A138265 -id:A138265 - cp's OEIS frontend

A194530 Number of unlabeled rigid interval posets with n non-maximal and 2 maximal elements.

0, 1, 3, 9, 32, 132, 623, 3314, 19628, 128126, 914005, 7074517, 59050739, 528741491, 5055414317, 51406084221, 553946196892, 6305737560455, 75610546284387, 952559077043183, 12579235034203780, 173759983171005721, 2505751777457313815, 37657189917162605826

Offset: 0

Alois P. Heinz, Aug 28 2011

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..200
Soheir Mohamed Khamis, Exact Counting of Unlabeled Rigid Interval Posets Regarding or Disregarding Height, Order (journal) (2011).

2nd column of A193344, Column k=2 of A218757.

Cf. A138265.

w:= proc(t) option remember;
      `if`(t=0, 1, expand(convert(series(series(z +z*(subs(
           z=z+y+y*z, w(t-1)) -w(t-1)), z, t+1), y, t+1), polynom)))
    end:
a:= n-> coeff(coeff(w(2+n), z, 2), y, n):
seq(a(n), n=0..50);

w[t_] := w[t] = If[t == 0, 1, Expand[Normal[Series[Series[z+z*((w[t-1] /. z -> z+y+y*z)-w[t-1]), {z, 0, t+1}], {y, 0, t+1}]]]]; a[n_] := a[n] = Coefficient[Coefficient[w[2+n], z, 2], y, n]; Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 0, 50}] (* Jean-François Alcover, Mar 05 2014, after Alois P. Heinz *)

a(n) = [ y^n z^2 ] W(y,z); W(y,z) = z + z*(W(y,y+z+yz) - W(y,z)).
From Peter Bala, Aug 21 2023: (Start)
Conjectural g.f.:
1) A(x) = Sum_{n >= 0} n*(Product_{i = 1..n} 1 - 1/(1+x)^i).
2) A(x) = (1/2)*Sum_{n >= 0} n*(n+1)/(1+x)^(n+1) * (Product_{i = 1..n} 1 - 1/(1+x)^i). Cf. A138265. (End)

A289312 The number of upper-triangular matrices with integer entries whose absolute sum is equal to n, and each row and column contains at least one nonzero entry.

Original entry on oeis.org

1, 2, 6, 26, 142, 946, 7446, 67658, 697118, 8031586, 102312486, 1427905658, 21666671534, 355138949394, 6253348428598, 117720540700842, 2359368991571518, 50157679523340994, 1127327559500923974, 26709016625807923418, 665292778385210384078

Offset: 0

Peter Bala, Jul 02 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 column contains a nonzero entry. See A022493.
Here we consider generalized Fishburn matrices where we allow the Fishburn matrices to have positive and negative nonzero entries. We define the size of a generalized Fishburn matrix to be the absolute sum of the matrix entries. This sequence gives the number of generalized Fishburn matrices of size n.
Alternatively, this sequence gives the number of 2-colored Fishburn matrices of size n, that is, ordinary 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 - 1/(1 + x)^i) is the o.g.f. for primitive Fishburn matrices A138265 (i.e., 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 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) = 6: The six upper triangular matrices of size 2 with no zero rows or columns are (+-2) and
  /+-1  0\
  |      |.
  \0  +-1/

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

Alois P. Heinz, Table of n, a(n) for n = 0..300
Ankush Goswami, Abhash Kumar Jha, Byungchan Kim, and Robert Osburn, Asymptotics and sign patterns for coefficients in expansions of Habiro elements, arXiv:2204.02628 [math.NT], 2022.
Hsien-Kuei Hwang and Emma Yu Jin, Asymptotics and statistics on Fishburn matrices and their generalizations, arXiv:1911.06690 [math.CO], 2019.

Cf. A022493, A138265, A289313.

