A245732 -id:A245732 - cp's OEIS frontend

A032032 Number of ways to partition n labeled elements into sets of sizes of at least 2 and order the sets.

1, 0, 1, 1, 7, 21, 141, 743, 5699, 42241, 382153, 3586155, 38075247, 428102117, 5257446533, 68571316063, 959218642651, 14208251423433, 223310418094785, 3699854395380371, 64579372322979335, 1182959813115161773, 22708472725269799933, 455643187943171348103

Offset: 0

Christian G. Bower

nonn

From Dennis P. Walsh, Apr 15 2013: (Start)
With m = floor(n/2), a(n) is the number of ways to distribute n different toys to m numbered children such that each child receiving a toy gets at least two toys and, if child k gets no toys, then each child numbered higher than k also gets no toys.
a(n) = sum of n-th row of triangle A200091 for n >= 2. (End)

			For n=5, a(5)=21 since there are 21 toy distributions satisfying the conditions above. Denoting a distribution by |kid_1 toys|kid_2 toys|, we have the distributions
  |t1,t2,t3,t4,t5|_|, |t1,t2,t3|t4,t5|, |t1,t2,t4|t3,t5|, |t1,t2,t5|t3,t4|, |t1,t3,t4|t2,t5|, |t1,t3,t5|t2,t4|, |t1,t4,t5|t2,t3|, |t2,t3,t4|t1,t5|, |t2,t3,t5|t1,t4|, |t2,t4,t5|t1,t3|, |t3,t4,t5|t1,t2|, |t1,t2|t3,t4,t5|, |t1,t3|t2,t4,t5|, |t1,t4|t2,t3,t5|, |t1,t5|t2,t3,t4|, |t2,t3|t1,t4,t5|, |t2,t4|t1,t3,t5|, |t2,t5|t1,t3,t4|, |t3,t4|t1,t2,t5|, |t3,t5|t1,t2,t4|, and |t4,t5|,t1,t2,t3|. - _Dennis P. Walsh_, Apr 15 2013

Alois P. Heinz, Table of n, a(n) for n = 0..400
C. G. Bower, Transforms (2).
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see p. 245.
I. Mezo, Periodicity of the last digits of some combinatorial sequences, arXiv preprint arXiv:1308.1637 [math.CO], 2013 and J. Int. Seq. 17 (2014) #14.1.1.
Robert A. Proctor, Let's Expand Rota's Twelvefold Way For Counting Partitions!, arXiv:math/0606404 [math.CO], 2006-2007.
Index entries for related partition-counting sequences

Cf. A102233, A232475.
Cf. column k=2 of A245732.
Cf. A200091.

spec := [ B, {B=Sequence(Set(Z,card>1))}, labeled ]; [seq(combstruct[count](spec, size=n), n=0..30)];
# second Maple program:
b:= proc(n) b(n):= `if`(n=0, 1, add(b(n-j)/j!, j=2..n)) end:
a:= n-> n!*b(n):
seq(a(n), n=0..25);  # Alois P. Heinz, Jul 29 2014

a[n_] := n! * Sum[ Binomial[k, j] * StirlingS2[n-k+j, j]*j! / (n-k+j)! * (-1)^(k-j), {k, 1, n}, {j, 0, k}]; a[0] = 1; Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Sep 05 2012, from given formula *)

x='x+O('x^66); Vec(serlaplace( 1/(2+x-exp(x)) ) ) \\ Joerg Arndt, Apr 16 2013

"AIJ" (ordered, indistinct, labeled) transform of 0, 1, 1, 1...
E.g.f.: 1/(2+x-exp(x)).
a(n) = n! * Sum_{k=1..n} Sum_{j=0..k} C(k,j) * Stirling2(n-k+j,j) * (j!/(n-k+j)!) *(-1)^(k-j); a(0)=1. - Vladimir Kruchinin, Feb 01 2011
a(n) ~ n! / ((r-1)*(r-2)^(n+1)), where r = -LambertW(-1,-exp(-2)) = 3.14619322062... - Vaclav Kotesovec, Oct 08 2013
a(0) = 1; a(n) = Sum_{k=2..n} binomial(n,k) * a(n-k). - Ilya Gutkovskiy, Feb 09 2020
a(n) = Sum_{s in S_n^0} Product_{i=1..n} binomial(i,s(i)-1), where s ranges over the set S_n^0 of derangements of [n], i.e., the permutations of [n] without fixed points. - Jose A. Rodriguez, Feb 02 2021

