A183109 -id:A183109 - cp's OEIS frontend

A019538 Triangle of numbers T(n,k) = k!*Stirling2(n,k) read by rows (n >= 1, 1 <= k <= n).

1, 1, 2, 1, 6, 6, 1, 14, 36, 24, 1, 30, 150, 240, 120, 1, 62, 540, 1560, 1800, 720, 1, 126, 1806, 8400, 16800, 15120, 5040, 1, 254, 5796, 40824, 126000, 191520, 141120, 40320, 1, 510, 18150, 186480, 834120, 1905120, 2328480, 1451520, 362880, 1, 1022, 55980, 818520, 5103000, 16435440, 29635200, 30240000, 16329600, 3628800

Offset: 1

N. J. A. Sloane and Manfred Goebel (goebel(AT)informatik.uni-tuebingen.de), Dec 11 1996

Number of ways n labeled objects can be distributed into k nonempty parcels. Also number of special terms in n variables with maximal degree k.
In older terminology these are called differences of 0. - Michael Somos, Oct 08 2003
Number of surjections (onto functions) from an n-element set to a k-element set.
Also coefficients (in ascending order) of so-called ordered Bell polynomials.
(k-1)!*Stirling2(n,k-1) is the number of chain topologies on an n-set having k open sets [Stephen].
Number of set compositions (ordered set partitions) of n items into k parts. Number of k dimensional 'faces' of the n dimensional permutohedron (see Simion, p. 162). - Mitch Harris, Jan 16 2007
Correction of comment before: Number of (n-k)-dimensional 'faces' of the permutohedron of order n (an (n-1)-dimensional polytope). - Tilman Piesk, Oct 29 2014
This array is related to the reciprocal of an e.g.f. as sketched in A133314. For example, the coefficient of the fourth-order term in the Taylor series expansion of 1/(a(0) + a(1) x + a(2) x^2/2! + a(3) x^3/3! + ...) is a(0)^(-5) * {24 a(1)^4 - 36 a(1)^2 a(2) a(0) + [8 a(1) a(3) + 6 a(2)^2] a(0)^2 - a(4) a(0)^3}. The unsigned coefficients characterize the P3 permutohedron depicted on page 10 in the Loday link with 24 vertices (0-D faces), 36 edges (1-D faces), 6 squares (2-D faces), 8 hexagons (2-D faces) and 1 3-D permutohedron. Summing coefficients over like dimensions gives A019538 and A090582. Compare to A133437 for the associahedron. - Tom Copeland, Sep 29 2008, Oct 07 2008
Further to the comments of Tom Copeland above, the permutohedron of type A_3 can be taken as the truncated octahedron. Its dual is the tetrakis hexahedron, a simplicial polyhedron, with f-vector (1,14,36,24) giving the fourth row of this triangle. See the Wikipedia entry and [Fomin and Reading p. 21]. The corresponding h-vectors of permutohedra of type A give the rows of the triangle of Eulerian numbers A008292. See A145901 and A145902 for the array of f-vectors for type B and type D permutohedra respectively. - Peter Bala, Oct 26 2008
Subtriangle of triangle in A131689. - Philippe Deléham, Nov 03 2008
Since T(n,k) counts surjective functions and surjective functions are "consistent", T(n,k) satisfies a binomial identity, namely, T(n,x+y) = Sum_{j=0..n} C(n,j)*T(j,x)*T(n-j,y). For definition of consistent functions and a generalized binomial identity, see "Toy stories and combinatorial identities" in the link section below. - Dennis P. Walsh, Feb 24 2012
T(n,k) is the number of labeled forests on n+k vertices satisfying the following two conditions: (i) each forest consists of exactly k rooted trees with roots labeled 1, 2, ..., k; (ii) every root has at least one child vertex. - Dennis P. Walsh, Feb 24 2012
The triangle is the inverse binomial transform of triangle A028246, deleting the left column and shifting up one row. - Gary W. Adamson, Mar 05 2012
See A074909 for associations among this array and the Bernoulli polynomials and their umbral compositional inverses. - Tom Copeland, Nov 14 2014
E.g.f. for the shifted signed polynomials is G(x,t) = (e^t-1)/[1+(1+x)(e^t-1)] = 1-(1+x)(e^t-1) + (1+x)^2(e^t-1)^2 - ... (see also A008292 and A074909), which has the infinitesimal generator g(x,u)d/du = [(1-x*u)(1-(1+x)u)]d/du, i.e., exp[t*g(x,u)d/du]u eval. at u=0 gives G(x,t), and dG(x,t)/dt = g(x,G(x,t)). The compositional inverse is log((1-xt)/(1-(1+x)t)). G(x,t) is a generating series associated to the generalized Hirzebruch genera. See the G. Rzadowski link for the relation of the derivatives of g(x,u) to solutions of the Riccatt differential equation, soliton solns. to the KdV equation, and the Eulerian and Bernoulli numbers. In addition A145271 connects products of derivatives of g(x,u) and the refined Eulerian numbers to the inverse of G(x,t), which gives the normalized, reverse face polynomials of the simplices (A135278, divided by n+1). See A028246 for the generator g(x,u)d/dx. - Tom Copeland, Nov 21 2014
For connections to toric varieties and Eulerian polynomials, see the Dolgachev and Lunts and the Stembridge links. - Tom Copeland, Dec 31 2015
See A008279 for a relation between the e.g.f.s enumerating the faces of permutahedra (this entry) and stellahedra. - Tom Copeland, Nov 14 2016
T(n, k) appears in a Worpitzky identity relating monomials to binomials: x^n = Sum_{k=1..n} T(n, k)*binomial(x,k), n >= 1. See eq. (11.) of the Worpitzky link on p. 209. The relation to the Eulerian numbers is given there in eqs. (14.) and (15.). See the formula below relating to A008292. See also Graham et al. eq. (6.10) (relating monomials to falling factorials) on p. 248 (2nd ed. p. 262). The Worpitzky identity given in the Graham et al. reference as eq. (6.37) (2nd ed. p. 269) is eq. (5.), p. 207, of Worpitzky. - Wolfdieter Lang, Mar 10 2017
T(n, m) is also the number of minimum clique coverings and minimum matchings in the complete bipartite graph K_{m,n}. - Eric W. Weisstein, Apr 26 2017
From the Hasan and Franco and Hasan papers: The m-permutohedra for m=1,2,3,4 are the line segment, hexagon, truncated octahedron and omnitruncated 5-cell. The first three are well-known from the study of elliptic models, brane tilings and brane brick models. The m+1 torus can be tiled by a single (m+2)-permutohedron. Relations to toric Calabi-Yau Kahler manifolds are also discussed. - Tom Copeland, May 14 2020
From Manfred Boergens, Jul 25 2021: (Start)
Number of n X k binary matrices with row sums = 1 and no zero columns. These matrices are a subset of the matrices defining A183109.
The distribution into parcels in the leading comment can be regarded as a covering of [n] by tuples (A_1,...,A_k) in P([n])^k with nonempty and disjoint A_j, with P(.) denoting the power set (corrected for clarity by Manfred Boergens, May 26 2024). For the non-disjoint case see A183109 and A218695.
For tuples with "nonempty" dropped see A089072. For tuples with "nonempty and disjoint" dropped see A092477 and A329943 (amendment by Manfred Boergens, Jun 24 2024). (End)

