A256067 -id:A256067

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A009490

Number of distinct orders of permutations of n objects; number of nonisomorphic cyclic subgroups of symmetric group S_n.

+0
15

1, 1, 2, 3, 4, 6, 6, 9, 11, 14, 16, 20, 23, 27, 31, 35, 43, 47, 55, 61, 70, 78, 88, 98, 111, 123, 136, 152, 168, 187, 204, 225, 248, 271, 296, 325, 356, 387, 418, 455, 495, 537, 581, 629, 678, 732, 787, 851, 918, 986, 1056, 1133, 1217, 1307, 1399, 1498, 1600, 1708, 1823

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

Also number of different LCM's of partitions of n.

a(n) <= A023893(n), which counts the nonisomorphic Abelian subgroups of S_n. - M. F. Hasler, May 24 2013

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)

L. Elliott, A. Levine, and J. D. Mitchell, Counting monogenic monoids and inverse monoids, arXiv:2303.12387 [math.GR], 2023.

Index entries for sequences related to lcm's

FORMULA

a(n) = Sum_{k=0..n} b(k), where b(k) is the number of partitions of k into distinct prime power parts (1 excluded) (A051613). - Vladeta Jovovic

G.f.: Product_{p prime} 1 + Sum(k >= 1, x^(p^k))) / (1-x). - David W. Wilson, Apr 19 2000

MAPLE

b:= proc(n, i) option remember; local p;

p:= `if`(i<1, 1, ithprime(i));

`if`(n=0 or i<1, 1, b(n, i-1)+

add(b(n-p^j, i-1), j=1..ilog[p](n)))

end:

a:= n-> b(n, numtheory[pi](n)):

seq(a(n), n=0..100); # Alois P. Heinz, Feb 15 2013

MATHEMATICA

Table[ Length[ Union[ Apply[ LCM, Partitions[ n ], 1 ] ] ], {n, 30} ]

f[n_] := Length@ Union[LCM @@@ IntegerPartitions@ n]; Array[f, 60, 0]

(* Caution, the following is Extremely Slow and Resource Intensive *) CoefficientList[ Series[ Expand[ Product[1 + Sum[x^(Prime@ i^k), {k, 4}], {i, 10}]/(1 - x)], {x, 0, 30}], x]

b[n_, i_] := b[n, i] = Module[{p}, p = If[i<1, 1, Prime[i]]; If[n == 0 || i<1, 1, b[n, i-1]+Sum[b[n-p^j, i-1], {j, 1, Log[p, n]}]]]; a[n_] := b[n, PrimePi[n]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Feb 03 2014, after Alois P. Heinz *)

PROG

(PARI) /* compute David W. Wilson's g.f., needs <1 sec for 1000 terms */

N=1000; x='x+O('x^N); /* N terms */

gf=1; /* generating function */

{ forprime(p=2, N,

sm = 1; pp=p; /* sum; prime power */

while ( pp<N, sm += x^pp; pp *= p; );

gf *= sm; /* update g.f. */

); }

gf/=(1-x); /* cumulative sums */

Vec(gf) /* show terms */ /* Joerg Arndt, Jan 19 2011 */

CROSSREFS

Cf. A051613 (first differences), A000792, A000793, A034891, A051625 (all cyclic subgroups), A256067.

KEYWORD

nonn,nice,easy

AUTHOR

David W. Wilson

STATUS

approved

A168532

Triangle read by rows, A054525 * A168021.

+0
23

1, 1, 1, 2, 0, 1, 3, 1, 0, 1, 6, 0, 0, 0, 1, 7, 2, 1, 0, 0, 1, 14, 0, 0, 0, 0, 0, 1, 17, 3, 0, 1, 0, 0, 0, 1, 27, 0, 2, 0, 0, 0, 0, 0, 1, 34, 6, 0, 0, 1, 0, 0, 0, 0, 1, 55, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 63, 7, 3, 2, 0, 1, 0, 0, 0, 0, 0, 1, 100, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,4

COMMENTS

Row sums = A000041 starting (1, 2, 3, 5, 7, 11, 15, ...).

T(n,k) is the number of partitions of n into parts with GCD = k. - Alois P. Heinz, Jun 06 2013

LINKS

Alois P. Heinz, Rows n = 1..141, flattened

FORMULA

Mobius transform of triangle A168021 = an infinite lower triangular matrix with aerated variants of A000837 in each column; where A000837 = the Mobius transform of the partition numbers, A000041.

