A023998 -id:A023998 - cp's OEIS frontend

A051921 Duplicate of A023998.

1, 1, 3, 16, 131, 1496, 22482, 426833, 9934563, 277006192, 9085194458

Offset: 0

dead

A275043 Number A(n,k) of set partitions of [k*n] such that within each block the numbers of elements from all residue classes modulo k are equal for k>0, A(n,0)=1; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 5, 1, 1, 1, 5, 16, 15, 1, 1, 1, 9, 64, 131, 52, 1, 1, 1, 17, 298, 1613, 1496, 203, 1, 1, 1, 33, 1540, 25097, 69026, 22482, 877, 1, 1, 1, 65, 8506, 461105, 4383626, 4566992, 426833, 4140, 1, 1, 1, 129, 48844, 9483041, 350813126, 1394519922, 437665649, 9934563, 21147, 1

Offset: 0

Alois P. Heinz, Jul 14 2016

			A(2,2) = 3: 1234, 12|34, 14|23.
A(2,3) = 5: 123456, 123|456, 126|345, 135|246, 156|234.
A(2,4) = 9: 12345678, 1234|5678, 1238|4567, 1247|3568, 1278|3456, 1346|2578, 1368|2457, 1467|2358, 1678|2345.
A(3,2) = 16: 123456, 1234|56, 1236|45, 1245|36, 1256|34, 12|3456, 12|34|56, 12|36|45, 1346|25, 1456|23, 14|2356, 14|23|56, 16|2345, 16|23|45, 14|25|36, 16|25|34.
Square array A(n,k) begins:
  1,   1,     1,       1,          1,            1,               1, ...
  1,   1,     1,       1,          1,            1,               1, ...
  1,   2,     3,       5,          9,           17,              33, ...
  1,   5,    16,      64,        298,         1540,            8506, ...
  1,  15,   131,    1613,      25097,       461105,         9483041, ...
  1,  52,  1496,   69026,    4383626,    350813126,     33056715626, ...
  1, 203, 22482, 4566992, 1394519922, 573843627152, 293327384637282, ...

Alois P. Heinz, Antidiagonals n = 0..60, flattened
J.-M. Sixdeniers, K. A. Penson and A. I. Solomon, Extended Bell and Stirling Numbers From Hypergeometric Exponentiation, J. Integer Seqs. Vol. 4 (2001), #01.1.4.
Wikipedia, Partition of a set

Columns k=0-10 give: A000012, A000110, A023998, A061684, A061685, A061686, A061687, A061688, A275097, A275098, A275099.
Rows n=0+1,2-5 give: A000012, A094373, A275100, A275101, A275102.
Main diagonal gives A275044.
Cf. A345400.

A:= proc(n, k) option remember; `if`(k*n=0, 1, add(
       binomial(n, j)^k*(n-j)*A(j, k), j=0..n-1)/n)
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);

A[n_, k_] := A[n, k] = If[k*n == 0, 1, Sum[Binomial[n, j]^k*(n-j)*A[j, k], {j, 0, n-1}]/n]; Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Jan 17 2017, translated from Maple *)

A023997 Number of block permutations on an n-set.

Original entry on oeis.org

1, 1, 3, 25, 339, 6721, 179643, 6166105, 262308819, 13471274401, 818288740923, 57836113793305, 4693153430067699, 432360767273547841, 44794795522199781243, 5176959027946049635225, 662704551840482536170579

Offset: 0

Des FitzGerald (D.FitzGerald(AT)utas.edu.au)

A block permutation of a set X is a bijection between two quotient sets of X (of necessarily equal rank).

Number of labeled partitions of (n,n) into pairs (i,j) where there are n black objects labeled 1..n and n white objects labeled 1..n. Each partition must have at least one black object and at least one white object. - Christian G. Bower, Jun 03 2005

			For n=3, there are the 3! ordinary permutations (of rank 3), 18 block permutations of rank 2 (2! for each pair of partitions of rank 2) and the single rank 1 one.

Vaclav Kotesovec, Table of n, a(n) for n = 0..288 (terms 0..45 from Vincenzo Librandi)
Zhanar Berikkyzy, Pamela E. Harris, Anna Pun, Catherine Yan, and Chenchen Zhao, Combinatorial Identities for Vacillating Tableaux, arXiv:2308.14183 [math.CO], 2023. See p. 22.
D. G. FitzGerald and Jonathan Leech, Dual symmetric inverse monoids and representation theory, J. Australian Mathematical Society (Series A), Vol. 64 (1998), pp. 345-367.

