A001044 -id:A001044 - cp's OEIS frontend

A192721 The number of pairs of permutations in the product group S_n X S_n with k common descents, n >= 1 and 0 <= k <= n-1.

1, 3, 1, 19, 16, 1, 211, 299, 65, 1, 3651, 7346, 3156, 246, 1, 90921, 237517, 160322, 28722, 917, 1, 3081513, 9903776, 9302567, 2864912, 245407, 3424, 1, 136407699, 520507423, 632274183, 288196659, 46261609, 2041965, 12861, 1

Offset: 1

Peter Bala, Jul 11 2011

Let S_n denote the symmetric group on {1,2,...,n}. A permutation p_1p_2...p_n in S_n has a descent at position i (1 <= i <= n-1) if p_i > p_(i+1). The Eulerian numbers A008292 (with an offset of 0 in the column indexing) enumerate permutations by descents. We define a pair of permutations p_1p_2...p_n and q_1q_2...q_n to have a common descent at position i (1 <= i <= n-1) if both p_i > p_(i+1) and q_i > q_(i+1). For example, the permutations (3241) and (4231) in S_4 have common descents at positions i = 1 and i = 3. The table entry T(n,k) gives the number of pairs of permutations in the Cartesian product S_n x S_n with k common descents.

The generalized Stirling numbers associated with this triangle is A061691. See also A192722.

			The triangle begins
n/k|.....0.......1.......2......3....4.....5
============================================
..1|.....1
..2|.....3.......1
..3|....19......16.......1
..4|...211.....299......65......1
..5|..3651....7346....3156....246....1
..6|.90921..237517..160322..28722..917.....1
..
Row 3 entries T(3,0) = 19, T(3,1) = 16 and T(3,2) = 1 can be read from the following table:
============================================
Number of common descents in S_3 x S_3
============================================
.
...|.123...132...213...231...312...321
======================================
123|..0.....0.....0.....0.....0.....0
132|..0.....1.....0.....1.....0.....1
213|..0.....0.....1.....0.....1.....1
231|..0.....1.....0.....1.....0.....1
312|..0.....0.....1.....0.....1.....1
321|..0.....1.....1.....1.....1.....2
Matrix identity A192721 * A007318 = row reverse of A192722:
/...1................\ /..1..............\
|...3.....1...........||..1....1..........|
|..19....16.....1.....||..1....2....1.....|
|.211...299....65....1||..1....3....3....1|
|.....................||..................|
=
/...1...................\
|...4......1.............|
|..36.....18......1......|
|.576....432.....68.....1|
|........................|

Alois P. Heinz, Rows n = 1..45, flattened
L. Carlitz, R. Scoville and T. Vaughan, Enumeration of pairs of permutations and sequences, Bull. Amer. Math. Soc., 80 (1974), 881-884.
L. Carlitz, R. Scoville, T. Vaughan, Enumeration of pairs of permutations, Discrete Math. 14, (1976) 215-239.
J-Marc Fedou and D. Rawlings, More statistics on permutation pairs, The Electronic Journal of Combinatorics, 1 (1994) #R11.
M. V. Koutras, Eulerian numbers associated with sequences of polynomials, Fibonacci Quart. 32 (1994) 44-57.
T. Mendes, J. Remmel, A. Riehl, A Generalization of the Generating Functions for Descent Statistic.
R. P. Stanley, Binomial posets, Möbius inversion and permutation enumeration, J. Combinat. Theory, A 20 (1976), 336-356.

Cf. A000275 (first column), A001044 (row sums), A008292, A008459, A061691, A192722.

#A192721
#J = sum {n>=0} z^n/n!^2
J := unapply(BesselJ(0, 2*I*sqrt(z)),z):
G := (1-x)/(-x + J(z*(x-1))):
Gser := simplify(series(G, z = 0, 12)):
for n from 1 to 10 do
P[n] := n!^2*sort(coeff(Gser, z, n)) od:
for n from 1 to 10 do seq(coeff(P[n],x,k), k = 0..n-1) od;
# gives sequence in triangular form

max = 9; j[z_] := BesselJ[0, 2 I*Sqrt[z]]; g = (1 - x)/(-x + j[z*(x - 1)]); gser = Series[g, {z, 0, max}]; p[n_] := n!^2 Coefficient[ gser, z, n]; a[n_, k_] := Coefficient[ p[n], x, k]; Flatten[ Table[ a[n, k], {n, 1, max-1}, {k, 0, n-1}]] (* Jean-François Alcover, Dec 13 2011, after Maple *)