			The triangle T(n, k) begins:
  n\k 1    2     3      4       5        6        7        8        9      10
  1:  1
  2:  1    2
  3:  1    6     6
  4:  1   14    36     24
  5:  1   30   150    240     120
  6:  1   62   540   1560    1800      720
  7:  1  126  1806   8400   16800    15120     5040
  8:  1  254  5796  40824  126000   191520   141120    40320
  9:  1  510 18150 186480  834120  1905120  2328480  1451520   362880
  10: 1 1022 55980 818520 5103000 16435440 29635200 30240000 16329600 3628800
  ... Reformatted and extended - _Wolfdieter Lang_, Oct 04 2014
---------------------------------------------------------------------------
T(4,1) = 1: {1234}. T(4,2) = 14: {1}{234} (4 ways), {12}{34} (6 ways), {123}{4} (4 ways). T(4,3) = 36: {12}{3}{4} (12 ways), {1}{23}{4} (12 ways), {1}{2}{34} (12 ways). T(4,4) = 1: {1}{2}{3}{4} (1 way).

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Row sums give A000670. Maximal terms in rows give A002869. Central terms T(2k-1,k) give A233734.
Diagonal is n! (A000142). 2nd diagonal is A001286. 3rd diagonal is A037960.
Cf. A000918, A000919, A001117, A001118, A008275, A008277, A008279, A048594, A059117, A059515, A074909, A084938, A089072, A092477, A183109, A218695, A329943.
Reflected version of A090582. A371568 is another version.
See also the two closely related triangles: A008277(n, k) = T(n, k)/k! (Stirling numbers of second kind) and A028246(n, k) = T(n, k)/k.
Cf. A033282 'faces' of the associahedron.
Cf. A008292, A047969, A145901, A145902. - Peter Bala, Oct 26 2008
Visible in the 3-D array in A249042.
See also A000182.

a019538 n k = a019538_tabl !! (n-1) !! (k-1)
a019538_row n = a019538_tabl !! (n-1)
a019538_tabl = iterate f [1] where
   f xs = zipWith (*) [1..] $ zipWith (+) ([0] ++ xs) (xs ++ [0])
-- Reinhard Zumkeller, Dec 15 2013

with(combinat): A019538 := (n,k)->k!*stirling2(n,k);

Table[k! StirlingS2[n, k], {n, 9}, {k, n}] // Flatten

{T(n, k) = if( k<0 || k>n, 0, sum(i=0, k, (-1)^i * binomial(k, i) * (k-i)^n))}; /* Michael Somos, Oct 08 2003 */

def T(n, k): return factorial(k)*stirling_number2(n,k) # Danny Rorabaugh, Oct 10 2015

