A104602 -id:A104602 - cp's OEIS frontend

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

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

A138178 Number of symmetric matrices with nonnegative integer entries and without zero rows or columns such that sum of all entries is equal to n.

Original entry on oeis.org

1, 1, 3, 9, 33, 125, 531, 2349, 11205, 55589, 291423, 1583485, 8985813, 52661609, 319898103, 2000390153, 12898434825, 85374842121, 580479540219, 4041838056561, 28824970996809, 210092964771637, 1564766851282299, 11890096357039749, 92151199272181629

Offset: 0

Vladeta Jovovic, Mar 03 2008

Number of normal semistandard Young tableaux of size n, where a tableau is normal if its entries span an initial interval of positive integers. - Gus Wiseman, Feb 23 2018

			a(4) = 33 because there are 1 such matrix of type 1 X 1, 7 matrices of type 2 X 2, 15 of type 3 X 3 and 10 of type 4 X 4, cf. A138177.
From _Gus Wiseman_, Feb 23 2018: (Start)
The a(3) = 9 normal semistandard Young tableaux:
1   1 2   1 3   1 2   1 1   1 2 3   1 2 2   1 1 2   1 1 1
2   3     2     2     2
3
(End)
From _Gus Wiseman_, Nov 14 2018: (Start)
The a(4) = 33 matrices:
[4]
.
[30][21][20][11][10][02][01]
[01][10][02][11][03][20][12]
.
[200][200][110][101][100][100][100][100][011][010][010][010][001][001][001]
[010][001][100][010][020][011][010][001][100][110][101][100][020][010][001]
[001][010][001][100][001][010][002][011][100][001][010][002][100][101][110]
.
[1000][1000][1000][1000][0100][0100][0010][0010][0001][0001]
[0100][0100][0010][0001][1000][1000][0100][0001][0100][0010]
[0010][0001][0100][0010][0010][0001][1000][1000][0010][0100]
[0001][0010][0001][0100][0001][0010][0001][0100][1000][1000]
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..500

Row sums of A138177.
Cf. A007716, A120733, A135588, A296188.
Cf. A057151, A104601, A104602, A120732, A316983, A320796, A321401, A321405, A321407.

gf:= proc(j) local k, n; add(add((-1)^(n-k) *binomial(n, k) *(1-x)^(-k) *(1-x^2)^(-binomial(k, 2)), k=0..n), n=0..j) end: a:= n-> coeftayl(gf(n+1), x=0, n): seq(a(n), n=0..25); # Alois P. Heinz, Sep 25 2008

Table[Sum[SeriesCoefficient[1/(2^(k+1)*(1-x)^k*(1-x^2)^(k*(k-1)/2)),{x,0,n}],{k,0,Infinity}],{n,0,20}]  (* Vaclav Kotesovec, Jul 03 2014 *)
multsubs[set_,k_]:=If[k==0,{{}},Join@@Table[Prepend[#,set[[i]]]&/@multsubs[Drop[set,i-1],k-1],{i,Length[set]}]]; Table[Length[Select[multsubs[Tuples[Range[n],2],n],And[Union[First/@#]==Range[Max@@First/@#],Union[Last/@#]==Range[Max@@Last/@#],Sort[Reverse/@#]==#]&]],{n,5}] (* Gus Wiseman, Nov 14 2018 *)

G.f.: Sum_{n>=0} Sum_{k=0..n} (-1)^(n-k)*C(n,k)*(1-x)^(-k)*(1-x^2)^(-C(k,2)).

G.f.: Sum_{n>=0} 2^(-n-1)*(1-x)^(-n)*(1-x^2)^(-C(n,2)). - Vladeta Jovovic, Dec 09 2009

More terms from Alois P. Heinz, Sep 25 2008

A120732 Number of square matrices with nonnegative integer entries and without zero rows or columns such that sum of all entries is equal to n.