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

m = 21; Sum[Product[1 - ((1-x)/(1+x))^k + O[x]^m, {k, 1, n}], {n, 0, m}] // CoefficientList[#, x]& (* Jean-François Alcover, Feb 28 2020 *)

G.f.: Sum_{n >= 0} Product_{k = 1..n} 1 - ((1 - x)/(1 + x))^k.
Alternative g.f.: Sum_{n >= 0} ((1 + x)/(1 - x))^(n+1) * Product_{k = 1..n} 1 - ((1 + x)/(1 - x))^k.
G.f.: B(2*x/(1+x)) where B(x) is the g.f. of A022493. - Michael D. Weiner, Feb 28 2019
a(n) ~ 2^(2*n + 5/2) * 3^(n + 3/2) * n^(n+1) / (exp(n) * Pi^(2*n+2)). - Vaclav Kotesovec, Aug 31 2023

A193344 Triangle read by rows: T(n,m) (n>=0, 1 <= m <= n+1) = number of unlabeled rigid interval posets with n non-maximal and m maximal elements.

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 2, 9, 13, 6, 5, 32, 72, 69, 24, 16, 132, 409, 605, 432, 120, 61, 623, 2480, 5016, 5498, 3120, 720, 271, 3314, 16222, 41955, 62626, 54370, 25560, 5040, 1372, 19628, 114594, 363123, 690935, 814690, 584580, 234360, 40320

Offset: 0

N. J. A. Sloane, Aug 26 2011

			Triangle begins
1
1 1
1 3 2
2 9 13 6
5 32 72 69 24
16 132 409 605 432 120
61 623 2480 5016 5498 3120 720
271 3314 16222 41955 62626 54370 25560 5040
1372 19628 114594 363123 690935 814690 584580 ...

Alois P. Heinz, Rows n = 0..50, flattened
Soheir Mohamed Khamis, Exact Counting of Unlabeled Rigid Interval Posets Regarding or Disregarding Height, Order 29, pp. 443-461 (2012).

First column is A138265, second column is A194530.

w:= proc(t) option remember;
      `if`(t=0, 1, expand(convert(series(series(z+z*(subs(
       z=z+y+y*z, w(t-1)) -w(t-1)), z, t+1), y, t+1), polynom)))
    end:
T:= (n,m)-> coeff(coeff(w(m+n), z, m), y, n):
seq(seq(T(n, m), m=1..n+1), n=0..10);  # Alois P. Heinz, Aug 27 2011

w[t_] := w[t] = If[t == 0, 1, Expand[Normal[Series[Series[z+z*((w[t-1] /. z -> z+y+y*z)-w[t-1]), {z, 0, t+1}], {y, 0, t+1}]]]]; T[n_, m_] := Coefficient[Coefficient[w[m+n], z, m], y, n]; Table[Table[T[n, m], {m, 1, n+1}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Mar 05 2014, after Alois P. Heinz *)

T(n,m) = [ y^n z^m ] W(y,z); W(y,z) = z + z*(W(y,y+z+yz) - W(y,z)).

A193357 Triangle read by rows: T(n,k) (n>=1, 1 <= k <= n) = number of n-element unlabeled rigid interval posets of height k.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 8, 6, 1, 0, 1, 20, 29, 10, 1, 0, 1, 51, 128, 75, 15, 1, 0, 1, 134, 555, 500, 160, 21, 1, 0, 1, 367, 2427, 3190, 1480, 301, 28, 1, 0, 1, 1048, 10864, 20129, 12835, 3661, 518, 36, 1, 0, 1, 3119, 50145, 127912, 108308, 41041, 7980, 834, 45, 1, 0, 1, 9655, 239581, 827053, 907923, 441553, 111671, 15828, 1275, 55, 1

Offset: 1

N. J. A. Sloane, Aug 26 2011

			Triangle begins
1
0 1
0 1 1
0 1 3 1
0 1 8 6 1
0 1 20 29 10 1
0 1 51 128 75 15 1
0 1 134 555 500 160 21 1
0 1 367 2427 3190 1480 301 28 1
0 1 1048 10864 20129 12835 3661 518 36 1
0 1 3119 50145 127912 108308 41041 7980 834 45 1
0 1 9655 239581 827053 907923 441553 111671 15828 1275 55 1
0 1 31024 1186941 5474614 7654921 4664814 1482110 26944 29175 1870 66 1
...

Soheir Mohamed Khamis, Exact Counting of Unlabeled Rigid Interval Posets Regarding or Disregarding Height, Order 29, pp. 443-461 (2012).

