A379614 -id:A379614 - cp's OEIS frontend

A365638 Triangular array read by rows: T(n, k) is the number of ways that a k-element transitive tournament can occur as a subtournament of a larger tournament on n labeled vertices.

1, 1, 1, 2, 4, 2, 8, 24, 24, 6, 64, 256, 384, 192, 24, 1024, 5120, 10240, 7680, 1920, 120, 32768, 196608, 491520, 491520, 184320, 23040, 720, 2097152, 14680064, 44040192, 55050240, 27525120, 5160960, 322560, 5040, 268435456, 2147483648, 7516192768, 11274289152, 7046430720, 1761607680, 165150720, 5160960, 40320

Offset: 0

Thomas Scheuerle, Sep 14 2023

A tournament is a directed digraph obtained by assigning a direction for each edge in an undirected complete graph. In a transitive tournament all nodes can be strictly ordered by their reachability.

			Triangle begins:
     1
     1,     1
     2,     4,     2
     8,    24,    24,     6
    64,   256,   384,   192,    24
  1024,  5120, 10240,  7680,  1920,  120

Paul Erdős, On a problem in graph theory, The Mathematical Gazette, 47: 220-223 (1963).

Cf. A002866, A006125, A052670, A095340, A103904, A008279, A117260, A379614 (row sums).

Cf. A122027, A224886, A259105, A350608, A350609, A350610.

T := (n, k) -> 2^(((n-1)*n - (k-1)*k)/2) * n! / (n-k)!:
seq(seq(T(n, k), k = 0..n), n = 0..8);  # Peter Luschny, Nov 02 2023

T(n, k) = binomial(n, k)*k!*2^(binomial(n, 2) - binomial(k, 2))

T(n, k) = binomial(n, k)*k!*2^(binomial(n, 2) - binomial(k, 2)).
T(n, 0) = A006125(n).
T(n, 1) = A095340(n).
T(n, 2) = A103904(n).
T(n, n) = n!.
T(n, n-1) = A002866(n-1).
T(n, n-2) = A052670(n).
T(n, k) = A008279(n, k) * A117260(n, k). - Peter Luschny, Dec 31 2024

A382240 a(n) = Sum_{k=0..n} 3^((n+k-1)(n-k)/2) n! / (n-k)!.

Original entry on oeis.org

1, 2, 11, 168, 7233, 889014, 314965899, 323989244676, 972969439627809, 8566667168429128842, 221877626825222187484203, 16949442370817602102051560384, 3827091229259231090623800852526113, 2558686452439976557585601153755243553406, 5072634396431144733070212976874036427346208619

Offset: 0

Vaclav Kotesovec, Mar 19 2025

nonn

In general, for m>1, Sum_{k=0..n} m^((n+k-1)*(n-k)/2) * n! / (n-k)! ~ sqrt(2*Pi/log(m)) * n^(log(n)/(2*log(m)) + 1/2) * m^((2*n - 1)^2/8).

Cf. A379614.

Table[Sum[3^((n+k-1)*(n-k)/2) * n!/(n-k)!, {k, 0, n}], {n, 0, 15}]

a(n) ~ sqrt(2*Pi/log(3)) * n^(log(n)/(2*log(3)) + 1/2) * 3^((2*n-1)^2/8).

cp's OEIS Frontend

A365638 Triangular array read by rows: T(n, k) is the number of ways that a k-element transitive tournament can occur as a subtournament of a larger tournament on n labeled vertices.

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A382240 a(n) = Sum_{k=0..n} 3^((n+k-1)(n-k)/2) n! / (n-k)!.

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cp's OEIS Frontend

A365638 Triangular array read by rows: T(n, k) is the number of ways that a k-element transitive tournament can occur as a subtournament of a larger tournament on n labeled vertices.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

PARI

Formula

A382240 a(n) = Sum_{k=0..n} 3^((n+k-1)*(n-k)/2) * n! / (n-k)!.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A382240 a(n) = Sum_{k=0..n} 3^((n+k-1)(n-k)/2) n! / (n-k)!.