A245733 Number T(n,k) of endofunctions on [n] such that at least one preimage with cardinality k exists and, if j is the largest value with a nonempty preimage, the preimage cardinality of i is >=k for all i<=j and equal to k for at least one i<=j; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 14, 12, 0, 1, 181, 68, 6, 0, 1, 2584, 520, 20, 0, 0, 1, 41973, 4542, 120, 20, 0, 0, 1, 776250, 46550, 672, 70, 0, 0, 0, 1, 16231381, 540136, 5516, 112, 70, 0, 0, 0, 1, 380333228, 7045020, 40140, 1848, 252, 0, 0, 0, 0, 1

Offset: 0

Alois P. Heinz, Jul 30 2014

T(0,0) = 1 by convention.

			T(2,0) = 1: (2,2).
T(2,1) = 2: (1,2), (2,1).
T(2,2) = 1: (1,1).
T(3,1) = 12: (1,1,2), (1,2,1), (1,2,2), (1,2,3), (1,3,2), (2,1,1), (2,1,2), (2,1,3), (2,2,1), (2,3,1), (3,1,2), (3,2,1).
T(3,3) = 1: (1,1,1).
Triangle T(n,k) begins:
0 :         1;
1 :         0,      1;
2 :         1,      2,    1;
3 :        14,     12,    0,   1;
4 :       181,     68,    6,   0,  1;
5 :      2584,    520,   20,   0,  0, 1;
6 :     41973,   4542,  120,  20,  0, 0, 1;
7 :    776250,  46550,  672,  70,  0, 0, 0, 1;
8 :  16231381, 540136, 5516, 112, 70, 0, 0, 0, 1;
     ...

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

Columns k=0-10 give: A133286 (for n>0), A245854, A245855, A245856, A245857, A245858, A245859, A245860, A245861, A245862, A245863.
Row sums give A000312.
T(2n,n) gives A000984(n).
Cf. A245732.

b:= proc(n, k) option remember; `if`(n=0, 1,
      add(b(n-j, k)*binomial(n, j), j=k..n))
    end:
g:= (n, k)-> `if`(k=0, n^n, `if`(n=0, 0, b(n, k))):
T:= (n, k)-> g(n, k) -g(n, k+1):
seq(seq(T(n, k), k=0..n), n=0..12);

b[n_, k_] := b[n, k] = If[n == 0, 1, Sum[b[n-j, k]*Binomial[n, j], {j, k, n}]]; g[n_, k_] := If[k == 0, n^n, If[n == 0, 0, b[n, k]]]; T[n_, k_] := g[n, k] - g[n, k+1]; T[0, 0] = 1; Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Jan 27 2015, after Alois P. Heinz *)

E.g.f. of column k=0: 1 +1/(1+LambertW(-x)) -1/(2-exp(x)); e.g.f. of column k>0: 1/(1-Sum_{j>=k} x^j/j!) - 1/(1-Sum_{j>=k+1} x^j/j!).

T(n,k) = A245732(n,k) - A245732(n,k+1).

A102233 Number of preferential arrangements of n labeled elements when at least k=3 elements per rank are required.

