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A023998
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Number of block permutations on an n-set which are uniform, i.e. corresponding blocks have same size.
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4
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1, 1, 3, 16, 131, 1496, 22482, 426833, 9934563, 277006192, 9085194458, 345322038293, 15024619744202, 740552967629021, 40984758230303149, 2527342803112928081, 172490568947825135203, 12952575262915522547136
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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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 (njas(AT)research.att.com), Dec 19, 1999.
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REFERENCES
| 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.
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LINKS
| 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.
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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 ). [From 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
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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 +...
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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 (pauldhanna(AT)juno.com), 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 */
/* or: */
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 */
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CROSSREFS
| Cf. A023997, A002720. A061691
Sequence in context: A121673 A051921 * A179119 A141628 A048802 A119392
Adjacent sequences: A023995 A023996 A023997 * A023999 A024000 A024001
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KEYWORD
| nonn,nice
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AUTHOR
| Des FitzGerald (D.FitzGerald(AT)utas.edu.au)
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EXTENSIONS
| More terms from Vladeta Jovovic (vladeta(AT)eunet.rs), Sep 03 2002
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