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A061691 Triangle of generalized Stirling numbers. 7
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

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]

LINKS

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, 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.

FORMULA

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

EXAMPLE

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 4x4 = 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.

MAPLE

#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

MATHEMATICA

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-Fran├žois Alcover, Apr 04 2012, after Maple *)

CROSSREFS

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

Sequence in context: A133174 A155545 A141618 * A235595 A061356 A141028

Adjacent sequences:  A061688 A061689 A061690 * A061692 A061693 A061694

KEYWORD

nonn,tabl

AUTHOR

N. J. A. Sloane, Jun 18 2001

EXTENSIONS

More terms from Vladeta Jovovic, Apr 23 2003

STATUS

approved

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Last modified May 28 15:33 EDT 2020. Contains 334684 sequences. (Running on oeis4.)