A289312 -id:A289312 - cp's OEIS frontend

A289313 The number of upper-triangular matrices with integer entries whose absolute sum is equal to n and such that each row contains a nonzero entry.

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

Offset: 0

Peter Bala, Jul 02 2017

A row-Fishburn matrix of size n is defined to be an upper-triangular matrix with nonnegative integer entries which sum to n and such that each row contains a nonzero entry. See A158691.
Here we consider generalized row-Fishburn matrices where we allow the row_Fishburn matrices to have positive and negative nonzero entries. We define the size of a generalized row-Fishburn matrix to be the absolute sum of the matrix entries. This sequence gives the number of generalized row-Fishburn matrices of size n.
Alternatively, this sequence gives the number of 2-colored row-Fishburn matrices of size n, that is, ordinary row-Fishburn 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 row-Fishburn matrices A179525 (i.e., row-Fishburn 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 row-Fishburn 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.

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

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

Seiichi Manyama, Table of n, a(n) for n = 0..200
Hsien-Kuei Hwang and Emma Yu Jin, Asymptotics and statistics on Fishburn matrices and their generalizations, arXiv:1911.06690 [math.CO], 2019.

Cf. A022493, A158691, A179525, A289312.

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

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*i-1) ) );
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 ).
a(n) ~ 2^(3*n+2) * 3^(n+1) * n^(n + 1/2) / (exp(n) * Pi^(2*n + 3/2)). - Vaclav Kotesovec, Aug 31 2023

A289316 The number of upper-triangular matrices whose nonzero entries are positive odd numbers summing to n and each row contains a nonzero entry.

Original entry on oeis.org

1, 1, 2, 8, 37, 219, 1557, 12994, 124427, 1344506, 16178891, 214522339, 3107144562, 48805300668, 826268787588, 14998055299920, 290550119360174, 5983278021430064, 130512410617529321, 3006012061455129053, 72900477505718600661

Offset: 0

Peter Bala, Jul 24 2017

A row-Fishburn matrix of size n is defined to be an upper-triangular matrix with nonnegative integer entries which sum to n and each row contains a nonzero entry. See A158691. Here we are considering row-Fishburn matrices where the nonzero entries are all odd.

The g.f. F(x) for primitive row_Fishburn matrices (i.e., row_Fishburn matrices with entries restricted to the set {0,1}), is F(x) = Sum_{n>=0} Product_{k=1..n} ( (1 + x)^k - 1 ). See A179525. Let C(x) = x/(1 - x^2) = x + x^3 + x^5 + x^7 + .... Then appplying Lemma 2.2.22 of Goulden and Jackson gives the g.f. for the present sequence as the composition F(C(x)).

			a(3) = 8: The eight row-Fishburn matrices of size 3 with odd nonzero entries are
(3) /1 1\
    \0 1/
/1 0 0\  /0 1 0\  /0 0 1\
|0 1 0|  |0 1 0|  |0 1 0|
\0 0 1/  \0 0 1/  \0 0 1/
/1 0 0\  /0 1 0\  /0 0 1\
|0 0 1|  |0 0 1|  |0 0 1|
\0 0 1/  \0 0 1/  \0 0 1/

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

Seiichi Manyama, Table of n, a(n) for n = 0..200
Hsien-Kuei Hwang, Emma Yu Jin, Asymptotics and statistics on Fishburn matrices and their generalizations, arXiv:1911.06690 [math.CO], 2019.

Cf. A158691, A179525, A289312, A289313, A289314, A289315, A289317.

C:= x -> x/(1 - x^2):
G:= add(mul( (1 + C(x))^k - 1, k=1..n),n=0..20):
S:= series(G,x,21):
seq(coeff(S,x,j),j=0..20);

G.f.: A(x) = Sum_{n >= 0} Product_{k = 1..n} ( (1 + x/(1 - x^2))^k - 1 ).

a(n) ~ 12^(n+1) * n^(n + 1/2) / (exp(n + Pi^2/24) * Pi^(2*n + 3/2)). - Vaclav Kotesovec, Aug 31 2023

A289317 The number of upper-triangular matrices whose nonzero entries are positive odd numbers summing to n and each row and each column contains a nonzero entry.

Original entry on oeis.org

1, 1, 1, 3, 7, 23, 84, 364, 1792, 9953, 61455, 417720, 3098515, 24902930, 215538825, 1998518430, 19761943208, 207571259703, 2307812703419, 27075591512866, 334263981931669

Offset: 0

Peter Bala, Jul 25 2017

A Fishburn matrix of size n is defined to be an upper-triangular matrix with nonnegative integer entries which sum to n and each row and each column contains a nonzero entry. See A022493. Here we are considering Fishburn matrices where the nonzero entries are all odd.

The g.f. for primitive Fishburn matrices (i.e., Fishburn matrices with entries restricted to the set {0,1}), is F(x) = Sum_{n>=0} Product_{k=1..n} ( 1 - 1/(1 + x)^k ). See A138265. Let C(x) = x/(1 - x^2) = x + x^3 + x^5 + x^7 + .... Then applying Lemma 2.2.22 of Goulden and Jackson gives the g.f. for this sequence as the composition F(C(x)).

			a(4) = 7: The Fishburn matrices of size 4 with odd nonzero entries are
/3 0\ /1 0\
\0 1/ \0 3/
/1 1 0\ /1 0 1\ /1 0 0\
|0 1 0| |0 1 0| |0 1 1|
\0 0 1/ \0 0 1/ \0 0 1/
/1 1 0\
|0 0 1|
\0 0 1/
/1 0 0 0\
|0 1 0 0|
|0 0 1 0|
\0 0 0 1/

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

Seiichi Manyama, Table of n, a(n) for n = 0..200
Hsien-Kuei Hwang, Emma Yu Jin, Asymptotics and statistics on Fishburn matrices and their generalizations, arXiv:1911.06690 [math.CO], 2019.

Cf. A022493, A138265, A289312, A289313, A289314, A289315, A289316.

C:= x -> x/(1 - x^2):
G:= add(mul( 1 - 1/(1 + C(x))^k, k=1..n), n=0..20):
S:= series(G,x,21):
seq(coeff(S,x,j),j=0..20);

G.f.: A(x) = Sum_{n >= 0} Product_{k = 1..n} ( 1 - 1/(1 + x/(1 - x^2))^k ).

a(n) ~ 2^(n + 5/2) * 3^(n + 3/2) * n^(n+1) / (exp(n + Pi^2/12) * Pi^(2*n + 2)). - Vaclav Kotesovec, Aug 31 2023

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A289313 The number of upper-triangular matrices with integer entries whose absolute sum is equal to n and such that each row contains a nonzero entry.

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A289316 The number of upper-triangular matrices whose nonzero entries are positive odd numbers summing to n and each row contains a nonzero entry.

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A289317 The number of upper-triangular matrices whose nonzero entries are positive odd numbers summing to n and each row and each column contains a nonzero entry.

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