Original entry on oeis.org

1, 1, 3, 15, 107, 991, 11267, 151721, 2360375, 41650861, 821881709, 17932031225, 428630422697, 11138928977049, 312680873171465, 9428701154866535, 303957777464447449, 10431949496859168189, 379755239311735494421

Offset: 0

Vladeta Jovovic, Aug 18 2006

nonn

			From _Gus Wiseman_, Nov 14 2018: (Start)
The a(3) = 15 matrices:
  [3]
.
  [2 0] [1 1] [1 1] [1 0] [1 0] [0 2] [0 1] [0 1]
  [0 1] [1 0] [0 1] [1 1] [0 2] [1 0] [2 0] [1 1]
.
  [1 0 0] [1 0 0] [0 1 0] [0 1 0] [0 0 1] [0 0 1]
  [0 1 0] [0 0 1] [1 0 0] [0 0 1] [1 0 0] [0 1 0]
  [0 0 1] [0 1 0] [0 0 1] [1 0 0] [0 1 0] [1 0 0]
(End)

G. C. Greubel, Table of n, a(n) for n = 0..250

Cf. A007716, A048291, A054976, A057149, A057150, A057151, A104601, A104602, A120733, A138178, A316983, A319616.

Table[1/n!*Sum[(-1)^(n-k)*StirlingS1[n,k]*Sum[(m!)^2*StirlingS2[k,m]^2,{m,0,k}],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, May 07 2014 *)
multsubs[set_,k_]:=If[k==0,{{}},Join@@Table[Prepend[#,set[[i]]]&/@multsubs[Drop[set,i-1],k-1],{i,Length[set]}]]; Table[Length[Select[multsubs[Tuples[Range[n],2],n],Union[First/@#]==Union[Last/@#]==Range[Max@@First/@#]&]],{n,5}] (* Gus Wiseman, Nov 14 2018 *)

a(n) = (1/n!)*Sum_{k=0..n} (-1)^(n-k)*Stirling1(n,k)*A048144(k).
G.f.: Sum_{n>=0} Sum_{j=0..n} (-1)^(n-j)*binomial(n,j)*((1-x)^(-j)-1)^n.
a(n) ~ c * n! / (sqrt(n) * (log(2))^(2*n)), where c = 0.4670932578797312973586879293426... . - Vaclav Kotesovec, May 07 2014
In closed form, c = 2^(log(2)/2-2) / (log(2) * sqrt(Pi*(1-log(2)))). - Vaclav Kotesovec, May 03 2015
G.f.: Sum_{n>=0} (1-x)^n * (1 - (1-x)^n)^n. - Paul D. Hanna, Mar 26 2018

A122400 Number of square (0,1)-matrices without zero rows and with exactly n entries equal to 1.

Original entry on oeis.org

1, 1, 4, 31, 338, 4769, 82467, 1687989, 39905269, 1069863695, 32071995198, 1062991989013, 38596477083550, 1523554760656205, 64961391010251904, 2975343608212835855, 145687881987604377815, 7594435556630244257213

Offset: 0

Vladeta Jovovic, Aug 31 2006

G. C. Greubel, Table of n, a(n) for n = 0..375

Cf. A104602, A220353, A301581, A301582, A301583, A301584.

A122399 := proc(n) option remember ; add( combinat[stirling2](n,k)*k^n*k!,k=0..n) ; end: A122400 := proc(n) option remember ; add( combinat[stirling1](n,k)*A122399(k),k=0..n)/n! ; end: for n from 0 to 30 do printf("%d, ",A122400(n)) ; od ; # R. J. Mathar, May 18 2007

max = 17; CoefficientList[ Series[ 1 + Sum[ ((1 + x)^n - 1)^n, {n, 1, max}], {x, 0, max}], x] (* Jean-François Alcover, Mar 26 2013, after Vladeta Jovovic *)

a(n) = (1/n!)* Sum_{k=0..n} Stirling1(n,k)*A122399(k).
G.f.: Sum_{n>=0} ((1+x)^n - 1)^n. - Vladeta Jovovic, Sep 03 2006
G.f.: Sum_{n>=0} (1+x)^(n^2) / (1 + (1+x)^n)^(n+1). - Paul D. Hanna, Mar 23 2018
a(n) ~ c * d^n * n! / sqrt(n), where d = A317855 = (1+exp(1/r))*r^2 = 3.161088653865428813830172202588132491726382774188556341627278..., r = 0.8737024332396683304965683047207192982139922672025395099... is the root of the equation exp(1/r)/r + (1+exp(1/r))*LambertW(-exp(-1/r)/r) = 0, and c = 0.2796968489586733500739737080739303725411427162653658... . - Vaclav Kotesovec, May 07 2014

More terms from R. J. Mathar, May 18 2007

A048144 a(n) = Sum_{k=0..n} (k!)^2 * Stirling_2(n,k)^2.