T(n, k) = k*(T(n-1, k-1)+T(n-1, k)) with T(0, 0) = 1 [or T(1, 1) = 1]. - Henry Bottomley, Mar 02 2001
E.g.f.: (y*(exp(x)-1) - exp(x))/(y*(exp(x)-1) - 1). - Vladeta Jovovic, Jan 30 2003
Equals [0, 1, 0, 2, 0, 3, 0, 4, 0, 5, ...] DELTA [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, ...] where DELTA is Deléham's operator defined in A084938.
T(n, k) = Sum_{j=0..k} (-1)^(k-j)*j^n*binomial(k, j). - Mario Catalani (mario.catalani(AT)unito.it), Nov 28 2003. See Graham et al., eq. (6.19), p. 251. For a proof see Bert Seghers, Jun 29 2013.
Sum_{k=0..n} T(n, k)(-1)^(n-k) = 1, Sum_{k=0..n} T(n, k)(-1)^k = (-1)^n. - Mario Catalani (mario.catalani(AT)unito.it), Dec 11 2003
O.g.f. for n-th row: polylog(-n, x/(1+x))/(x+x^2). - Vladeta Jovovic, Jan 30 2005
E.g.f.: 1 / (1 + t*(1-exp(x))). - Tom Copeland, Oct 13 2008
From Peter Bala, Oct 26 2008: (Start)
O.g.f. as a continued fraction: 1/(1 - x*t/(1 - (x + 1)*t/(1 - 2*x*t/(1 - 2*(x + 1)*t/(1 - ...))))) = 1 + x*t + (x + 2*x^2)*t^2 + (x + 6*x^2 + 6*x^3)*t^3 + ... .
The row polynomials R(n,x), which begin R(1,x) = x, R(2,x) = x + 2*x^2, R(3,x) = x + 6*x^2 + 6*x^3, satisfy the recurrence x*d/dx ((x + 1)*R(n,x)) = R(n+1,x). It follows that the zeros of R(n,x) are real and negative (apply Corollary 1.2 of [Liu and Wang]).
Since this is the triangle of f-vectors of the (simplicial complexes dual to the) type A permutohedra, whose h-vectors form the Eulerian number triangle A008292, the coefficients of the polynomial (x-1)^n*R(n,1/(x-1)) give the n-th row of A008292. For example, from row 3 we have x^2 + 6*x + 6 = 1 + 4*y + y^2, where y = x + 1, producing [1,4,1] as the third row of A008292. The matrix product A008292 * A007318 gives the mirror image of this triangle (see A090582).
For n,k >= 0, T(n+1,k+1) = Sum_{j=0..k} (-1)^(k-j)*binomial(k,j)*[(j+1)^(n+1) - j^(n+1)]. The matrix product of Pascal's triangle A007318 with the current array gives (essentially) A047969. This triangle is also related to triangle A047969 by means of the S-transform of [Hetyei], a linear transformation of polynomials whose value on the basis monomials x^k is given by S(x^k) = binomial(x,k). The S-transform of the shifted n-th row polynomial Q(n,x) := R(n,x)/x is S(Q(n,x)) = (x+1)^n - x^n. For example, from row 3 we obtain S(1 + 6*x + 6*x^2) = 1 + 6*x + 6*x*(x-1)/2 = 1 + 3*x + 3*x^2 = (x+1)^3 - x^3. For fixed k, the values S(Q(n,k)) give the nonzero entries in column (k-1) of the triangle A047969 (the Hilbert transform of the Eulerian numbers). (End)
E.g.f.: (exp(x)-1)^k = sum T(n,k)x^n/n!. - Vladimir Kruchinin, Aug 10 2010
T(n,k) = Sum_{i=1..k} A(n,i)*Binomial(n-i,k-i) where A(n,i) is the number of n-permutations that have i ascending runs, A008292.
From Tom Copeland, Oct 11 2011: (Start)
With e.g.f. A(x,t) = -1 + 1/(1+t*(1-exp(x))), the comp. inverse in x is B(x,t) = log(((1+t)/t) - 1/(t(1+x))).
With h(x,t) = 1/(dB/dx)= (1+x)((1+t)(1+x)-1), the row polynomial P(n,t) is given by (h(x,t)*d/dx)^n x, eval. at x=0, A=exp(x*h(y,t)*d/dy) y, eval. at y=0, and dA/dx = h(A(x,t),t), with P(0,t)=0.
(A factor of -1/n! was removed by Copeland on Aug 25 2016.) (End)
The term linear in x of [x*h(d/dx,t)]^n 1 gives the n-th row polynomial. (See A134685.) - Tom Copeland, Nov 07 2011
Row polynomials are given by D^n(1/(1-x*t)) evaluated at x = 0, where D is the operator (1+x)*d/dx. - Peter Bala, Nov 25 2011
T(n,x+y) = Sum_{j=0..n} binomial(n,j)*T(j,x)*T(n-j,y). - Dennis P. Walsh, Feb 24 2012
Let P be a Rota-Baxter operator of weight 1 satisfying the identity P(x)*P(y) = P(P(x)*y) + P(x*P(y)) + P(x*y). Then P(1)^2 = P(1) + 2*P^2(1). More generally, Guo shows that P(1)^n = Sum_{k=1..n} T(n,k)*P^k(1). - Peter Bala, Jun 08 2012
Sum_{i=1..n} (-1)^i*T(n,i)/i = 0, for n > 1. - Leonid Bedratyuk, Aug 09 2012
T(n, k) = Sum_{j=0..k} (-1)^j*binomial(k, j)*(k-j)^n. [M. Catalani's re-indexed formula from Nov 28 2003] Proof: count the surjections of [n] onto [k] with the inclusion-exclusion principle, as an alternating sum of the number of functions from [n] to [k-j]. - Bert Seghers, Jun 29 2013
n-th row polynomial = 1/(1 + x)*( Sum_{k>=0} k^n*(x/(1 + x))^k ), valid for x in the open interval (-1/2, inf). See Tanny link. Cf. A145901. - Peter Bala, Jul 22 2014
T(n,k) = k * A141618(n,k-1) / binomial(n,k-1). - Tom Copeland, Oct 25 2014
Sum_{n>=0} n^k*a^n = Sum_{i=1..k} (a / (1 - a))^i * T(k, i)/(1-a) for |a| < 1. - David A. Corneth, Mar 09 2015
From Peter Bala, May 26 2015: (Start)
The row polynomials R(n,x) satisfy (1 + x)*R(n,x) = (-1)^n*x*R(n,-(1 + x)).
For a fixed integer k, the expansion of the function A(k,z) := exp( Sum_{n >= 1} R(n,k)*z^n/n ) has integer coefficients and satisfies the functional equation A(k,z)^(k + 1) = BINOMIAL(A(k,z))^k, where BINOMIAL(F(z))= 1/(1 - z)*F(z/(1 - z)) denotes the binomial transform of the o.g.f. F(z). Cf. A145901. For cases see A084784 (k = 1), A090352 (k = 2), A090355 (k = 3), A090357 (k = 4), A090362 (k = 5) and A084785 (k = -2 with z -> -z).
A(k,z)^(k + 1) = A(-(k + 1),-z)^k and hence BINOMIAL(A(k,z)) = A(-(k + 1),-z). (End)
From Tom Copeland, Oct 19 2016: (Start)
Let a(1) = 1 + x + B(1) = x + 1/2 and a(n) = B(n) = (B.)^n, where B(n) are the Bernoulli numbers defined by e^(B.t) = t / (e^t-1), then t / e^(a.t) = t / [(x + 1) * t + exp(B.t)] = (e^t - 1) /[ 1 + (x + 1) (e^t - 1)] = exp(p.(x)t), where (p.(x))^n = p_n(x) are the shifted, signed row polynomials of this array: p_0(x) = 0, p_1(x) = 1, p_2(x) = -(1 + 2 x), p_3(x) = 1 + 6 x + 6 x^2, ... and p_n(x) = n * b(n-1), where b(n) are the partition polynomials of A133314 evaluated with these a(n).
Sum_{n > 0} R(n,-1/2) x^n/n! = 2 * tanh(x/2), where R(n,x) = Sum_{k = 1..n} T(n,k) x^(k-1) are the shifted row polynomials of this entry, so R(n,-1/2) = 4 * (2^(n+1)-1) B(n+1)/(n+1). (Cf. A000182.)
(End)
Also the Bernoulli numbers are given by B(n) = Sum_{k =1..n} (-1)^k T(n,k) / (k+1). - Tom Copeland, Nov 06 2016
G.f. for column k: k! x^k / Product_{i=1..k} (1-i*x). - Robert A. Russell, Sep 25 2018
a(j) <= A183109(j). - Manfred Boergens, Jul 25 2021

