A063841 Table T(n,k) giving number of k-multigraphs on n nodes (n >= 1, k >= 0) read by antidiagonals.

1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 10, 11, 1, 1, 5, 20, 66, 34, 1, 1, 6, 35, 276, 792, 156, 1, 1, 7, 56, 900, 10688, 25506, 1044, 1, 1, 8, 84, 2451, 90005, 1601952, 2302938, 12346, 1, 1, 9, 120, 5831, 533358, 43571400, 892341888, 591901884, 274668, 1

Offset

1,5

Comments

The first five rows admit the g.f. 1/(1-x), 1/(1-x)^2, 1/(1-x)^4 and those given in A063842, A063843. Is it known that the n-th row admits a rational g.f. with denominator (1-x)^A000124(n)? - M. F. Hasler, Jan 19 2012

T(n+1,k-1) is the number of unoriented ways to color the edges of a regular n-dimensional simplex using up to k colors. - Robert A. Russell, Aug 21 2019

Links

Andrew Howroyd, Table of n, a(n) for n = 1..820

Harald Fripertinger, The cycle type of the induced action on 2-subsets

Vladeta Jovovic, Formulae for the number T(n,k) of n-multigraphs on k nodes

Formula

T(n,k) = A327084(n-1,k+1) for n > 1. - Robert A. Russell, Aug 21 2019

Example

Table begins
===========================================================
n\k| 0    1       2         3           4             5
---|-------------------------------------------------------
1  | 1    1       1         1           1             1 ...
2  | 1    2       3         4           5             6 ...
3  | 1    4      10        20          35            56 ...
4  | 1   11      66       276         900          2451 ...
5  | 1   34     792     10688       90005        533358 ...
6  | 1  156   25506   1601952    43571400     661452084 ...
7  | 1 1044 2302938 892341888 95277592625 4364646955812 ...
...
T(3,2)=10 because there are 10 unlabeled graphs with 3 nodes with at most 2 edges connecting any pair.
(. . .),(.-. .),(.-.-.),(.-.-.-),(.=. .),(.=.=.),(.=.=.=),(.-.=.),(.-.-.=),(.=.=.-). - Geoffrey Critzer, Jan 23 2012

Mathematica

(* This code gives the array T(n, k). *) Needs["Combinatorica`"]; Transpose[Table[Table[PairGroupIndex[SymmetricGroup[n], s]/.Table[s[i]->k+1, {i, 0, Binomial[n, 2]}], {n, 1, 7}], {k, 0, 6}]]//Grid (* Geoffrey Critzer, Jan 23 2012 *)
permcount[v_] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t*k; s += t]; s!/m];
edges[v_] := Sum[GCD[v[[i]], v[[j]]], {i, 2, Length[v]}, {j, 1, i - 1}] + Total[Quotient[v, 2]];
T[n_, k_] := (s=0; Do[s += permcount[p]*(k+1)^edges[p], {p, IntegerPartitions[n]}]; s/n!);
Table[T[n-k, k], {n, 1, 10}, {k, n-1, 0, -1}] // Flatten (* Jean-François Alcover, Jul 08 2018, after Andrew Howroyd *)

Prog

(PARI)
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
edges(v) = {sum(i=2, #v, sum(j=1, i-1, gcd(v[i], v[j]))) + sum(i=1, #v, v[i]\2)}
T(n, k) = {my(s=0); forpart(p=n, s+=permcount(p)*(k+1)^edges(p)); s/n!} \\ Andrew Howroyd, Oct 22 2017

Crossrefs

Columns give A000088, A004102, A053400, A053420, A053421.

Rows (4th and 5th) are listed in A063842, A063843.

Cf. A327084 (unoriented simplex edge colorings).

Sequence in context: A073165 A137153 A340814 * A256161 A137596 A111669

Adjacent sequences: A063838 A063839 A063840 * A063842 A063843 A063844

Keyword

nonn,nice,tabl

Author

N. J. A. Sloane, Aug 25 2001

Extensions

More terms from Vladeta Jovovic, Sep 03 2001

Status

approved