A060491 -id:A060491 - cp's OEIS frontend

A331571 Array read by antidiagonals: A(n,k) is the number of binary matrices with k columns and any number of distinct nonzero rows with n ones in every column and columns in nonincreasing lexicographic order.

1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 4, 3, 0, 1, 1, 8, 23, 0, 0, 1, 1, 16, 290, 184, 0, 0, 1, 1, 32, 4298, 17488, 840, 0, 0, 1, 1, 64, 79143, 2780752, 771305, 0, 0, 0, 1, 1, 128, 1702923, 689187720, 1496866413, 21770070, 0, 0, 0, 1, 1, 256, 42299820, 236477490418, 5261551562405, 585897733896, 328149360, 0, 0, 0, 1

Offset: 0

Andrew Howroyd, Jan 20 2020

The condition that the columns be in nonincreasing order is equivalent to considering nonequivalent matrices up to permutation of columns.

			Array begins:
===============================================================
n\k | 0 1 2   3         4               5                 6
----+----------------------------------------------------------
  0 | 1 1 1   1         1               1                 1 ...
  1 | 1 1 2   4         8              16                32 ...
  2 | 1 0 3  23       290            4298             79143 ...
  3 | 1 0 0 184     17488         2780752         689187720 ...
  4 | 1 0 0 840    771305      1496866413     5261551562405 ...
  5 | 1 0 0   0  21770070    585897733896 30607728081550686 ...
  6 | 1 0 0   0 328149360 161088785679360 ...
  ...
The A(2,2) = 3 matrices are:
   [1 1]  [1 0]  [1 0]
   [1 0]  [1 1]  [0 1]
   [0 1]  [0 1]  [1 1]

Andrew Howroyd, Table of n, a(n) for n = 0..209

Rows n=0..4 are A000012, A011782, A060090, A060491, A331652.

Cf. A330942, A331567, A331569, A331570, A331572, A331653.

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}
D(p, n, k)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); binomial(WeighT(v)[n] + k - 1, k)/prod(i=1, #v, i^v[i]*v[i]!)}
T(n, k)={ my(m=n*k+1, q=Vec(exp(intformal(O(x^m) - x^n/(1-x)))), f=Vec(serlaplace(1/(1+x) + O(x*x^m))/(x-1))); if(n==0, 1, sum(j=1, m, my(s=0); forpart(p=j, s+=(-1)^#p*D(p, n, k), [1, n]); s*sum(i=j, m, q[i-j+1]*f[i]))); }

A(n, k) = Sum_{j=0..k} abs(Stirling1(k, j))*A331567(n, j)/k!.
A(n, k) = Sum_{j=0..k} binomial(k-1, k-j)*A331569(n, j).
A(n, k) = 0 for k > 0, n > 2^(k-1).
A331653(n) = Sum_{d|n} A(n/d, d).

A060492 Triangle T(n,k) of k-block ordered tricoverings of an unlabeled n-set (n >= 3, k = 4..2n).

Original entry on oeis.org

4, 60, 120, 13, 375, 3030, 9030, 5040, 28, 1392, 24552, 207900, 838320, 1345680, 362880, 50, 4020, 130740, 2208430, 20334720, 101752560, 257065200, 261122400, 46569600, 80, 9960, 551640, 16365410, 274814760, 2709457128, 15812198640

Offset: 3

Vladeta Jovovic, Mar 20 2001

A covering of a set is a tricovering if every element of the set is covered by exactly three blocks of the covering.

All columns are polynomials of order binomial(k, 3). - Andrew Howroyd, Jan 30 2020

			Triangle begins:
  [4, 60, 120],
  [13, 375, 3030, 9030, 5040],
  [28, 1392, 24552, 207900, 838320, 1345680, 362880],
  [50, 4020, 130740, 2208430, 20334720, 101752560, 257065200, 261122400, 46569600], [80, 9960, 551640, 16365410, 274814760, 2709457128, 15812198640, 52897521600, 91945022400, 64778313600, 8043235200],
   ...
There are 184 ordered tricoverings of an unlabeled 3-set: 4 4-block, 60 5-block and 120 6-block tricoverings (cf. A060491).

Andrew Howroyd, Table of n, a(n) for n = 3..1522 (rows n=3..40)

Row sums are A060491.
Columns k=4..6 are A060488, A060489, A060490.
Cf. A059443, A059530, A060052, A060092, A060487, A331571.

