1 (a single vertex) and K^loop_1 (a single looped vertex), and is closed under taking unions and adding looped dominating vertices (looped, and adjacent to everything previously added).

Examples

			Triangle begins:
        2;
        3,        4;
       16,       18,       8;
      133,      155,      72,      16;
     1521,     1810,     910,     240,     32;
    22184,    26797,   14145,    4180,    720,    64;
   393681,   480879,  262514,   83230,  16520,  2016,  128;
  8233803, 10144283, 5675866, 1888873, 409360, 58912, 5376, 256;
  ...
		

Links

• D. Galvin, G. Wesley and B. Zacovic, Enumerating threshold graphs and some related graph classes, arXiv:2110.08953 [math.CO], 2021.

Crossrefs

Row sums are A038052.

Except at n=1, the first column is A048802 (A048802 takes value 1 at n=1).

Programs

Mathematica

qltconn[0] = 0; qltconn[1] = 2; qltconn[n_] := qltconn[n] = Sum[StirlingS2[n, k]*(k^(k - 1)), {k, 1, n}] (*qltconn is the number of connected quasi loop threshold graphs on n vertices*); T[n_, l_] := T[n, l] := (Factorial[n]/Factorial[l])*Coefficient[(Sum[(qltconn[k]*(x^k))/Factorial[k], {k, 1, n}])^l, x, n]; Table[T[n, l], {n, 1, 12}, {l, 1, n}]

Formula

See Section 1.4 of Galvin, Wesley and Zacovic link for two methods to compute T(n,k).