A048291 Number of {0,1} n X n matrices with no zero rows or columns.

Original entry on oeis.org

1, 1, 7, 265, 41503, 24997921, 57366997447, 505874809287625, 17343602252913832063, 2334958727565749108488321, 1243237913592275536716800402887, 2630119877024657776969635243647463625, 22170632855360952977731028744522744983195423

Offset: 0

Joe Keane (jgk(AT)jgk.org)

Number of relations on n labeled points such that for every point x there exists y and z such that xRy and zRx.
Also the number of edge covers in the complete bipartite graph K_{n,n}. - Eric W. Weisstein, Apr 24 2017
Counts labeled digraphs (loops allowed, no multiarcs) on n nodes where each indegree and each outdegree is >= 1. The corresponding sequence for unlabeled digraphs (1, 5, 55, 1918,... for n >= 1) seems not to be in the OEIS. - R. J. Mathar, Nov 21 2023
These relations form a subsemigroup of the semigroup of all binary relations on [n].  The zero element is the universal relation (all 1's matrix).  See Schwarz link. - Geoffrey Critzer, Jan 15 2024

			a(2) = 7:  |01|  |01|  |10|  |10|  |11|  |11|  |11|
           |10|  |11|  |01|  |11|  |01|  |10|  |11|.

Brendan McKay, Posting to sci.math.research, Jun 14 1999.

T. D. Noe, Table of n, a(n) for n = 0..32
H. Cheballah, S. Giraudo, and R. Maurice, Combinatorial Hopf algebra structure on packed square matrices, arXiv preprint arXiv:1306.6605 [math.CO], 2013.
David Dolžan and Gabriel Verret, The automorphism group of the zero-divisor digraph of matrices over an antiring, arXiv:1908.04614 [math.AC], 2019.
R. J. Mathar, The number of nXm matrices with at most k 1's in each row or column, (2014).
Steven Schlicker, Roman Vasquez, and Rachel Wofford, Integer Sequences from Configurations in the Hausdorff Metric Geometry via Edge Covers of Bipartite Graphs, J. Int. Seq. (2023) Vol. 26, Art. 23.6.6.
Stefan Schwarz, The semigroup of fully indecomposable relations and Hall relations, Czechoslovak Mathematical Journal, 23 (1973), 151-163.
R. Tauraso, Edge cover time for regular graphs, JIS 11 (2008) 08.4.4.
Eric Weisstein's World of Mathematics, Complete Bipartite Graph.
Eric Weisstein's World of Mathematics, Edge Cover.

Cf. A054976, A104602, A283624, A287065, A173403.
Cf. A055601, A055599, A104601, A086193 (traceless, no loops), A086206, A322661 (adj. matr. undirected edges).
Diagonal of A183109.

seq(add((-1)^(n+k)*binomial(n, k)*(2^k-1)^n, k=0..n), n=0..15); # Robert FERREOL, Mar 10 2017

Flatten[{1,Table[Sum[Binomial[n,k]*(-1)^k*(2^(n-k)-1)^n,{k,0,n}],{n,1,15}]}] (* Vaclav Kotesovec, Jul 02 2014 *)

a(n)=sum(k=0,n,binomial(n,k)*(-1)^k*(2^(n-k)-1)^n)

import math
f = math.factorial
def A048291(n): return sum([(f(n)/f(s)/f(n - s))*(-1)**s*(2**(n - s) - 1)**n for s in range(0, n+1)]) # Indranil Ghosh, Mar 14 2017

a(n) = Sum_{s=0..n} binomial(n, s)*(-1)^s*2^((n-s)*n)*(1-2^(-n+s))^n.
From Vladeta Jovovic, Feb 23 2008: (Start)
E.g.f.: Sum_{n>=0} (2^n-1)^n*exp((1-2^n)*x)*x^n/n!.
a(n) = Sum_{i=0..n} Sum_{j=0..n} (-1)^(i+j)*binomial(n,i)*binomial(n,j)*2^(i*j). (End)
a(n) ~ 2^(n^2). - Vaclav Kotesovec, Jul 02 2014
a(n) = Sum_{s=0..n-1} binomial(n,s)*(-1)^s*A092477(n,n-s), n > 0. - R. J. Mathar, Nov 18 2023

A286189 Number of connected induced (non-null) subgraphs of the n X n rook graph.

