A052849 -id:A052849 - cp's OEIS frontend

A138770 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} such that there are exactly k entries between the entries 1 and 2 (n>=2, 0<=k<=n-2).

2, 4, 2, 12, 8, 4, 48, 36, 24, 12, 240, 192, 144, 96, 48, 1440, 1200, 960, 720, 480, 240, 10080, 8640, 7200, 5760, 4320, 2880, 1440, 80640, 70560, 60480, 50400, 40320, 30240, 20160, 10080, 725760, 645120, 564480, 483840, 403200, 322560, 241920, 161280, 80640

Offset: 2

Emeric Deutsch, Apr 06 2008

Sum of row n = n! = A000142(n).

The expected value of k is (n-2)/3. [Geoffrey Critzer, Dec 19 2009]

			T(4,2)=4 because we have 1342, 1432, 2341 and 2431.
Triangle starts:
  2;
  4,2;
  12,8,4;
  48,36,24,12;
  240,192,144,96,48;
  ...

Cf. A000142, A052489, A052582, A052609, A005990, A061206, A052849, A090672.

T:=proc(n,k) if n-2 < k then 0 else (2*n-2*k-2)*factorial(n-2) end if end proc; for n from 2 to 10 do seq(T(n, k),k=0..n-2) end do; # yields sequence in triangular form

Table[Table[2 (n - r) (n - 2)!, {r, 1, n - 1}], {n, 1, 10}] // Grid (* Geoffrey Critzer, Dec 19 2009 *)

T(n,k) = 2*(n-k-1)*(n-2)!.
T(n,0) = 2(n-1)! = A052849(n-1).
T(n,1) = A052582(n-2).
T(n,2) = A052609(n-2).
T(n,3) = 12*A005990(n-3).
T(n,4) = 48*A061206(n-5).
T(n,n-2) = 2(n-2)! (A052849).
Sum_{k=0..n-2} k*T(n,k) = n!*(n-2)/3 = A090672(n-1).

A159038 a(n) = 8 * n!.

Original entry on oeis.org

8, 16, 48, 192, 960, 5760, 40320, 322560, 2903040, 29030400, 319334400, 3832012800, 49816166400, 697426329600, 10461394944000, 167382319104000, 2845499424768000, 51218989645824000, 973160803270656000

Offset: 1

Zerinvary Lajos, Apr 03 2009

Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014.

Cf. A000142, A052560, A052578, A052648, A052849, A062098.

A159038 := proc(n)
    8*n! ;
end proc:
seq(A159038(n),n=1..10) ; # R. J. Mathar, Sep 30 2013

a(n) = 8 * A000142(n) for n > 0.

A256031 Number of irreducible idempotents in partial Brauer monoid PB_n.

Original entry on oeis.org

2, 3, 12, 30, 240, 840, 10080, 45360, 725760, 3991680, 79833600, 518918400, 12454041600, 93405312000, 2615348736000, 22230464256000, 711374856192000, 6758061133824000, 243290200817664000, 2554547108585472000, 102181884343418880000, 1175091669949317120000

Offset: 1

N. J. A. Sloane, Mar 14 2015

nonn

Table 2 in chapter 7 of the preprint contains a typo: a(9) is not 725860. - R. J. Mathar, Mar 14 2015

I. Dolinka, J. East, A. Evangelou, D. FitzGerald, N. Ham, J. Hyde and N. Loughlin, Enumeration of idempotents in diagram semigroups and algebras, arXiv preprint arXiv:1408.2021 [math.GR], 2014. See Prop. 22.

Cf. A052612, A052616, A052849, A098558, A208529, A256032, A174549 (bisection).

A256031 := proc(n)
    if type(n,'odd') then
        2*n! ;
    else
        (n+1)*(n-1)! ;
    end if;
end proc:
seq(A256031(n),n=1..20) ; # R. J. Mathar, Mar 14 2015

a[n_] := If[OddQ[n], 2*n!, (n + 1)*(n - 1)!];
Array[a, 20] (* Jean-François Alcover, Nov 24 2017, from Maple *)

There are simple formulas for the two bisections - see Dolinka et al.
a(2n-1) = A052612(2n-1) = A052616(2n-1) = A052849(2n-1) = A098558(2n-1) = A208529(2n+1). - Omar E. Pol, Mar 14 2015
Sum_{n>=1} 1/a(n) = (e^2+3)/(4*e) = 1/e + sinh(1)/2. - Amiram Eldar, Feb 02 2023

A298881 a(0) = 0; for n>0, a(n) = 6*n!.