Cf. A023998, A002720, A014235, A111420.

Table[Sum[StirlingS2[n,k]^2k!,{k,0,n}],{n,0,100}] (* Emanuele Munarini, Jul 04 2011 *)

makelist(sum(stirling2(n,k)^2*k!,k,0,n),n,0,24); /* Emanuele Munarini, Jul 04 2011 */

a(n) = if (n==0, 1, sum(k=1, n, k!*stirling(n, k, 2)^2)); \\ Michel Marcus, Jun 18 2019

a(0)=1, a(n) = Sum_{k=1..n} k! * S2(n,k)^2, S2(n,k) are the Stirling numbers of the second kind.

More terms from Christian G. Bower, Jun 03 2005

A061691 Triangle of generalized Stirling numbers.

Original entry on oeis.org

1, 1, 2, 1, 9, 6, 1, 34, 72, 24, 1, 125, 650, 600, 120, 1, 461, 5400, 10500, 5400, 720, 1, 1715, 43757, 161700, 161700, 52920, 5040, 1, 6434, 353192, 2361016, 4116000, 2493120, 564480, 40320, 1, 24309, 2862330, 33731208, 96960024, 97161120, 39372480, 6531840, 362880

Offset: 1

N. J. A. Sloane, Jun 18 2001

The Eulerian-type number triangle associated with this triangle of generalized Stirling numbers is A192721. The table entry T(n,k) gives the number of uniform block permutations of the set {1,2,...,n} partitioned into k blocks. An example is given below. T(n,k) also gives the number of games of simple patience with n cards resulting in k piles (adapt Algorithm 1.1.22 of Lankham). [Peter Bala, Jul 14 2011]

			Triangle begins:
  1;
  1,2;
  1,9,6;
  1,34,72,24;
  1,125,650,600,120;
  ...
T(4,2) = 34:
There are 7 partitions of the set {1,2,3,4} into 2 blocks. The four partitions {1,2,3}{4}, {1,2,4}{3}, {1,3,4}{2} and {2,3,4}{1} give rise to 4*4 = 16 uniform block permutations while the remaining 3 partitions {1,2}{3,4}, {1,3}{2,4} and {1,4}{2,3} give 2!*3*3 = 18 uniform block permutations : thus in total there are 16+18 = 34 block permutations between the set partitions of {1,2,3,4} into 2 blocks.

Alois P. Heinz, Rows n = 1..141, flattened
M. Aguiar and R. C. Orellana, The Hopf algebra of uniform block permutations, 17th International Conference on Formal Power Series and Algebraic Combinatorics, Taormina, July 2005.
D. Aldous and P. Diaconis, Longest increasing subsequences: from patience sorting to the Baik-Deift-Johansson theorem, Bull. Amer. Math. Soc. 36 (1999), 413-432.
D. G. Fitzgerald, A presentation for the monoid of uniform block permutations, Bull. Austral. Math. Soc. 68 (2003), 317-324.
A. T. Irish, F. Quitin, U. Madhow, and M. Rodwell, Achieving multiple degrees of freedom in long-range mm-wave MIMO channels using randomly distributed relays, 2014.
I. P. Lankham, Patience Sorting and Its Generalizations, arXiv:0705.4524 [math.CO], 2007.
J.-M. Sixdeniers, K. A. Penson and A. I. Solomon, Extended Bell and Stirling Numbers From Hypergeometric Exponentiation, J. Integer Seqs. Vol. 4 (2001), #01.1.4.

Diagonals give A010763, A061690, A000142, A001809, A061689. Cf. A061692. A023998 (row sums), A192721, A192722.

#A061691
#J = sum {n>=0} z^n/n!^2
J := BesselJ(0, 2*i*sqrt(z)):
G := exp(x*(J(z)-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 = 1..n) od;
# yields sequence in triangular form
# second Maple program:
b:= proc(n) option remember; expand(`if`(n=0, 1,
      add(x*b(n-i)*binomial(n, i)/i!, i=1..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i)/i!, i=1..n))(b(n)*n!):
seq(T(n), n=1..12);  # Alois P. Heinz, Sep 10 2019

max = 9; g := Exp[x*(BesselI[0, 2*Sqrt[z]] - 1)]; gser = Series[g, {z, 0, max}, {x, 0, max}]; t[n_, k_] := n!^2*SeriesCoefficient[ gser // Normal, {z, 0, n}, {x, 0, k}]; Flatten[ Table[ t[n, k], {n, 1, max}, {k, 1, n}]] (* Jean-François Alcover, Apr 04 2012, after Maple *)