Original entry on oeis.org

1, 13, 397, 55933, 31450861, 67253507293, 559182556492477, 18408476382988290493, 2416307646576708948065581, 1267404418454077249779938768413, 2658301080374793666228695738368407037, 22300360304310794054520197736231374212892413

Offset: 1

Giovanni Resta, May 04 2017

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..50
Eric Weisstein's World of Mathematics, Rook Graph
Eric Weisstein's World of Mathematics, Vertex-Induced Subgraph

Main diagonal of A360873.
Cf. A262307, A183109.
Cf. A020873 (wheel), A059020 (ladder), A059525 (grid), A286139 (king), A286182 (prism), A286183 (antiprism), A286184 (helm), A286185 (Möbius ladder), A286186 (friendship), A286187 (web), A286188 (gear), A285765 (queen).

{1} ~ Join ~ Table[g = GraphData[{"Rook", {n,n}}]; -1 + ParallelSum[ Boole@ ConnectedGraphQ@ Subgraph[g, s], {s, Subsets@ Range[n^2]}], {n, 2, 4}]
(* Second program: *)
(* b = A183109, T = A262307 *)
b[n_, m_] := Sum[(-1)^j*Binomial[m, j]*(2^(m - j) - 1)^n, {j, 0, m}];
T[m_, n_] := T[m, n] = b[m, n] - Sum[T[i, j]*b[m - i, n - j] Binomial[m - 1, i - 1]*Binomial[n, j], {i, 1, m - 1}, {j, 1, n - 1}];
a[n_] := Sum[Binomial[n, i]*Binomial[n, j]*T[i, j], {i, 1, n}, {j, 1, n}];
Array[a, 12] (* Jean-François Alcover, Oct 11 2017, after Andrew Howroyd *)

G(N)={my(S=matrix(N,N), T=matrix(N,N), U=matrix(N,N));
\\ S is A183109, T is A262307, U is mxn variant of this sequence.
for(m=1,N,for(n=1,N,
S[m,n]=sum(j=0, m, (-1)^j*binomial(m, j)*(2^(m - j) - 1)^n);
T[m,n]=S[m,n]-sum(i=1, m-1, sum(j=1, n-1, T[i,j]*S[m-i,n-j]*binomial(m-1,i-1)*binomial(n,j)));
U[m,n]=sum(i=1,m,sum(j=1,n,binomial(m,i)*binomial(n,j)*T[i,j])) ));U}
a(n)=G(n)[n,n]; \\ Andrew Howroyd, May 22 2017

a(n) = Sum_{i=1..n} Sum_{j=1..n} binomial(n,i)*binomial(n,j)*A262307(i,j). - Andrew Howroyd, May 22 2017

a(n) ~ 2^(n^2). - Vaclav Kotesovec, Oct 12 2017

Terms a(7) and beyond from Andrew Howroyd, May 22 2017

A262307 Array read by antidiagonals: T(m,n) = number of m X n binary matrices with all 1's connected and no zero rows or columns.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 19, 19, 1, 1, 65, 205, 65, 1, 1, 211, 1795, 1795, 211, 1, 1, 665, 14221, 36317, 14221, 665, 1, 1, 2059, 106819, 636331, 636331, 106819, 2059, 1, 1, 6305, 778765, 10365005, 23679901, 10365005, 778765, 6305, 1

Offset: 1

N. J. A. Sloane, Oct 04 2015

Two 1's are connected if they are in the same row or column. The requirement is for them to form a single connected set.
The number of m X n binary matrices with no zero rows or columns is given by A183109(m, n). If there are multiple components (not connected) then they cannot share either rows or columns. For i < n and j < m there are T(i,j) ways of creating an i X j component that occupies the first row. Its remaining i-1 rows may be on any of the remaining m-1 rows with its j columns on any of the n columns. The m-i rows and n-j columns not used by this component can be any matrix with no zero rows or columns.
T(m,n) is also the number of bipartite connected labeled graphs with parts of size m and n. (See A005333, A227322.)
This is the array a(m,n) in Kreweras (1969). Kreweras describes this as a symmetric triangle read by rows, giving numbers of connected relations.
The companion array b(m,n) (and the first few of its diagonals) in Kreweras (1969) should also be added to the OEIS if they are not already present.

			Table starts:
==========================================================================
m\n| 1    2      3         4           5             6               7
---|----------------------------------------------------------------------
1  | 1    1      1         1           1             1               1 ...
2  | 1    5     19        65         211           665            2059 ...
3  | 1   19    205      1795       14221        106819          778765 ...
4  | 1   65   1795     36317      636331      10365005       162470155 ...
5  | 1  211  14221    636331    23679901     805351531     26175881341 ...
6  | 1  665 106819  10365005   805351531   56294206205   3735873535339 ...
7  | 1 2059 778765 162470155 26175881341 3735873535339 502757743028605 ...
...
As a triangle, this begins:
  1;
  1,    1;
  1,    5,      1;
  1,   19,     19,      1;
  1,   65,    205,     65,      1;
  1,  211,   1795,   1795,    211,      1;
  1,  665,  14221,  36317,  14221,    665,    1;
  1, 2059, 106819, 636331, 636331, 106819, 2059, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..780
G. Kreweras, Inversion des polynômes de Bell bidimensionnels et application au dénombrement des relations binaires connexes, C. R. Acad. Sci. Paris Ser. A-B 268 1969 A577-A579.