Generating function (Carlitz et al. 1976): Let J(z) = sum {n>=0} z^n/n!^2. Then (1-x)/(J(z*(x-1))-x) = 1 + sum {n>=1} (sum {k = 0..n-1} T(n,k)*x^k)*z^n/n!^2 = 1 + z + (3+x)*z^2/2!^2 + (19+16*x+x^2)*z^3/3!^2 + .... Define a polynomial sequence {p(n,x) }n>=0 by means of the generating function J(z)^x = sum {n>=0} p(n,x)*z^n/n!^2. The generalized Eulerian polynomials associated with the sequence {p(n,x)} as defined by [Koutras, 1994] are the polynomials sum {k = 0..n-1} T(n,k)*x^(n-k).
Relations with other sequences: The first column of the array (x*I-A008459)^-1 (I the identity matrix) is a sequence of rational functions whose numerator polynomials are the row generating polynomials for the present triangle. The change of variable x -> (x+1)/x followed by z -> x*z transforms the above bivariate generating function (1-x)/(J(z*(x-1))-x) into 1/(1+x-x*J(z)), which is the generating function for A192722. Equivalently, if we postmultiply the present triangle by Pascal's triangle A007318 we obtain the row reversed form of A192722: A192721 * A007318 = row reverse of A192722.
Row n sum = n!^2 = A001044(n).
First column [1,3,19,211,3651,...] = A000275 (apart from initial term).

A088021 a(n) = (n^2)!/(n!)^2.

Original entry on oeis.org

1, 1, 6, 10080, 36324288000, 1077167364120207360000, 717579719887926731226850787328000000, 23946596436219275985459662514223331478629410406400000000

Offset: 0

Hugo Pfoertner, Sep 18 2003

nonn

Based on an observation of Hugo Pfoertner, W. Edwin Clark conjectured and Xiang-dong Hou proved that (n^2)!/(n!)^2 gives the number of distinct determinants of the generic n X n matrix whose entries are n^2 different indeterminates under all (n^2)! permutations of the entries.
Using J. T. Schwarz's Sparse Zeros Lemma this implies that for any positive integer n there is an n X n matrix A with positive integer entries such that the set of determinant values obtained from A by permuting the elements of A is (n^2)!/(n!)^2.
Moreover, for any entries, no larger number of determinants can be obtained. In fact, by the Sparse Zeros Lemma one can select the entries of A from any sufficiently large subset of real numbers.

Vincenzo Librandi, Table of n, a(n) for n = 0..21

Cf. A001044, A088020.

[Factorial(n^2)/Factorial(n)^2: n in [0..10]]; // Vincenzo Librandi, May 31 2011

a(n) = A088020(n)/A001044(n).

A104344 a(n) = Sum_{k=1..n} k!^2.

Original entry on oeis.org

1, 5, 41, 617, 15017, 533417, 25935017, 1651637417, 133333531817, 13301522971817, 1606652445211817, 231049185247771817, 39006837228880411817, 7639061293780877851817, 1717651314017980301851817, 439480788011413032845851817, 126953027293558583218061851817

Offset: 1

Eric W. Weisstein, Mar 02 2005

Seiichi Manyama, Table of n, a(n) for n = 1..253
Eric Weisstein's World of Mathematics, Factorial Sums

Cf. A001044, A061062, A100289.

Sum_{k=1..n} (k!)^m: A007489 (m=1), this sequence (m=2), A138564 (m=3), A289945 (m=4), A316777 (m=5), A289946 (m=6).

Table[Sum[(k!)^2,{k,n}],{n,15}] (* Harvey P. Dale, Jul 21 2011 *)
Accumulate[(Range[20]!)^2] (* Much more efficient than the above program. *) (* Harvey P. Dale, Aug 15 2022 *)

a(n) = sum(k=1, n, k!^2); \\ Michel Marcus, Jul 16 2017

a(n) = A061062(n) - 1. - Michel Marcus, Feb 28 2014

More terms from Vladimir Joseph Stephan Orlovsky, Sep 24 2009

A134366 a(n) = (n!)^(n-1).

