A001499 -id:A001499 - cp's OEIS frontend

A058527 Number of 2n X 2n 0-1 matrices with n ones in each row and each column.

1, 2, 90, 297200, 116963796250, 6736218287430460752, 64051375889927380035549804336, 108738182111446498614705217754614976371200, 34812290428176298285394893936773707951192224124239796250, 2188263032066768922535710968724036448759525154977348944382853301460850000

Offset: 0

David desJardins, Dec 22 2000

nonn

Vladeta Jovovic, Nov 12 2006, Table of n, a(n) for n = 0..15
A Conflitti, C. M. Da Fonseca, and R. Mamede, The maximal length of a chain in the Bruhat order for a chain of binary matrices., Lin. Algebra Applic. (2011)
Alessandro Conflitti, C. M. da Fonseca and Ricardo Mamede, On the largest size of an antichain in the Bruhat order for A(2k, k).
Alessandro Conflitti, C. M. da Fonseca and Ricardo Mamede, On the Largest Size of an Antichain in the Bruhat Order for A(2k,k), ORDER, 2011, DOI: 10.1007/s11083-011-9241-1.
Jonathan Jedwab and Tabriz Popatia, A new representation of mutually orthogonal frequency squares, Simon Fraser University (Burnaby, BC, Canada, 2020).
M. A. Khojastepour and M. Farajzadeh-Tehrani, Characterizing per Node Degrees of Freedom in an Interference Network, 2014; also 2014 IEEE International Symposium on Information Theory, pp. 1016-1020.
B. D. McKay, 0-1 matrices with constant row and column sums
Michael Penn, A not so magic square..., YouTube video, 2021.
Wikipedia, Dynamic programming

Central coefficients of A008300.
Main diagonal of A376935.
Cf. A001499, A001501, A253316.

More terms (using dynamic programming in Python) from Greg Kuperberg, Feb 08 2001

More terms from Vladeta Jovovic, Nov 12 2006

A174580 Let J_n be an n X n matrix which contains 1's only, I = I_n be the n X n identity matrix, and P = P_n be the incidence matrix of the cycle (1,2,3,...,n). Then a(n) is the number of (0,1,2) n X n matrices A <= 2(J_n - I - P) with exactly one 1 and one 2 in every row and column.

Original entry on oeis.org

0, 2, 36, 1462, 83600, 5955474

Offset: 3

Vladimir Shevelev, Mar 23 2010

V. S. Shevelev, Development of the rook technique for calculating the cyclic indicators of (0,1)-matrices, Izvestia Vuzov of the North-Caucasus region, Nature sciences 4 (1996), 21-28 (in Russian).
S. E. Grigorchuk, V. S. Shevelev, An algorithm of computing the cyclic indicator of couples discordant permutations with restricted position, Izvestia Vuzov of the North-Caucasus region, Nature sciences 3 (1997), 5-13 (in Russian).

Cf. A001499, A007107, A082491, A000186, A174564.

A174581 Let J_n be an n X n all-1's matrix, I = I_n the n X n identity matrix and P = P_n the incidence matrix of the cycle (1,2,3,...,n). Then a(n) is the number of (0,1) n X n matrices A <= J_n - I - P - P^2 with exactly two 1's in every row and column.

Original entry on oeis.org

0, 1, 20, 1266, 102574, 9746472

Offset: 4

Vladimir Shevelev, Mar 23 2010

V. S. Shevelev, Development of the rook technique for calculating the cyclic indicators of (0,1)-matrices, Izvestia Vuzov of the North-Caucasus region, Nature sciences 4 (1996), 21-28 (in Russian).
S. E. Grigorchuk, V. S. Shevelev, An algorithm of computing the cyclic indicator of couples discordant permutations with restricted position, Izvestia Vuzov of the North-Caucasus region, Nature sciences 3 (1997), 5-13 (in Russian).

Cf. A001499, A007107, A082491, A000186, A174564, A174580.

A174586 Number of n X n (0,1) matrices with two 1's in each row having positive permanent.

Original entry on oeis.org

0, 1, 24, 954, 59040, 5295150, 651354480, 105393619800, 21717404916480, 5554438422838200, 1726882980691176000, 641506478978753110800, 280659563041747649760000, 142843312073975729801785200, 83684308104396267184700784000, 55915646244745131440225950320000

Offset: 1

Vladimir Shevelev, Mar 23 2010

nonn

a(n) is the normalized volume of the convex hull of (classical) parking functions of length n. - Andrés R. Vindas-Meléndez, Jan 13 2023

Vladimir Shevelev, On the permanent of the stochastic (0,1)-matrices with equal row sums, Izvestia Vuzov of the North-Caucasus region, Nature sciences 1 (1997), 21-38 (in Russian).

Aruzhan Amanbayeva and Danielle Wang, The convex hull of parking functions of length n, Enumer. Comb. Appl. 2 (2022), no. 2, Paper No. S2R10, 1.
Mitsuki Hanada, John Lentfer, and Andrés R. Vindas-Meléndez, Generalized parking function polytopes, arXiv:2212.06885 [math.CO], 2022.

