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A348436 Triangle read by rows. T(n,k) is the number of labeled threshold graphs on n vertices with k components, for 1 <= k <= n. 0
1, 1, 1, 4, 3, 1, 23, 16, 6, 1, 166, 115, 40, 10, 1, 1437, 996, 345, 80, 15, 1, 14512, 10059, 3486, 805, 140, 21, 1, 167491, 116096, 40236, 9296, 1610, 224, 28, 1, 2174746, 1507419, 522432, 120708, 20916, 2898, 336, 36, 1, 31374953, 21747460, 7537095, 1741440, 301770, 41832, 4830, 480, 45, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,4
COMMENTS
The class of threshold graphs is the smallest class of graphs that includes K1 and is closed under adding isolated vertices and dominating vertices.
LINKS
D. Galvin, G. Wesley and B. Zacovic, Enumerating threshold graphs and some related graph classes, arXiv:2110.08953 [math.CO], 2021.
Sam Spiro, Counting Threshold Graphs with Eulerian Numbers, arXiv:1909.06518 [math.CO], 2019.
FORMULA
T(1,1) = 1; for n >= 2, T(n,1) = A005840(n)/2; for n >= 3 and 2 <= k <= n-1, T(n,k) = binomial(n,k-1)*T(n-k+1,1); and for n >= 2, T(n,n)=1.
T(n, k) = binomial(n, k-1)*A053525(n - k + 1) if k != n, otherwise 1. - Peter Luschny, Oct 24 2021
EXAMPLE
Triangle begins:
1;
1, 1;
4, 3, 1;
23, 16, 6, 1;
166, 115, 40, 10, 1;
1437, 996, 345, 80, 15, 1;
14512, 10059, 3486, 805, 140, 21, 1;
167491, 116096, 40236, 9296, 1610, 224, 28, 1;
2174746, 1507419, 522432, 120708, 20916, 2898, 336, 36, 1;
31374953, 21747460, 7537095, 1741440, 301770, 41832, 4830, 480, 45, 1;
...
MAPLE
T := (n, k) -> `if`(n = k, 1, binomial(n, k-1)*A053525(n-k+1)):
for n from 1 to 10 do seq(T(n, k), k=1..n) od; # Peter Luschny, Oct 24 2021
MATHEMATICA
eulerian[0, 0] := 1; eulerian[n_, m_] := eulerian[n, m] =
Sum[((-1)^k)*Binomial[n + 1, k]*((m + 1 - k)^n), {k, 0, m + 1}];
(* t[n] counts the labeled threshold graphs on n vertices *)
t[0] = 1; t[1] = 1;
t[n_] := t[n] = Sum[(n - k)*eulerian[n - 1, k - 1]*(2^k), {k, 1, n - 1}];
T[1, 1] := 1; T[n_, 1] := T[n, 1] = (1/2)*t[n]; T[n_, n_] := T[n, n] = 1;
T[n_, k_] := T[n, k] = Binomial[n, k - 1]*T[n - k + 1, 1];
Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten
CROSSREFS
Cf. A005840 (row sums), A317057 (column k=1), A053525.
Sequence in context: A181355 A128320 A189507 * A350528 A208057 A298673
KEYWORD
nonn,tabl
AUTHOR
David Galvin, Oct 18 2021
STATUS
approved

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Last modified December 3 19:10 EST 2023. Contains 367540 sequences. (Running on oeis4.)