A005321 -id:A005321 - cp's OEIS frontend

A259971 Triangle read by rows: coefficients xi(n,k) arising from the study of completely transitive graphs on n nodes.

1, 1, 1, 2, 5, 3, 10, 40, 51, 21, 122, 644, 1236, 1029, 315, 3346, 21496, 54060, 66780, 40635, 9765, 196082, 1471460, 4527228, 7328580, 6596100, 3134565, 615195

Offset: 1

N. J. A. Sloane, Jul 11 2015

			Triangle begins:
1,
1,1,
2,5,3,
10,40,51,21,
122,644,1236,1029,315,
3346,21496,54060,66780,40635,9765,
196082,1471460,4527228,7328580,6596100,3134565,615195,
...

E. Andresen, K. Kjeldsen, On certain subgraphs of a complete transitively directed graph, Discrete Math. 14 (1976), no. 2, 103-119.
Hsien-Kuei Hwang, Emma Yu Jin, Michael J. Schlosser, Asymptotics and statistics on Fishburn Matrices: dimension distribution and a conjecture of Stoimenow, arXiv:2012.13570 [math.CO], Dec 25 2020, p. 29.

Cf. A259971, A259970.
Diagonals include A005321, A005330
Row sums are also A005321.

A369415 Number A(n,k) of n X n Fishburn matrices with entries in the set {0,1,...,k}; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 12, 10, 0, 1, 4, 36, 264, 122, 0, 1, 5, 80, 2052, 19632, 3346, 0, 1, 6, 150, 9280, 505764, 4606752, 196082, 0, 1, 7, 252, 30750, 5684480, 511718148, 3311447232, 23869210, 0, 1, 8, 392, 83160, 39378750, 17672135680, 2088275673636, 7202118117504, 5939193962, 0

Offset: 0

Alois P. Heinz, Jan 22 2024

Number of upper triangular n X n {0,1,...,k}-matrices with no zero rows or columns.

			A(2,3) = 3*3*4 = 36:
  [10] [10] [10]  [20] [20] [20]  [30] [30] [30]
  [ 1] [ 2] [ 3]  [ 1] [ 2] [ 3]  [ 1] [ 2] [ 3]
.
  [11] [11] [11]  [21] [21] [21]  [31] [31] [31]
  [ 1] [ 2] [ 3]  [ 1] [ 2] [ 3]  [ 1] [ 2] [ 3]
.
  [12] [12] [12]  [22] [22] [22]  [32] [32] [32]
  [ 1] [ 2] [ 3]  [ 1] [ 2] [ 3]  [ 1] [ 2] [ 3]
.
  [13] [13] [13]  [23] [23] [23]  [33] [33] [33]
  [ 1] [ 2] [ 3]  [ 1] [ 2] [ 3]  [ 1] [ 2] [ 3]
.
Square array A(n,k) begins:
  1,    1,       1,         1,           1,            1, ...
  0,    1,       2,         3,           4,            5, ...
  0,    2,      12,        36,          80,          150, ...
  0,   10,     264,      2052,        9280,        30750, ...
  0,  122,   19632,    505764,     5684480,     39378750, ...
  0, 3346, 4606752, 511718148, 17672135680, 305416893750, ...
  ...

Alois P. Heinz, Antidiagonals n = 0..53, flattened
Hsien-Kuei Hwang, Emma Yu Jin, and Michael J. Schlosser, Asymptotics and statistics on Fishburn Matrices: dimension distribution and a conjecture of Stoimenow, arXiv:2012.13570 [math.CO], 2020.
Vít Jelínek, Counting general and self-dual interval orders, Journal of Combinatorial Theory, Series A, Volume 119, Issue 3, April 2012, pp. 599-614; arXiv preprint, arXiv:1106.2261 [math.CO], 2011.
Wikipedia, Peter C. Fishburn

Columns k=0-3 give: A000007, A005321, A289314, A289315.
Rows n=0-3 give: A000012, A001477, A011379, A369423.
Main diagonal gives A369336.