Cf. A001499, A007107, A082491.

Table[n!/2^n * Sum[(2*i-1)*(2*i-1)!!*Binomial[n,i]*(2n-1)^(n-i-1),{i,0,n}],{n,1,20}] (* Vaclav Kotesovec, Nov 30 2017 *)

a(2)=1, for n>=3, a(n) = A001499(n) + Sum_{k=1..n-2} (-1)^(k+1)*k!*(C(n,k))^2*(n-k)^k*a(n-k).
a(n) = n!*((n-1)/2^(n-1))*Sum_{i=0..n-2} (2i+1)!!*C(n-2,i)*(2n-1)^(n-i-2). [corrected by John Lentfer, Oct 05 2022]
For n>=2, a(n) = (n!/2^n)*Sum_{i=0..n} (2i-1)*(2i-1)!!*C(n,i)*(2n-1)^(n-i-1).
a(n) = Gamma(3/4)*(sqrt(2)*Pi*e)^(-1/2)*n!*n^(n-1/4)*(1+O(n^((-1/4)+epsilon) with arbitrary small epsilon>0 for sufficiently large n.

A066300 Number of n X n matrices with exactly 2 1's in each row, other entries 0.

Original entry on oeis.org

0, 1, 27, 1296, 100000, 11390625, 1801088541, 377801998336, 101559956668416, 34050628916015625, 13931233916552734375, 6831675453247426400256, 3955759092264800909058048, 2670419511272061205254504361

Offset: 1

Sharon Sela (sharonsela(AT)hotmail.com), Jan 01 2002

nonn

Cf. A001499.

Mathematica
```
Table[ Binomial[n, 2]^n, {n, 1, 16} ]
```

a(n) = binomial(n,2)^n \\ Charles R Greathouse IV, Dec 16 2016

a(n) = binomial(n,2)^n.

More terms from Robert G. Wilson v, Jan 03 2002

A112579 Number of 3-D arrays of 0's and 1's with plane sums 2.

Original entry on oeis.org

0, 8, 900, 366336, 378028800, 833156928000, 3477528928742400, 25183876050321408000, 296058177312000019660800, 5362158372805111867637760000, 143458227395428379364635443200000

Offset: 1

Peter J. Cameron, Sep 14 2005

			a(2)=8: six pairs of opposite edges and two inscribed tetrahedra.

P. J. Cameron and T. W. Mueller, Decomposable functors and the exponential principle, II, in preparation

P. J. Cameron and T. W. Mueller, Decomposable functors and the exponential principle, II, arXiv:0911.3760 [math.CO]

Cf. A001499 (2-D case), A112578, A112580.

a(n) = b(n) + Sum (k/n)*(n choose k)^3*b(k)*a(n-k), where b(n) counts indecomposable arrays (A112578).

A112580 Number of 3-D arrays of nonnegative integers with plane sums 2.

Original entry on oeis.org

1, 12, 1152, 431424, 427723200, 920031955200, 3777894212198400, 27039993414897254400, 315084437077115278540800, 5667616936309704095784960000, 150796432741520745587273564160000

Offset: 1

Peter J. Cameron, Sep 14 2005

			a(2)=12: eight 0-1 arrays and four with 2s at opposite vertices.

P. J. Cameron and T. W. Mueller, Decomposable functors and the exponential principle, II, in preparation

P. J. Cameron and T. W. Mueller, Decomposable functors and the exponential principle, II, arXiv:0911.3760 [math.CO]

Cf. A001499 (2-D case), A112578, A112579.

a(n) = b(n) + Sum (k/n)*(n choose k)^3*b(k)*a(n-k), where b(n) counts indecomposable arrays (A112578 with first term 1).

A174582 Let J_n be n X n matrix which contains 1's only, I=I_n be the n X n identity matrix and P=P_n be the incidence matrix of the cycle (1,2,3,...,n). Then a(n) is the number of (0,1,2) n X n matrices A<=2(J_n-I-P-P^2) with exactly one 1 and one 2 in every row and column.

Original entry on oeis.org

0, 2, 72, 3722, 329192, 32842446

Offset: 4

Vladimir Shevelev, Mar 23 2010

V. S. Shevelev, Development of the rook technique for calculating the cyclic indicators of (0,1)-matrices, Izvestia Vuzov of the North-Caucasus region, Nature sciences 4 (1996), 21-28 (in Russian).
S. E. Grigorchuk, V. S. Shevelev, An algorithm of computing the cyclic indicator of couples discordant permutations with restricted position, Izvestia Vuzov of the North-Caucasus region, Nature sciences 3 (1997), 5-13 (in Russian).

A001499 A007107 A082491 A000186 A174564 A174580 A174581

A225623 Number of ways to arrange 2n queens on an n X n chessboard, with no more than 2 queens in each row, column or diagonal.