T(n, k) = 1/k!*Sum multinomial(n, n_1, n_2, ..n_k)^2, where the sum extends over all compositions (n_1, n_2, .., n_k) of n into exactly k nonnegative parts. - Vladeta Jovovic, Apr 23 2003
From Peter Bala, Jul 14 2011: (Start)
The table entry T(n,k) may also be expressed as a sum over (unordered) partitions of n into k parts:
T(n,k) = sum {partitions m_1*1+...+m_n*n = n, m_1+...+m_n = k} 1/(m_1!*...*m_n!)*{n!/(1!^(m_1)*...*n!^(m_n))}^2.
Generating function:
Let J(z) = sum {n>=0} z^n/n!^2. Then
exp(x*(J(z)-1)) = 1 + x*z + (x + 2*x^2)*z^2/2!^2 + (x + 9*x^2 + 6*x^3)*z^3/3!^2 + ....
Relations with other sequences:
T(n,k) = 1/k!*A192722(n,k).
Row sums [1,3,16,131,...] = A023998. (End)
The row polynomials R(n,x) satisfy the recurrence equation R(n,x) = x*( sum {k = 0..n-1} binomial(n,k)*binomial(n-1,k)*R(k,x) ) with R(0,x) = 1. Also R(n,x + y) = sum {k = 0..n} binomial(n,k)^2*R(k,x)*R(n-k,y). - Peter Bala, Sep 17 2013

More terms from Vladeta Jovovic, Apr 23 2003

A322670 Number T(n,k) of colored set partitions of [n] where colors of the elements of subsets are distinct and in increasing order and exactly k colors are used; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 3, 0, 1, 12, 16, 0, 1, 41, 156, 131, 0, 1, 140, 1155, 2460, 1496, 0, 1, 497, 8020, 32600, 47355, 22482, 0, 1, 1848, 55629, 385420, 1004360, 1098678, 426833, 0, 1, 7191, 394884, 4396189, 18304510, 34625304, 30259712, 9934563

Offset: 0

Alois P. Heinz, Aug 29 2019

			T(3,2) = 12: 1a|2a3b, 1b|2a3b, 1a3b|2a, 1a3b|2b, 1a2b|3a, 1a2b|3b, 1a|2a|3b, 1a|2b|3a, 1b|2a|3a, 1a|2b|3b, 1b|2a|3b, 1b|2b|3a.
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1,    3;
  0, 1,   12,    16;
  0, 1,   41,   156,    131;
  0, 1,  140,  1155,   2460,    1496;
  0, 1,  497,  8020,  32600,   47355,   22482;
  0, 1, 1848, 55629, 385420, 1004360, 1098678, 426833;
  ...

Alois P. Heinz, Rows n = 0..140, flattened
Wikipedia, Partition of a set

Columns k=0-2 give: A000007, A057427, A325482.
Main diagonal gives A023998.
Row sums give A325478.
T(2n,n) gives A325481.
Cf. A321296, A323128, A325930.

A:= proc(n, k) option remember; `if`(n=0, 1, add(A(n-j, k)*
      binomial(n-1, j-1)*binomial(k, j), j=1..min(k, n)))
    end:
T:= (n, k)-> add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..10);

A[n_, k_] := A[n, k] = If[n == 0, 1, Sum[A[n-j, k] Binomial[n-1, j-1]* Binomial[k, j], {j, 1, Min[k, n]}]];
T[n_, k_] := Sum[A[n, k-i] (-1)^i Binomial[k, i], {i, 0, k}];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 30 2020, after Alois P. Heinz *)

Sum_{k=1..n} k * T(n,k) = A325930(n).

A193161 E.g.f. A(x) satisfies: A(x/(1-x))/(1-x) = d/dx x*A(x).

Original entry on oeis.org

1, 1, 3, 17, 152, 1944, 33404, 738212, 20316288, 679237248, 27050017152, 1262790237312, 68193683598336, 4212508572109824, 294822473048043264, 23184842446161993984, 2033884583922970558464, 197767395237549512097792, 21194678534706844531458048

Offset: 0

Paul D. Hanna, Jul 16 2011

nonn

In Cellarosi and Sinai (2011) on page 257, m_k denotes a(k)/k!. - Michael Somos, Dec 28 2012

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 17*x^3/3! + 152*x^4/4! + 1944*x^5/5! + ...
Related expansions:
A(x/(1-x))/(1-x) = 1 + 2*x + 9*x^2/2! + 68*x^3/3! + 760*x^4/4! + ...
A(x) + x*A'(x) = 1 + 2*x + 9*x^2/2! + 68*x^3/3! + 760*x^4/4! + ...
Also, a(n) appears in the expansion:
B(x) = 1 + x + 3*x^2/2!^2 + 17*x^3/3!^2 + 152*x^4/4!^2 + 1944*x^5/5!^2 + ...
where
log(B(x)) = x + x^2/(2*2!) + x^3/(3*3!) + x^4/(4*4!) + x^5/(5*5!) + ...