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 21, 71, 183, 2101, 13513, 64285, 629949, 5762615, 41992107, 427215283, 4789958371, 47283346849, 540921904725, 6980052633257, 85901272312905, 1129338979629643, 16398293425501375, 238339738265039119, 3588600147767147775, 58124879519314730741

Offset: 0

Thomas Wieder, Jan 01 2005

nonn

The labeled case for at least k=2 elements per rank is given by A032032 = Partition n labeled elements into sets of sizes of at least 2 and order the sets. The unlabeled case for at least k=3 elements per rank is given by A000930 = A Lamé sequence of higher order. The unlabeled case for at least k=2 elements per rank is given by A000045 = Fibonacci numbers.
With m = floor(n/3), a(n) is the number of ways to distribute n different toys to m numbered children such that each child receiving a toy gets at least three toys and, if child k gets no toys, then each child numbered higher than k also gets no toys. Furthermore, a(n)= row sums of triangle A200092 for n>=3. - Dennis P. Walsh, Apr 15 2013
Row sums of triangle A200092. - Dennis P. Walsh, Apr 15 2013

			Let 1,2,3,4,5,6 denote six labeled elements. Let | denote a separation between two ranks. E.g., if elements 1,2 and 3 are on rank (also called level) one and elements 3,4 and 5 are on rank two, then we have the ranking 123|456.
For n=9 we have a(9)=2101 rankings. The order within a rank does not count. Six examples are: 123|456|789; 123456789; 12345|6789; 129|345678; 1235|46789; 789|123456.

Alois P. Heinz, Table of n, a(n) for n = 0..400
Vladimir Kruchinin, D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.
I. Mezo, Periodicity of the last digits of some combinatorial sequences, arXiv preprint arXiv:1308.1637 [math.CO], 2013 and J. Int. Seq. 17 (2014) #14.1.1.

Cf. A000640, A102232, A032032, A232475.

Cf. column k=3 of A245732.

seq (n! *coeff (series (1- (z^2-2*exp(z)+2+2*z) /(4-2*exp(z)+2*z+z^2), z=0, n+1), z, n), n=0..30);
with(combstruct): SeqSetL := [S, {S=Sequence(U), U=Set(Z, card >= 3)}, labeled]: seq(count(SeqSetL, size=j), j=0..23); # Zerinvary Lajos, Oct 19 2006
# third Maple program:
b:= proc(n) b(n):= `if`(n=0, 1, add(b(n-j)/j!, j=3..n)) end:
a:= n-> n!*b(n):
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 29 2014

CoefficientList[Series[1-(x^2-2*E^x+2+2*x)/(4-2*E^x+2*x+x^2), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 29 2013 *)
b[n_] := b[n] = If[n==0, 1, Sum[b[n-j]/j!, {j, 3, n}]]; a[n_] := n!*b[n]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jan 31 2016, after Alois P. Heinz *)

z='z+O('z^66); Vec(serlaplace( 1-(z^2-2*exp(z)+2+2*z) / (4-2*exp(z)+2*z+z^2) ) ) \\ Joerg Arndt, Apr 16 2013

E.g.f.: 1-(z^2-2*exp(z)+2+2*z)/(4-2*exp(z)+2*z+z^2).
a(n) = n! * sum(m=1..n, sum(k=0..m, k!*(-1)^(m-k) *binomial(m,k) *sum(i=0..n-m, stirling2(i+k,k) *binomial(m-k,n-m-i) *2^(-n+m+i) /(i+k)!))); a(0)=1. - Vladimir Kruchinin, Feb 01 2011
a(n) ~ 2*n!/((2+r^2)*r^(n+1)), where r = 1.56811999239... is the root of the equation 4+2*r+r^2 = 2*exp(r). - Vaclav Kotesovec, Sep 29 2013
a(0) = 1; a(n) = Sum_{k=3..n} binomial(n,k) * a(n-k). - Ilya Gutkovskiy, Feb 09 2020
E.g.f.: 1/(2 + x + x^2/2 - exp(x)). - Christian Sievers, Oct 27 2024

a(0) changed to 1 at the suggestion of Zerinvary Lajos, Oct 26 2006

A232475 Number of preferential arrangements of n labeled elements when at least k=4 elements per rank are required.

