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Search: a104601 -id:a104601
Displaying 1-10 of 11 results found.
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A048291
Number of {0,1} n X n matrices with no zero rows or columns.
+10
71
1, 1, 7, 265, 41503, 24997921, 57366997447, 505874809287625, 17343602252913832063, 2334958727565749108488321, 1243237913592275536716800402887, 2630119877024657776969635243647463625, 22170632855360952977731028744522744983195423
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
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
REFERENCES
Brendan McKay, Posting to sci.math.research, Jun 14 1999.
LINKS
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.
FORMULA
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
EXAMPLE
a(2) = 7: |01| |01| |10| |10| |11| |11| |11|
|10| |11| |01| |11| |01| |10| |11|.
MAPLE
seq(add((-1)^(n+k)*binomial(n, k)*(2^k-1)^n, k=0..n), n=0..15); # Robert FERREOL, Mar 10 2017
MATHEMATICA
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 *)
PROG
(PARI) a(n)=sum(k=0, n, binomial(n, k)*(-1)^k*(2^(n-k)-1)^n)
(Python)
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
CROSSREFS
Cf. A054976, A104602, A283624, A287065, A173403.
Cf. A055601, A055599, A104601, A086193 (traceless, no loops), A086206, A322661 (adj. matr. undirected edges).
Diagonal of A183109.
KEYWORD
nonn,easy,nice,changed
AUTHOR
Joe Keane (jgk(AT)jgk.org)
STATUS
approved
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.
+10
45
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
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
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..500
FORMULA
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
EXAMPLE
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)
MAPLE
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
MATHEMATICA
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 *)
CROSSREFS
Row sums of A138177.
Cf. A007716, A120733, A135588, A296188.
Cf. A057151, A104601, A104602, A120732, A316983, A320796, A321401, A321405, A321407.
KEYWORD
easy,nonn
AUTHOR
Vladeta Jovovic, Mar 03 2008
EXTENSIONS
More terms from Alois P. Heinz, Sep 25 2008
STATUS
approved
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.
+10
27
1, 1, 3, 15, 107, 991, 11267, 151721, 2360375, 41650861, 821881709, 17932031225, 428630422697, 11138928977049, 312680873171465, 9428701154866535, 303957777464447449, 10431949496859168189, 379755239311735494421
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..250
FORMULA
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
EXAMPLE
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)
MATHEMATICA
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 *)
CROSSREFS
Cf. A007716, A048291, A054976, A057149, A057150, A057151, A104601, A104602, A120733, A138178, A316983, A319616.
KEYWORD
nonn
AUTHOR
Vladeta Jovovic, Aug 18 2006
STATUS
approved
A104602
Number of square (0,1)-matrices with exactly n entries equal to 1 and no zero row or columns.
+10
21
1, 1, 2, 10, 70, 642, 7246, 97052, 1503700, 26448872, 520556146, 11333475922, 270422904986, 7016943483450, 196717253145470, 5925211960335162, 190825629733950454, 6543503207678564364, 238019066600097607402, 9153956822981328930170, 371126108428565106918404
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Number of square (0,1)-matrices with exactly n entries equal to 1 and no zero row or columns, up to row and column permutation, is A057151(n). - Vladeta Jovovic, Mar 25 2006
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..400
H. Cheballah, S. Giraudo, R. Maurice, Combinatorial Hopf algebra structure on packed square matrices, arXiv preprint arXiv:1306.6605 [math.CO], 2013-2015.
M. Maia and M. Mendez, On the arithmetic product of combinatorial species, arXiv:math/0503436 [math.CO], 2005.
FORMULA
a(n) = (1/n!)*Sum_{k=0..n} Stirling1(n,k)*A048144(k). - Vladeta Jovovic, Mar 25 2006
G.f.: Sum_{n>=0} Sum_{j=0..n} (-1)^(n-j)*binomial(n,j)*((1+x)^j-1)^n. - Vladeta Jovovic, Mar 25 2006
a(n) ~ c * n! / (sqrt(n) * (log(2))^(2*n)), where c = 0.28889864564457451375789435201798... . - Vaclav Kotesovec, May 07 2014
In closed form, c = 1 / (log(2) * 2^(log(2)/2+2) * sqrt(Pi*(1-log(2)))). - Vaclav Kotesovec, May 03 2015
G.f.: Sum_{n>=0} ((1+x)^n - 1)^n / (1+x)^(n*(n+1)). - Paul D. Hanna, Mar 26 2018
EXAMPLE
From Gus Wiseman, Nov 14 2018: (Start)
The a(3) = 10 matrices:
[1 1] [1 1] [1 0] [0 1]
[1 0] [0 1] [1 1] [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)
MATHEMATICA
Table[1/n!*Sum[StirlingS1[n, k]*Sum[(m!)^2*StirlingS2[k, m]^2, {m, 0, k}], {k, 0, n}], {n, 1, 20}] (* Vaclav Kotesovec, May 07 2014 *)
Table[Length[Select[Subsets[Tuples[Range[n], 2], {n}], Union[First/@#]==Union[Last/@#]==Range[Max@@First/@#]&]], {n, 5}] (* Gus Wiseman, Nov 14 2018 *)
CROSSREFS
Row sums of triangle A104601.
