A082491 -id:A082491 - cp's OEIS frontend

A221438 T(n,k) is the number of n X k 1..(max n,k) arrays with each row and column having unrepeated values.

1, 2, 2, 6, 2, 6, 24, 12, 12, 24, 120, 216, 12, 216, 120, 720, 5280, 576, 576, 5280, 720, 5040, 190800, 66240, 576, 66240, 190800, 5040, 40320, 9344160, 15321600, 161280, 161280, 15321600, 9344160, 40320, 362880, 598066560, 5411750400, 283046400

Offset: 1

R. H. Hardin, Jan 16 2013

Table starts:
....1.......2..........6...........24............120............720
....2.......2.........12..........216...........5280.........190800
....6......12.........12..........576..........66240.......15321600
...24.....216........576..........576.........161280......283046400
..120....5280......66240.......161280.........161280......812851200
..720..190800...15321600....283046400......812851200......812851200
.5040.9344160.5411750400.782137036800.20449013760000.61479419904000

			Some solutions for n=3 and k=4:
..4..1..3..2....3..1..4..2....1..2..4..3....3..4..2..1....2..3..4..1
..2..3..4..1....1..3..2..4....3..4..1..2....1..2..3..4....1..4..3..2
..1..4..2..3....2..4..3..1....2..1..3..4....2..1..4..3....3..2..1..4

R. H. Hardin, Table of n, a(n) for n = 1..83

Diagonal is A002860.
Column 1 is A000142.
Column 2 is A082491.

A174564 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 (2,3,...,n,1). Then a(n) is the number of (0,1) n X n matrices A<=J_n-I-P with exactly two 1's in every row and column.

Original entry on oeis.org

0, 1, 13, 522, 27828, 1867363

Offset: 3

Vladimir Shevelev, Mar 22 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

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.

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

A174584 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) n X n matrices A<=J_n-I-P-P^2-P^3 with exactly two 1's in every row and column.

Original entry on oeis.org

0, 1, 31, 3114, 381022

Offset: 5

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 A174582

A346409 a(n) = (n!)^2 * Sum_{k=0..n-1} (-1)^k / ((n-k)^2 * k!).

Original entry on oeis.org

0, 1, -3, 13, -52, 476, 1344, 156192, 6935424, 470168064, 38948065920, 3979380286080, 489922581219840, 71586095491054080, 12249193741572372480, 2426646293132502067200, 551096248249459158220800, 142236660450422499604070400, 41404182857569072540171468800

Offset: 0

Ilya Gutkovskiy, Jul 15 2021

sign

Cf. A002741, A009940, A082491, A142999, A346405.

Table[(n!)^2 Sum[(-1)^k/((n - k)^2 k!), {k, 0, n - 1}], {n, 0, 18}]
nmax = 18; CoefficientList[Series[PolyLog[2, x] Exp[-x], {x, 0, nmax}], x] Range[0, nmax]!^2

Sum_{n>=0} a(n) * x^n / (n!)^2 = polylog(2,x) * exp(-x).

A174585 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-P^3) with exactly one 1 and one 2 in every row and column.

Original entry on oeis.org

0, 2, 132, 9800, 1309928

Offset: 5

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 A174582 A174584

cp's OEIS Frontend

A221438 T(n,k) is the number of n X k 1..(max n,k) arrays with each row and column having unrepeated values.

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A174564 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 (2,3,...,n,1). Then a(n) is the number of (0,1) n X n matrices A<=J_n-I-P with exactly two 1's in every row and column.

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Author

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References

Crossrefs

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.

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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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Mathematica

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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

A174584 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) n X n matrices A<=J_n-I-P-P^2-P^3 with exactly two 1's in every row and column.

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Author

Keywords

References

Crossrefs

A346409 a(n) = (n!)^2 * Sum_{k=0..n-1} (-1)^k / ((n-k)^2 * k!).

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Author

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Crossrefs

Programs

Mathematica

Formula

A174585 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-P^3) with exactly one 1 and one 2 in every row and column.

Views

Author

Keywords

References

Crossrefs