Essentially same table as triangle A227322 (where the antidiagonals only go halfway).
Main diagonal is A005333.
Initial diagonals give A001047, A002501, A002502.
Cf. A226658, A123260, A286189, A183109, A048291, A287274.

A183109[n_, m_] := Sum[(-1)^j*Binomial[m, j]*(2^(m-j) - 1)^n, {j, 0, m}];
T[m_, n_] := A183109[m, n] - Sum[T[i, j]*A183109[m - i, n - j] Binomial[m - 1, i - 1]*Binomial[n, j], {i, 1, m - 1}, {j, 1, n - 1}];
Table[T[m - n + 1, n], {m, 1, 9}, {n, 1, m}] // Flatten (* Jean-François Alcover, Oct 08 2017, after Andrew Howroyd *)

G(N)={my(S=matrix(N,N), T=matrix(N,N));
for(m=1,N,for(n=1,N,
S[m,n]=sum(j=0, m, (-1)^j*binomial(m, j)*(2^(m - j) - 1)^n);
T[m,n]=S[m,n]-sum(i=1, m-1, sum(j=1, n-1, T[i,j]*S[m-i,n-j]*binomial(m-1,i-1)*binomial(n,j)));
));T}
G(7) \\ Andrew Howroyd, May 22 2017

T(m,n) = A183109(m,n) - Sum_{i=1..m-1} Sum_{j=1..n-1} T(i,j)*A183109(m-i, n-j)*binomial(m-1,i-1)*binomial(n,j). - Andrew Howroyd, May 22 2017

Revised by N. J. A. Sloane, May 26 2017, to incorporate material from Andrew Howroyd's May 22 2017 submission (formerly A287297), which was essentially identical to this, although giving an alternative description and more information.

A287274 Array read by antidiagonals: T(m,n) = number of dominating sets in the lattice (rook) graph K_m X K_n.

Original entry on oeis.org

1, 3, 3, 7, 11, 7, 15, 51, 51, 15, 31, 227, 421, 227, 31, 63, 963, 3615, 3615, 963, 63, 127, 3971, 30517, 59747, 30517, 3971, 127, 255, 16131, 252231, 989295, 989295, 252231, 16131, 255, 511, 65027, 2054941, 16219187, 32260381, 16219187, 2054941, 65027, 511

Offset: 1

Andrew Howroyd, May 22 2017

A set of vertices can be represented as an m X n binary matrix. If all rows contain at least one 1 then regardless of what is in each row the set will form a dominating set giving (2^n-1)^m solutions. Otherwise if only iA183109(i,n) solutions.

			Array begins:
=============================================================================
m\n|   1     2       3         4           5             6               7
---|-------------------------------------------------------------------------
1  |   1     3       7        15          31            63             127...
2  |   3    11      51       227         963          3971           16131...
3  |   7    51     421      3615       30517        252231         2054941...
4  |  15   227    3615     59747      989295      16219187       263425695...
5  |  31   963   30517    989295    32260381    1048220463     33884452717...
6  |  63  3971  252231  16219187  1048220463   67680006971   4358402146791...
7  | 127 16131 2054941 263425695 33884452717 4358402146791 559876911043381...
...

Andrew Howroyd, Table of n, a(n) for n = 1..780
Stephan Mertens, Domination Polynomial of the Rook Graph, JIS 27 (2024) 24.3.7; arXiv:2401.00716 [math.CO], 2024.
Eric Weisstein's World of Mathematics, Dominating Set
Eric Weisstein's World of Mathematics, Rook Graph

Main diagonal is A287065.
Row 2 is A191341.
Cf. A183109, A088699 (independent vertex sets), A290632.

b[m_, n_] := Sum[(-1)^j*Binomial[m, j]*(2^(m - j) - 1)^n, {j, 0, m}];
a[m_, n_] := (2^n - 1)^m + Sum[ b[i, n]*Binomial[m, i], {i, 1, m - 1}];
Table[a[m - n + 1, n], {m, 1, 9}, {n, 1, m}] // Flatten (* Jean-François Alcover, Jun 12 2017, adapted from PARI *)

b(m,n)=sum(j=0, m, (-1)^j*binomial(m, j)*(2^(m - j) - 1)^n);
a(m,n)=(2^n-1)^m + sum(i=1,m-1,b(i,n)*binomial(m,i));
for (i=1,7,for(j=1,7, print1(a(i,j), ",")); print);

T(m, n) = (2^n-1)^m + Sum_{i=1..m-1} binomial(m,i) * A183109(i,n).