Original entry on oeis.org

0, 6, 12, 36, 144, 720, 4320, 30240, 241920, 2177280, 21772800, 239500800, 2874009600, 37362124800, 523069747200, 7846046208000, 125536739328000, 2134124568576000, 38414242234368000, 729870602452992000, 14597412049059840000, 306545653030256640000

Offset: 0

Vincenzo Librandi, Feb 13 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Cf. A274266.

Cf. sequences of the type k*n!: A000142 (k=1), A052849 (k=2), A052560 (k=3), A052578 (k=4), A052648 (k=5), this sequence (k=6), A062098 (k=7), A159038 (k=8), A174183 (k=10).

Concatenation([0], List([1..25], n -> 6*Factorial(n))); # Bruno Berselli, Feb 13 2018

[n eq 0 select 0 else 6*Factorial(n): n in [0..25]];

Mathematica
```
Join[{0}, 6 Range[25]!]
```

a(n) = if (n, 6*n!, 0); \\ Michel Marcus, Feb 15 2018

E.g.f.: 6*x/(1-x).

a(n) = n*a(n-1) = 6*A000142(n) for n>0.

Edited by Bruno Berselli, Feb 13 2018

A324225 Total number T(n,k) of 1's in falling diagonals with index k in all n X n permutation matrices; triangle T(n,k), n>=1, 1-n<=k<=n-1, read by rows.

Original entry on oeis.org

1, 1, 2, 1, 2, 4, 6, 4, 2, 6, 12, 18, 24, 18, 12, 6, 24, 48, 72, 96, 120, 96, 72, 48, 24, 120, 240, 360, 480, 600, 720, 600, 480, 360, 240, 120, 720, 1440, 2160, 2880, 3600, 4320, 5040, 4320, 3600, 2880, 2160, 1440, 720, 5040, 10080, 15120, 20160, 25200, 30240, 35280, 40320, 35280, 30240, 25200, 20160, 15120, 10080, 5040

Offset: 1

Alois P. Heinz, Feb 18 2019

T(n,k) is the number of occurrences of k in all (signed) displacement lists [p(i)-i, i=1..n] of permutations p of [n].

			The 6 permutations p of [3]: 123, 132, 213, 231, 312, 321 have (signed) displacement lists [p(i)-i, i=1..3]: [0,0,0], [0,1,-1], [1,-1,0], [1,1,-2], [2,-1,-1], [2,0,-2], representing the indices of falling diagonals of 1's in the permutation matrices
  [1    ]  [1    ]  [  1  ]  [  1  ]  [    1]  [    1]
  [  1  ]  [    1]  [1    ]  [    1]  [1    ]  [  1  ]
  [    1]  [  1  ]  [    1]  [1    ]  [  1  ]  [1    ] , respectively. Indices -2 and 2 occur twice, -1 and 1 occur four times, and 0 occurs six times. So row n=3 is [2, 4, 6, 4, 2].
Triangle T(n,k) begins:
  :                             1                           ;
  :                        1,   2,   1                      ;
  :                   2,   4,   6,   4,   2                 ;
  :              6,  12,  18,  24,  18,  12,   6            ;
  :        24,  48,  72,  96, 120,  96,  72,  48,  24       ;
  :  120, 240, 360, 480, 600, 720, 600, 480, 360, 240, 120  ;

Alois P. Heinz, Rows n = 1..100, flattened
Nadir Samos Sáenz de Buruaga, Rafał Bistroń, Marcin Rudziński, Rodrigo Miguel Chinita Pereira, Karol Życzkowski, and Pedro Ribeiro, Fidelity decay and error accumulation in quantum volume circuits, arXiv:2404.11444 [quant-ph], 2024. See p. 18.
Wikipedia, Permutation
Wikipedia, Permutation matrix

Columns k=0-6 give (offsets may differ): A000142, A001563, A062119, A052571, A052520, A282822, A052521.
Row sums give A001563.
T(n+1,n) gives A000142.
T(n+1,n-1) gives A052849.
T(n+1,n-2) gives A052560 for n>1.
Cf. A152883 (right half of this triangle without center column), A162608 (left half of this triangle), A306461, A324224.
Cf. A001710.