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 1, 1, 71, 253, 673, 1585, 38149, 277707, 1402831, 5923503, 85577571, 937629969, 7475614341, 48939413477, 587610659505, 7906296686903, 87384175023995, 804959532778571, 9729015122635103, 144711323234918941, 2009073351016603121

Offset: 0

N. J. A. Sloane, Nov 27 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..400
I. Mezo, Periodicity of the last digits of some combinatorial sequences, arXiv preprint arXiv:1308.1637 [math.CO], 2013.

Cf. A032032, A102233.

Cf. column k=4 of A245732.

b:= proc(n) b(n):= `if`(n=0, 1, add(b(n-j)/j!, j=4..n)) end:
a:= n-> n!*b(n):
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 29 2014

CoefficientList[Series[1/(2 + x - E^x + x^2/2 + x^3/6),{x,0,20}],x]*Range[0,20]! (* Vaclav Kotesovec, Aug 02 2014 *)

E.g.f.: 1/(2 + x - exp(x) + x^2/2 + x^3/6). - Vaclav Kotesovec, Aug 02 2014
a(n) ~ n! / ((1+r^3/6) * r^(n+1)), where r = 1.97615974210650519398... is the root of the equation 2 + r - exp(r) + r^2/2 + r^3/6 = 0. - Vaclav Kotesovec, Aug 02 2014
a(0) = 1; a(n) = Sum_{k=4..n} binomial(n,k) * a(n-k). - Ilya Gutkovskiy, Feb 09 2020

More terms from Alois P. Heinz, Jul 29 2014

A244174 Number of compositions of 3n in which the minimal multiplicity of parts equals n.

Original entry on oeis.org

1, 3, 7, 21, 71, 253, 925, 3433, 12871, 48621, 184757, 705433, 2704157, 10400601, 40116601, 155117521, 601080391, 2333606221, 9075135301, 35345263801, 137846528821, 538257874441, 2104098963721, 8233430727601, 32247603683101, 126410606437753, 495918532948105

Offset: 0

Alois P. Heinz, Jun 21 2014

nonn

			a(2) = 7: [1,1,2,2], [1,2,1,2], [1,2,2,1], [2,1,1,2], [2,1,2,1], [2,2,1,1], [3,3].

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

Cf. A000984, A007318, A065567, A242451, A245732.

a:= proc(n) option remember;
      `if`(n<3, 2^(n+1)-1, ((15*n^2-31*n+12) *a(n-1)
       -2*(3*n-2)*(2*n-3) *a(n-2)) / ((3*n-5)*n))
    end:
seq(a(n), n=0..30);

a[n_] := a[n] = If[n < 3, 2^(n+1) - 1, ((15*n^2 - 31*n + 12)*a[n-1] - 2*(3*n - 2)*(2*n - 3)*a[n-2])/((3*n - 5)*n)]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Nov 07 2014, after Alois P. Heinz *)

A244174 = lambda m: SetPartitions(2*m,[2*m]).cardinality()+2*SetPartitions(2*m,[m,m]).cardinality()
[1] + [A244174(m) for m in (1..26)] # Peter Luschny, Aug 02 2015

a(n) = A242451(3n,n).
Recurrence: see Maple program.
For n>0, a(n) = 1 + C(2n,n) = 1 + A000984(n). - Vaclav Kotesovec, Jun 21 2014
G.f.: 1/(sqrt(1-4*x)) + x/(1-x). - Alois P. Heinz, Jun 22 2014
a(n) = A245732(2n,n). - Alois P. Heinz, Jul 30 2014
a(n) = A065567(2n,n) for n>=1. - Alois P. Heinz, Sep 05 2023