A218695 Square array A(h,k) = (2^h-1)A(h,k-1) + Sum_{i=1..h-1} binomial(h,h-i)2^i*A(i,k-1), with A(1,k) = A(h,1) = 1; read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 7, 1, 1, 25, 25, 1, 1, 79, 265, 79, 1, 1, 241, 2161, 2161, 241, 1, 1, 727, 16081, 41503, 16081, 727, 1, 1, 2185, 115465, 693601, 693601, 115465, 2185, 1, 1, 6559, 816985, 10924399, 24997921, 10924399, 816985, 6559, 1

Offset: 1

M. F. Hasler, Nov 04 2012

This symmetric table is defined in the Kreweras papers, used also in A223911. Its upper or lower triangular part equals A183109, which might provide a simpler formula.
Number of h X k binary matrices with no zero rows or columns. - Andrew Howroyd, Mar 29 2023
A(h,k) is the number of coverings of [h] by tuples (A_1,...,A_k) in P([h])^k with nonempty A_j, with P(.) denoting the power set. For the disjoint case see A019538. For tuples with "nonempty" omitted see A092477 and A329943 (amendment by Manfred Boergens, Jun 24 2024). - Manfred Boergens, May 26 2024

			Array A(h,k) begins:
=====================================================
h\k | 1   2      3        4         5           6 ...
----+------------------------------------------------
  1 | 1   1      1        1         1           1 ...
  2 | 1   7     25       79       241         727 ...
  3 | 1  25    265     2161     16081      115465 ...
  4 | 1  79   2161    41503    693601    10924399 ...
  5 | 1 241  16081   693601  24997921   831719761 ...
  6 | 1 727 115465 10924399 831719761 57366997447 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 antidiagonals)
G. Kreweras, Dénombrements systématiques de relations binaires externes, Math. Sci. Humaines 26 (1969) 5-15.
G. Kreweras, Dénombrement des ordres étagés, Discrete Math., 53 (1985), 147-149.

Columns 1..3 are A000012, A058481, A058482.
Main diagonal is A048291.
Cf. A019538, A056152 (unlabeled case), A052332, A092477, A183109, A223911, A329943.

c(h,k)={(h<2 || k<2) & return(1); sum(i=1,h-1,binomial(h,h-i)*2^i*c(i,k-1))+(2^h-1)*c(h,k-1)}
/* For better performance when h and k are large, insert the following memoization code before "sum(...)": cM=='cM & cM=matrix(h,k); my(s=matsize(cM));
s[1] >= h & s[2] >= k & cM[h,k] & return(cM[h,k]);
s[1]

A(m, n) = sum(k=0, m, (-1)^(m-k) * binomial(m, k) * (2^k-1)^n ) \\ Andrew Howroyd, Mar 29 2023

From Andrew Howroyd, Mar 29 2023: (Start)
A(h, k) = Sum_{i=0..h} (-1)^(h-i) * binomial(h, i) * (2^i-1)^k.
A052332(n) = Sum_{i=1..n-1} binomial(n,i)*A(i, n-i) for n > 0. (End)

A287065 Number of dominating sets on the n X n rook graph.

Original entry on oeis.org

1, 11, 421, 59747, 32260381, 67680006971, 559876911043381, 18412604442711949187, 2416403019417984915336061, 1267413006543912045144741284411, 2658304092145691708492995820522716981, 22300364428188338185156192161829091442585827

Offset: 1

Eric W. Weisstein, May 19 2017

nonn

Number of {0,1} n X n matrices with no zero rows or no zero columns. - Geoffrey Critzer, Jan 15 2024

Andrew Howroyd, Table of n, a(n) for n = 1..50
Eric Weisstein's World of Mathematics, Dominating Set
Eric Weisstein's World of Mathematics, Rook Graph

Main diagonal of A287274.
Row sums of A368831.
Cf. A183109, A002720, A048291, A173403.

Table[(2^n - 1)^n + Sum[Binomial[n, i] Sum[(-1)^j (-1 + 2^(n - j))^i Binomial[n, j], {j, 0, n}], {i, n - 1}], {n, 20}] (* Eric W. Weisstein, May 27 2017 *)

b(m,n)=sum(j=0, m, (-1)^j*binomial(m, j)*(2^(m - j) - 1)^n);
a(n)=(2^n-1)^n + sum(i=1,n-1,b(n,i)*binomial(n,i)); \\ Andrew Howroyd, May 22 2017

a(n) = (2^n-1)^n + Sum_{i=1..n-1} binomial(n,i) * A183109(n,i). - Andrew Howroyd, May 22 2017

a(6)-a(12) from Andrew Howroyd, May 22 2017

A360873 Array read by antidiagonals: T(m,n) is the number of (non-null) connected induced subgraphs in the rook graph K_m X K_n.

Original entry on oeis.org

1, 3, 3, 7, 13, 7, 15, 51, 51, 15, 31, 205, 397, 205, 31, 63, 843, 3303, 3303, 843, 63, 127, 3493, 27877, 55933, 27877, 3493, 127, 255, 14451, 233751, 943095, 943095, 233751, 14451, 255, 511, 59485, 1938517, 15678925, 31450861, 15678925, 1938517, 59485, 511

Offset: 1

Andrew Howroyd, Feb 24 2023

			Array begins:
=======================================================
m\n|  1    2      3        4          5           6 ...
---+---------------------------------------------------
1  |  1    3      7       15         31          63 ...
2  |  3   13     51      205        843        3493 ...
3  |  7   51    397     3303      27877      233751 ...
4  | 15  205   3303    55933     943095    15678925 ...
5  | 31  843  27877   943095   31450861  1033355223 ...
6  | 63 3493 233751 15678925 1033355223 67253507293 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 antidiagonals).
Eric Weisstein's World of Mathematics, Rook Graph.
Eric Weisstein's World of Mathematics, Vertex-Induced Subgraph.

Main diagonal is A286189.
Rows 1..2 are A000225, A360874.
Cf. A360850, A360851, A360853, A360875.

