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Identifier
Values
=>
Cc0002;cc-rep
[1]=>0 [2]=>0 [1,1]=>1 [3]=>0 [2,1]=>1 [1,1,1]=>0 [4]=>0 [3,1]=>1 [2,2]=>1 [2,1,1]=>1 [1,1,1,1]=>1 [5]=>0 [4,1]=>1 [3,2]=>2 [3,1,1]=>2 [2,2,1]=>2 [2,1,1,1]=>2 [1,1,1,1,1]=>0 [6]=>0 [5,1]=>1 [4,2]=>3 [4,1,1]=>3 [3,3]=>2 [3,2,1]=>6 [3,1,1,1]=>4 [2,2,2]=>2 [2,2,1,1]=>4 [2,1,1,1,1]=>2 [1,1,1,1,1,1]=>1 [7]=>0 [6,1]=>1 [5,2]=>4 [5,1,1]=>4 [4,3]=>5 [4,2,1]=>12 [4,1,1,1]=>7 [3,3,1]=>8 [3,2,2]=>8 [3,2,1,1]=>14 [3,1,1,1,1]=>6 [2,2,2,1]=>6 [2,2,1,1,1]=>6 [2,1,1,1,1,1]=>3 [1,1,1,1,1,1,1]=>0 [8]=>0 [7,1]=>1 [6,2]=>5 [6,1,1]=>5 [5,3]=>9 [5,2,1]=>20 [5,1,1,1]=>11 [4,4]=>5 [4,3,1]=>25 [4,2,2]=>20 [4,2,1,1]=>33 [4,1,1,1,1]=>13 [3,3,2]=>16 [3,3,1,1]=>22 [3,2,2,1]=>28 [3,2,1,1,1]=>26 [3,1,1,1,1,1]=>9 [2,2,2,2]=>6 [2,2,2,1,1]=>12 [2,2,1,1,1,1]=>9 [2,1,1,1,1,1,1]=>3 [1,1,1,1,1,1,1,1]=>1 [9]=>0 [8,1]=>1 [7,2]=>6 [7,1,1]=>6 [6,3]=>14 [6,2,1]=>30 [6,1,1,1]=>16 [5,4]=>14 [5,3,1]=>54 [5,2,2]=>40 [5,2,1,1]=>64 [5,1,1,1,1]=>24 [4,4,1]=>30 [4,3,2]=>61 [4,3,1,1]=>80 [4,2,2,1]=>81 [4,2,1,1,1]=>72 [4,1,1,1,1,1]=>22 [3,3,3]=>16 [3,3,2,1]=>66 [3,3,1,1,1]=>48 [3,2,2,2]=>34 [3,2,2,1,1]=>66 [3,2,1,1,1,1]=>44 [3,1,1,1,1,1,1]=>12 [2,2,2,2,1]=>18 [2,2,2,1,1,1]=>21 [2,2,1,1,1,1,1]=>12 [2,1,1,1,1,1,1,1]=>4 [1,1,1,1,1,1,1,1,1]=>0 [10]=>0 [9,1]=>1 [8,2]=>7 [8,1,1]=>7 [7,3]=>20 [7,2,1]=>42 [7,1,1,1]=>22 [6,4]=>28 [6,3,1]=>98 [6,2,2]=>70 [6,2,1,1]=>110 [6,1,1,1,1]=>40 [5,5]=>14 [5,4,1]=>98 [5,3,2]=>155 [5,3,1,1]=>198 [5,2,2,1]=>185 [5,2,1,1,1]=>160 [5,1,1,1,1,1]=>46 [4,4,2]=>91 [4,4,1,1]=>110 [4,3,3]=>77 [4,3,2,1]=>288 [4,3,1,1,1]=>200 [4,2,2,2]=>115 [4,2,2,1,1]=>219 [4,2,1,1,1,1]=>138 [4,1,1,1,1,1,1]=>34 [3,3,3,1]=>82 [3,3,2,2]=>100 [3,3,2,1,1]=>180 [3,3,1,1,1,1]=>92 [3,2,2,2,1]=>118 [3,2,2,1,1,1]=>131 [3,2,1,1,1,1,1]=>68 [3,1,1,1,1,1,1,1]=>16 [2,2,2,2,2]=>18 [2,2,2,2,1,1]=>39 [2,2,2,1,1,1,1]=>33 [2,2,1,1,1,1,1,1]=>16 [2,1,1,1,1,1,1,1,1]=>4 [1,1,1,1,1,1,1,1,1,1]=>1 [11]=>0 [10,1]=>1 [9,2]=>8 [9,1,1]=>8 [8,3]=>27 [8,2,1]=>56 [8,1,1,1]=>29 [7,4]=>48 [7,3,1]=>160 [7,2,2]=>112 [7,2,1,1]=>174 [7,1,1,1,1]=>62 [6,5]=>42 [6,4,1]=>224 [6,3,2]=>323 [6,3,1,1]=>406 [6,2,2,1]=>365 [6,2,1,1,1]=>310 [6,1,1,1,1,1]=>86 [5,5,1]=>112 [5,4,2]=>344 [5,4,1,1]=>406 [5,3,3]=>232 [5,3,2,1]=>826 [5,3,1,1,1]=>558 [5,2,2,2]=>300 [5,2,2,1,1]=>564 [5,2,1,1,1,1]=>344 [5,1,1,1,1,1,1]=>80 [4,4,3]=>168 [4,4,2,1]=>489 [4,4,1,1,1]=>310 [4,3,3,1]=>447 [4,3,2,2]=>503 [4,3,2,1,1]=>887 [4,3,1,1,1,1]=>430 [4,2,2,2,1]=>452 [4,2,2,1,1,1]=>488 [4,2,1,1,1,1,1]=>240 [4,1,1,1,1,1,1,1]=>50 [3,3,3,2]=>182 [3,3,3,1,1]=>262 [3,3,2,2,1]=>398 [3,3,2,1,1,1]=>403 [3,3,1,1,1,1,1]=>160 [3,2,2,2,2]=>136 [3,2,2,2,1,1]=>288 [3,2,2,1,1,1,1]=>232 [3,2,1,1,1,1,1,1]=>100 [3,1,1,1,1,1,1,1,1]=>20 [2,2,2,2,2,1]=>57 [2,2,2,2,1,1,1]=>72 [2,2,2,1,1,1,1,1]=>49 [2,2,1,1,1,1,1,1,1]=>20 [2,1,1,1,1,1,1,1,1,1]=>5 [1,1,1,1,1,1,1,1,1,1,1]=>0 [12]=>0 [11,1]=>1 [10,2]=>9 [10,1,1]=>9 [9,3]=>35 [9,2,1]=>72 [9,1,1,1]=>37 [8,4]=>75 [8,3,1]=>243 [8,2,2]=>168 [8,2,1,1]=>259 [8,1,1,1,1]=>91 [7,5]=>90 [7,4,1]=>432 [7,3,2]=>595 [7,3,1,1]=>740 [7,2,2,1]=>651 [7,2,1,1,1]=>546 [7,1,1,1,1,1]=>148 [6,6]=>42 [6,5,1]=>378 [6,4,2]=>891 [6,4,1,1]=>1036 [6,3,3]=>555 [6,3,2,1]=>1920 [6,3,1,1,1]=>1274 [6,2,2,2]=>665 [6,2,2,1,1]=>1239 [6,2,1,1,1,1]=>740 [6,1,1,1,1,1,1]=>166 [5,5,2]=>456 [5,5,1,1]=>518 [5,4,3]=>744 [5,4,2,1]=>2065 [5,4,1,1,1]=>1274 [5,3,3,1]=>1505 [5,3,2,2]=>1629 [5,3,2,1,1]=>2835 [5,3,1,1,1,1]=>1332 [5,2,2,2,1]=>1316 [5,2,2,1,1,1]=>1396 [5,2,1,1,1,1,1]=>664 [5,1,1,1,1,1,1,1]=>130 [4,4,4]=>168 [4,4,3,1]=>1104 [4,4,2,2]=>992 [4,4,2,1,1]=>1686 [4,4,1,1,1,1]=>740 [4,3,3,2]=>1132 [4,3,3,1,1]=>1596 [4,3,2,2,1]=>2240 [4,3,2,1,1,1]=>2208 [4,3,1,1,1,1,1]=>830 [4,2,2,2,2]=>588 [4,2,2,2,1,1]=>1228 [4,2,2,1,1,1,1]=>960 [4,2,1,1,1,1,1,1]=>390 [4,1,1,1,1,1,1,1,1]=>70 [3,3,3,3]=>182 [3,3,3,2,1]=>842 [3,3,3,1,1,1]=>665 [3,3,2,2,2]=>534 [3,3,2,2,1,1]=>1089 [3,3,2,1,1,1,1]=>795 [3,3,1,1,1,1,1,1]=>260 [3,2,2,2,2,1]=>481 [3,2,2,2,1,1,1]=>592 [3,2,2,1,1,1,1,1]=>381 [3,2,1,1,1,1,1,1,1]=>140 [3,1,1,1,1,1,1,1,1,1]=>25 [2,2,2,2,2,2]=>57 [2,2,2,2,2,1,1]=>129 [2,2,2,2,1,1,1,1]=>121 [2,2,2,1,1,1,1,1,1]=>69 [2,2,1,1,1,1,1,1,1,1]=>25 [2,1,1,1,1,1,1,1,1,1,1]=>5 [1,1,1,1,1,1,1,1,1,1,1,1]=>1
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Description
The number of standard desarrangement tableaux of shape equal to the given partition.
A standard desarrangement tableau is a standard tableau whose first ascent is even. Here, an ascent of a standard tableau is an entry $i$ such that $i+1$ appears to the right or above $i$ in the tableau (with respect to English tableau notation).
This is also the nullity of the random-to-random operator (and the random-to-top) operator acting on the simple module of the symmetric group indexed by the given partition. See also:
  • St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition.: The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition
  • St000500Eigenvalues of the random-to-random operator acting on the regular representation.: Eigenvalues of the random-to-random operator acting on the regular representation.
Code
def tableau_ascents(t):
    r"""
    The (sorted list) of ascents of the standard tableau `t`.

    An *ascent* of a standard tableau `t` is an entry `i`
    such that `i+1` apears to the right or above `i` in `t`
    (in English notation for tableaux).
    """
    locations = {}
    for (i, row) in enumerate(t):
        for (j, entry) in enumerate(row):
            locations[entry] = (i, j)
    ascents = [t.size()]
    for i in range(1, t.size()):
        # ascent means i+1 appears to the right or above
        x, _ = locations[i]
        u, _ = locations[i+1]
        if u <= x:
            ascents.append(i)
    return sorted(ascents)

def is_desarrangement_tableau(t):
    r"""
    Test whether a tableau is a desarrangement tableau.

    A *desarrangement tableau* is a standard tableau
    whose first ascent is even.
    """
    return min(tableau_ascents(Tableau(t))) % 2 == 0

def statistic(la):
    return len([t for t in StandardTableaux(la) if is_desarrangement_tableau(t)])

Created
May 24, 2016 at 23:10 by Franco Saliola
Updated
Jun 11, 2016 at 01:03 by Martin Rubey