Row sums = A138265. Cf. A193387.

A193548 Decimal expansion of exp(Pi^2/12).

Original entry on oeis.org

2, 2, 7, 6, 1, 0, 8, 1, 5, 1, 6, 2, 5, 7, 3, 4, 0, 9, 4, 7, 9, 1, 0, 6, 1, 4, 1, 2, 0, 3, 1, 4, 9, 7, 4, 4, 6, 6, 9, 7, 9, 7, 4, 2, 6, 0, 3, 0, 0, 2, 3, 7, 7, 5, 6, 1, 5, 5, 1, 6, 1, 7, 0, 9, 8, 2, 7, 5, 0, 6, 3, 7, 2, 8, 6, 3, 0, 1, 4, 3, 1, 8, 6, 6, 8, 4, 6, 5, 7

Offset: 1

John M. Campbell, Jul 30 2011

Cf. A001113, A022493, A122214, A122215, A122216, A122217, A138265, A207651, A242153, A242154, A242155, A242156, A242157, A242158, A242159, A242160, A242161, A242162, A242163, A242164.

Product[Product[k^((1/(n+1))*(-1)^(k)*Binomial[n,k-1]*HarmonicNumber[n]),{k,1,n+1}],{n,1,Infinity}]
RealDigits[E^(Pi^2/12), 10, 100]

exp(Pi^2/12) \\ Charles R Greathouse IV, Jul 30 2011

exp(Pi^2/12) = Product_{n>=1} Product_{k=1..n+1} k^(1/(n+1)) * H(n) * (-1)^k * binomial(n, k-1) where H(n) is the n-th harmonic number.

exp(Pi^2/12) = lim_{n -> infinity} Product_{k=1..n} (1 + k/n)^(1/k). - Peter Bala, Feb 14 2015

A218757 Triangle read by rows: T(n,k) is the number of length-n ascent sequences without flat steps, containing k zeros.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 2, 3, 0, 0, 0, 5, 9, 2, 0, 0, 0, 16, 32, 13, 0, 0, 0, 0, 61, 132, 72, 6, 0, 0, 0, 0, 271, 623, 409, 69, 0, 0, 0, 0, 0, 1372, 3314, 2480, 605, 24, 0, 0, 0, 0, 0, 7795, 19628, 16222, 5016, 432, 0, 0, 0, 0, 0, 0, 49093, 128126, 114594, 41955, 5498, 120, 0, 0, 0, 0, 0

Offset: 0

Joerg Arndt, Nov 05 2012

An ascent sequence is a sequence [d(1), d(2), ..., d(n)] where d(1)=0, d(k)>=0, and d(k) <= asc([d(1), d(2), ..., d(k-1)]) and asc(.) gives the number of ascents of its argument.  Here we consider only those where adjacent digits are unequal.
The rows are the upward diagonals of A193344.
Row sums are A138265.
The column for k=1 is A138265 (i.e. the sum of row n equals the element for k=1 of the row n+1): the length-(n+1) sequences with one zero (which must be at the initial position) are formed by incrementing each digit of the length-n sequences and prepending zero.
The second column is A194530.

			Triangle starts:
1,
0, 1,
0, 1, 0,
0, 1, 1, 0,
0, 2, 3, 0, 0,
0, 5, 9, 2, 0, 0,
0, 16, 32, 13, 0, 0, 0,
0, 61, 132, 72, 6, 0, 0, 0,
0, 271, 623, 409, 69, 0, 0, 0, 0,
0, 1372, 3314, 2480, 605, 24, 0, 0, 0, 0,
0, 7795, 19628, 16222, 5016, 432, 0, 0, 0, 0, 0,
0, 49093, 128126, 114594, 41955, 5498, 120, 0, 0, 0, 0, 0,
0, 339386, 914005, 872336, 363123, 62626, 3120, 0, 0, 0, 0, 0, 0,
...
The A138265(5) = 16 length-5 ascent sequences without flat steps are (dots for zeros):
[ #]     ascent-seq.  #zeros
[ 1]    [ . 1 . 1 . ]  3
[ 2]    [ . 1 . 1 2 ]  2
[ 3]    [ . 1 . 1 3 ]  2
[ 4]    [ . 1 . 2 . ]  3
[ 5]    [ . 1 . 2 1 ]  2
[ 6]    [ . 1 . 2 3 ]  2
[ 7]    [ . 1 2 . 1 ]  2
[ 8]    [ . 1 2 . 2 ]  2
[ 9]    [ . 1 2 . 3 ]  2
[10]    [ . 1 2 1 . ]  2
[11]    [ . 1 2 1 2 ]  1
[12]    [ . 1 2 1 3 ]  1
[13]    [ . 1 2 3 . ]  2
[14]    [ . 1 2 3 1 ]  1
[15]    [ . 1 2 3 2 ]  1
[16]    [ . 1 2 3 4 ]  1
There are 5 sequences with 1 zero, 9 with two zeros and 2 with three zeros, so the row for n==5 is 0, 5, 9, 2, 0, 0.

