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A138177
Triangle T(n,k) read by rows: number of k X k symmetric matrices with nonnegative integer entries and without zero rows or columns such that sum of all entries is equal to n, n>=1, 1<=k<=n.
6
1, 1, 2, 1, 4, 4, 1, 7, 15, 10, 1, 10, 36, 52, 26, 1, 14, 74, 176, 190, 76, 1, 18, 132, 460, 810, 696, 232, 1, 23, 222, 1060, 2705, 3756, 2674, 764, 1, 28, 347, 2180, 7565, 15106, 17262, 10480, 2620, 1, 34, 525, 4204, 19013, 51162, 83440, 80816, 42732, 9496, 1, 40
OFFSET
1,3
COMMENTS
See the Brualdi/Ma reference for the connection to A161126. - Joerg Arndt, Nov 02 2014
T(n,k) is also the number of semistandard Young tableaux of size n whose entries span the interval 1..k. See also Gus Wiseman's comment in A138178. The T(4,2) = 7 semi-standard Young tableaux of size 4 spanning the interval 1..2 are:
11 122 112 111 1222 1122 1112
22 2 2 2 . - Jacob Post, Jun 15 2018
LINKS
Richard A. Brualdi, Shi-Mei Ma, Enumeration of involutions by descents and symmetric matrices, European Journal of Combinatorics, vol.43, pp.220-228, (January 2015).
FindStat - Combinatorial Statistic Finder, Semistandard Young tableaux
Samantha Dahlberg, Combinatorial Proofs of Identities Involving Symmetric Matrices, arXiv:1410.7356 [math.CO], (27-October-2014)
FORMULA
T(n,k) = Sum_{i=0..k} (-1)^i * binomial(k,i) * A210391(n,k-i). - Alois P. Heinz, Apr 06 2015
EXAMPLE
Triangle T(n,k) begins:
1;
1, 2;
1, 4, 4;
1, 7, 15, 10;
1, 10, 36, 52, 26;
1, 14, 74, 176, 190, 76;
1, 18, 132, 460, 810, 696, 232;
1, 23, 222, 1060, 2705, 3756, 2674, 764;
...
MAPLE
gf:= k-> 1/((1-x)^k*(1-x^2)^(k*(k-1)/2)):
A:= (n, k)-> coeff(series(gf(k), x, n+1), x, n):
T:= (n, k)-> add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k):
seq(seq(T(n, k), k=1..n), n=1..12); # Alois P. Heinz, Apr 06 2015
MATHEMATICA
gf[k_] := 1/((1-x)^k*(1-x^2)^(k*(k-1)/2)); A[n_, k_] := Coefficient[ Series [gf[k], {x, 0, n+1}], x, n]; T[n_, k_] := Sum[(-1)^j*Binomial[k, j]*A[n, k-j], {j, 0, k}]; Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 31 2016, after Alois P. Heinz *)
CROSSREFS
Cf. (row sums) A138178, A135589, A135588, A161126, A210391.
Main diagonal gives A000085. - Alois P. Heinz, Apr 06 2015
T(2n,n) gives A266305.
T(n^2,n) gives A268309.
Sequence in context: A214984 A118976 A210235 * A101559 A220537 A229717
KEYWORD
nonn,tabl
AUTHOR
Vladeta Jovovic, Mar 03 2008
STATUS
approved