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Identifier

St000036: Permutations ⟶ ℤ

Values

[1] => 1

[1,2] => 1

[2,1] => 1

[1,2,3] => 1

[1,3,2] => 1

[2,1,3] => 1

[2,3,1] => 1

[3,1,2] => 1

[3,2,1] => 1

[1,2,3,4] => 1

[1,2,4,3] => 1

[1,3,2,4] => 1

[1,3,4,2] => 1

[1,4,2,3] => 1

[1,4,3,2] => 1

[2,1,3,4] => 1

[2,1,4,3] => 1

[2,3,1,4] => 1

[2,3,4,1] => 1

[2,4,1,3] => 1

[2,4,3,1] => 1

[3,1,2,4] => 1

[3,1,4,2] => 1

[3,2,1,4] => 1

[3,2,4,1] => 1

[3,4,1,2] => 2

[3,4,2,1] => 1

[4,1,2,3] => 1

[4,1,3,2] => 1

[4,2,1,3] => 1

[4,2,3,1] => 2

[4,3,1,2] => 1

[4,3,2,1] => 1

[1,2,3,4,5] => 1

[1,2,3,5,4] => 1

[1,2,4,3,5] => 1

[1,2,4,5,3] => 1

[1,2,5,3,4] => 1

[1,2,5,4,3] => 1

[1,3,2,4,5] => 1

[1,3,2,5,4] => 1

[1,3,4,2,5] => 1

[1,3,4,5,2] => 1

[1,3,5,2,4] => 1

[1,3,5,4,2] => 1

[1,4,2,3,5] => 1

[1,4,2,5,3] => 1

[1,4,3,2,5] => 1

[1,4,3,5,2] => 1

[1,4,5,2,3] => 2

[1,4,5,3,2] => 1

[1,5,2,3,4] => 1

[1,5,2,4,3] => 1

[1,5,3,2,4] => 1

[1,5,3,4,2] => 2

[1,5,4,2,3] => 1

[1,5,4,3,2] => 1

[2,1,3,4,5] => 1

[2,1,3,5,4] => 1

[2,1,4,3,5] => 1

[2,1,4,5,3] => 1

[2,1,5,3,4] => 1

[2,1,5,4,3] => 1

[2,3,1,4,5] => 1

[2,3,1,5,4] => 1

[2,3,4,1,5] => 1

[2,3,4,5,1] => 1

[2,3,5,1,4] => 1

[2,3,5,4,1] => 1

[2,4,1,3,5] => 1

[2,4,1,5,3] => 1

[2,4,3,1,5] => 1

[2,4,3,5,1] => 1

[2,4,5,1,3] => 2

[2,4,5,3,1] => 1

[2,5,1,3,4] => 1

[2,5,1,4,3] => 1

[2,5,3,1,4] => 1

[2,5,3,4,1] => 2

[2,5,4,1,3] => 1

[2,5,4,3,1] => 1

[3,1,2,4,5] => 1

[3,1,2,5,4] => 1

[3,1,4,2,5] => 1

[3,1,4,5,2] => 1

[3,1,5,2,4] => 1

[3,1,5,4,2] => 1

[3,2,1,4,5] => 1

[3,2,1,5,4] => 1

[3,2,4,1,5] => 1

[3,2,4,5,1] => 1

[3,2,5,1,4] => 1

[3,2,5,4,1] => 1

[3,4,1,2,5] => 2

[3,4,1,5,2] => 2

[3,4,2,1,5] => 1

[3,4,2,5,1] => 1

[3,4,5,1,2] => 3

[3,4,5,2,1] => 1

[3,5,1,2,4] => 2

[3,5,1,4,2] => 2

>>> Load all 1581 entries. <<<

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Description

The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation.
These are multiplicities of Verma modules.

References

[1] Kazhdan, D., Lusztig, G. Representations of Coxeter groups and Hecke algebras MathSciNet:0560412
[2] Björner, A., Brenti, F. Combinatorics of Coxeter groups MathSciNet:2133266
[3] Humphreys, J. E. Reflection groups and Coxeter groups MathSciNet:1066460
[4] Number of permutations w in S_n whose Kazhdan-Lusztig polynomial evaluated at q=1 is less than or equal to 2; P_id,w(1) <= 2. OEIS:A224255
[5] Number of permutations w in S_n whose Kazhdan-Lusztig polynomial P_id,w(1) <= 3. OEIS:A224316
[6] Woo, A. Permutations with Kazhdan-Lusztig polynomial $P_{id,w}(q)=1+q^h$ MathSciNet:2515773

Code

# takes a long time!
def statistic(x):
    if x.size() == 1:
        return 1
    G = WeylGroup(['A',x.size()-1])
    R.=LaurentPolynomialRing(QQ)
    KL=KazhdanLusztigPolynomial(G,q)
    w = G.from_reduced_word(x.reduced_word())
    return KL.P(1,w).substitute({q:1})

# alternative, with coxeter3 installed:
def statistic_alt(x):
    if x.size() == 1:
        return 1
    G = CoxeterGroup(['A',x.size()-1], implementation="coxeter3")
    R. = PolynomialRing(QQ)
    P = G.kazhdan_lusztig_polynomial([],x.reduced_word())
    return P.substitute({q:1})

Created

Feb 14, 2013 at 00:43 by Sara Billey

Updated

Dec 28, 2017 at 20:18 by Martin Rubey