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A023998 Number of block permutations on an n-set which are uniform, i.e., corresponding blocks have same size. 6
1, 1, 3, 16, 131, 1496, 22482, 426833, 9934563, 277006192, 9085194458, 345322038293, 15024619744202, 740552967629021, 40984758230303149, 2527342803112928081, 172490568947825135203, 12952575262915522547136, 1064521056888312620947794, 95305764621957309071404877 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Number of games of simple patience with n cards. Take a shuffled deck of n cards labeled 1..n; as each card is dealt it is placed either on a higher-numbered card or starts a new pile to the right. Cards are not moved once they are placed. Suggested by reading Aldous and Diaconis. - N. J. A. Sloane, Dec 19 1999

Number of set partitions of [2n] such that within each block the numbers of odd and even elements are equal. a(2) = 3: 1234, 12|34, 14|23; a(3) = 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. - Alois P. Heinz, Jul 14 2016

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 0..300

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.

H. Cheballah, S. Giraudo, R. Maurice, Combinatorial Hopf algebra structure on packed square matrices, arXiv preprint arXiv:1306.6605 [math.CO], 2013.

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.

FORMULA

a(n) = Sum_{k=0..n-1} C(n,k)*C(n-1,k)*a(k) for n>0 with a(0)=1. - Paul D. Hanna, Aug 15 2007

G.f.: Sum_{n>=0} a(n)*x^n/n!^2 = exp( Sum_{n>=1} x^n/n!^2 ). [Paul D. Hanna, Jan 04 2011; merged from duplicate entry A179119]

Row sums of A061691.

Generating function: Let J(z) = sum {n>=0} z^n/n!^2. Then exp(x*(J(z)-1) = sum {n>=0} a(n)*z^n/n!^2 = 1 + z + 3*z^2/2!^2 + 36*z^3/3!^2 + .... - Peter Bala, Jul 11 2011

EXAMPLE

For n=3 there are 25 block permutations, of which 9 of the form ({1} maps to {1,2}; {2,3} maps to {3}), are not uniform. Hence a(3) = 25 - 9 = 16.

Alternatively, for n=3 the 6 permutations of 3 cards produce 16 games, as follows: 123 -> {1,2,3}; 132 -> {1,32}, {1,3,2}; 213 -> {21,3}, {2,1,3}; 231 -> {21,3}, {2,31}, {2,3,1}; 312 -> {31,2}, {32,1}, {3,1,2}; 321 -> {321}, {32,1}, {31,2}, {3,21}, {3,2,1}.

G.f.: A(x) = 1 + x + 3*x^2/2!^2 + 16*x^3/3!^2 + 131*x^4/4!^2 + 1496*x^5/5!^2 + ...

log(A(x)) = x + x^2/2!^2 + x^3/3!^2 + x^4/4!^2 + x^5/5!^2 + ...

MAPLE

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

MATHEMATICA

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] Binomial[n-1, k] a[k], {k, 0, n-1}];

Array[a, 25, 0] (* Jean-Fran├žois Alcover, Jul 28 2016 *)

PROG

(PARI) a(n)=if(n==0, 1, sum(k=0, n-1, binomial(n, k)*binomial(n-1, k)*a(k))) \\ Paul D. Hanna, Aug 15 2007

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

(PARI)  N=66; x='x+O('x^N); /* that many terms */

Vec(serlaplace(serlaplace(exp(sum(n=1, N, x^n/n!^2))))) /* show terms */

/* Joerg Arndt, Jul 12 2011 */

(PARI)

v=vector(N); v[1]=1;

for (n=1, N-1, v[n+1]=sum(k=0, n-1, binomial(n, k)*binomial(n-1, k)*v[k+1]) );

v /* show terms */

/* Joerg Arndt, Jul 12 2011 */

(Haskell)

a023998 n = a023998_list !! n

a023998_list = 1 : f 2 [1] a132813_tabl where

   f x ys (zs:zss) = y : f (x + 1) (ys ++ [y]) zss where

                     y = sum $ zipWith (*) ys zs

-- Reinhard Zumkeller, Apr 04 2014

CROSSREFS

Cf. A023997, A002720, A061691.

Cf. A132813.

Column k=2 of A275043.

Sequence in context: A121673 A273997 A051921 * A241464 A141628 A048802

Adjacent sequences:  A023995 A023996 A023997 * A023999 A024000 A024001

KEYWORD

nonn,nice

AUTHOR

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

EXTENSIONS

More terms from Vladeta Jovovic, Sep 03 2002

STATUS

approved

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Last modified August 16 08:52 EDT 2017. Contains 290623 sequences.