Identifier
Identifier
Values
[1] generating graphics... => 0
[2] generating graphics... => 0
[1,1] generating graphics... => 1
[3] generating graphics... => 0
[2,1] generating graphics... => 1
[1,1,1] generating graphics... => 0
[4] generating graphics... => 0
[3,1] generating graphics... => 1
[2,2] generating graphics... => 1
[2,1,1] generating graphics... => 1
[1,1,1,1] generating graphics... => 1
[5] generating graphics... => 0
[4,1] generating graphics... => 1
[3,2] generating graphics... => 2
[3,1,1] generating graphics... => 2
[2,2,1] generating graphics... => 2
[2,1,1,1] generating graphics... => 2
[1,1,1,1,1] generating graphics... => 0
[6] generating graphics... => 0
[5,1] generating graphics... => 1
[4,2] generating graphics... => 3
[4,1,1] generating graphics... => 3
[3,3] generating graphics... => 2
[3,2,1] generating graphics... => 6
[3,1,1,1] generating graphics... => 4
[2,2,2] generating graphics... => 2
[2,2,1,1] generating graphics... => 4
[2,1,1,1,1] generating graphics... => 2
[1,1,1,1,1,1] generating graphics... => 1
[7] generating graphics... => 0
[6,1] generating graphics... => 1
[5,2] generating graphics... => 4
[5,1,1] generating graphics... => 4
[4,3] generating graphics... => 5
[4,2,1] generating graphics... => 12
[4,1,1,1] generating graphics... => 7
[3,3,1] generating graphics... => 8
[3,2,2] generating graphics... => 8
[3,2,1,1] generating graphics... => 14
[3,1,1,1,1] generating graphics... => 6
[2,2,2,1] generating graphics... => 6
[2,2,1,1,1] generating graphics... => 6
[2,1,1,1,1,1] generating graphics... => 3
[1,1,1,1,1,1,1] generating graphics... => 0
[8] generating graphics... => 0
[7,1] generating graphics... => 1
[6,2] generating graphics... => 5
[6,1,1] generating graphics... => 5
[5,3] generating graphics... => 9
[5,2,1] generating graphics... => 20
[5,1,1,1] generating graphics... => 11
[4,4] generating graphics... => 5
[4,3,1] generating graphics... => 25
[4,2,2] generating graphics... => 20
[4,2,1,1] generating graphics... => 33
[4,1,1,1,1] generating graphics... => 13
[3,3,2] generating graphics... => 16
[3,3,1,1] generating graphics... => 22
[3,2,2,1] generating graphics... => 28
[3,2,1,1,1] generating graphics... => 26
[3,1,1,1,1,1] generating graphics... => 9
[2,2,2,2] generating graphics... => 6
[2,2,2,1,1] generating graphics... => 12
[2,2,1,1,1,1] generating graphics... => 9
[2,1,1,1,1,1,1] generating graphics... => 3
[1,1,1,1,1,1,1,1] generating graphics... => 1
[9] generating graphics... => 0
[8,1] generating graphics... => 1
[7,2] generating graphics... => 6
[7,1,1] generating graphics... => 6
[6,3] generating graphics... => 14
[6,2,1] generating graphics... => 30
[6,1,1,1] generating graphics... => 16
[5,4] generating graphics... => 14
[5,3,1] generating graphics... => 54
[5,2,2] generating graphics... => 40
[5,2,1,1] generating graphics... => 64
[5,1,1,1,1] generating graphics... => 24
[4,4,1] generating graphics... => 30
[4,3,2] generating graphics... => 61
[4,3,1,1] generating graphics... => 80
[4,2,2,1] generating graphics... => 81
[4,2,1,1,1] generating graphics... => 72
[4,1,1,1,1,1] generating graphics... => 22
[3,3,3] generating graphics... => 16
[3,3,2,1] generating graphics... => 66
[3,3,1,1,1] generating graphics... => 48
[3,2,2,2] generating graphics... => 34
[3,2,2,1,1] generating graphics... => 66
[3,2,1,1,1,1] generating graphics... => 44
[3,1,1,1,1,1,1] generating graphics... => 12
[2,2,2,2,1] generating graphics... => 18
[2,2,2,1,1,1] generating graphics... => 21
[2,2,1,1,1,1,1] generating graphics... => 12
[2,1,1,1,1,1,1,1] generating graphics... => 4
[1,1,1,1,1,1,1,1,1] generating graphics... => 0