A:= (n, k)-> coeff(series(add(x^j*mul(((k+1)^i-1)/(1+x*
           ((k+1)^i-1)), i=1..j), j=0..n), x, n+1), x, n):
seq(seq(A(n, d-n), n=0..d), d=0..10);

A(n,k) = [x^n] Sum_{j=0..n} x^j * Product_{i=1..j} ((k+1)^i-1)/(1+x*((k+1)^i-1)).

A369336 Number of n X n Fishburn matrices with entries in the set {0,1,...,n}.

Original entry on oeis.org

1, 1, 12, 2052, 5684480, 305416893750, 391129148721673152, 14286237711414132094989064, 17309880507327972883933887341789184, 792117985317303404452447777723478865406570410, 1534214120588806182890487155420702132205591283310000000000

Offset: 0

Alois P. Heinz, Jan 20 2024

nonn

Number of upper triangular n X n {0,1,...,n}-matrices with no zero rows or columns.

			a(0) = 1: [].
a(1) = 1: [1].
a(2) = 12:
  [10] [10] [20] [20]  [11] [11] [21] [21]  [12] [12] [22] [22]
  [ 1] [ 2] [ 1] [ 2]  [ 1] [ 2] [ 1] [ 2]  [ 1] [ 2] [ 1] [ 2].

Alois P. Heinz, Table of n, a(n) for n = 0..35
Hsien-Kuei Hwang, Emma Yu Jin, and Michael J. Schlosser, Asymptotics and statistics on Fishburn Matrices: dimension distribution and a conjecture of Stoimenow, arXiv:2012.13570 [math.CO], 2020.
Vít Jelínek, Counting general and self-dual interval orders, Journal of Combinatorial Theory, Series A, Volume 119, Issue 3, April 2012, pp. 599-614; arXiv preprint, arXiv:1106.2261 [math.CO], 2011.
Wikipedia, Peter C. Fishburn

Cf. A000007, A005321, A289314, A289315.

Main diagonal of A369415.

a:= n-> coeff(series(add(x^j*mul(((n+1)^i-1)/(1+x*
    ((n+1)^i-1)), i=1..j), j=0..n), x, n+1), x, n):
seq(a(n), n=0..10);

a(n) = [x^n] Sum_{j=0..n} x^j * Product_{i=1..j} ((n+1)^i-1)/(1+x*((n+1)^i-1)).

A005016 Certain subgraphs of a directed graph.

Original entry on oeis.org

1, 1, 3, 15, 159, 3903, 214143, 25098495, 6110517759, 3040867308543, 3064498377754623, 6220489664197758975, 25354161321592779612159, 207142125428402158677213183, 3388838467537660347660899221503

Offset: 0

N. J. A. Sloane

nonn

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

E. Andresen and K. Kjeldsen, On certain subgraphs of a complete transitively directed graph, Discrete Math. 14 (1976), no. 2, 103-119.
William T. Dugan, On the f-vectors of flow polytopes for the complete graph, Sém. Lotharingien Comb., Proc. 36th Conf. Formal Power Series Alg. Comb. (2024) Vol. 91B, Art. No. 101. See p. 3.
Hsien-Kuei Hwang, Emma Yu Jin, and Michael J. Schlosser, Asymptotics and statistics on Fishburn Matrices: dimension distribution and a conjecture of Stoimenow, arXiv:2012.13570 [math.CO], 2020.

Cf. A005321.

G.f.: Sum(x^n*Product((2^i-1)/(1+(2^i-2)*x),i = 1 .. n),n = 0 .. infinity). - Vladeta Jovovic, Mar 10 2008

More terms from Vladeta Jovovic, Mar 10 2008

a(0), a(14) from Max Alekseyev, May 04 2010

A072925 Probably an erroneous version of A002845.

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 17, 36, 78, 171, 379, 888, 1944

Offset: 1

dead

The old entry with this sequence number was a duplicate of A037444.

J. Q. Longyear, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press, 1995; entry M1139.

F. Goebel and R. P. Nederpelt, The number of numerical outcomes of iterated powers, Amer. Math. Monthly, 80 (1971), 1097-1103.
R. K. Guy and J. L. Selfridge, The nesting and roosting habits of the laddered parenthesis, Amer. Math. Monthly 80 (8) (1973), 868-876.
J. Longyear, T. Trotter, N. J. A. Sloane, Correspondence

A259876 Triangle of numbers S(n,k) (0 <= k <= n) arising in the enumeration of interval orders without duplicated holdings.

