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A120733 Number of matrices with nonnegative integer entries and without zero rows or columns such that sum of all entries is equal to n. 8
1, 1, 5, 33, 281, 2961, 37277, 546193, 9132865, 171634161, 3581539973, 82171451025, 2055919433081, 55710251353953, 1625385528173693, 50800411296363617, 1693351638586070209, 59966271207156833313, 2248276994650395873861, 88969158875611127548481 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

The number of such matrices up to rows/columns permutations are given in A007716.

Dimensions of the graded components of the Hopf algebra MQSym (Matrix quasi-symmetric functions). - Jean-Yves Thibon (jyt(AT)univ-mlv.fr), Oct 23 2006

From Kyle Petersen, Aug 10 2016: (Start)

Number of cells in the two-sided Coxeter complex of the symmetric group. Inclusion of faces corresponds to refinement of matrices, see Section 6 of Petersen paper. The number of cells in the type B analog is given by A275787.

Also known as "two-way contingency tables" in the Diaconis-Gangolli reference. (End)

REFERENCES

Sara C. Billey, M Konvalinka, TK Petersen, W Slofstra, B. E. Tenner, Parabolic double cosets in Coxeter groups, Discrete Mathematics and Theoretical Computer Science, Submitted, 2016; http://www.math.washington.edu/~billey/papers/DoubleCosets.pdf

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..400

P. Diaconis and A. Gangolli, Rectangular arrays with fixed margins, Discrete probability and algorithms (Minneapolis, MN, 1993), 15-41, IMA Vol. Math. Appl., 72, Springer, New York, 1995.

G. Duchamp, F. Hivert and J.-Y. Thibon, Noncommutative symmetric functions VI: Free quasi-symmetric functions and related algebras, Internat. J. Alg. Comp. 12 (2002), 671-717

Vaclav Kotesovec, Asymptotics of the sequence A120733

E. Munarini, M. Poneti, S. Rimaldi, Matrix compositions, JIS 12 (2009) 09.4.8, Remark 30.

T. K. Petersen, A two-sided analogue of the Coxeter complex, arXiv:1607.00086 [math.CO], (2016).

FORMULA

a(n) = (1/n!)*Sum_{k=0..n} (-1)^(n-k)*Stirling1(n,k)*A000670(k)^2.

G.f.: Sum_{m>=0,n>=0} Sum_{j=0..n} (-1)^(n-j)*C(n,j)*((1-x)^(-j)-1)^m.

a(n) = Sum_{r>=0,s>=0} binomial(r*s+n-1,n)/2^(r+s+2).

G.f.: Sum_{n>=0} 1/(2-(1-x)^(-n))/2^(n+1). - Vladeta Jovovic, Oct 30 2006

a(n) ~ 2^(log(2)/2-2) * n! / (log(2))^(2*n+2). - Vaclav Kotesovec, May 07 2014

EXAMPLE

a(2) = 5:

[1 0]   [0 1]   [1]   [1 1]   [2]

[0 1]   [1 0]   [1]

MAPLE

t1 := M -> add( add( add( (-1)^(n-j)*binomial(n, j)*((1-x)^(-j)-1)^m, j=0..n), n=0..M), m=0..M); s := series(t1(20), x, 20); gfun[seriestolist](%); # N. J. A. Sloane, Jan 14 2009

MATHEMATICA

a[n_] := Sum[2^(-2-r-s)*Binomial[n+r*s-1, n], {r, 0, Infinity}, {s, 0, Infinity}]; Table[Print[an = a[n]]; an, {n, 0, 19}] (* Jean-Fran├žois Alcover, May 15 2012, after Vladeta Jovovic *)

Flatten[{1, Table[1/n!*Sum[(-1)^(n-k)*StirlingS1[n, k]*Sum[m!*StirlingS2[k, m], {m, 1, k}]^2, {k, 1, n}], {n, 1, 20}]}] (* Vaclav Kotesovec, May 07 2014 *)

CROSSREFS

Cf. A007322 (partial sums), A007716, A101370, A120732, A261838.

Row sums of A261781.

Sequence in context: A215671 A049377 A129890 * A218496 A144792 A291846

Adjacent sequences:  A120730 A120731 A120732 * A120734 A120735 A120736

KEYWORD

nonn

AUTHOR

Vladeta Jovovic, Aug 18 2006, Aug 21 2006

EXTENSIONS

More terms from N. J. A. Sloane, Jan 14 2009

STATUS

approved

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Last modified September 26 06:54 EDT 2017. Contains 292502 sequences.