login
Dimension of the space spanned by the symmetric functions L_lambda of Gessel and Reutenauer, where lambda ranges over all partitions of n.
0

%I #8 Mar 25 2022 09:28:56

%S 1,1,2,3,4,6,10,13,19,26,38,52,70,91,123,161

%N Dimension of the space spanned by the symmetric functions L_lambda of Gessel and Reutenauer, where lambda ranges over all partitions of n.

%D R. P. Stanley, Enumerative Combinatorics, vol. 2 (Exercise 7.89).

%H I. M. Gessel and C. Reutenauer, <a href="https://doi.org/10.1016/0097-3165(93)90095-P">Counting permutations with given cycle structure and descent set</a>, J. Combin. Theory, Ser. A, 64, 1993, 189-215.

%e In terms of Schur functions we have:

%e L[4] = s[3,1] + s[2,1,1],

%e L[3,1] = s[3,1] + s[2,2] + s[1,1,1,1],

%e L[2,2] = s[2,2] + s[1,1,1,1],

%e L[2,1,1] = s[3,1] + s[2,1,1],

%e L[4] = s[4].

%e There is one linear dependence relation, viz., L[4] = L[2,1,1],

%e so for n=4 we get the value 5-1=4.

%K nonn,more

%O 0,3

%A _Richard Stanley_, Aug 12 2006