Original entry on oeis.org

1, 1, 5, 73, 2069, 95401, 6487445, 610093513, 75796724309, 12020754177001, 2369364111428885, 568128719132038153, 162835627057766030549, 54975855375379966645801, 21593185551426744571090325, 9762238510837560633366673993, 5033241437347149354018370856789

Offset: 0

N. J. A. Sloane

Number of digraphs with loops, with labeled vertices and labeled arcs, with n arcs and with no vertex of indegree 0 or outdegree 0, cf. A121936, A122418, A122399. - Vladeta Jovovic, Sep 06 2006
Chromatic invariant of the complete bipartite graph K_{n+1,n+1}. - Eric W. Weisstein, Jul 11 2011
Generally, for p >= 1, Sum_{k=0..n} (k!*StirlingS2(n,k))^p is asymptotic to n^(p*n+1/2) * sqrt(Pi/(2*p*(1-log(2))^(p-1))) / (exp(p*n) * log(2)^(p*n+1)). - Vaclav Kotesovec, May 10 2014

Alois P. Heinz, Table of n, a(n) for n = 0..100
Eric Weisstein's World of Mathematics, Chromatic Invariant
Eric Weisstein's World of Mathematics, Complete Bipartite Graph

Cf. A000670, A242280, A212084, A120732, A104602, A272644, A371761.

a := proc(n) local A, j; A := proc(n, k) option remember; if n = 0 then n^k else add(binomial(k + `if`(j>0, 1, 0), j+1) * A(n-1, k-j), j = 0..k) fi end: A(n,n) end:
seq(a(n), n = 0..16);  # Peter Luschny, Nov 20 2024

Table[Sum[(k!)^2*StirlingS2[n,k]^2,{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, May 07 2014 *)

a(n) = sum(k=0, n, k!^2*stirling(n, k, 2)^2); \\ Michel Marcus, Mar 07 2020

from functools import cache
from math import comb as binomial
@cache
def A(n, k): return int(k == 0) if n == 0 else sum(binomial(k + int(j > 0), j + 1) * A(n - 1, k - j) for j in range(k + 1))
a = lambda n: A(n, n)
print([a(n) for n in range(17)])  # Peter Luschny, Nov 20 2024

E.g.f.: Sum_{n>=0} Sum_{j=0..n} (-1)^(n-j)*binomial(n,j)*(exp(j*x)-1)^n. a(n) = Sum_{k=0..n} Stirling2(n,k)*k!*A104602(k). - Vladeta Jovovic, Mar 25 2006
a(n) ~ sqrt(Pi/(1-log(2))) * n^(2*n+1/2) / (2*exp(2*n) * (log(2))^(2*n+1)). - Vaclav Kotesovec, May 09 2014
E.g.f.: Sum_{n>=0} (1 - exp(-n*x))^n * exp(-n*x). - Paul D. Hanna, Mar 26 2018
E.g.f.: Sum_{n>=0} (exp(n*x) - 1)^n * exp(-n*(n+1)*x). - Paul D. Hanna, Mar 26 2018
a(n) = A272644(2n,n). - Alois P. Heinz, Oct 17 2024
a(n) = A371761(n, n). - Peter Luschny, Nov 20 2024
a(n) = (n!)^2 * [(x*y)^n] 1 / (exp(x) + exp(y) - exp(x + y)). - Ilya Gutkovskiy, Apr 24 2025

A054976 Number of binary n X n matrices with no zero rows or columns, up to row and column permutation.

