A008279 -id:A008279 - cp's OEIS frontend

A329943 Square array read by antidiagonals: T(n,k) is the number of right total relations between set A with n elements and set B with k elements.

1, 3, 1, 7, 9, 1, 15, 49, 27, 1, 31, 225, 343, 81, 1, 63, 961, 3375, 2401, 243, 1, 127, 3969, 29791, 50625, 16807, 729, 1, 255, 16129, 250047, 923521, 759375, 117649, 2187, 1, 511, 65025, 2048383, 15752961, 28629151, 11390625, 823543, 6561, 1

Offset: 1

Roy S. Freedman, Nov 24 2019

A relation R between set A with n elements and set B with k elements is a subset of the Cartesian product A x B.  A relation R is right total if for each b in B there exists an a in A such that (a,b) in R.  T(n,k) is the number of right total relations and T(k,n) is the number of left total relations: relation R is left total if for each a in A there exists a b in B such that (a,b) in R.
From Manfred Boergens, Jun 23 2024: (Start)
T(n,k) is the number of k X n binary matrices with no 0 rows.
T(n,k) is the number of coverings of [k] by tuples (A_1,...,A_n) in P([k])^n, with P(.) denoting the power set.
Swapping n,k gives A092477 (with k<=n).
For nonempty A_j see A218695 (n,k swapped).
For disjoint A_j see A089072 (n,k swapped).
For nonempty and disjoint A_j see A019538 (n,k swapped). (End)

			T(n,k) begins, for 1 <= n,k <= 9:
    1,     1,       1,         1,           1,             1,               1
    3,     9,      27,        81,         243,           729,            2187
    7,    49,     343,      2401,       16807,        117649,          823543
   15,   225,    3375,     50625,      759375,      11390625,       170859375
   31,   961,   29791,    923521,    28629151,     887503681,     27512614111
   63,  3969,  250047,  15752961,   992436543,   62523502209,   3938980639167
  127, 16129, 2048383, 260144641, 33038369407, 4195872914689, 532875860165503

Roy S. Freedman, Some New Results on Binary Relations, arXiv:1501.01914 [cs.DM], 2015.

Cf. A218695.
The diagonal T(n,n) is A055601.
A092477 = T(k,n) is the number of left total relations between A and B.
A053440 is the number of relations that are both right unique (see A329940) and right total.
A089072 is the number of functions from A to B: relations between A and B that are both right unique and left total.
A019538 is the number of surjections between A and B: relations that are right unique, right total, and left total.
A008279 is the number of injections: relations that are right unique, left total, and left unique.
A000142 is the number of bijections: relations that are right unique, left total, right total, and left unique.

Maple
```
rt:=(n,k)->(2^n-1)^k:
```

T[n_, k_] := (2^n - 1)^k; Table[T[n - k + 1, k], {n, 1, 9}, {k, 1, n}] // Flatten (* Amiram Eldar, Nov 25 2019 *)

MuPAD
```
rt:=(n,k)->(2^n-1)^k:
```

T(n,k) = (2^n - 1)^k.

A049411 Triangle read by rows, the Bell transform of n!*binomial(5,n) (without column 0).

Original entry on oeis.org

1, 5, 1, 20, 15, 1, 60, 155, 30, 1, 120, 1300, 575, 50, 1, 120, 9220, 8775, 1525, 75, 1, 0, 55440, 114520, 36225, 3325, 105, 1, 0, 277200, 1315160, 730345, 112700, 6370, 140, 1, 0, 1108800, 13428800, 13000680, 3209745, 291060, 11130, 180, 1, 0, 3326400

Offset: 1

Wolfdieter Lang

Previous name was: A triangle of numbers related to triangle A049327.
a(n,1) = A008279(5,n-1). a(n,m) =: S1(-5; n,m), a member of a sequence of lower triangular Jabotinsky matrices, including S1(1; n,m) = A008275 (signed Stirling first kind), S1(2; n,m) = A008297(n,m) (signed Lah numbers). a(n,m) matrix is inverse to signed matrix ((-1)^(n-m))*A013988(n,m).
The monic row polynomials E(n,x) := sum(a(n,m)*x^m,m=1..n), E(0,x) := 1 are exponential convolution polynomials (see A039692 for the definition and a Knuth reference).
For the definition of the Bell transform see A264428 and the link. - Peter Luschny, Jan 16 2016

			Row polynomial E(3,x) = 20*x + 15*x^2 + x^3.
Triangle starts:
{  1}
{  5,    1}
{ 20,   15,   1}
{ 60,  155,  30,  1}
{120, 1300, 575, 50, 1}

W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
Peter Luschny, The Bell transform

Cf. A049327.

