A003822 Number of commutative elements in Coxeter group E_n.

10, 42, 167, 662, 2670, 10846, 44199, 180438, 737762, 3021000, 12387990, 50864885, 209095841, 860447494, 3544046278, 14608974346, 60261567146, 248726602105, 1027143932653, 4243640251368, 17539577253151, 72518982292559, 299928724501455

Offset

3,1

References

C. K. Fan, A Hecke Algebra Quotient and Properties of Commutative Elements of a Weyl Group, MIT Ph.D. Thesis 1995.

J. R. Stembridge, Abstracts Amer. Math. Soc., 18 (1) (1997), p. 17, #918-05-495.

Links

Table of n, a(n) for n=3..25.

Boothby, T.; Burkert, J.; Eichwald, M.; Ernst, D. C.; Green, R. M.; Macauley, M.  On the cyclically fully commutative elements of Coxeter groups, J. Algebr. Comb. 36, No. 1, 123-148 (2012), Table 1 Type E.

C. K. Fan, A Hecke algebra quotient and some combinatorial applications, J. Algebraic Combin. 5 (1996), no. 3, 175-189.

C. K. Fan, Structure of a Hecke algebra quotient, J. Amer. Math. Soc. 10 (1997), no. 1, 139-167.

J. R. Stembridge, The enumeration of fully commutative elements of Coxeter groups, J. Algebraic Combin. 7 (1998), no. 3, 291-320.

Formula

G.f.: x^3 * ((16-52*x+45*x^2-x^(-1)*(R(x)-1))/(1-7*x+14*x^2-9*x^3) - (6-14*x+12*x^2)/(1-4*x+5*x^2-3*x^3) + (1-x^3-x^4)/(1-x-x^2+x^5)) where R(x)=(1-sqrt(1-4*x))/(2*x) is the generating function for the Catalan numbers. [From Stembridge (1998)] - Sean A. Irvine, Sep 01 2015

Prog

(PARI) x='x+O('x^33);  R(x)=(1-sqrt(1-4*x))/(2*x);
Vec( x^3 * ((16-52*x+45*x^2-x^(-1)*(R(x)-1))/(1-7*x+14*x^2-9*x^3) - (6-14*x+12*x^2)/(1-4*x+5*x^2-3*x^3) + (1-x^3-x^4)/(1-x-x^2+x^5)) ) \\ Joerg Arndt, Sep 02 2015

Crossrefs

Cf. A000108.

Sequence in context: A226988 A358249 A027171 * A087120 A222358 A321314

Adjacent sequences: A003819 A003820 A003821 * A003823 A003824 A003825

Keyword

nonn

Author

Ken Fan

Extensions

More terms from Sean A. Irvine, Sep 01 2015

Status

approved