\\ gives g.f. of k-th column.
ColGf(k) = k!*polcoef(exp(-x + x^2/2 + x^3*y/(3*(1-y)) + O(x*x^k) )*sum(j=0, k, 1/(1-y)^binomial(j, 3)*exp((-x^2/2)/(1-y)^j + O(x*x^k))*x^j/j!), k) \\ Andrew Howroyd, Jan 30 2020

T(n)={my(m=2*n, y='y + O('y^(n+1))); my(g=serlaplace(exp(-x + x^2/2 + x^3*y/(3*(1-y)) + O(x*x^m))*sum(k=0, m, 1/(1-y)^binomial(k, 3)*exp((-x^2/2)/(1-y)^k + O(x*x^m))*x^k/k!))); Mat([Col(p/y^3, -n) | p<-Vec(g)[2..m+1]])}
{ my(A=T(8)); for(n=3, matsize(A)[1], print(A[n, 4..2*n])) } \\ Andrew Howroyd, Jan 30 2020

E.g.f. for ordered k-block tricoverings of an unlabeled n-set is exp(-x+x^2/2+x^3/3*y/(1-y))*Sum_{k=0..inf}1/(1-y)^binomial(k, 3)*exp(-x^2/2*1/(1-y)^n)*x^k/k!.

A060488 Number of 4-block ordered tricoverings of an unlabeled n-set.

Original entry on oeis.org

4, 13, 28, 50, 80, 119, 168, 228, 300, 385, 484, 598, 728, 875, 1040, 1224, 1428, 1653, 1900, 2170, 2464, 2783, 3128, 3500, 3900, 4329, 4788, 5278, 5800, 6355, 6944, 7568, 8228, 8925, 9660, 10434, 11248, 12103, 13000, 13940, 14924, 15953, 17028, 18150, 19320

Offset: 3

Vladeta Jovovic, Mar 20 2001

A covering of a set is a tricovering if every element of the set is covered by exactly three blocks of the covering.
If Y is a 4-subset of an n-set X then, for n>=6, a(n-3) is the number of 3-subsets of X having at most one element in common with Y. - Milan Janjic, Dec 08 2007
Also the number of balls in a triangular pyramid of which all balls located on the edges have been removed such that the remaining pyramid's edges each consist of two adjacent balls. The layers of pyramids of this form start (from the top) 3, 7, 12, 18, 25, 33,... (A055998) with one smaller additional layer 1, 3, 6, 10, 15, 21,... (A000217) at the bottom. Thus, a(n) = A000217(n) + Sum_{k=1..n} A055998(k). Example: a(4) = (3+7+12+18)+10 = 50. - K. G. Stier, Dec 12 2012

Vincenzo Librandi, Table of n, a(n) for n = 3..1000
Amya Luo, Pattern Avoidance in Nonnesting Permutations, Undergraduate Thesis, Dartmouth College (2024). See p. 11.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Essentially the same as A026054. - Vladeta Jovovic, Jun 15 2006
Column k=4 of A060492.
Cf. A060091, A060491.
Fourth column (m=3) of (1, 4)-Pascal triangle A095666.

[(n-2)*(n-1)*(n+9)/6: n in [3..60]]; // Vincenzo Librandi, Jun 15 2011

Table[ 3 (n - 1) (n - 2)/2! + n (n - 1) (n - 2)/3!, {n, 3, 62}] (* Vladimir Joseph Stephan Orlovsky, Jun 14 2011 *)
Table[(n-2)(n-1)(n+9)/6,{n,3,50}] (* or *) LinearRecurrence[{4,-6,4,-1}, {4,13,28,50},50] (* Harvey P. Dale, Jul 21 2012 *)

a(n)=(n-2)*(n-1)*(n+9)/6 \\ Charles R Greathouse IV, Jun 14 2011

a(n) = binomial(n+3, 3) - 6*binomial(n+1, 1) + 8*binomial(n, 0) - 3*binomial(n-1, -1).
G.f.: -y^3*(-4+3*y)/(-1+y)^4.
E.g.f. for ordered k-block tricoverings of an unlabeled n-set is exp(-x+x^2/2+x^3/3*y/(1-y)) * sum(k>=0, 1/(1-y)^binomial(k, 3)*exp(-x^2/2*1/(1-y)^n)*x^k/k! ).
a(n) = (n+9)*binomial(n-1, 2)/3.
a(n) = (n-2)*(n-1)*(n+9)/6. - Zak Seidov, Jun 15 2006
a(3)=4, a(4)=13, a(5)=28, a(6)=50, a(n) = 4*a(n-1)-6*a(n-2)+ 4*a(n-3)- a(n-4). - Harvey P. Dale, Jul 21 2012

A060489 Number of 5-block ordered tricoverings of an unlabeled n-set.