Original entry on oeis.org

1, 3, 17, 179, 3835, 200082, 29610804, 13702979132, 20677458750966, 103609939177198046, 1745061194503344181714, 99860890306900024150675406, 19611238933283757244479826044874, 13340750149227624084760722122669739026, 31706433098827528779057124372265863803044450

Offset: 1

Vladeta Jovovic, May 27 2000

Also the number of non-isomorphic set multipartitions (multisets of sets) with n parts and n vertices. - Gus Wiseman, Nov 18 2018

			From _Gus Wiseman_, Nov 18 2018: (Start)
Inequivalent representatives of the a(3) = 17 matrices:
  100 100 100 100 100 010 010 001 001 001 001 110 101 101 011 011 111
  100 010 001 011 011 001 101 001 101 011 111 101 011 011 011 111 111
  011 001 011 011 111 111 011 111 011 111 111 011 011 111 111 111 111
Non-isomorphic representatives of the a(1) = 1 through a(3) = 17 set multipartitions:
  {{1}}  {{1},{2}}      {{1},{2},{3}}
         {{2},{1,2}}    {{1},{1},{2,3}}
         {{1,2},{1,2}}  {{1},{3},{2,3}}
                        {{1},{2,3},{2,3}}
                        {{2},{1,3},{2,3}}
                        {{2},{3},{1,2,3}}
                        {{3},{1,3},{2,3}}
                        {{3},{3},{1,2,3}}
                        {{1,2},{1,3},{2,3}}
                        {{1},{2,3},{1,2,3}}
                        {{1,3},{2,3},{2,3}}
                        {{3},{2,3},{1,2,3}}
                        {{1,3},{2,3},{1,2,3}}
                        {{2,3},{2,3},{1,2,3}}
                        {{3},{1,2,3},{1,2,3}}
                        {{2,3},{1,2,3},{1,2,3}}
                        {{1,2,3},{1,2,3},{1,2,3}}
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..50

Column sums of A057150.

Cf. A007716, A048291 (labeled), A049311, A057149, A057151, A104601, A104602, A319616, A320808.

A002724 = Cases[Import["https://oeis.org/A002724/b002724.txt", "Table"], {, }][[All, 2]];
A002725 = Cases[Import["https://oeis.org/A002725/b002725.txt", "Table"], {, }][[All, 2]];
a[n_] := A002724[[n + 1]] - 2 A002725[[n]] + A002724[[n]];
a /@ Range[1, 13] (* Jean-François Alcover, Sep 14 2019 *)

a(n) = A002724(n) - 2*A002725(n-1) + A002724(n-1).

More terms from David Wasserman, Mar 06 2002

Terms a(14) and beyond from Andrew Howroyd, Apr 11 2020

A138265 Number of upper triangular zero-one matrices with n ones and no zero rows or columns.

Original entry on oeis.org

1, 1, 1, 2, 5, 16, 61, 271, 1372, 7795, 49093, 339386, 2554596, 20794982, 182010945, 1704439030, 17003262470, 180011279335, 2015683264820, 23801055350435, 295563725628564, 3850618520827590, 52514066450469255, 748191494586458700, 11115833059268126770

Offset: 0

Vladeta Jovovic, Mar 10 2008, Mar 11 2008

Row sums of A193357.
This is also the number of rigid unlabeled interval orders with n points (see Brightwell-Keller, Theorem 2; or Dukes-Kitaev-Remmel-Steingrímsson, Theorem 8). - N. J. A. Sloane, Dec 04 2011  [Corrected by Vít Jelínek, Sep 04 2014.]
Number of length-n ascent sequences without flat steps (i.e., no two adjacent digits are equal).  An ascent sequence is a sequence [d(1), d(2), ..., d(n)] where d(k)>=0 and d(k) <= 1 + asc([d(1), d(2), ..., d(k-1)]) and asc(.) gives the number of ascents of its argument. [Joerg Arndt, Nov 05 2012]