Row sums give A049428.

rows = 10;
a[n_, m_] := BellY[n, m, Table[k! Binomial[5, k], {k, 0, rows}]];
Table[a[n, m], {n, 1, rows}, {m, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018 *)

# uses[bell_matrix from A264428]
# Adds 1,0,0,0,... as column 0 at the left side of the triangle.
bell_matrix(lambda n: factorial(n)*binomial(5, n), 8) # Peter Luschny, Jan 16 2016

a(n, m) = n!*A049327(n, m)/(m!*6^(n-m));
a(n, m) = (6*m-n+1)*a(n-1, m) + a(n-1, m-1), n >= m >= 1;
a(n, m) = 0, nE.g.f. for m-th column: (((-1+(1+x)^6)/6)^m)/m!.

New name from Peter Luschny, Jan 16 2016

A049424 Triangle read by rows, the Bell transform of n!*binomial(4,n) (without column 0).

Original entry on oeis.org

1, 4, 1, 12, 12, 1, 24, 96, 24, 1, 24, 600, 360, 40, 1, 0, 3024, 4200, 960, 60, 1, 0, 12096, 40824, 17640, 2100, 84, 1, 0, 36288, 338688, 270144, 55440, 4032, 112, 1, 0, 72576, 2407104, 3580416, 1212624, 144144, 7056, 144, 1, 0, 72576, 14515200, 41791680

Offset: 1

Wolfdieter Lang

Previous name was: A triangle of numbers related to triangle A049326.
a(n,1) = A008279(4,n-1). a(n,m) =: S1(-4; n,m), a member of a sequence of lower triangular Jabotinsky matrices, including S1(1; n,m) = A008275 (signed Stirling first kind), S1(2; n,m) = A008297(n,m) (signed Lah numbers). a(n,m) matrix is inverse to signed matrix ((-1)^(n-m))*A011801(n,m). The monic row polynomials E(n,x) := Sum_{m=1..n} a(n,m)*x^m, E(0,x) := 1 are exponential convolution polynomials (see A039692 for the definition and a Knuth reference).
For the definition of the Bell transform see A264428 and the link. - Peter Luschny, Jan 16 2016

			E.g., row polynomial E(3,x) = 12*x + 12*x^2 + x^3.
Triangle starts:
   1;
   4,   1;
  12,  12,   1;
  24,  96,  24,   1;
  24, 600, 360,  40,   1;

W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
Peter Luschny, The Bell transform

Cf. A049326.

Row sums give A049427.

rows = 10;
a[n_, m_] := BellY[n, m, Table[k! Binomial[4, k], {k, 0, rows}]];
Table[a[n, m], {n, 1, rows}, {m, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018 *)

# uses[bell_matrix from A264428]
# Adds 1, 0, 0, 0, ... as column 0 at the left side of the triangle.
bell_matrix(lambda n: factorial(n)*binomial(2, n), 8) # Peter Luschny, Jan 16 2016

a(n, m) = n!*A049326(n, m)/(m!*5^(n-m));
a(n, m) = (5*m-n+1)*a(n-1, m) + a(n-1, m-1), n >= m >= 1;
a(n, m) = 0, nE.g.f. for m-th column: (((-1+(1+x)^5)/5)^m)/m!.

New name from Peter Luschny, Jan 16 2016

A070531 Generalized Bell numbers B_{4,3}.

Original entry on oeis.org

1, 73, 16333, 8030353, 7209986401, 10541813012041, 23227377813664333, 72925401604382826913, 312727862321385812968033, 1772004571987390827615327241, 12917715377912025572750844722221, 118521774439119390334062953438350513, 1343761301099219856651740487814621053313

Offset: 1

Karol A. Penson, May 02 2002

nonn

G. C. Greubel, Table of n, a(n) for n = 1..200
P. Blasiak, Karol A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004.
P. Blasiak, Karol A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
M. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665.