EXAMPLE

First few rows of the triangle:

1, 1;

2, 0, 1;

3, 1, 0, 1;

6, 0, 0, 0, 1;

7, 2, 1, 0, 0, 1;

14, 0, 0, 0, 0, 0, 1;

17, 3, 0, 1, 0, 0, 0, 1;

27, 0, 2, 0, 0, 0, 0, 0, 1;

34, 6, 0, 0, 1, 0, 0, 0, 0, 1;

55, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;

63, 7, 3, 2, 0, 1, 0, 0, 0, 0, 0, 1;

100, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;

119, 14, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1;

167, 0, 6, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;

209, 17, 0, 3, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1;

296, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;

...

MAPLE

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i=1, x,

b(n, i-1)+(p-> add(coeff(p, x, t)*x^igcd(t, i),

t=0..degree(p)))(add(b(n-i*j, i-1), j=1..n/i))))

end:

T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n$2)):

seq(T(n), n=1..17); # Alois P. Heinz, Mar 29 2015

MATHEMATICA

b[n_, i_] := b[n, i] = If[n==0, 1, If[i==1, x, b[n, i-1] + Function[{p}, Sum[Coefficient[p, x, t]*x^GCD[t, i], {t, 0, Exponent[p, x]}]][Sum[b[n - i*j, i-1], {j, 1, n/i}]]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 1, n}]][b[n, n]]; Table[T[n], {n, 1, 17}] // Flatten (* Jean-François Alcover, Jan 08 2016, after Alois P. Heinz *)

CROSSREFS

Cf. A168021, A000837.

Cf. A256067 (the same for LCM).

KEYWORD

nonn,tabl

AUTHOR

Gary W. Adamson, Nov 28 2009

EXTENSIONS

Corrected and extended by Alois P. Heinz, Jun 06 2013

STATUS

approved

A256553

Triangle T(n,k) in which the n-th row contains the increasing list of distinct orders of degree-n permutations; n>=0, 1<=k<=A009490(n).

+0
4

1, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 7, 10, 12, 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 15, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 20, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 20, 21, 30

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

LINKS

Alois P. Heinz, Rows n = 0..60, flattened

FORMULA

Sum_{k>=0} T(n,k)*A256554(n,k) = A181844(n).

T(n,k) = k for n>0 and 1<=k<=n.

EXAMPLE

Triangle T(n,k) begins:

1, 2;

1, 2, 3;

1, 2, 3, 4;

1, 2, 3, 4, 5, 6;

1, 2, 3, 4, 5, 6, 7, 10, 12;

1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 15;

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 20;

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 20, 21, 30;

MAPLE

b:= proc(n, i) option remember; `if`(n=0 or i=1, x,

b(n, i-1)+(p-> add(coeff(p, x, t)*x^ilcm(t, i),

t=1..degree(p)))(add(b(n-i*j, i-1), j=1..n/i)))

end:

T:= n->(p->seq((h->`if`(h=0, [][], i))(coeff(p, x, i))

, i=1..degree(p)))(b(n$2)):

seq(T(n), n=0..12);

MATHEMATICA

b[n_, i_] := b[n, i] = If[n == 0 || i == 1, x,

b[n, i - 1] + Function[p, Sum[Coefficient[p, x, t]*x^LCM[t, i],

{t, 1, Exponent[p, x]}]][Sum[b[n - i*j, i - 1], {j, 1, n/i}]]];

T[n_] := Function[p, Table[Function[h, If[h == 0, Nothing, i]][

Coefficient[p, x, i]], {i, 1, Exponent[p, x]}]][b[n, n]];

Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Jul 15 2021, after Alois P. Heinz *)

CROSSREFS

Row sums give A060179.

Row lengths give A009490.

Last elements of rows give A000793.

Main diagonal gives A000027.

Cf. A181844, A256067, A256554.

KEYWORD

nonn,look,tabf

AUTHOR

Alois P. Heinz, Apr 01 2015

STATUS

approved

A074752

Number of combinatorially inequivalent cyclic subgroups of S_n of order 6. Number of partitions of n of order 6.

