0,3

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. - njas, Dec 19, 1999.

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.

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.

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 (pauldhanna(AT)juno.com), Aug 15 2007

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}.

(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

Cf. A023997, A002720.

Sequence in context: A131490 A121673 A051921 this_sequence A141628 A048802 A119392

Adjacent sequences: A023995 A023996 A023997 this_sequence A023999 A024000 A024001

nonn,nice

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

More terms from Vladeta Jovovic (vladeta(AT)Eunet.yu), Sep 03 2002