\\ S is A183109, T is A262307, U is this sequence.
G(M,N=M)={ my(S=matrix(M, N), T=matrix(M, N), U=matrix(M, N));
for(m=1, M, for(n=1, N,
  S[m, n]=sum(j=0, m, (-1)^j*binomial(m, j)*(2^(m - j) - 1)^n);
  T[m, n]=S[m, n]-sum(i=1, m-1, sum(j=1, n-1, T[i, j]*S[m-i, n-j]*binomial(m-1, i-1)*binomial(n, j)));
  U[m, n]=sum(i=1, m, sum(j=1, n, binomial(m, i)*binomial(n, j)*T[i, j])) )); U
}
{ my(A=G(7)); for(n=1, #A~, print(A[n,])) }

T(m,n) = Sum_{i=1..m} Sum_{j=1..n} binomial(m, i) * binomial(n, j) * A262307(i, j).

T(m,n) = T(n,m).

A360875 Array read by antidiagonals: T(m,n) is the number of connected dominating sets in the rook graph K_m X K_n.

Original entry on oeis.org

1, 3, 3, 7, 9, 7, 15, 39, 39, 15, 31, 177, 325, 177, 31, 63, 783, 2931, 2931, 783, 63, 127, 3369, 26077, 51465, 26077, 3369, 127, 255, 14199, 225459, 894675, 894675, 225459, 14199, 255, 511, 58977, 1901725, 15195897, 30331861, 15195897, 1901725, 58977, 511

Offset: 1

Andrew Howroyd, Feb 24 2023

			Array begins:
=======================================================
m\n|  1    2      3        4          5           6 ...
---+---------------------------------------------------
1  |  1    3      7       15         31          63 ...
2  |  3    9     39      177        783        3369 ...
3  |  7   39    325     2931      26077      225459 ...
4  | 15  177   2931    51465     894675    15195897 ...
5  | 31  783  26077   894675   30331861  1010163363 ...
6  | 63 3369 225459 15195897 1010163363 66273667449 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 antidiagonals).
Eric Weisstein's World of Mathematics, Connected Dominating Set
Eric Weisstein's World of Mathematics, Rook Graph

Main diagonal is A289196.
Rows 1..2 are A000225, A360876.
Cf. A287274, A360873.

\\ S is A183109, T is A262307, U is this sequence.
G(M,N=M)={S=matrix(M, N); T=matrix(M, N); U=matrix(M, N);
for(m=1, M, for(n=1, N,
  S[m, n]=sum(j=0, m, (-1)^j*binomial(m, j)*(2^(m - j) - 1)^n);
  T[m, n]=S[m, n]-sum(i=1, m-1, sum(j=1, n-1, T[i, j]*S[m-i, n-j]*binomial(m-1, i-1)*binomial(n, j)));
  U[m, n]=sum(i=1, m, binomial(m, i)*T[i, n])+sum(j=1, n, binomial(n, j)*T[m, j])-T[m, n] )); U
}
{ my(A=G(7)); for(n=1, #A~, print(A[n,])) }

T(m,n) = (Sum_{i=1..m} binomial(m,i) * A262307(n,i)) + (Sum_{j=1..n} binomial(n,j) * A262307(m,j)) - A262307(m,n).

T(m,n) = T(n,m).

A058482 Number of 3 X n binary matrices with no zero rows or columns.

Original entry on oeis.org

1, 25, 265, 2161, 16081, 115465, 816985, 5745121, 40294561, 282298105, 1976795305, 13839692881, 96884227441, 678208723945, 4747518463225, 33232801429441, 232630126566721, 1628412435648985, 11398891698588745, 79792255837258801, 558545832702224401

Offset: 1

Vladeta Jovovic, Nov 26 2000

Index entries for linear recurrences with constant coefficients, signature (11,-31,21).

Cf. A055602, A024206, A055609 (unlabeled case), A058481, column 3 of A183109 and A218695.

LinearRecurrence[{11,-31,21},{1,25,265},30] (* Harvey P. Dale, Aug 15 2014 *)

a(n) = 7^n-3*3^n+3 \\ Charles R Greathouse IV, Feb 10 2017

Number of m X n binary matrices with no zero rows or columns is Sum_{j=0..m}(-1)^j*C(m, j)*(2^(m-j)-1)^n.
a(n) = 7^n-3*3^n+3.
a(n) = 11*a(n-1)-31*a(n-2)+21*a(n-3). G.f.: -x*(21*x^2+14*x+1) / ((x-1)*(3*x-1)*(7*x-1)). - Colin Barker, Jul 10 2013

More terms from Larry Reeves (larryr(AT)acm.org), Dec 04 2000

More terms from Colin Barker, Jul 10 2013

cp's OEIS Frontend

A019538 Triangle of numbers T(n,k) = k!*Stirling2(n,k) read by rows (n >= 1, 1 <= k <= n).

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A048291 Number of {0,1} n X n matrices with no zero rows or columns.

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A286189 Number of connected induced (non-null) subgraphs of the n X n rook graph.

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A262307 Array read by antidiagonals: T(m,n) = number of m X n binary matrices with all 1's connected and no zero rows or columns.

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A287274 Array read by antidiagonals: T(m,n) = number of dominating sets in the lattice (rook) graph K_m X K_n.

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A218695 Square array A(h,k) = (2^h-1)*A(h,k-1) + Sum_{i=1..h-1} binomial(h,h-i)*2^i*A(i,k-1), with A(1,k) = A(h,1) = 1; read by antidiagonals.

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A218695 Square array A(h,k) = (2^h-1)A(h,k-1) + Sum_{i=1..h-1} binomial(h,h-i)2^i*A(i,k-1), with A(1,k) = A(h,1) = 1; read by antidiagonals.