

A289313


The number of uppertriangular matrices with integer entries whose absolute sum is equal to n and such that each row contains a nonzero entry.


3



1, 2, 10, 74, 722, 8786, 128218, 2182554, 42456226, 929093538, 22590839466, 604225121258, 17630145814898, 557285515817970, 18970857530674554, 691929648113663802, 26919562120779248962, 1112769248605003393858, 48704349211392743606602
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OFFSET

0,2


COMMENTS

A rowFishburn matrix of size n is defined to be an uppertriangular matrix with nonnegative integer entries which sum to n and such that each row contains a nonzero entry. See A158691.
Here we consider generalized rowFishburn matrices where we allow the row_Fishburn matrices to have positive and negative nonzero entries. We define the size of a generalized rowFishburn matrix to be the absolute sum of the matrix entries. This sequence gives the number of generalized rowFishburn matrices of size n.
Alternatively, this sequence gives the number of 2colored rowFishburn matrices of size n, that is, ordinary rowFishburn matrices of size n where each nonzero entry in the matrix can have one of two different colors.
More generally, if F(x) = Sum_{n >= 0} ( Product_{i = 1..n} (1 + x)^i  1 ) is the o.g.f. for primitive rowFishburn matrices A179525 (i.e., rowFishburn matrices with entries restricted to the set {0,1}) and C(x) := c_1*x + c_2*x^2 + ..., where c_i is a sequence of nonnegative integers, then the composition F(C(x)) is the o.g.f. for colored rowFishburn matrices where entry i in the matrix can have one of c_i different colors: c_i = 0 for some i means i does not appear as an entry in the Fishburn matrix. This result is an application of Lemma 2.2.22 of Goulden and Jackson.


REFERENCES

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, p. 42.


LINKS

Table of n, a(n) for n=0..18.
HsienKuei Hwang, Emma Yu Jin, Asymptotics and statistics on Fishburn matrices and their generalizations, arXiv:1911.06690 [math.CO], 2019.


FORMULA

O.g.f.: Sum_{n >= 0} ( Product_{i = 1..n} ((1 + x)/(1  x))^i  1 ).
The o.g.f. has several alternative forms:
Sum_{n >= 0} ( Product_{i = 1..n} ( 1  ((1  x)/(1 + x))^(2*i1) ) );
Sum_{n >= 0} ((1  x)/(1 + x))^(n+1) * ( Product_{i = 1..n} 1  ((1  x)/(1 + x))^(2*i) );
1/2*( 1 + Sum_{n >= 0} ((1 + x)/(1  x))^((n+1)*(n+2)/2) * Product_{i = 1..n} ( 1  ((1  x)/(1 + x))^i ) ).
Conjectural g.f.: Sum_{n >= 0} ((1 + x)/(1  x))^((n+1)*(2*n+1)) * Product_{i = 1..2*n} ( ((1  x)/(1 + x))^i  1 ).


EXAMPLE

a(2) = 10: The ten generalized rowFishburn matrices of size 2 are
(+2),
/+1 0\ and /0 +1\
   
\0 +1/ \0 +1/.


MAPLE

G:= add(mul( ((1 + x)/(1  x))^i  1, i=1..n), n=0..20):
S:= series(G, x, 21):
seq(coeff(S, x, j), j=0..20);
#  Peter Bala, Jul 24 2017


CROSSREFS

Cf. A022493, A158691, A179525, A289312.
Sequence in context: A185971 A000698 A301932 * A092881 A004123 A086352
Adjacent sequences: A289310 A289311 A289312 * A289314 A289315 A289316


KEYWORD

nonn,easy


AUTHOR

Peter Bala, Jul 02 2017


STATUS

approved