Alois P. Heinz, Table of n, a(n) for n = 0..293
F. Cellarosi and Ya. G. Sinai, The Möbius function and statistical mechanics, Bull. Math. Sci., 2011.

Cf. A023998, A193160, A209884.

b:= proc(n) option remember; `if`(n=0, 1,
      add(b(n-i)*binomial(n-1, i-1)/i, i=1..n))
    end:
a:= n-> b(n)*n!:
seq(a(n), n=0..25);  # Alois P. Heinz, May 11 2016

a[ n_] := If[ n<0, 0, n!^2 Assuming[ x>0, SeriesCoefficient[ Exp[ Integrate[ (Exp[t] - 1)/t, {t, 0, x}]], {x, 0, n}]]]; (* Michael Somos, Dec 28 2012 *)
a[ n_] := If[ n<0, 0, n!^2 Assuming[ x>0, SeriesCoefficient[ Exp[ ExpIntegralEi[x] - Log[x] - EulerGamma], {x, 0, n}]]]; (* Michael Somos, Dec 28 2012 *)
Table[Sum[BellY[n, k, 1/Range[n]], {k, 0, n}] n!, {n, 0, 20}] (* Vladimir Reshetnikov, Nov 09 2016 *)

{a(n)=local(A=1+x,B);for(i=1,n,B=subst(A,x,x/(1-x+x*O(x^n)))/(1-x);A=1+intformal((B-A)/x));n!*polcoeff(A,n)}

{a(n)=if(n<0,0,if(n==0,1,(n-1)!*sum(k=0,n-1,binomial(n,k)*a(k)/k!)))}

{a(n)=n!^2*polcoeff(exp(sum(m=1,n,x^m/(m*m!))+x*O(x^n)),n)}

a(n) = (n-1)!* Sum_{k=0..n-1} binomial(n,k)*a(k)/k! for n>0 with a(0)=1.
a(n) = A193160(n+1)/(n+1).
E.g.f.: exp( Sum_{n>=1} x^n/(n*n!) ) = Sum_{n>=0} a(n)*x^n/n!^2.
a(n) = n! * A177208(n) / A177209(n) for n>=1 (see comment from Michael Somos).

A321296 Number T(n,k) of colored set partitions of [n] where colors of the elements of subsets are in (weakly) increasing order and exactly k colors are used; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 2, 3, 0, 5, 20, 16, 0, 15, 122, 237, 131, 0, 52, 774, 2751, 3524, 1496, 0, 203, 5247, 30470, 68000, 65055, 22482, 0, 877, 38198, 341244, 1181900, 1913465, 1462320, 426833, 0, 4140, 298139, 3949806, 19946654, 48636035, 61692855, 39282229, 9934563

Offset: 0

Alois P. Heinz, Aug 29 2019

			T(3,2) = 20: 1a2a3b, 1a2b3b, 1a|2a3b, 1a|2b3b, 1b|2a3a, 1b|2a3b, 1a3b|2a, 1b3b|2a, 1a3a|2b, 1a3b|2b, 1a2b|3a, 1b2b|3a, 1a2a|3b, 1a2b|3b, 1a|2a|3b, 1a|2b|3a, 1b|2a|3a, 1a|2b|3b, 1b|2a|3b, 1b|2b|3a.
Triangle T(n,k) begins:
  1;
  0,   1;
  0,   2,     3;
  0,   5,    20,     16;
  0,  15,   122,    237,     131;
  0,  52,   774,   2751,    3524,    1496;
  0, 203,  5247,  30470,   68000,   65055,   22482;
  0, 877, 38198, 341244, 1181900, 1913465, 1462320, 426833;
  ...

Alois P. Heinz, Rows n = 0..140, flattened
Wikipedia, Partition of a set

Columns k=0-2 give: A000007, A000110 (for n>0), A325890.
Main diagonal gives A023998.
Row sums give A325888.
T(2n,n) gives A325889.
Cf. A322670.