			From _Joerg Arndt_, Nov 05 2012: (Start)
The a(4) = 5 such matrices with 4 ones are (dots for zeros):
  1 . . .      1 1 .      1 . 1      1 1 .      1 . .
  . 1 . .      . . 1      . 1 .      . 1 .      . 1 1
  . . 1 .      . . 1      . . 1      . . 1      . . 1
  . . . 1
The a(5)=16 ascent sequences without flat steps are (dots for zeros):
  [ 1]   [ . 1 . 1 . ]
  [ 2]   [ . 1 . 1 2 ]
  [ 3]   [ . 1 . 1 3 ]
  [ 4]   [ . 1 . 2 . ]
  [ 5]   [ . 1 . 2 1 ]
  [ 6]   [ . 1 . 2 3 ]
  [ 7]   [ . 1 2 . 1 ]
  [ 8]   [ . 1 2 . 2 ]
  [ 9]   [ . 1 2 . 3 ]
  [10]   [ . 1 2 1 . ]
  [11]   [ . 1 2 1 2 ]
  [12]   [ . 1 2 1 3 ]
  [13]   [ . 1 2 3 . ]
  [14]   [ . 1 2 3 1 ]
  [15]   [ . 1 2 3 2 ]
  [16]   [ . 1 2 3 4 ]
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..300
G. E. Andrews and J. A. Sellers, Congruences for the Fishburn Numbers, arXiv preprint arXiv:1401.5345 [math.NT], 2014 (see final paragraph of text).
George E. Andrews and Vít Jelínek, On q-Series Identities Related to Interval Orders, arXiv:1309.6669 [math.CO], 2013.
George E. Andrews and Vít Jelínek, On q-Series Identities Related to Interval Orders, European Journal of Combinatorics, Volume 39, July 2014, 178-187.
Graham Brightwell and Mitchel T. Keller, Asymptotic Enumeration of Labelled Interval Orders, arXiv 1111.6766 [math.CO], 2011.
Kathrin Bringmann, Yingkun Li, and Robert C. Rhoades, Asymptotics for the number of row-Fishburn matrices, European Journal of Combinatorics, vol.41, pp.183-196, (October-2014); preprint.
Matthieu Dien, Antoine Genitrini, and Frederic Peschanski, A Combinatorial Study of Async/Await Processes, Conf.: 19th Int'l Colloq. Theor. Aspects of Comp. (2022), (Analytic) Combinatorics of concurrent systems.
M. Dukes, S. Kitaev, J. Remmel, and E. Steingrímsson, Enumerating (2+2)-free posets by indistinguishable elements, arXiv:1006.2696 [math.CO], 2010.
M. Dukes, S. Kitaev, J. Remmel, and E. Steingrímsson, Enumerating (2+2)-free posets by indistinguishable elements, Journal of Combinatorics, Volume 2, Issue 1, 2011, pp. 139-163.
F. Garvan, Congruences and relations for the r-Fishburn numbers, arXiv:1406.5611 [math.NT], 2014.
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.
V. Jelínek, Counting self-dual interval orders, arXiv:1106.2261 [math.CO], 2011.
V. Jelínek, Counting general and self-dual interval orders, Journal of Combinatorial Theory, Series A, Volume 119, Issue 3, April 2012, pp. 599-614.
V. Jelinek, Catalan pairs and Fishburn triples, Adv. Appl. Math. 70 (2015) 1-31
Soheir Mohamed Khamis, Exact Counting of Unlabeled Rigid Interval Posets Regarding or Disregarding Height, Order (journal), published on-line, 2011.
Yan X. Zhang, Four Variations on Graded Posets, arXiv preprint arXiv:1508.00318 [math.CO], 2015.

Cf. A005321, A104602, A135588, A193548, A194530.
Column k=0 of A242153.
Column k=1 of A264909.
Row sums of A137252.

g:=sum(product(1-1/(1+x)^i,i=1..n),n=0..35): gser:=series(g,x=0,30): seq(coeff(gser,x,n),n=0..22);  # Emeric Deutsch, Mar 23 2008
# second Maple program:
b:= proc(n, i, t) option remember; `if`(n<1, 1, add(
     `if`(i=j, 0, b(n-1, j, t+`if`(j>i, 1, 0))), j=0..t+1))
    end:
a:= n-> b(n-1, 0$2):
seq(a(n), n=0..30);  # Alois P. Heinz, Nov 09 2012, Jan 14 2015

max = 25; g = Sum[Product[1 - 1/(1 - x)^i, {i, 1, n}], {n, 0, max}]; gser = Series[g, {x, 0, max}]; a[n_] := SeriesCoefficient[gser, {x, 0, n}]; Table[a[n] // Abs, {n, 0, max-1}] (* Jean-François Alcover, Jan 24 2014, after Emeric Deutsch *)