A245790 Number of preferential arrangements of n labeled elements when at least k=5 elements per rank are required.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 253, 925, 2509, 6007, 13443, 785643, 6114551, 31980469, 138704361, 539262713, 13685913105, 170996304653, 1442111683785, 9802624250281, 58233700998845, 939069565583991, 15109164547164171, 181402703206632211, 1758154702415920051

Offset: 0

Alois P. Heinz, Aug 01 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..400

Cf. column k=5 of A245732.

a:= proc(n) option remember; `if`(n=0, 1,
       add(a(n-j)*binomial(n, j), j=5..n))
    end:
seq(a(n), n=0..30);

CoefficientList[Series[1/(2 + x - E^x + x^2/2! + x^3/3! + x^4/4!),{x,0,30}],x]*Range[0,30]! (* Vaclav Kotesovec, Aug 02 2014 *)

E.g.f.: 1/(2 + x - exp(x) + x^2/2! + x^3/3! + x^4/4!). - Vaclav Kotesovec, Aug 02 2014

a(n) ~ n! / ((1+r^4/4!) * r^(n+1)), where r = 2.376178375424367122... is the root of the equation 2 + r - exp(r) + r^2/2! + r^3/3! + r^4/4! = 0. - Vaclav Kotesovec, Aug 02 2014

A245791 Number of preferential arrangements of n labeled elements when at least k=6 elements per rank are required.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 925, 3433, 9439, 22881, 51767, 112269, 17390049, 140166497, 749266977, 3311021321, 13091222301, 48138992687, 2477067794573, 33549609515571, 292811657874791, 2040445353211231, 12382874543793451, 68436110449556971

Offset: 0

Alois P. Heinz, Aug 01 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..400

Cf. column k=6 of A245732.

a:= proc(n) option remember; `if`(n=0, 1,
       add(a(n-j)*binomial(n, j), j=6..n))
    end:
seq(a(n), n=0..35);

CoefficientList[Series[1/(2 + x - E^x + x^2/2! + x^3/3! + x^4/4! + x^5/5!),{x,0,30}],x]*Range[0,30]! (* Vaclav Kotesovec, Aug 02 2014 *)

E.g.f.: 1/(2 + x - exp(x) + x^2/2! + x^3/3! + x^4/4! + x^5/5!). - Vaclav Kotesovec, Aug 02 2014

a(n) ~ n! / ((1+r^5/5!) * r^(n+1)), where r = 2.77092853312194416389... is the root of the equation 2 + r - exp(r) + r^2/2! + r^3/3! + r^4/4! + r^5/5! = 0. - Vaclav Kotesovec, Aug 02 2014

A245792 Number of preferential arrangements of n labeled elements when at least k=7 elements per rank are required.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 3433, 12871, 35751, 87517, 199785, 436697, 927657, 401005793, 3296326113, 17887397621, 80157730101, 321127444171, 1195366208091, 4226755326331, 486914893507831, 6899197122043711, 61532746814157691, 436349292456987871

Offset: 0

Alois P. Heinz, Aug 01 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..400

Cf. column k=7 of A245732.

a:= proc(n) option remember; `if`(n=0, 1,
       add(a(n-j)*binomial(n, j), j=7..n))
    end:
seq(a(n), n=0..35);

CoefficientList[Series[1/(2 + x - E^x + x^2/2! + x^3/3! + x^4/4! + x^5/5! + x^6/6!),{x,0,40}],x]*Range[0,40]! (* Vaclav Kotesovec, Aug 02 2014 *)

E.g.f.: 1/(2 + x - exp(x) + x^2/2! + x^3/3! + x^4/4! + x^5/5! + x^6/6!). - Vaclav Kotesovec, Aug 02 2014

a(n) ~ n! / ((1+r^6/6!) * r^(n+1)), where r = 3.161936258680679649... is the root of the equation 2 + r - exp(r) + r^2/2! + r^3/3! + r^4/4! + r^5/5! + r^6/6! = 0. - Vaclav Kotesovec, Aug 02 2014

