login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A007717 Number of symmetric polynomial functions of degree n of a symmetric matrix (of indefinitely large size) under joint row and column permutations. Also number of multigraphs with n edges (allowing loops) on an infinite set of nodes. 42
1, 2, 7, 23, 79, 274, 1003, 3763, 14723, 59663, 250738, 1090608, 4905430, 22777420, 109040012, 537401702, 2723210617, 14170838544, 75639280146, 413692111521, 2316122210804, 13261980807830, 77598959094772, 463626704130058, 2826406013488180, 17569700716557737 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Euler transform of A007719.

Also the number of non-isomorphic multiset partitions of {1, 1, 2, 2, 3, 3, ..., n, n}. - Gus Wiseman, Jul 18 2018

Number of distinct n X 2n matrices with integer entries and rows sums 2, up to row and column permutations. - Andrew Howroyd, Sep 06 2018

REFERENCES

Huaien Li and David C. Torney, Enumerations of Multigraphs, 2002.

LINKS

Table of n, a(n) for n=0..25.

Huaien Li and David C. Torney, Enumeration of unlabelled multigraphs, Ars Combin. 75 (2005) 171-188. MR2133219.

R. J. Mathar, Statistics on Small Graphs, arXiv:1709.09000 [math.CO] (2017) table 67.

EXAMPLE

a(2) = 7 (here - denotes an edge, = denotes a pair of parallel edges and o is a loop):

oo

o o

o-

o -

=

--

- -

From Gus Wiseman, Jul 18 2018: (Start)

Non-isomorphic representatives of the a(2) = 7 multiset partitions of {1, 1, 2, 2}:

  (1122),

  (1)(122), (11)(22), (12)(12),

  (1)(1)(22), (1)(2)(12),

  (1)(1)(2)(2).

(End)

MATHEMATICA

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];

Kq[q_, t_, k_] := SeriesCoefficient[1/Product[g = GCD[t, q[[j]]]; (1 - x^(q[[j]]/g))^g, {j, 1, Length[q]}], {x, 0, k}];

RowSumMats[n_, m_, k_] := Module[{s=0}, Do[s += permcount[q]* SeriesCoefficient[Exp[Sum[Kq[q, t, k]/t x^t, {t, 1, n}]], {x, 0, n}], {q, IntegerPartitions[m]}]; s/m!];

a[n_] := RowSumMats[n, 2n, 2];

Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 25}] (* Jean-Fran├žois Alcover, Oct 27 2018, after Andrew Howroyd *)

PROG

(PARI) \\ See A318951 for RowSumMats

a(n)=RowSumMats(n, 2*n, 2); \\ Andrew Howroyd, Sep 06 2018

CROSSREFS

Cf. A000664, A002620, A007716, A007719, A020555, A050531, A050532, A050535, A052171, A053418, A053419, A094574, A316972, A316974, A318951.

Sequence in context: A068593 A198944 A112657 * A130567 A091514 A143629

Adjacent sequences:  A007714 A007715 A007716 * A007718 A007719 A007720

KEYWORD

nonn

AUTHOR

Colin Mallows

EXTENSIONS

More terms from Vladeta Jovovic, Jan 26 2000

a(0)=1 prepended and a(16)-a(25) added by Max Alekseyev, Jun 21 2011

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified November 19 14:30 EST 2018. Contains 317352 sequences. (Running on oeis4.)