A115068 Triangle read by rows: T(n,k) = number of elements in the Coxeter group D_n with descent set contained in {s_k}, for 0<=k<=n-1.

1, 2, 2, 4, 6, 3, 8, 16, 12, 4, 16, 40, 40, 20, 5, 32, 96, 120, 80, 30, 6, 64, 224, 336, 280, 140, 42, 7, 128, 512, 896, 896, 560, 224, 56, 8, 256, 1152, 2304, 2688, 2016, 1008, 336, 72, 9, 512, 2560, 5760, 7680, 6720, 4032, 1680, 480, 90, 10, 1024, 5632, 14080, 21120

Offset: 1

Author

Elizabeth Morris (epmorris(AT)math.washington.edu), Mar 01 2006

Keywords

• easy
• nonn
• tabl

Comments

A115068 is the fission of the polynomial sequence (p(x,n)) by the polynomial sequence ((2x+1)^n), where p(n,x)=x^n+x^(n-1)+...+x+1, n>=0. See A193842 for the definition of fission. - Clark Kimberling, Aug 07 2011

Examples

			First six rows:
1
2...2
4...6....3
8...16...12...4
16..40...40...20...5
32..96...120..80...30...6
		

References

• A. Bjorner and F. Brenti, Combinatorics of Coxeter Groups, Springer, New York, 2005.
• J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge University Press, Cambridge, 1990.

Links

• Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened

Crossrefs

Cf. A007318, A038207, A193862.

Cf. A105728, A013609.

Programs

Haskell

a115068 n k = a115068_tabl !! (n-1) !! (k-1)
a115068_row n = a115068_tabl !! (n-1)
a115068_tabl = iterate (\row -> zipWith (+) (row ++ [1]) $
                                zipWith (+) (row ++ [0]) ([0] ++ row)) [1]
-- Reinhard Zumkeller, Jul 22 2013

Mathematica

z = 11;
p[0, x_] := 1; p[n_, x_] := x*p[n - 1, x] + 1;
q[n_, x_] := (2 x + 1)^n;
p1[n_, k_] := Coefficient[p[n, x], x^k];
p1[n_, 0] := p[n, x] /. x -> 0;
d[n_, x_] := Sum[p1[n, k]*q[n - 1 - k, x], {k, 0, n - 1}]
h[n_] := CoefficientList[d[n, x], {x}]
TableForm[Table[Reverse[h[n]], {n, 0, z}]]
Flatten[Table[Reverse[h[n]], {n, -1, z}]]  (* A115068 *)
TableForm[Table[h[n], {n, 0, z}]]
Flatten[Table[h[n], {n, -1, z}]]   (* A193862 *)

Formula

T(n,k)=binomial(n,k)*2^(n-k-1).

T(n,1) = 2^(n-1), T(n,n) = n, for n > 1: T(n,k) = T(n-1,k-1) + 2*T(n-1,k), 1 < k < n. - Reinhard Zumkeller, Jul 22 2013