A245793 Number of preferential arrangements of n labeled elements when at least k=8 elements per rank are required.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 12871, 48621, 136137, 335921, 772617, 1700273, 3633105, 7607297, 9481216677, 78911366771, 433024685291, 1961914734031, 7943932891111, 29871106149031, 106624217245891, 366332387265871, 100783979161693411

Offset: 0

Alois P. Heinz, Aug 01 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..400

Cf. column k=8 of A245732.

a:= proc(n) option remember; `if`(n=0, 1,
       add(a(n-j)*binomial(n, j), j=8..n))
    end:
seq(a(n), n=0..35);

CoefficientList[Series[1/(2 + x - E^x + x^2/2! + x^3/3! + x^4/4! + x^5/5! + x^6/6! + x^7/7!),{x,0,40}],x]*Range[0,40]! (* Vaclav Kotesovec, Aug 02 2014 *)

E.g.f.: 1/(2 + x - exp(x) + x^2/2! + x^3/3! + x^4/4! + x^5/5! + x^6/6! + x^7/7!). - Vaclav Kotesovec, Aug 02 2014

a(n) ~ n! / ((1+r^7/7!) * r^(n+1)), where r = 3.550140591759854453327299... is the root of the equation 2 + r - exp(r) + r^2/2! + r^3/3! + r^4/4! + r^5/5! + r^6/6! + r^7/7! = 0. - Vaclav Kotesovec, Aug 02 2014

A245794 Number of preferential arrangements of n labeled elements when at least k=9 elements per rank are required.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 48621, 184757, 520677, 1293293, 2993565, 6626669, 14233965, 29938871, 62040891, 228000637831, 1914395677411, 10597881432571, 48446119334191, 197900630004571, 750527665784311, 2700730064112181

Offset: 0

Alois P. Heinz, Aug 01 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..400

Cf. column k=9 of A245732.

a:= proc(n) option remember; `if`(n=0, 1,
       add(a(n-j)*binomial(n, j), j=9..n))
    end:
seq(a(n), n=0..40);

CoefficientList[Series[1/(2 + x - E^x + x^2/2! + x^3/3! + x^4/4! + x^5/5! + x^6/6! + x^7/7! + x^8/8!),{x,0,40}],x]*Range[0,40]! (* Vaclav Kotesovec, Aug 02 2014 *)

E.g.f.: 1/(2 + x - exp(x) + x^2/2! + x^3/3! + x^4/4! + x^5/5! + x^6/6! + x^7/7! + x^8/8!). - Vaclav Kotesovec, Aug 02 2014

a(n) ~ n! / ((1+r^8/8!) * r^(n+1)), where r = 3.93616250913523371282009... is the root of the equation 2 + r - exp(r) + r^2/2! + r^3/3! + r^4/4! + r^5/5! + r^6/6! + r^7/7! + r^8/8! = 0. - Vaclav Kotesovec, Aug 02 2014

cp's OEIS Frontend

A032032 Number of ways to partition n labeled elements into sets of sizes of at least 2 and order the sets.

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A245733 Number T(n,k) of endofunctions on [n] such that at least one preimage with cardinality k exists and, if j is the largest value with a nonempty preimage, the preimage cardinality of i is >=k for all i<=j and equal to k for at least one i<=j; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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Maple

Mathematica

Formula

A102233 Number of preferential arrangements of n labeled elements when at least k=3 elements per rank are required.

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Maple

Mathematica

PARI

Formula

Extensions

A232475 Number of preferential arrangements of n labeled elements when at least k=4 elements per rank are required.

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Maple

Mathematica

Formula

Extensions

A244174 Number of compositions of 3n in which the minimal multiplicity of parts equals n.

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Maple

Mathematica

Sage

Formula

A245790 Number of preferential arrangements of n labeled elements when at least k=5 elements per rank are required.

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Maple

Mathematica

Formula

A245791 Number of preferential arrangements of n labeled elements when at least k=6 elements per rank are required.

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