+0
4

1, 2, 3, 5, 7, 9, 12, 16, 19, 24, 29, 34, 40, 48, 54, 63, 72, 81, 91, 104, 114, 128, 142, 156, 171, 190, 205, 225, 245, 265, 286, 312, 333, 360, 387, 414, 442, 476, 504, 539, 574, 609, 645, 688, 724, 768, 812, 856, 901, 954, 999, 1053, 1107, 1161, 1216, 1280

(list; graph; refs; listen; history; text; internal format)

OFFSET

5,2

COMMENTS

Two permutation groups are combinatorially equivalent iff they have the same cycle index. Order of partition is lcm of its parts.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 5..1000

Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1,-1,2,-1,-1,0,1,1,-1).

FORMULA

G.f.: x^5*(1+x-x^6)/((x-1)*(x^2-1)*(x^3-1)*(x^6-1)). More generally, g.f. for number of partitions of order d is Sum_{i divides d} mu(d/i)*1/Product_{j divides i} (1-x^j).

MATHEMATICA

LinearRecurrence[{1, 1, 0, -1, -1, 2, -1, -1, 0, 1, 1, -1}, {1, 2, 3, 5, 7, 9, 12, 16, 19, 24, 29, 34}, 60] (* Harvey P. Dale, May 23 2020 *)

CROSSREFS

Column k=6 of A256067, A256554.

KEYWORD

easy,nonn

AUTHOR

Vladeta Jovovic, Sep 28 2002

STATUS

approved

A181844

Sum over all partitions of n of the LCM of the parts.

+0
9

1, 1, 3, 6, 12, 23, 38, 73, 118, 198, 318, 530, 819, 1298, 1974, 2975, 4516, 6698, 9980, 14550, 21186, 30304, 43503, 62030, 87908, 123292, 172543, 239720, 331688, 458198, 629376, 860332, 1168172, 1583176, 2138438, 2876283, 3859770, 5159886, 6863702, 9112356

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

Old name was: Row sums of A181842.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..188 (terms n=1..80 from Vincenzo Librandi)

FORMULA

a(n) = Sum_{k>=0} k * A256067(n,k) = Sum_{k>=0} A256553(n,k)*A256554(n,k). - Alois P. Heinz, Apr 02 2015

MAPLE

with(combstruct):

a181844 := proc(n) local k, L, l, R, part;

R := NULL; L := 0;

for k from 1 to n do

part := iterstructs(Partition(n), size=k):

while not finished(part) do

l := nextstruct(part);

L := L + ilcm(op(l));

od;

L end:

# second Maple program:

b:= proc(n, i, r) option remember; `if`(n=0, r, `if`(i<1, 0,

b(n, i-1, r)+b(n-i, min(i, n-i), ilcm(i, r))))

end:

a:= n-> b(n$2, 1):

seq(a(n), n=0..42); # Alois P. Heinz, Mar 18 2019

MATHEMATICA

t[n_, k_] := LCM @@@ IntegerPartitions[n, {n - k + 1}] // Total; a[n_] := Sum[t[n, k], {k, 1, n}]; Table[a[n], {n, 1, 32}] (* Jean-François Alcover, Jul 26 2013 *)

CROSSREFS

Cf. A078392 (the same for GCD), A181843, A181842, A256067, A256553, A256554, A306956.

KEYWORD

nonn

AUTHOR

Peter Luschny, Dec 07 2010

EXTENSIONS

a(0)=1 prepended by Alois P. Heinz, Mar 29 2015

New name from Alois P. Heinz, Mar 18 2019

STATUS

approved

A074761

Number of partitions of n of order n.

+0
37

1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 9, 1, 4, 5, 1, 1, 12, 1, 27, 7, 6, 1, 81, 1, 7, 1, 54, 1, 407, 1, 1, 11, 9, 13, 494, 1, 10, 13, 423, 1, 981, 1, 137, 115, 12, 1, 1309, 1, 59, 17, 193, 1, 240, 21, 1207, 19, 15, 1, 47274, 1, 16, 239, 1, 25, 3284, 1, 333, 23, 3731, 1, 42109, 1, 19

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,6

COMMENTS

Order of partition is lcm of its parts.

a(n) is the number of conjugacy classes of the symmetric group S_n such that a representative of the class has order n. Here order means the order of an element of a group. Note that a(n) = 1 if and only if n is a prime power. - W. Edwin Clark, Aug 05 2014

LINKS

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 1..4000 (first 1025 terms from Joerg Arndt)

FORMULA

Coefficient of x^n in expansion of Sum_{i divides n} A008683(n/i)*1/Product_{j divides i} (1-x^j).