Original entry on oeis.org

1, 1, -1, 3, -3, 1, 21, -21, 7, -1, 315, -315, 105, -15, 1, 9765, -9765, 3255, -465, 31, -1, 615195, -615195, 205065, -29295, 1953, -63, 1, 78129765, -78129765, 26043255, -3720465, 248031, -8001, 127, -1, 19923090075, -19923090075, 6641030025, -948718575, 63247905, -2040255, 32385, -255, 1

Offset: 0

N. J. A. Sloane, Jul 09 2015

			Triangle begins:
     1;
     1,    -1;
     3,    -3,    1;
    21,   -21,    7,   -1;
   315,  -315,  105,  -15,  1;
  9765, -9765, 3255, -465, 31, -1;
  ...

T. L. Greenough, Enumeration of interval orders without duplicated holdings, Preprint, circa 1976.

T. L. Greenough, Enumeration of interval orders without duplicated holdings, Preprint, circa 1976. [Annotated scanned copy]
T. L. Greenough, Enumeration of interval orders without duplicated holdings, Notices of the AMS, Vol 23-2, February 1976, Issue 168, pages A-314 and A-315. [Mentions this paper]

Row sums give A005327.
Column k=0 gives A005329.
Main diagonal gives A033999.
T(n+1,n) gives A225883(n+1).

T(n,k) = qfactorial(n)/qfactorial(k)*(-1)^(k), n>=k, where qfactorial(n) is A005329. - Vladimir Kruchinin, Feb 17 2020

More terms from Alois P. Heinz, Feb 17 2020

A318301 Triangle T(n, k) read by rows: T(0, 0) = 1 and T(n, k) = Sum_{i=0..k-1} T(n, i) + Sum_{i=k..n-1} T(n-1, i).

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 10, 18, 33, 61, 122, 234, 450, 867, 1673, 3346, 6570, 12906, 25362, 49857, 98041, 196082, 388818, 771066, 1529226, 3033090, 6016323, 11934605, 23869210, 47542338, 94695858, 188620650, 275712074, 748391058, 1490765793, 2969596981, 5939193962, 11854518714

Offset: 0

Nicolas Nagel, Aug 24 2018

The left edge of the triangle appears to be A005321.

			Triangle begins:
     1
     1    1
     2    3     5
    10   18    33    61
   122  234   450   867  1673
  3346 6570 12906 25362 49857 98041
  ...
T(5, 2) = (3346 + 6570) + (450 + 867 + 1673) = 12906;
T(5, 2) = 2 * 6570 - 234 = 12906.

Cf. A005321.

T(n, k) = if (k == 0, if (n <= 1, 1, 2 * T(n-1, n-1)), 2 * T(n, k-1) - T(n-1, k-1));
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n,k), ", ")); print); \\ Michel Marcus, Aug 25 2018

def T(n, k):
    if k == 0:
        if n == 0 or n == 1:
            return 1
        return 2 * T(n-1, n-1)
    return 2 * T(n, k-1) - T(n-1, k-1)

An equivalent recursion: T(0, 0) = T(1, 0) = 1, T(n, 0) = 2*T(n-1, n-1) if n>=2, T(n, k) = 2*T(n, k-1) - T(n-1, k-1) if n>=k>=1.

cp's OEIS Frontend

A259971 Triangle read by rows: coefficients xi(n,k) arising from the study of completely transitive graphs on n nodes.

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A369415 Number A(n,k) of n X n Fishburn matrices with entries in the set {0,1,...,k}; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A369336 Number of n X n Fishburn matrices with entries in the set {0,1,...,n}.

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A005016 Certain subgraphs of a directed graph.

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A072925 Probably an erroneous version of A002845.

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A259876 Triangle of numbers S(n,k) (0 <= k <= n) arising in the enumeration of interval orders without duplicated holdings.

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A318301 Triangle T(n, k) read by rows: T(0, 0) = 1 and T(n, k) = Sum_{i=0..k-1} T(n, i) + Sum_{i=k..n-1} T(n-1, i).

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