Original entry on oeis.org

0, 1, 2, 11, 92, 1097, 19448, 477136, 14244856, 537809179, 24194010708, 1317062528249

Offset: 1

Vaclav Kotesovec, Aug 04 2013

This problem is slightly different from A000769 or A219760. In the first example on an 8 x 8 board, the queens c7, d5 and e3 (or queens a2, c5 and e8) are in a line, but such case is allowed. The elementary step can be only [0,1], [1,0] or [1,1], not for example [1,2] or [2,3].

Vaclav Kotesovec, Examples

Cf. A000170, A000769, A001499, A001501, A058528, A075754, A172544, A172541.

Definition clarified by Vaclav Kotesovec, Dec 18 2014

a(10)-a(12) from Martin Ehrenstein, Jan 09 2022

A156430 Number of n X n arrays of squares of integers, symmetric about both diagonal and antidiagonal, with all rows summing to 2.

Original entry on oeis.org

1, 2, 10, 12, 84, 132, 954, 1728, 13290, 26820, 217500, 481320, 4086600, 9783480, 86549820, 221921280, 2037627900, 5552479800, 52745205240, 151802154000, 1487961422640, 4500041903280, 45412066438200, 143712079822080, 1490217165997560, 4917227802767280

Offset: 2

R. H. Hardin, Feb 09 2009

nonn

Albert Zhou, Proof of recurrence

Cf. A001499, A000986.

# Even-dim bisymmetric
A = [1, 1, 10]
B = [0, 2, 6]
C = [0, 1, 6]
for n in range(3, 13):
    a_next = A[-1] + (n-1)*A[-2] + 4*(n-1)*B[-1] + 2*(n-1)*(n-2)*C[-1]
    b_next = 2*A[-1] + 2*(n-1)*B[-1]
    c_next = 4*B[-1] - 2*A[-2] + 4*(n-2)*A[-3] + 4*(n-2)*(n-3)*C[-2]
    A.append(a_next)
    B.append(b_next)
    C.append(c_next)
# Odd-dim bisymmetric
A_odd = [B[n]*n for n in range(len(B))]
# Albert Zhou, Jan 26 2025

From Albert Zhou, Jan 26 2025: (Start)
a(2*n) = a(2*(n-1)) + (n-1)*a(2*(n-2)) + 4*(n-1)*b(2*(n-1)) + 2*(n-1)*(n-2)*c(2*(n-1)), where
b(2*n) = 2*a(2*(n-1)) + 2*(n-1)*b(2*(n-1)), and
c(2*n) = 4*b(2*(n-1)) - 2*a(2*(n-2)) + 4*(n-2)*a(2*(n-3)) + 4*(n-2)*(n-3)*c(2*(n-2)), with
a(0) = 1, a(2) = 1, a(4) = 10, and
b(0) = 0, b(2) = 2, b(4) = 6, and
c(0) = 0, c(2) = 1, c(6) = 6.
a(2*n+1) = n*b(2*n).
Proof attached. (End)

a(26)-a(27) from Albert Zhou, Jan 26 2025

cp's OEIS Frontend

A058527 Number of 2n X 2n 0-1 matrices with n ones in each row and each column.

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A174580 Let J_n be an n X n matrix which contains 1's only, I = I_n be the n X n identity matrix, and P = P_n be the incidence matrix of the cycle (1,2,3,...,n). Then a(n) is the number of (0,1,2) n X n matrices A <= 2(J_n - I - P) with exactly one 1 and one 2 in every row and column.

Views

Author

Keywords

References

Crossrefs

A174581 Let J_n be an n X n all-1's matrix, I = I_n the n X n identity matrix and P = P_n the incidence matrix of the cycle (1,2,3,...,n). Then a(n) is the number of (0,1) n X n matrices A <= J_n - I - P - P^2 with exactly two 1's in every row and column.

Views

Author

Keywords

References

Crossrefs

A174586 Number of n X n (0,1) matrices with two 1's in each row having positive permanent.

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Programs

Mathematica

Formula

A066300 Number of n X n matrices with exactly 2 1's in each row, other entries 0.

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Mathematica

PARI

Formula

Extensions

A112579 Number of 3-D arrays of 0's and 1's with plane sums 2.

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Examples

References

Links

Crossrefs

Formula

A112580 Number of 3-D arrays of nonnegative integers with plane sums 2.

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Author

Keywords

Examples

References

Links

Crossrefs

Formula

A174582 Let J_n be n X n matrix which contains 1's only, I=I_n be the n X n identity matrix and P=P_n be the incidence matrix of the cycle (1,2,3,...,n). Then a(n) is the number of (0,1,2) n X n matrices A<=2(J_n-I-P-P^2) with exactly one 1 and one 2 in every row and column.

Views

Author

Keywords

References

Crossrefs

A225623 Number of ways to arrange 2n queens on an n X n chessboard, with no more than 2 queens in each row, column or diagonal.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A156430 Number of n X n arrays of squares of integers, symmetric about both diagonal and antidiagonal, with all rows summing to 2.

Views

Author

Keywords

Links

Crossrefs

Programs