EXAMPLE

The a(15) = 5 partitions are (15), (5,3,3,3,1), (5,5,3,1,1), (5,3,3,1,1,1,1), (5,3,1,1,1,1,1,1,1). - Gus Wiseman, Aug 01 2018

MAPLE

A:= proc(n)

uses numtheory;

local S;

S:= add(mobius(n/i)*1/mul(1-x^j, j=divisors(i)), i=divisors(n));

coeff(series(S, x, n+1), x, n);

end proc:

seq(A(n), n=1..100); # Robert Israel, Aug 06 2014

MATHEMATICA

a[n_] := With[{s = Sum[MoebiusMu[n/i]*1/Product[1-x^j, {j, Divisors[i]}], {i, Divisors[n]}]}, SeriesCoefficient[s, {x, 0, n}]]; Array[a, 80}] (* Jean-François Alcover, Feb 29 2016 *)

Table[Length[Select[IntegerPartitions[n], LCM@@#==n&]], {n, 50}] (* Gus Wiseman, Aug 01 2018 *)

PROG

(PARI)

pr(k, x)={my(t=1); fordiv(k, d, t *= (1-x^d) ); return(t); }

a(n) =

{

my( x = 'x+O('x^(n+1)) );

polcoeff( Pol( sumdiv(n, i, moebius(n/i) / pr(i, x) ) ), n );

}

vector(66, n, a(n) )

\\ Joerg Arndt, Aug 06 2014

CROSSREFS

Cf. A018818, A074351, A074752.

Main diagonal of A256067, A256554.

Cf. A000837, A074761, A285572, A290103, A305566, A316429, A316431, A316432, A316433, A317624.

KEYWORD

easy,nonn

AUTHOR

Vladeta Jovovic, Sep 28 2002

STATUS

approved

A256554

Number T(n,k) of cycle types of degree-n permutations having the k-th smallest possible order; triangle T(n,k), n>=0, 1<=k<=A009490(n), read by rows.

+0
6

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 2, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 1, 1, 1, 4, 2, 4, 1, 5, 1, 1, 1, 1, 1, 1, 4, 3, 4, 1, 7, 1, 1, 1, 2, 2, 1, 1, 1, 1, 5, 3, 6, 2, 9, 1, 2, 1, 3, 4, 1, 1, 1, 1, 1, 1, 5, 3, 6, 2, 12, 1, 2, 1, 4, 1, 6, 2, 2, 1, 2, 1, 1, 1, 2

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,9

COMMENTS

Sum_{k>=0} A256553(n,k)*T(n,k) = A181844(n).

LINKS

Alois P. Heinz, Rows n = 0..60, flattened

EXAMPLE

Triangle T(n,k) begins:

1, 1;

1, 1, 1;

1, 2, 1, 1;

1, 2, 1, 1, 1, 1;

1, 3, 2, 2, 1, 2;

1, 3, 2, 2, 1, 3, 1, 1, 1;

1, 4, 2, 4, 1, 5, 1, 1, 1, 1, 1;

1, 4, 3, 4, 1, 7, 1, 1, 1, 2, 2, 1, 1, 1;

1, 5, 3, 6, 2, 9, 1, 2, 1, 3, 4, 1, 1, 1, 1, 1;

MAPLE

b:= proc(n, i) option remember; `if`(n=0 or i=1, x,

b(n, i-1)+(p-> add(coeff(p, x, t)*x^ilcm(t, i),

t=1..degree(p)))(add(b(n-i*j, i-1), j=1..n/i)))

end:

T:= n->(p->seq((h->`if`(h=0, [][], h))(coeff(p, x, i))

, i=1..degree(p)))(b(n$2)):

seq(T(n), n=0..12);

MATHEMATICA

b[n_, i_] := b[n, i] = If[n == 0 || i == 1, x, b[n, i - 1] + Function[p, Sum[Coefficient[p, x, t]*x^LCM[t, i], {t, 1, Exponent[p, x]}]][Sum[b[n - i*j, i - 1], {j, 1, n/i}]]]; T[n_] := Function[p, Table[Function[h, If[h == 0, {{}, {}}, h]][Coefficient[p, x, i]], {i, 1, Exponent[p, x]}]][b[n, n]]; Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Jan 23 2017, translated from Maple *)

CROSSREFS

Row sums give A000041.