b:= proc(s, c) option remember; (n-> `if`(n=0, c,
      add(b(s minus {i}, c+x^(n-i)), i=s)))(nops(s))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1-n..n-1))(b({$1..n}, 0)):
seq(T(n), n=1..8);
# second Maple program:
egf:= k-> (t-> x^t/t*hypergeom([2, t], [t+1], x))(abs(k)+1):
T:= (n, k)-> n! * coeff(series(egf(k), x, n+1), x, n):
seq(seq(T(n, k), k=1-n..n-1), n=1..8);
# third Maple program:
T:= (n, k)-> (t-> `if`(t

T[n_, k_] := With[{t = Abs[k]}, If[tJean-François Alcover, Mar 25 2021, after 3rd Maple program *)

T(n,k) = T(n,-k).
T(n,k) = (n-t)*(n-1)! if t < n with t = |k|, T(n,k) = 0 otherwise.
T(n,k) = |k|! * A324224(n,k).
E.g.f. of column k: x^t/t * hypergeom([2, t], [t+1], x) with t = |k|+1.
|T(n,k)-T(n,k-1)| = (n-1)! for k = 1-n..n.
Sum_{k=0..n-1} T(n,k) = A001710(n+1).

A064482 Triangle read by rows: T(n,k) (n >= 2, 1<=k<=n-1) is the number of permutations p of 1,...,n with max(|p(i)-p(i-1)|, i=2..n) = k.

Original entry on oeis.org

2, 2, 4, 2, 10, 12, 2, 18, 52, 48, 2, 32, 146, 300, 240, 2, 54, 372, 1204, 1968, 1440, 2, 86, 954, 4082, 10476, 14640, 10080, 2, 134, 2376, 13348, 46012, 97968, 122400, 80640, 2, 206, 5704, 44274, 186202, 536652, 990960, 1139040, 725760, 2, 312, 13278, 145216, 742940, 2655004, 6562128, 10847520, 11692800, 7257600

Offset: 2

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Oct 05 2001

T(n,n-1) = A052849; sum(T(n,k),k=1..n-1) = A000142.

			Triangle T(n,k) begins:
  2;
  2,   4;
  2,  10,   12;
  2,  18,   52,    48;
  2,  32,  146,   300,   240;
  2,  54,  372,  1204,  1968,  1440;
  2,  86,  954,  4082, 10476, 14640,  10080;
  2, 134, 2376, 13348, 46012, 97968, 122400, 80640;

Alois P. Heinz, Rows n = 2..18, flattened

Cf. A052849, A000142, A129534.

More terms from Naohiro Nomoto, Dec 04 2001

More terms from R. J. Mathar, Oct 11 2007

A256881 a(n) = n!/ceiling(n/2).

Original entry on oeis.org

1, 2, 3, 12, 40, 240, 1260, 10080, 72576, 725760, 6652800, 79833600, 889574400, 12454041600, 163459296000, 2615348736000, 39520825344000, 711374856192000, 12164510040883200, 243290200817664000, 4644631106519040000, 102181884343418880000, 2154334728240414720000

Offset: 1

M. F. Hasler, Apr 22 2015

nonn

Original name was: n!/round(n/2). - Robert Israel, Sep 03 2018

Robert Israel, Table of n, a(n) for n = 1..450
Pierre-Alain Sallard, Sum of repeated integrals of sinh.

Cf. A009445, A052612, A052616, A052849, A081457, A208529, A098558, A107991, A110468, A229244, A256031.

[Factorial(n)/Round(n/2): n in [1..30]]; // Vincenzo Librandi, Apr 23 2015

Maple
```
A256881 := n!/round(n/2);
```

Function[x, 1/x] /@
CoefficientList[Series[(Sinh[x] + x*Exp[x])/2, {x, 0, 20}], x] (* Pierre-Alain Sallard, Dec 15 2018 *)

PARI
```
A256881(n)=n!/round(n/2)
```

a(2n) = 2*A009445(n) = A052612(2n-1) = A052616(2n-1) = A052849(2n-1) = A098558(2n-1) = A081457(3n-1) = A208529(2n+1) = A256031(2n-1).
a(2n+1) = A110468(n) = A107991(2n+2) = A229244(2n+1), n>=0.
From Robert Israel, Sep 03 2018: (Start)
E.g.f.: -(1+1/x)*log(1-x^2).
n*(n+1)*(n+2)*a(n)+(n+2)*a(n+1)-(n+3)*a(n+2)=0. (End)
a(n) = 2/([x^n](sinh(x) + x*exp(x))). - Pierre-Alain Sallard, Dec 15 2018
Sum_{n>=1} 1/a(n) = (3*e-1/e)/4 = (e + sinh(1))/2. - Amiram Eldar, Feb 02 2023