Cf. A048291, A049311, A054976, A057150, A057151, A101370, A120732, A120733, A138178, A316983, A319616.
KEYWORD
nonn
AUTHOR
Ralf Stephan, Mar 27 2005
EXTENSIONS
More terms from Vladeta Jovovic, Mar 25 2006
a(0)=1 prepended by Alois P. Heinz, Jan 14 2015
STATUS
approved
A054976
Number of binary n X n matrices with no zero rows or columns, up to row and column permutation.
+10
19
1, 3, 17, 179, 3835, 200082, 29610804, 13702979132, 20677458750966, 103609939177198046, 1745061194503344181714, 99860890306900024150675406, 19611238933283757244479826044874, 13340750149227624084760722122669739026, 31706433098827528779057124372265863803044450
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Also the number of non-isomorphic set multipartitions (multisets of sets) with n parts and n vertices. - Gus Wiseman, Nov 18 2018
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..50
FORMULA
a(n) = A002724(n) - 2*A002725(n-1) + A002724(n-1).
EXAMPLE
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)
MATHEMATICA
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 *)
CROSSREFS
Column sums of A057150.
Cf. A007716, A048291 (labeled), A049311, A057149, A057151, A104601, A104602, A319616, A320808.
KEYWORD
easy,nonn
AUTHOR
Vladeta Jovovic, May 27 2000
EXTENSIONS
More terms from David Wasserman, Mar 06 2002
Terms a(14) and beyond from Andrew Howroyd, Apr 11 2020
STATUS
approved
A057151
Number of square binary matrices with n ones, with no zero rows or columns, up to row and column permutation.
+10
15
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
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
LINKS
Max Alekseyev, Table of n, a(n) for n = 1..30
EXAMPLE
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)
CROSSREFS
Cf. A049311, A056037, A056079, A056080, A057149, A057150, A057152.
Cf. A054976, A101370, A104601, A104602, A120732, A283877, A319616.
KEYWORD
nonn
AUTHOR
Vladeta Jovovic, Aug 14 2000
EXTENSIONS
More terms from Max Alekseyev, May 31 2007
STATUS
approved
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.
+10
12
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
(list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,9
COMMENTS
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
LINKS
Table of n, a(n) for n=1..78.
EXAMPLE
[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)
MATHEMATICA
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 *)
PROG
(PARI) \\ 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
CROSSREFS
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.
KEYWORD
nonn,tabl
AUTHOR
Vladeta Jovovic, Aug 14 2000
EXTENSIONS
Duplicate seventh row removed by Gus Wiseman, Nov 14 2018
STATUS
approved
A135588
Number of symmetric (0,1)-matrices with exactly n entries equal to 1 and no zero rows or columns.
+10
11
1, 1, 2, 6, 20, 74, 302, 1314, 6122, 29982, 154718, 831986, 4667070, 27118610, 163264862, 1013640242, 6488705638, 42687497378, 288492113950, 1998190669298, 14177192483742, 102856494496050, 762657487965086, 5771613810502002, 44555989658479726, 350503696871063138
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..721 (first 51 terms from Vincenzo Librandi)
FORMULA
G.f.: Sum_{n>=0} (1+x)^n*(1+x^2)^binomial(n,2)/2^(n+1).
G.f.: Sum_{n>=0} (Sum_{k=0..n} (-1)^(n-k)*binomial(n,k)*(1+x)^k*(1+x^2)^binomial(k,2)).
EXAMPLE
From Gus Wiseman, Nov 14 2018: (Start)
The a(4) = 20 matrices:
[11]
[11]
.
[110][101][100][100][011][010][010][001][001]
[100][010][011][001][100][110][101][010][001]
[001][100][010][011][100][001][010][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)
MATHEMATICA
Table[Sum[SeriesCoefficient[(1+x)^k*(1+x^2)^(k*(k-1)/2)/2^(k+1), {x, 0, n}], {k, 0, Infinity}], {n, 0, 20}] (* Vaclav Kotesovec, Jul 02 2014 *)
Join[{1}, Table[Length[Select[Subsets[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 *)
CROSSREFS
Row sums of A135589.