Cf. A091028 (alternating row sums of A090440).

ff[n_, k_] = Pochhammer[n - k + 1, k]; a[1, 3] = 1; a[n_, k_] := a[n, k] = Sum[Binomial[3, p]*ff[(n - 1 - p + k), 3 - p]*a[n - 1, k - p], {p, 0, 3} ]; a[n_ /; n < 2, ] = 0; Table[Sum[a[n, k] , {k, 3, 3 n}], {n, 1, 9}] (* _Jean-François Alcover, Sep 01 2011 *)

In Maple notation, a(n) = (1/12)*n!*(n+1)!*(n+2)!*hypergeom([n+1, n+2, n+3], [2, 3, 4], 1)/exp(1).

a(n) = Sum_{k=3..3*n} A090440(n, k) = (Sum_{k>=3} (1/k!)*Product_{j=1..n} fallfac(k+(j-1)*(4-3), 3))/exp(1), n>=1. From eq.(9) of the Blasiak et al. reference with r=4, s=3. fallfac(n, m) := A008279(n, m) (falling factorials triangle). a(0) := 1 may be added.

Edited by Wolfdieter Lang, Dec 23 2003

A089515 Triangle of signed numbers used for the computation of the column sequences of triangle A090215.

Original entry on oeis.org

1, -1, 5, 1, -35, 90, -3, 595, -6885, 12005, 143, -150535, 6175845, -39484445, 52245760, -58201, 316465625, -42458934375, 772604284375, -3322503800000, 3547818864576, 216931, -6012846875, 2544269990625, -120371747505625, 1294115230100000, -4145626343257056, 3713894747640000

Offset: 1

Wolfdieter Lang, Dec 01 2003

A090215(n+m,m)= sum(a(m,p)*((p+3)*(p+2)*(p+1)*p)^n,p=1..m)/D(m) with D(m) := A089516(m); m=1,2,..., n>=0.

			Triangle begins:
   1;
  -1,   5;
   1, -35,   90;
  -3, 595,-6885, 12005;
  ...
A090215(2+3,3) = 199296 = (1*(4*3*2*1)^2 - 35*(5*4*3*2)^2 + 90*(6*5*4*3)^2)/56.
a(3,2)= -35 = 56*(-1)*((5*4*3*2)^2)/((5*4*3*2-4*3*2*1)*(6*5*4*3-5*4*3*2)).

Wolfdieter Lang, First 7 rows.

a(n, m)= D(n)*((-1)^(n-m))*(fallfac(m+3, 4)^(n-1))/(product(fallfac(m+3, 4)-fallfac(r+3, 4), r=1..m-1)*product(fallfac(r+3, 4)-fallfac(m+3, 4), r=m+1..n)), with D(n) := A089516(n) and fallfac(n, m) := A008279(n, m) (falling factorials), 1<=m<=n else 0. (Replace in the denominator the first product by 1 if m=1 and the second one by 1 if m=n.)

A090211 Alternating row sums of array A078739 ((2,2)-Stirling2).

Original entry on oeis.org

1, -1, -1, 41, -375, -3001, 177063, -990543, -144800527, 3644593711, 214013895023, -12488200175463, -553322483517383, 61495192102867639, 2469939623420627543, -448608666325921194271, -19104207797417792353951, 4742067751530355028847327

Offset: 1

Wolfdieter Lang, Dec 01 2003

P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
M. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665.

P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem.

Cf. -A000587(n) from Stirling2 case A008277 with a(0) := -1. A020556 (non-alternating sum, generalized Bell-numbers).

a[n_] := Sum[(-1)^k FactorialPower[k, 2]^n/k!, {k, 2, Infinity}]*E; Array[a, 18] (* Jean-François Alcover, Sep 01 2016 *)

a(n) := sum( A078739(n, m)*(-1)^m, m=2..2*n), n>=1. a(0) := +1 may be added.
a(n) = sum(((-1)^k)*(fallfac(k, 2)^n)/k!, k=2..infinity)*exp(1), with fallfac(k, 2)=A008279(k, 2)=k*(k-1) and n>=1. This produces also a(0)=1.
E.g.f. if a(0)=1 is added: exp(1)*(sum(((-1)^k)*exp(k*(k-1)*x)/k!, k=2..infinity)). Similar to derivation on top p. 4656 of the Schork reference.