Definition clarified by Robert Israel, Sep 03 2018

A344901 Triangle read by rows: T(n,k) is the number of permutations of length n that have k same elements at the same positions with its inverse permutation for 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 2, 0, 0, 4, 6, 8, 0, 0, 10, 24, 30, 40, 0, 0, 26, 160, 144, 180, 160, 0, 0, 76, 1140, 1120, 1008, 840, 700, 0, 0, 232, 8988, 9120, 8960, 5376, 4200, 2912, 0, 0, 764, 80864, 80892, 82080, 53760, 30240, 19656, 12768, 0, 0, 2620, 809856, 808640, 808920, 547200, 336000, 157248, 95760, 55680, 0, 0, 9496

Offset: 0

Mikhail Kurkov, Jun 01 2021

			Triangle T(n,k) begins:
     1;
     0,    1;
     0,    0,    2;
     2,    0,    0,    4;
     6,    8,    0,    0,   10;
    24,   30,   40,    0,    0,   26;
   160,  144,  180,  160,    0,    0, 76;
  1140, 1120, 1008,  840,  700,    0,  0, 232;
  8988, 9120, 8960, 5376, 4200, 2912,  0,   0, 764;
  ...

Alois P. Heinz, Rows n = 0..140, flattened

Columns k=0-1 give: A038205, A221145.
Row sums give A000142.
Main diagonal gives A000085.
Cf. A000023, A007318, A052571, A052849, A098558, A137482.

b:= proc(n, t) option remember; `if`(n=0, 1, add(b(n-j, t)*
      binomial(n-1, j-1)*(j-1)!, j=`if`(t=1, 1..min(2, n), 3..n)))
    end:
T:= (n, k)-> binomial(n, k)*b(k, 1)*b(n-k, 0):
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Oct 28 2024

b[n_, t_] := b[n, t] = If[n == 0, 1, Sum[b[n-j, t]* Binomial[n-1, j-1]*(j-1)!, {j, If[t == 1, Range @ Min[2, n], Range[3, n]]}]];
T[n_, k_] := Binomial[n, k]*b[k, 1]*b[n-k, 0];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Apr 24 2025, after Alois P. Heinz *)

T(n,k) = binomial(n,k)*A000085(k)*A038205(n-k).
From Alois P. Heinz, Oct 28 2024: (Start)
Sum_{k=0..n} k * T(n,k) = A052849(n) = A098558(n) for n>=2.
Sum_{k=0..n} (n-k) * T(n,k) = A052571(n).
Sum_{k=0..n} (-1)^k * T(n,k) = A000023(n).
T(n,0) + T(n,1) = A137482(n). (End)

A360205 Triangle read by rows. T(n, k) = (-1)^(n-k)(k+1)binomial(n, k)*pochhammer(1-n, n-k).

Original entry on oeis.org

1, 0, 2, 0, 4, 3, 0, 12, 18, 4, 0, 48, 108, 48, 5, 0, 240, 720, 480, 100, 6, 0, 1440, 5400, 4800, 1500, 180, 7, 0, 10080, 45360, 50400, 21000, 3780, 294, 8, 0, 80640, 423360, 564480, 294000, 70560, 8232, 448, 9, 0, 725760, 4354560, 6773760, 4233600, 1270080, 197568, 16128, 648, 10

Offset: 0

Peter Luschny, Feb 08 2023

A refinement of the number of partial permutations of an n-set (A002720).

Also the coefficients of a shifted derivative of the unsigned Lah polynomials (A271703).

			Triangle T(n, k) starts:
[0] 1;
[1] 0,     2;
[2] 0,     4,      3;
[3] 0,    12,     18,      4;
[4] 0,    48,    108,     48,      5;
[5] 0,   240,    720,    480,    100,     6;
[6] 0,  1440,   5400,   4800,   1500,   180,    7;
[7] 0, 10080,  45360,  50400,  21000,  3780,  294,   8;
[8] 0, 80640, 423360, 564480, 294000, 70560, 8232, 448, 9;

Cf. A052849 (column 1), A045991 (subdiagonal), A002720 (row sums), A271703.