Original entry on oeis.org

0, 0, 60, 375, 1392, 4020, 9960, 22200, 45730, 88543, 163000, 287650, 489610, 807625, 1295944, 2029165, 3108220, 4667690, 6884660, 9989345, 14277740, 20126570, 28010840, 38524310, 52403246, 70553825, 94083600, 124337460, 162938550, 211834647, 273350520, 350246835

Offset: 1

Vladeta Jovovic, Mar 20 2001

nonn

A covering of a set is a tricovering if every element of the set is covered by exactly three blocks of the covering.

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Column k=5 of A060492.

Cf. A060093, A060483, A060491.

a(n) = binomial(n+9, 9) - 15*binomial(n+3, 3) + 45*binomial(n+1, 1) - 40*binomial(n, 0) + 9*binomial(n-1, -1).
G.f.: y^3*(-225*y^3 + 60 - 225*y + 342*y^2 + 90*y^5 - 50*y^6 + 9*y^7)/(-1+y)^10.
E.g.f. for ordered k-block tricoverings of an unlabeled n-set is exp(-x+x^2/2+x^3/3*y/(1-y))*Sum_{k>=0} 1/(1-y)^binomial(k, 3)*exp(-x^2/2*1/(1-y)^n)*x^k/k!.

a(1)=a(2)=0 prepended and terms a(30) and beyond from Andrew Howroyd, Jan 30 2020

A060490 Number of 6-block ordered tricoverings of an unlabeled n-set.

Original entry on oeis.org

0, 0, 120, 3030, 24552, 130740, 551640, 1997415, 6470420, 19219462, 53187840, 138658760, 343297780, 812249250, 1845669776, 4044119530, 8573706300, 17637474350, 35294157340, 68850086745, 131179071560, 244518601660, 446576824800, 800201972990, 1408466719120

Offset: 1

Vladeta Jovovic, Mar 20 2001

nonn

A covering of a set is a tricovering if every element of the set is covered by exactly three blocks of the covering.

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Column k=6 of A060492.

Cf. A060094, A060484, A060491.

a(n) = binomial(n + 19, 19) - 6*binomial(n + 9, 9) - 15*binomial(n + 7, 7) + 135*binomial(n + 3, 3) - 310*binomial(n + 1, 1) + 240*binomial(n, 0) - 45*binomial(n - 1, -1).
G.f.: -y^3*( -78600*y^3 + 271080*y^4 - 120 - 630*y + 13248*y^2 - 635805*y^5 + 4300*y^15 - 15840*y^14 + 32760*y^13 - 18240*y^12 - 114120*y^11 + 442800*y^10 - 915315*y^9 - 1371804*y^7 + 1305540*y^8 + 1081360*y^6 + 45*y^17 - 660*y^16)/(-1 + y)^20.
E.g.f. for ordered k-block tricoverings of an unlabeled n-set is exp( -x + x^2/2 + x^3/3*y/(1 - y))*Sum_{k>=0} 1/(1 - y)^binomial(k, 3)*exp( -x^2/2*1/(1 - y)^n)*x^k/k!.

a(1)=a(2)=0 prepended and terms a(23) and beyond from Andrew Howroyd, Jan 30 2020

cp's OEIS Frontend

A331571 Array read by antidiagonals: A(n,k) is the number of binary matrices with k columns and any number of distinct nonzero rows with n ones in every column and columns in nonincreasing lexicographic order.

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A060492 Triangle T(n,k) of k-block ordered tricoverings of an unlabeled n-set (n >= 3, k = 4..2n).

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A060488 Number of 4-block ordered tricoverings of an unlabeled n-set.

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A060489 Number of 5-block ordered tricoverings of an unlabeled n-set.

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A060490 Number of 6-block ordered tricoverings of an unlabeled n-set.

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