Original entry on oeis.org

1, 1, 2, 36, 13824, 207360000, 193491763200000, 16390160963076096000000, 173238200573946282828103680000000, 300679807141675805997423113304381849600000000

Offset: 0

Artur Jasinski, Oct 22 2007

nonn

Cf. A000142, A001044, A000442, A134366. A134367, A134368, A134369, A134370, A134371, A134374, A134375.

a:=n->mul(n!/k, k=1..n): seq(a(n), n=0..9); # Zerinvary Lajos, Jan 22 2008
restart:with (combinat):a:=n->mul(stirling1(n,1), j=3..n): seq(a(n), n=1..10); # Zerinvary Lajos, Jan 01 2009

Mathematica
```
Table[(n!)^(n - 1), {n, 0, 10}]
```

a(n) = (n!)^(n-1); \\ Michel Marcus, Dec 23 2015

a(n) ~ exp(1/12 + n - n^2) * n^((n-1)*(2*n+1)/2) * (2*Pi)^((n-1)/2). - Vaclav Kotesovec, Oct 26 2017

Offset corrected to 0 by Michel Marcus, Dec 23 2015

A134369 a(n) = ((2n+1)!)^(n+1).

Original entry on oeis.org

1, 36, 1728000, 645241282560000, 6292383221978976013516800000, 4045146997974190235742848547815424000000000000, 363046466970952735968096996065196818096105852014637875200000000000000

Offset: 0

Artur Jasinski, Oct 22 2007

nonn

Cf. A000142, A001044, A000442, A036740, A134366, A134367, A134368, A134370, A134371, A134374, A134375.

Mathematica
```
Table[((2n+1)!)^(n + 1), {n, 0, 10}]
```

a(n) ~ 2^(2*(n+1)^2) * exp(13/24 - 2*n*(n+1)) * n^((n+1)*(4*n+3)/2) * Pi^((n+1)/2). - Vaclav Kotesovec, Oct 26 2017

A134371 a(n) = ((2n)!)^n.

Original entry on oeis.org

1, 2, 576, 373248000, 2642908293365760000, 629238322197897601351680000000000, 12078744213598964456884373878200091017216000000000000

Offset: 0

Artur Jasinski, Oct 22 2007

nonn

Cf. A000142, A001044, A000442, A036740, A134366, A134367, A134368, A134369, A134370, A134374, A134375.

Mathematica
```
Table[((2n)!)^(n), {n, 0, 10}]
```

a(n) ~ 2^(n*(2*n+1)) * exp(1/24 - 2*n^2) * n^(n*(4*n+1)/2) * Pi^(n/2). - Vaclav Kotesovec, Oct 26 2017

A225816 Square array A(n,k) = (k!)^n, n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 6, 4, 1, 1, 1, 24, 36, 8, 1, 1, 1, 120, 576, 216, 16, 1, 1, 1, 720, 14400, 13824, 1296, 32, 1, 1, 1, 5040, 518400, 1728000, 331776, 7776, 64, 1, 1, 1, 40320, 25401600, 373248000, 207360000, 7962624, 46656, 128, 1, 1

Offset: 0

Alois P. Heinz, Jul 29 2013

A(n,k) is the determinant of the k X k matrix M = [Stirling2(n+i,j)] for 1<=i,j<=k.  A(2,3) = det([1,3,1; 1,7,6; 1,15,25]) = 36.
A(n,k) is the determinant of the symmetric k X k matrix M = [sigma_n(gcd(i,j))] for 1<=i,j<=k.  A(2,3) = det([1,1,1; 1,5,1; 1,1,10]) = 36.
A(n,k) is (-1)^(n*k) times the determinant of the n X n matrix M = [Stirling1(k+i,j)] for 1<=i,j<=n.  A(2,3) = (-1)^(2+3) * det([-6,11; 24,-50]) = 36.
A(n,k) is the number of lattice paths from {n}^k to {0}^k using steps that decrement one component by 1 such that for each point (p_1,p_2,...,p_k) we have abs(p_i-p_j) <= 1 for 1<=i,j<=k.  A(2,3) = 36:
          (1,2,2)-(1,1,2)         (0,1,1)-(0,0,1)
         /       X       \       /       X       \
  (2,2,2)-(2,1,2) (1,2,1)-(1,1,1)-(1,0,1) (0,1,0)-(0,0,0).
         \       X       /       \       X       /
          (2,2,1) (2,1,1)         (1,1,0) (1,0,0)
A(n,k) is the number of set partitions of [k*(n+1)] into k blocks of size n+1 such that the elements of each block are distinct mod n+1.  A(2,3) = 36: 123|456|789, 126|345|789, ..., 189|234|567, 189|246|357.