Cf. A069138 (Stirling2 counterpart), A360174 (Stirling1 counterpart).

T := (n, k) -> (-1)^(n - k)*(k + 1)*binomial(n, k)*pochhammer(1 - n, n - k):
seq(seq(T(n, k), k = 0..n), n = 0..9);

A064484 Triangle T(n,k), n >= 2, n+1 <= k <= 2*n-1, number of permutations p of 1,...,n, with max(p(i)+p(i-1), i=2..n) = k.

Original entry on oeis.org

2, 2, 4, 4, 8, 12, 4, 32, 36, 48, 8, 64, 216, 192, 240, 8, 208, 648, 1536, 1200, 1440, 16, 416, 3024, 6144, 12000, 8640, 10080, 16, 1280, 9072, 37632, 60000, 103680, 70560, 80640, 32, 2560, 38880, 150528, 456000, 622080, 987840, 645120, 725760

Offset: 2

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Oct 05 2001

			For n=3 we have:
T(3,4)=2 with the permutations {312, 213} and
T(3,5)=4 with {123, 321, 132, 231}.

Andrew Woods, Rows n = 2..101 of triangle, flattened

Cf. A052849, A000142.

T[n_ /; n >= 2, k_] /; n+1 <= k <= 2n-1 := T[n, k] = If[EvenQ[k], (k-n)* T[n-1, k-1], (k-n+1)*T[n-1, k-1] + 2*Sum[T[n-1, i], {i, n, k-2}]];
T[1, 2] = 1; T[, ] = 0;
Table[T[n, k], {n, 2, 10}, {k, n+1, 2n-1}] // Flatten (* Jean-François Alcover, Jul 19 2022 *)

# Generate n-th row (n>1) by checking all n! permutations
from itertools import permutations
def onerow(n):
  row=[0]*(n-1)
  for i in permutations(range(1, n+1)):
    row[max([j[0]+j[1] for j in zip(i, i[1:])])-n-1]+=1
  return row
# Andrew Woods, Jun 18 2013

# Generate first twenty rows using recurrence
rows=[[2]]; row=[2]
for i in range(19):
  row=[(row[j]*(j+2)+sum(row[:j])*2) if (i+j)%2==1 else row[j]*(j+1) for j in range(i+1)]+[row[-1]*(i+2)]
  rows.append(row)
# Andrew Woods, Jun 18 2013

Sum_{k=n+1..2*n-1} T(n,k) = n! = A000142(n).
T(n,2*n-1) = 2*(n-1)! = A052849(n-1).
From Andrew Woods, Jun 16 2013: (Start)
T(n, even k) = (k-n)*T(n-1,k-1);
T(n, odd k)  = (k-n+1)*T(n-1,k-1)+2*sum(T(n-1,i) for i=n..k-2);
T(n,2*n-1) = 2*(n-1)!;
T(n,2*n-2) = 2*(n-1)!-2*(n-2)! for n>2;
T(n,2*n-3) = 4*(n-1)!-12*(n-2)!+4*(n-3)! for n>3;
T(n,2*n-4) = 4*(n-1)!-24*(n-2)!+28*(n-3)!-4*(n-4)! for n>4;
T(n,2*n-5) = 6*(n-1)!-60*(n-2)!+152*(n-3)!-96*(n-4)!+8*(n-5)! for n>5.
(End)

More terms from Naohiro Nomoto, Nov 26 2001

cp's OEIS Frontend

A138770 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} such that there are exactly k entries between the entries 1 and 2 (n>=2, 0<=k<=n-2).

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Maple

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A159038 a(n) = 8 * n!.

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Maple

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A256031 Number of irreducible idempotents in partial Brauer monoid PB_n.

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Maple

Mathematica

Formula

A298881 a(0) = 0; for n>0, a(n) = 6*n!.

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A324225 Total number T(n,k) of 1's in falling diagonals with index k in all n X n permutation matrices; triangle T(n,k), n>=1, 1-n<=k<=n-1, read by rows.

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A064482 Triangle read by rows: T(n,k) (n >= 2, 1<=k<=n-1) is the number of permutations p of 1,...,n with max(|p(i)-p(i-1)|, i=2..n) = k.

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A256881 a(n) = n!/ceiling(n/2).

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Magma

Maple

Mathematica

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Extensions

A360205 Triangle read by rows. T(n, k) = (-1)^(n-k)(k+1)binomial(n, k)*pochhammer(1-n, n-k).