# Adaptation of the Maple program by Alois P. Heinz:
@CachedFunction
def b(n, i, t):
    if n<1: return 1
    return sum(b(n-1, j, t+(j>i)) for j in range(t+2))
def a(n):
    if n<1: return 1
    return sum((-1)^(n-k)*binomial(n-1, k-1)*b(k-1, 0, 0) for k in range(n+1))
[a(n) for n in range(33)]
# Joerg Arndt, Feb 26 2014

G.f.: Sum_{n>=0} (Product_{i=1..n} 1-1/(1+x)^i).
G.f.: Sum_{n>=0} (1+x)^(n+1)*Product_{i=1..n} (1-(1+x)^i)^2. Proved by Bringmann-Li-Rhoades, and by Andrews-Jelínek. - Vít Jelínek, Sep 04 2014
a(n) = (1/n!)*Sum_{k=0..n} Stirling1(n,k)*A079144(k). a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n-1,k-1)*A022493(k).
G.f.: B(x/(1+x)) where B(x) is the g.f. of A022493; g.f.: Q(0,u) where u=x/(1+x), Q(k,u) = 1 + (1 - (1-x)^(2*k+1))/(1 - (1-(1-x)^(2*k+2))/(1 -(1-x)^(2*k+2) + 1/Q(k+1,u) )); (continued fraction). - Sergei N. Gladkovskii, Oct 03 2013
Asymptotics (Brightwell and Keller, 2011): a(n) ~ 12*sqrt(3)/(exp(Pi^2/12)*Pi^(5/2)) * n!*sqrt(n)*(6/Pi^2)^n. - Vaclav Kotesovec, May 03 2014
From Vít Jelínek, Sep 04 2014: (Start)
For each m, a(5m+4) mod 5 = 0. Conjectured by Andrews-Sellers, and proved by Garvan (see Remark 1.4(ii) in Garvan's paper).
For each m, a(5m+1) mod 5 = a(5m+2) mod 5 = 3*a(5m+3) mod 5. Proved by Garvan (see (1.17) in Garvan's paper).
The limit a(n)/A022493(n) is equal to exp(-Pi^2/6). This corresponds to the asymptotic probability that a random unlabeled interval order is rigid (See Brightwell-Keller; or Jelínek, Fact 5.2). (End)
Conjectural g.f.:  1 + Sum_{n >= 0} n/(1+x)^(n+1) * (Product_{i = 1..n} 1 - 1/(1+x)^i). Cf. A194530. - Peter Bala, Aug 21 2023

More terms from Emeric Deutsch, Mar 23 2008

A057151 Number of square binary matrices with n ones, with no zero rows or columns, up to row and column permutation.

Original entry on oeis.org

1, 1, 2, 4, 8, 18, 41, 102, 252, 666, 1789, 5031, 14486, 43280, 132777, 420267, 1366307, 4566966, 15661086, 55081118, 198425478, 731661754, 2758808581, 10629386376, 41814350148, 167830018952, 686822393793, 2864024856054, 12162059027416, 52564545391789

Offset: 1

Vladeta Jovovic, Aug 14 2000

nonn

Number of square binary matrices with n ones and with no zero rows or columns is A104602(n). - Vladeta Jovovic, Mar 25 2006

Also the number of non-isomorphic square set multipartitions (multisets of sets) of weight n. A multiset partition or hypergraph is square if its length (number of blocks or edges) is equal to its number of vertices. The weight of a multiset partition is the sum of sizes of its parts. - Gus Wiseman, Nov 16 2018

			There are 666 square binary matrices with 10 ones, with no zero rows or columns, up to row and column permutation: 33 of size 4 X 4, 248 of size 5 X 5, 288 of size 6 X 6, 79 of size 7 X 7, 15 of size 8 X 8, 2 of size 9 X 9 and 1 of size 10 X 10. Cf. A057150.
From _Gus Wiseman_, Nov 16 2018: (Start)
Non-isomorphic representatives of the a(1) = 1 through a(6) = 18 square set multipartitions:
  {1}  {1}{2}  {2}{12}    {12}{12}      {1}{23}{23}      {12}{13}{23}
               {1}{2}{3}  {1}{1}{23}    {2}{13}{23}      {1}{23}{123}
                          {1}{3}{23}    {2}{3}{123}      {13}{23}{23}
                          {1}{2}{3}{4}  {3}{13}{23}      {3}{23}{123}
                                        {3}{3}{123}      {1}{1}{1}{234}
                                        {1}{2}{2}{34}    {1}{1}{24}{34}
                                        {1}{2}{4}{34}    {1}{1}{4}{234}
                                        {1}{2}{3}{4}{5}  {1}{2}{34}{34}
                                                         {1}{3}{24}{34}
                                                         {1}{3}{4}{234}
                                                         {1}{4}{24}{34}
                                                         {1}{4}{4}{234}
                                                         {2}{4}{12}{34}
                                                         {3}{4}{12}{34}
                                                         {4}{4}{12}{34}
                                                         {1}{2}{3}{3}{45}
                                                         {1}{2}{3}{5}{45}
                                                         {1}{2}{3}{4}{5}{6}
(End)