Row lengths give A009490.

Columns k=1-9 give: A000012, A004526, A002264, A008642(n-4), A002266, A074752, A132270, A008643(n-8) for n>7, A008649(n-9) for n>8.

Last elements of rows give A074064.

Main diagonal gives A074761.

Cf. A181844, A256067, A256553.

KEYWORD

nonn,look,tabf

AUTHOR

Alois P. Heinz, Apr 01 2015

STATUS

approved

A074064

Number of cycle types of degree-n permutations having the maximum possible order.

+0
7

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 1, 1, 1, 1, 2, 3, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 2, 4, 1, 1, 1, 1, 2, 3, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 2, 4

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,7

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..180

FORMULA

Coefficient of x^n in expansion of Sum_{i divides A000793(n)} mu(A000793(n)/i)*1/Product_{j divides i} (1-x^j).

EXAMPLE

For n = 22 we have 4 such cycle types: [1, 1, 1, 3, 4, 5, 7], [1, 2, 3, 4, 5, 7], [3, 3, 4, 5, 7], [4, 5, 6, 7].

MAPLE

A000793 := proc(n) option remember; local l, p, i ; l := 1: p := combinat[partition](n): for i from 1 to combinat[numbpart](n) do if ilcm( p[i][j] $ j=1..nops(p[i])) > l then l := ilcm( p[i][j] $ j=1..nops(p[i])) ; fi: od: RETURN(l) ; end proc:

taylInv := proc(i, n) local resul, j, idiv, k ; resul := 1 ; idiv := numtheory[divisors](i) ; for k from 1 to nops(idiv) do j := op(k, idiv) ; resul := resul*taylor(1/(1-x^j), x=0, n+1) ; resul := convert(taylor(resul, x=0, n+1), polynom) ; od ; coeftayl(resul, x=0, n) ; end proc:

A074064 := proc(n) local resul, a793, dvs, i, k ; resul := 0: a793 := A000793(n) ; dvs := numtheory[divisors](a793) ; for k from 1 to nops(dvs) do i := op(k, dvs) ; resul := resul+numtheory[mobius](a793/i)*taylInv(i, n) ; od : RETURN(resul) ; end proc: # R. J. Mathar, Mar 30 2007

MATHEMATICA

b[n_, i_] := b[n, i] = Module[{p}, p = If[i < 1, 1, Prime[i]]; If[n == 0 || i < 1, 1, Max[b[n, i-1], Table[p^j*b[n-p^j, i-1], {j, 1, Log[p, n] // Floor}]]]];

g[n_] := g[n] = b[n, If[n < 8, 3, PrimePi[Ceiling[1.328*Sqrt[n*Log[n] // Floor]]]]];

a[n_] := a[n] = SeriesCoefficient[Sum[MoebiusMu[g[n]/i]/Product[1-x^j, {j, Divisors[i]}], {i, Divisors[g[n]]}] + O[x]^(n+1), n];

Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 0, 100}] (* Jean-François Alcover, Apr 25 2017, after Alois P. Heinz *)

CROSSREFS

Cf. A000793, A074859, A256067, A256554.

KEYWORD

easy,nonn

AUTHOR

Vladeta Jovovic, Sep 15 2002

EXTENSIONS

More terms from R. J. Mathar, Mar 30 2007

More terms from Sean A. Irvine, Oct 04 2011

More terms from Alois P. Heinz, Mar 29 2015

STATUS

approved

A258470

Number of partitions of n of order 10.

+0
2

0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 4, 6, 7, 9, 11, 13, 16, 18, 21, 25, 28, 33, 36, 41, 46, 51, 57, 62, 68, 76, 82, 91, 97, 106, 115, 124, 134, 143, 153, 166, 176, 190, 200, 214, 228, 242, 257, 271, 286, 305, 320, 340, 355, 375, 395, 415, 436, 456, 477, 503, 524

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,10

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

FORMULA

G.f.: -(x^10-x^3-1)*x^7/((x-1)*(x^2-1)*(x^5-1)*(x^10-1)).

CROSSREFS

Column k=10 of A256067.

KEYWORD

nonn

AUTHOR

Alois P. Heinz, May 31 2015

STATUS

approved

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