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A138178
Number of symmetric matrices with nonnegative integer entries and without zero rows or columns such that sum of all entries is equal to n.
45
1, 1, 3, 9, 33, 125, 531, 2349, 11205, 55589, 291423, 1583485, 8985813, 52661609, 319898103, 2000390153, 12898434825, 85374842121, 580479540219, 4041838056561, 28824970996809, 210092964771637, 1564766851282299, 11890096357039749, 92151199272181629
OFFSET
0,3
COMMENTS
Number of normal semistandard Young tableaux of size n, where a tableau is normal if its entries span an initial interval of positive integers. - Gus Wiseman, Feb 23 2018
LINKS
FORMULA
G.f.: Sum_{n>=0} Sum_{k=0..n} (-1)^(n-k)*C(n,k)*(1-x)^(-k)*(1-x^2)^(-C(k,2)).
G.f.: Sum_{n>=0} 2^(-n-1)*(1-x)^(-n)*(1-x^2)^(-C(n,2)). - Vladeta Jovovic, Dec 09 2009
EXAMPLE
a(4) = 33 because there are 1 such matrix of type 1 X 1, 7 matrices of type 2 X 2, 15 of type 3 X 3 and 10 of type 4 X 4, cf. A138177.
From Gus Wiseman, Feb 23 2018: (Start)
The a(3) = 9 normal semistandard Young tableaux:
1 1 2 1 3 1 2 1 1 1 2 3 1 2 2 1 1 2 1 1 1
2 3 2 2 2
3
(End)
From Gus Wiseman, Nov 14 2018: (Start)
The a(4) = 33 matrices:
[4]
.
[30][21][20][11][10][02][01]
[01][10][02][11][03][20][12]
.
[200][200][110][101][100][100][100][100][011][010][010][010][001][001][001]
[010][001][100][010][020][011][010][001][100][110][101][100][020][010][001]
[001][010][001][100][001][010][002][011][100][001][010][002][100][101][110]
.
[1000][1000][1000][1000][0100][0100][0010][0010][0001][0001]
[0100][0100][0010][0001][1000][1000][0100][0001][0100][0010]
[0010][0001][0100][0010][0010][0001][1000][1000][0010][0100]
[0001][0010][0001][0100][0001][0010][0001][0100][1000][1000]
(End)
MAPLE
gf:= proc(j) local k, n; add(add((-1)^(n-k) *binomial(n, k) *(1-x)^(-k) *(1-x^2)^(-binomial(k, 2)), k=0..n), n=0..j) end: a:= n-> coeftayl(gf(n+1), x=0, n): seq(a(n), n=0..25); # Alois P. Heinz, Sep 25 2008
MATHEMATICA
Table[Sum[SeriesCoefficient[1/(2^(k+1)*(1-x)^k*(1-x^2)^(k*(k-1)/2)), {x, 0, n}], {k, 0, Infinity}], {n, 0, 20}] (* Vaclav Kotesovec, Jul 03 2014 *)
multsubs[set_, k_]:=If[k==0, {{}}, Join@@Table[Prepend[#, set[[i]]]&/@multsubs[Drop[set, i-1], k-1], {i, Length[set]}]]; Table[Length[Select[multsubs[Tuples[Range[n], 2], n], And[Union[First/@#]==Range[Max@@First/@#], Union[Last/@#]==Range[Max@@Last/@#], Sort[Reverse/@#]==#]&]], {n, 5}] (* Gus Wiseman, Nov 14 2018 *)
KEYWORD
easy,nonn
AUTHOR
Vladeta Jovovic, Mar 03 2008
EXTENSIONS
More terms from Alois P. Heinz, Sep 25 2008
STATUS
approved