Max Alekseyev, Table of n, a(n) for n = 1..30

Cf. A049311, A056037, A056079, A056080, A057149, A057150, A057152.

Cf. A054976, A101370, A104601, A104602, A120732, A283877, A319616.

More terms from Max Alekseyev, May 31 2007

A057150 Triangle read by rows: T(n,k) = number of k X k binary matrices with n ones, with no zero rows or columns, up to row and column permutation.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 0, 5, 2, 1, 0, 0, 4, 11, 2, 1, 0, 0, 3, 21, 14, 2, 1, 0, 0, 1, 34, 49, 15, 2, 1, 0, 0, 1, 33, 131, 69, 15, 2, 1, 0, 0, 0, 33, 248, 288, 79, 15, 2, 1, 0, 0, 0, 19, 410, 840, 420, 82, 15, 2, 1, 0, 0, 0, 14, 531, 2144, 1744, 497, 83, 15, 2, 1

Offset: 1

Vladeta Jovovic, Aug 14 2000

Also the number of non-isomorphic set multipartitions (multisets of sets) of weight n with k parts and k vertices. - Gus Wiseman, Nov 14 2018

			[1], [0,1], [0,1,1], [0,1,2,1], [0,0,5,2,1], [0,0,4,11,2,1], ...;
There are 8 square binary matrices with 5 ones, with no zero rows or columns, up to row and column permutation: 5 of size 3 X 3:
[0 0 1] [0 0 1] [0 0 1] [0 0 1] [0 0 1]
[0 0 1] [0 1 0] [0 1 1] [0 1 1] [1 1 0]
[1 1 1] [1 1 1] [1 0 1] [1 1 0] [1 1 0]
2 of size 4 X 4:
[0 0 0 1] [0 0 0 1]
[0 0 0 1] [0 0 1 0]
[0 0 1 0] [0 1 0 0]
[1 1 0 0] [1 0 0 1]
and 1 of size 5 X 5:
[0 0 0 0 1]
[0 0 0 1 0]
[0 0 1 0 0]
[0 1 0 0 0]
[1 0 0 0 0].
From _Gus Wiseman_, Nov 14 2018: (Start)
Triangle begins:
   1
   0   1
   0   1   1
   0   1   2   1
   0   0   5   2   1
   0   0   4  11   2   1
   0   0   3  21  14   2   1
   0   0   1  34  49  15   2   1
   0   0   1  33 131  69  15   2   1
   0   0   0  33 248 288  79  15   2   1
Non-isomorphic representatives of the multiset partitions counted in row 6 {0,0,4,11,2,1} are:
  {{12}{13}{23}}  {{1}{1}{1}{234}}  {{1}{2}{3}{3}{45}}  {{1}{2}{3}{4}{5}{6}}
  {{1}{23}{123}}  {{1}{1}{24}{34}}  {{1}{2}{3}{5}{45}}
  {{13}{23}{23}}  {{1}{1}{4}{234}}
  {{3}{23}{123}}  {{1}{2}{34}{34}}
                  {{1}{3}{24}{34}}
                  {{1}{3}{4}{234}}
                  {{1}{4}{24}{34}}
                  {{1}{4}{4}{234}}
                  {{2}{4}{12}{34}}
                  {{3}{4}{12}{34}}
                  {{4}{4}{12}{34}}
(End)

Row sums give A057151.
Cf. A049311, A056037, A056079, A056080, A057149, A057151, A057152.
Cf. A007716, A048291, A054976, A101370, A104601, A104602, A120732, A120733, A135588, A319616, A321609, A321615.