Joerg Arndt and Alois P. Heinz, Rows n = 0..65, flattened (rows 0..15 from Joerg Arndt)

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

A238425 Number of descent sequences of length n without two consecutive identical elements (descent sequences without flat steps).

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 11, 34, 124, 512, 2380, 12294, 69972, 435399, 2942672, 21478882, 168473955, 1413823577, 12644505883, 120097766639, 1207617481139, 12818915877849, 143278176040760, 1682262113899134, 20704109403389717, 266568690074855277, 3583926627760681407

Offset: 0

Joerg Arndt and Alois P. Heinz, Feb 26 2014

nonn

A descent sequence is a sequence [d(1), d(2), ..., d(n)] where d(1)=0, d(k)>=0, and d(k) <= 1 + desc([d(1), d(2), ..., d(k-1)]) where desc(.) gives the number of descents of its argument, see A225588.

			The a(6) = 11 such descent sequences are (dots denote zeros):
01:  [ . 1 . 1 . 1 ]
02:  [ . 1 . 1 . 2 ]
03:  [ . 1 . 1 . 3 ]
04:  [ . 1 . 1 2 . ]
05:  [ . 1 . 1 2 1 ]
06:  [ . 1 . 2 . 1 ]
07:  [ . 1 . 2 . 2 ]
08:  [ . 1 . 2 . 3 ]
09:  [ . 1 . 2 1 . ]
10:  [ . 1 . 2 1 2 ]
11:  [ . 1 . 2 1 3 ]

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 0..200

Cf. A138265 (ascent sequence without two consecutive identical elements).

Cf. A225588 (all descent sequences).

# b(n, i, t): number of length-n postfixes of these sequences with a
#             valid prefix having t descents and rightmost element i.
b:= proc(n, i, t) option remember; `if`(n<1, 1,
      add(`if`(j=i, 0, b(n-1, j, t+`if`(j b(n-1, 0, 0):
seq(a(n), n=0..30);

b[n_, i_, t_] := b[n, i, t] = If[n < 1, 1, Sum[If[j == i, 0, b[n - 1, j, t + If[j < i, 1, 0]]], {j, 0, t + 1}]];
a[n_] := b[n - 1, 0, 0];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 07 2022, after Alois P. Heinz *)

@CachedFunction
def b(n, i, t):
    if n<1:
        return 1
    return sum(b(n-1, j, t+(j

cp's OEIS Frontend

A194530 Number of unlabeled rigid interval posets with n non-maximal and 2 maximal elements.

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A289312 The number of upper-triangular matrices with integer entries whose absolute sum is equal to n, and each row and column contains at least one nonzero entry.

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A193344 Triangle read by rows: T(n,m) (n>=0, 1 <= m <= n+1) = number of unlabeled rigid interval posets with n non-maximal and m maximal elements.

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A193357 Triangle read by rows: T(n,k) (n>=1, 1 <= k <= n) = number of n-element unlabeled rigid interval posets of height k.

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A193548 Decimal expansion of exp(Pi^2/12).

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A218757 Triangle read by rows: T(n,k) is the number of length-n ascent sequences without flat steps, containing k zeros.

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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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A238425 Number of descent sequences of length n without two consecutive identical elements (descent sequences without flat steps).

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Maple