			Square array A(n,k) begins:
  1, 1,  1,    1,       1,           1, ...
  1, 1,  2,    6,      24,         120, ...
  1, 1,  4,   36,     576,       14400, ...
  1, 1,  8,  216,   13824,     1728000, ...
  1, 1, 16, 1296,  331776,   207360000, ...
  1, 1, 32, 7776, 7962624, 24883200000, ...

Alois P. Heinz, Antidiagonals n = 0..36, flattened

Columns k=0+1, 2-4 give: A000012, A000079, A000400, A009968.
Rows n=0-4 give: A000012, A000142, A001044, A000442, A134375.
Main diagonal gives: A036740.
Cf. A008275, A048994, A008277, A048993, A003989, A109004, A109974.

A:= (n, k)-> k!^n:
seq(seq(A(n,d-n), n=0..d), d=0..12);

A(n,k) = (k!)^n.
A(n,k) = k^n * A(n,k-1) for k>0, A(n,0) = 1.
A(n,k) = k!  * A(n-1,k) for n>0, A(0,k) = 1.
G.f. of column k: 1/(1-k!*x).

A288580 Array read by upwards antidiagonals: T(n,k) = Product_{ 0 < |n-ki| <= n} (n-ki), with n >= 0, k >= 1.

Original entry on oeis.org

1, 1, -1, 1, -1, 4, 1, 1, -4, -36, 1, 1, -2, 9, 576, 1, 1, -4, -9, 64, -14400, 1, 1, 2, -3, -8, -225, 518400, 1, 1, 2, -6, -16, 40, -2304, -25401600, 1, 1, 2, -9, -4, -15, 324, 11025, 1625702400, 1, 1, 2, 3, -8, -25, 144, 280, 147456, -131681894400, 1, 1, 2, 3, -12, -5, -24, 105, -2240, -893025, 13168189440000

Offset: 0

N. J. A. Sloane, Jul 03 2017

			Array begins:
1, -1, 4, -36, 576, -14400, 518400, -25401600, 1625702400, -131681894400,  ...
1, -1, -4, 9, 64, -225, -2304, 11025, 147456, -893025, -14745600, 108056025, ...
1, 1, -2, -9, -8, 40, 324, 280, -2240, -26244, -22400, 246400, 3779136, ...
1, 1, -4, -3, -16, -15, 144, 105, 1024, 945, -14400, -10395, -147456, ...
1, 1, 2, -6, -4, -25, -24, -42, 336, 216, 2500, 2376, 4032, ...
1, 1, 2, -9, -8, -5, -36, -35, -64, 729, 640, 385, 5184, ...
1, 1, 2, 3, -12, -10, -6, -49, -48, -90, -120, 1320, 1080, ...
1, 1, 2, 3, -16, -15, -12, -7, -64, -63, -120, -165, 2304, ...
1, 1, 2, 3, 4, -20, -18, -14, -8, -81, -80, -154, -216, ...
1, 1, 2, 3, 4, -25, -24, -21, -16, -9, -100, -99, -192, ...
...

F. Smarandache, Back and Forth Factorials, Arizona State Univ., Special Collections, 1972.

J. Dezert, ed., Smarandacheials (1), Mathematics Magazine for Grades 1-12, No. 4, 2004.
J. Dezert, ed., Smarandacheials (2), Mathematics Magazine for Grades 1-12, No. 4, 2004.

Rows k=1 through 9 are signed A001044 or A092396, signed A184877 or A092397, A092398, A092399, A092971, A092972, A092973, A092974,

T:=proc(n,k)  local i,p;
p:=1;
for i from 0 to floor(2*n/k) do
if n-k*i <> 0 then p:=p*(n-k*i) fi; od:
p;
end;
scan1:=proc(a,M1) local lis,n,k; lis:=[]; for n from 1 to M1 do for k from 0 to n-1 do
lis:=[op(lis),a(k,n-k)]; od: od: lis; end:
scan1(T,12);

T[n_, k_] := Module[{i, p = 1}, For[i = 0, i <= Floor[2n/k], i++, If[n - k i != 0, p *= (n - k i)]]; p]; T[_, 0] = 1;
Table[T[k, n - k + 1], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 05 2020, after Maple *)

A027837 Number of subgroups of index n in free group of rank 3.

Original entry on oeis.org

1, 7, 97, 2143, 68641, 3011263, 173773153, 12785668351, 1169623688353, 130305512589247, 17376934722756577, 2733655173624167551, 501034099176714373921, 105847486567006696384831

Offset: 1

Valery A. Liskovets

P. de la Harpe, Topics in Geometric Group Theory, Univ. Chicago Press, 2000, p. 23.
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.13(b).