permcount[v_List] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t*k; s += t]; s!/m];
c[p_List, q_List, k_] := SeriesCoefficient[Product[Product[(1 + O[x]^(k + 1) + x^LCM[p[[i]], q[[j]]])^GCD[p[[i]], q[[j]]], {j, 1, Length[q]}], {i, 1, Length[p]}], {x, 0, k}];
M[m_, n_, k_] := M[m, n, k] = Module[{s = 0}, Do[Do[s += permcount[p]* permcount[q]*c[p, q, k], {q, IntegerPartitions[n]}], {p, IntegerPartitions[m]}]; s/(m!*n!)];
T[n_, k_] := M[k, k, n] - 2*M[k, k - 1, n] + M[k - 1, k - 1, n];
Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Sep 10 2019, after Andrew Howroyd *)

\\ See A321609 for M.
T(n,k) = M(k,k,n) - 2*M(k,k-1,n) + M(k-1,k-1,n); \\ Andrew Howroyd, Nov 14 2018

Duplicate seventh row removed by Gus Wiseman, Nov 14 2018

A104601 Triangle T(r,n) read by rows: number of n X n (0,1)-matrices with exactly r entries equal to 1 and no zero row or columns.

Original entry on oeis.org

1, 0, 2, 0, 4, 6, 0, 1, 45, 24, 0, 0, 90, 432, 120, 0, 0, 78, 2248, 4200, 720, 0, 0, 36, 5776, 43000, 43200, 5040, 0, 0, 9, 9066, 222925, 755100, 476280, 40320, 0, 0, 1, 9696, 727375, 6700500, 13003620, 5644800, 362880, 0, 0, 0, 7480, 1674840

Offset: 1

Ralf Stephan, Mar 27 2005

			1
0,2
0,4,6
0,1,45,24
0,0,90,432,120
0,0,78,2248,4200,720
0,0,36,5776,43000,43200,5040
0,0,9,9066,222925,755100,476280,40320
0,0,1,9696,727375,6700500,13003620,5644800,362880
0,0,0,7480,1674840,37638036,179494350,226262400,71850240,3628800

M. Maia and M. Mendez, On the arithmetic product of combinatorial species, arXiv:math/0503436 [math.CO], 2005.

Right-edge diagonals include A000142, A055602, A055603. Row sums are in A104602.
Column sums are in A048291. The triangle read by columns = A055599.
Cf. A049311, A054976, A057150, A057151, A101370, A120732, A120733, A138178, A316983, A319616.

t[r_, n_] := Sum[ Sum[ (-1)^(2n - d - k/d)*Binomial[n, d]*Binomial[n, k/d]*Binomial[k, r], {d, Divisors[k]}], {k, r, n^2}]; Flatten[ Table[t[r, n], {r, 1, 10}, {n, 1, r}]] (* Jean-François Alcover, Jun 27 2012, from formula *)
Table[Length[Select[Subsets[Tuples[Range[n],2],{n}],Union[First/@#]==Union[Last/@#]==Range[k]&]],{n,6},{k,n}] (* Gus Wiseman, Nov 14 2018 *)

T(r, n) = Sum{l>=r, Sum{d|l, (-1)^(2n-d-l/d)*C(n, d)*C(n, l/d)*C(l, r) }}.

E.g.f.: Sum(((1+x)^n-1)^n*exp((1-(1+x)^n)*y)*y^n/n!,n=0..infinity). - Vladeta Jovovic, Feb 24 2008

cp's OEIS Frontend

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

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A138178 Number of symmetric matrices with nonnegative integer entries and without zero rows or columns such that sum of all entries is equal to n.

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A120732 Number of square matrices with nonnegative integer entries and without zero rows or columns such that sum of all entries is equal to n.

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A122400 Number of square (0,1)-matrices without zero rows and with exactly n entries equal to 1.

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A048144 a(n) = Sum_{k=0..n} (k!)^2 * Stirling_2(n,k)^2.

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A054976 Number of binary n X n matrices with no zero rows or columns, up to row and column permutation.

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A138265 Number of upper triangular zero-one matrices with n ones and no zero rows or columns.

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