Cf. A049311, A054976, A101370, A104601, A104602, A120733, A138178, A283877, A316983, A320796, A321401, A321405.
KEYWORD
nonn
AUTHOR
Vladeta Jovovic, Feb 25 2008, Mar 03 2008, Mar 04 2008
STATUS
approved
A055602
Number of n X n binary matrices with no 0 rows or columns and with n+1 1's.
+10
9
0, 4, 45, 432, 4200, 43200, 476280, 5644800, 71850240, 979776000, 14270256000, 221298739200, 3642807168000, 63465795993600, 1167099373440000, 22596613079040000, 459548157100032000, 9795631769763840000
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Robert Israel, Table of n, a(n) for n = 1..445
FORMULA
Number of m X n binary matrices with no zero rows or columns and with k=0..m*n ones is Sum_{i=0..n} (-1)^i*binomial(n, i)*a(m, n-i, k) where a(m, n, k)=Sum_{i=0..m} (-1)^i*binomial(m, i)*binomial((m-i)*n, k).
a(n) = n*(n-1)*(n+2)*n!/4. - Vladeta Jovovic, Mar 25 2006
From Robert Israel, May 04 2021: (Start)
E.g.f.: x^2*(4-x)/(2*(1-x)^2).
D-finite with recurrence 4*(n-2)*a(n)-n*(4*n+3)*a(n-1)-(n-1)^2*a(n-2)=0.
(End)
MAPLE
f:= n -> n*(n-1)*(n+2)*n!/4:
map(f, [$1..30]); # Robert Israel, May 04 2021
CROSSREFS
A diagonal of triangle A104601.
Cf. A055603.
KEYWORD
nonn
AUTHOR
Vladeta Jovovic, Jun 01 2000
EXTENSIONS
More terms from David Wasserman, Apr 28 2002
STATUS
approved
A321615
Triangle read by rows: T(n,k) is the number of k X k integer matrices with sum of elements n, with no zero rows or columns, up to row and column permutation.
+10
5
1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 6, 3, 1, 0, 1, 9, 13, 3, 1, 0, 1, 17, 38, 20, 3, 1, 0, 1, 23, 97, 82, 23, 3, 1, 0, 1, 36, 217, 311, 126, 24, 3, 1, 0, 1, 46, 453, 968, 624, 151, 24, 3, 1, 0, 1, 65, 868, 2825, 2637, 933, 162, 24, 3, 1, 0, 1, 80, 1585, 7394, 10098, 4942, 1132, 165, 24, 3, 1
(list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,9
COMMENTS
Also the number of non-isomorphic multiset partitions of weight n with k parts and k vertices, where the weight of a multiset partition is the sum of sizes of its parts. - Gus Wiseman, Nov 18 2018
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
EXAMPLE
Triangle begins:
1
0 1
0 1 1
0 1 2 1
0 1 6 3 1
0 1 9 13 3 1
0 1 17 38 20 3 1
0 1 23 97 82 23 3 1
0 1 36 217 311 126 24 3 1
0 1 46 453 968 624 151 24 3 1
0 1 65 868 2825 2637 933 162 24 3 1
MATHEMATICA
(* See A318795 for M[m, n, k]. *)
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, 11}, {k, 1, n}] // Flatten (* Jean-François Alcover, Nov 24 2018, from PARI *)
PROG
(PARI) \\ See A318795 for M.
T(n, k) = if(k==0, n==0, M(k, k, n) - 2*M(k, k-1, n) + M(k-1, k-1, n));
(PARI) \\ See A340652 for G.
T(n)={[Vecrev(p) | p<-Vec(1 + sum(k=1, n, y^k*(polcoef(G(k, n, n, y), k, y) - polcoef(G(k-1, n, n, y), k, y))))]}
{ my(A=T(10)); for(i=1, #A, print(A[i])) } \\ Andrew Howroyd, Jan 16 2024
CROSSREFS
Columns k=0..3 are A000007, A000012, A054974, A054975.
Row sums are A319616.
Cf. A007716, A048291, A054976, A057149, A057150, A104601, A120732, A318795, A320808.
KEYWORD
nonn,tabl
AUTHOR
Andrew Howroyd, Nov 14 2018
EXTENSIONS
Column k=0 inserted by Andrew Howroyd, Jan 17 2024
STATUS
approved
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