Reinhard Zumkeller, Table of n, a(n) for n = 1..250
M. Hall, Subgroups of finite index in free groups, Canad. J. Math., 1 (1949), 187-190.
V. A. Liskovets and A. Mednykh, Enumeration of subgroups in the fundamental groups of orientable circle bundles over surfaces, Commun. in Algebra, 28, No. 4 (2000), 1717-1738.

Cf. A003319, A049290-A049295.

Cf. A001044.

a027837 n = a027837_list !! (n-1)
a027837_list = f 1 [] where
   f x ys = y : f (x + 1) (y : ys) where
            y = a001044 x * x - sum (zipWith (*) ys $ tail a001044_list)
-- Reinhard Zumkeller, Sep 05 2015

a[n_] := a[n] = n*n!^2 - Sum [k!^2*a[n-k], {k, 1, n-1}]; Table[ a[n], {n, 1, 14}] (* Jean-François Alcover, Dec 13 2011, after formula *)

{a(n)=n*polcoeff(log(sum(k=0,n,k!^2*x^k)+x*O(x^n)),n)} \\ Paul D. Hanna, Apr 13 2009

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

L.g.f.: Sum_{n>=1} a(n)*x^n/n = log( Sum_{n>=1} (n-1)!^2*x^n ). - Paul D. Hanna, Apr 13 2009

More terms from Larry Reeves (larryr(AT)acm.org), Oct 05 2000

Further terms from Naohiro Nomoto, Jun 18 2001

A061062 Sum of squared factorials: (0!)^2 + (1!)^2 + (2!)^2 + (3!)^2 +...+ (n!)^2.

Original entry on oeis.org

1, 2, 6, 42, 618, 15018, 533418, 25935018, 1651637418, 133333531818, 13301522971818, 1606652445211818, 231049185247771818, 39006837228880411818, 7639061293780877851818, 1717651314017980301851818

Offset: 0

Jason Earls, May 27 2001

nonn

There is a Kurepa-like conjecture (see A049782) for this sequence: for primes p>3, p does not divide a(p-1). However, the conjecture fails for p=20879. - T. D. Noe, Dec 08 2004

			a(2) = 0!*0! + 1!*1! + 2!*2! = 6.

Seiichi Manyama, Table of n, a(n) for n = 0..253 (terms 0..100 from Harry J. Smith)

Cf. A001044, A100288 (primes of the form (1!)^2 + (2!)^2 + (3!)^2 +...+ (k!)^2), A104344 (if sum starts at k=1), A049782.

A061062:=n->sum((k!)^2, k=0..n): seq(A061062(n), n=0..15); # Zerinvary Lajos, Jan 22 2008

s=0; Table[s=s+(n!)^2, {n, 0, 20}]
Accumulate[(Range[0,20]!)^2] (* Harvey P. Dale, Apr 19 2015 *)

{ a=0; for (n=0, 100, write("b061062.txt", n, " ", a+=(n!)^2) ) } \\ Harry J. Smith, Jul 17 2009

a(n) = sum(k=0...n, (n-k)!^2 ). - Ross La Haye, Sep 21 2004

Recurrence: a(0) = 1, a(1) = 2, a(n) = (n^2+1)*a(n-1) - n^2*a(n-2). - Vladimir Reshetnikov, Oct 28 2015

More terms from T. D. Noe, Dec 08 2004

cp's OEIS Frontend

A192721 The number of pairs of permutations in the product group S_n X S_n with k common descents, n >= 1 and 0 <= k <= n-1.

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Maple

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A088021 a(n) = (n^2)!/(n!)^2.

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Magma

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A104344 a(n) = Sum_{k=1..n} k!^2.

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PARI

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A134366 a(n) = (n!)^(n-1).

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Maple

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PARI

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A134369 a(n) = ((2n+1)!)^(n+1).

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Mathematica

Formula

A134371 a(n) = ((2n)!)^n.

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Mathematica

Formula

A225816 Square array A(n,k) = (k!)^n, n>=0, k>=0, read by antidiagonals.

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Comments

Examples

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Programs

Maple

Formula

A288580 Array read by upwards antidiagonals: T(n,k) = Product_{ 0 < |n-k*i| <= n} (n-k*i), with n >= 0, k >= 1.

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A288580 Array read by upwards antidiagonals: T(n,k) = Product_{ 0 < |n-ki| <= n} (n-ki), with n >= 0, k >= 1.