A090222 Array used for numerators of g.f.s for column sequences of array A090216 ((5,5)-Stirling2).

Original entry on oeis.org

1, 600, 600, 648000, 200, 2592000, 1270080000, 25, 2871000, 13592880000, 4267468800000, 1, 1294920, 36462182400, 100221504768000, 23228686172160000, 284800, 38559024000, 551224880640000, 1056582600192000000

Offset: 5

Wolfdieter Lang, Dec 01 2003

The row length sequence for this array is A090223(k-5)+1= floor(4*(k-5)/5)+1, k>=5: [1, 1, 2, 3, 4, 4, 5, 6, 7, 7, 8, 9, 10, 10, 11, ...].

The g.f. G(k,x) for the k-th column (with leading zeros) of array A090216 is given there. The recurrence is G(k,x) = x*sum(binomial(k-r,5-r)*fallfac(5,5-r)*G(k-r,x),r=1..5))/(1-fallfac(k,5)*x), k>=5, with inputs G(k,x)=0 for k=1,2,3,4 and G(5,x)=x/(1-5!*x); where fallfac(n,m) := A008279(n,m) (falling factorials with fallfac(n,0) := 1). Computed from the Blasiak et al. reference, eqs. (20) and (21) with r=5: recurrence for S_{5,5}(n,k).

			[1]; [600]; [648000,200]; [2592000,1270080000,25]; ...
G(6,x)/x^2 = 600/((1-5!*x)*(1-6*5*4*3*2*x)). kmax(6)=0, hence P(6,x)=a(6,0)=600; x^2 from x^ceiling(6/5).

W. Lang, First 7 rows.

a(k, n) from: sum(a(k, n)*x^n, n=0..kmax(k)) = G(k, x)* product(1-fallfac(p, 5)*x, p=5..k)/x^ceiling(k/5), k>=5, with G(k, x) defined from the recurrence given above and kmax(k) := floor(4*(k-5)/5)= A090223(k-5).

A090435 Triangle of signed numbers used for the computation of the column sequences of triangle A090217.

Original entry on oeis.org

1, -1, 6, 1, -48, 147, -5, 1584, -24255, 50176, 1, -1980, 121275, -1003520, 1571724, -41, 496980, -113458275, 2950635520, -16174611684, 20412000000, 45182, -3322062810, 2744728561050, -206756932157440, 3081396966348393, -12443694076800000, 13160600037440625, -1294492177294

Offset: 1

Wolfdieter Lang, Dec 01 2003

A090217(n+m,m)= sum(a(m,p)*((p+4)*(p+3)*(p+2)*(p+1)*p)^n,p=1..m)/D(m) with D(m) := A090436(m); m=1,2,..., n>=0.

			[1]; [ -1,6]; [1,-48,147]; [ -5,1584,-24255,50176]; ...
A090217(2+3,3) = 9086400 = (1*(5*4*3*2*1)^2 - 48*(6*5*4*3*2)^2 + 147*(7*6*5*4*3)^2)/100.
a(3,2)= -48 = 100*(-1)*((6*5*4*3*2)^2)/((6*5*4*3*2-5*4*3*2*1)*(7*6*5*4*3-6*5*4*3*2)).

W. Lang, First 8 rows.

a(n, m)= D(n)*((-1)^(n-m))*(fallfac(m+4, 5)^(n-1))/(product(fallfac(m+4, 5)-fallfac(r+4, 5), r=1..m-1)*product(fallfac(r+4, 5)-fallfac(m+4, 5), r=m+1..n)), with D(n) := A090436(n) and fallfac(n, m) := A008279(n, m) (falling factorials), 1<=m<=n else 0. (Replace in the denominator the first product by 1 if m=1 and the second one by 1 if m=n.)

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A181511 Triangle T(n,k) = n!/(n-k)! read by rows, 0 <= k < n.

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A295181 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(-k*x)/(1 - x)^k.

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A329943 Square array read by antidiagonals: T(n,k) is the number of right total relations between set A with n elements and set B with k elements.

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A049411 Triangle read by rows, the Bell transform of n!*binomial(5,n) (without column 0).

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A049424 Triangle read by rows, the Bell transform of n!*binomial(4,n) (without column 0).

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A070531 Generalized Bell numbers B_{4,3}.

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A089515 Triangle of signed numbers used for the computation of the column sequences of triangle A090215.

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