A:= proc(n, k) option remember; `if`(n=0, 1, add(A(n-j, k)*
      binomial(n-1, j-1)*binomial(k+j-1, j), j=1..n))
    end:
T:= (n, k)-> add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..10);

A[n_, k_] := A[n, k] = If[n == 0, 1, Sum[A[n-j, k] Binomial[n-1, j-1]* Binomial[k + j - 1, j], {j, n}]];
T[n_, k_] := Sum[A[n, k - i] (-1)^i Binomial[k, i], {i, 0, k}];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 30 2020, after Alois P. Heinz *)

A336227 a(0) = 1; a(n) = n * Sum_{k=0..n-1} binomial(n-1,k)^2 * a(k).

Original entry on oeis.org

1, 1, 4, 27, 292, 4425, 89106, 2280901, 71928872, 2728450017, 122145511510, 6354868381521, 379376236939404, 25710543779239501, 1960001963705060926, 166753195643254805565, 15724259680648667902096, 1633462474351643785483457, 185931510605274506452763166

Offset: 0

Ilya Gutkovskiy, Jul 12 2020

nonn

Cf. A000248, A023998.

a[0] = 1; a[n_] := a[n] = n Sum[Binomial[n - 1, k]^2 a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 18}]
nmax = 18; CoefficientList[Series[Exp[Sqrt[x] BesselI[1, 2 Sqrt[x]]], {x, 0, nmax}], x] Range[0, nmax]!^2

a(n) = (n!)^2 * [x^n] exp(sqrt(x) * BesselI(1,2*sqrt(x))).

A061696 Generalized Bell numbers.

Original entry on oeis.org

1, 0, 1, 1, 19, 101, 1776, 23717, 515971, 11893597, 346475728, 11497161545, 444592761746, 19536147771219, 970739908493421, 54010183143383066, 3341831947578263267, 228462339968313577341, 17160142419913160027448, 1409008382280004776187961

Offset: 0

N. J. A. Sloane, Jun 19 2001

nonn

J.-M. Sixdeniers, K. A. Penson and A. I. Solomon, Extended Bell and Stirling Numbers From Hypergeometric Exponentiation, J. Integer Seqs. Vol. 4 (2001), #01.1.4.

Cf. A023998, A061697.

Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(BesselI(0,2*sqrt(x)) - 1 - x). - Ilya Gutkovskiy, Jul 12 2020

More terms from Ilya Gutkovskiy, Jul 12 2020

A336209 a(0) = 1; a(n) = -(1/n) * Sum_{k=0..n-1} binomial(n,k)^2 * (n-k) * a(k).

Original entry on oeis.org

1, -1, 1, 2, -15, -46, 880, 5837, -132783, -2109238, 35966256, 1440196097, -8909037720, -1504006716551, -16097564749643, 2021100230840147, 83130656159529937, -2475528081920694566, -331363460045748820376, -3874344448291316066455, 1255007424437046915956520

Offset: 0

Ilya Gutkovskiy, Jul 12 2020

sign

Cf. A000587, A023998, A336210.

a[0] = 1; a[n_] := a[n] = -(1/n) Sum[Binomial[n, k]^2 (n - k) a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 20}]
nmax = 20; CoefficientList[Series[Exp[-Sum[x^k/(k!)^2, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!^2

a(n) = (n!)^2 * [x^n] exp(-Sum_{k>=1} x^k / (k!)^2).

a(n) = (n!)^2 * [x^n] exp(1 - BesselI(0,2*sqrt(x))).

cp's OEIS Frontend

A051921 Duplicate of A023998.

Views

Author

Keywords

A275043 Number A(n,k) of set partitions of [k*n] such that within each block the numbers of elements from all residue classes modulo k are equal for k>0, A(n,0)=1; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A023997 Number of block permutations on an n-set.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Maxima

PARI

Formula

Extensions

A061691 Triangle of generalized Stirling numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A322670 Number T(n,k) of colored set partitions of [n] where colors of the elements of subsets are distinct and in increasing order and exactly k colors are used; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A193161 E.g.f. A(x) satisfies: A(x/(1-x))/(1-x) = d/dx x*A(x).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

PARI

Formula

A321296 Number T(n,k) of colored set partitions of [n] where colors of the elements of subsets are in (weakly) increasing order and exactly k colors are used; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A336227 a(0) = 1; a(n) = n * Sum_{k=0..n-1} binomial(n-1,k)^2 * a(k).

Views