[10] generating graphics... => 0
[9,1] generating graphics... => 1
[8,2] generating graphics... => 7
[8,1,1] generating graphics... => 7
[7,3] generating graphics... => 20
[7,2,1] generating graphics... => 42
[7,1,1,1] generating graphics... => 22
[6,4] generating graphics... => 28
[6,3,1] generating graphics... => 98
[6,2,2] generating graphics... => 70
[6,2,1,1] generating graphics... => 110
[6,1,1,1,1] generating graphics... => 40
[5,5] generating graphics... => 14
[5,4,1] generating graphics... => 98
[5,3,2] generating graphics... => 155
[5,3,1,1] generating graphics... => 198
[5,2,2,1] generating graphics... => 185
[5,2,1,1,1] generating graphics... => 160
[5,1,1,1,1,1] generating graphics... => 46
[4,4,2] generating graphics... => 91
[4,4,1,1] generating graphics... => 110
[4,3,3] generating graphics... => 77
[4,3,2,1] generating graphics... => 288
[4,3,1,1,1] generating graphics... => 200
[4,2,2,2] generating graphics... => 115
[4,2,2,1,1] generating graphics... => 219
[4,2,1,1,1,1] generating graphics... => 138
[4,1,1,1,1,1,1] generating graphics... => 34
[3,3,3,1] generating graphics... => 82
[3,3,2,2] generating graphics... => 100
[3,3,2,1,1] generating graphics... => 180
[3,3,1,1,1,1] generating graphics... => 92
[3,2,2,2,1] generating graphics... => 118
[3,2,2,1,1,1] generating graphics... => 131
[3,2,1,1,1,1,1] generating graphics... => 68
[3,1,1,1,1,1,1,1] generating graphics... => 16
[2,2,2,2,2] generating graphics... => 18
[2,2,2,2,1,1] generating graphics... => 39
[2,2,2,1,1,1,1] generating graphics... => 33
[2,2,1,1,1,1,1,1] generating graphics... => 16
[2,1,1,1,1,1,1,1,1] generating graphics... => 4
[1,1,1,1,1,1,1,1,1,1] generating graphics... => 1
[11] generating graphics... => 0
[10,1] generating graphics... => 1
[9,2] generating graphics... => 8
[9,1,1] generating graphics... => 8
[8,3] generating graphics... => 27
[8,2,1] generating graphics... => 56
[8,1,1,1] generating graphics... => 29
[7,4] generating graphics... => 48
[7,3,1] generating graphics... => 160
[7,2,2] generating graphics... => 112
[7,2,1,1] generating graphics... => 174
[7,1,1,1,1] generating graphics... => 62
[6,5] generating graphics... => 42
[6,4,1] generating graphics... => 224
[6,3,2] generating graphics... => 323
[6,3,1,1] generating graphics... => 406
[6,2,2,1] generating graphics... => 365
[6,2,1,1,1] generating graphics... => 310
[6,1,1,1,1,1] generating graphics... => 86
[5,5,1] generating graphics... => 112
[5,4,2] generating graphics... => 344
[5,4,1,1] generating graphics... => 406
[5,3,3] generating graphics... => 232
[5,3,2,1] generating graphics... => 826
[5,3,1,1,1] generating graphics... => 558
[5,2,2,2] generating graphics... => 300
[5,2,2,1,1] generating graphics... => 564
[5,2,1,1,1,1] generating graphics... => 344
[5,1,1,1,1,1,1] generating graphics... => 80
[4,4,3] generating graphics... => 168
[4,4,2,1] generating graphics... => 489
[4,4,1,1,1] generating graphics... => 310
[4,3,3,1] generating graphics... => 447
[4,3,2,2] generating graphics... => 503
[4,3,2,1,1] generating graphics... => 887
[4,3,1,1,1,1] generating graphics... => 430
[4,2,2,2,1] generating graphics... => 452
[4,2,2,1,1,1] generating graphics... => 488
[4,2,1,1,1,1,1] generating graphics... => 240
[4,1,1,1,1,1,1,1] generating graphics... => 50
[3,3,3,2] generating graphics... => 182
[3,3,3,1,1] generating graphics... => 262
[3,3,2,2,1] generating graphics... => 398
[3,3,2,1,1,1] generating graphics... => 403
[3,3,1,1,1,1,1] generating graphics... => 160
[3,2,2,2,2] generating graphics... => 136
[3,2,2,2,1,1] generating graphics... => 288
[3,2,2,1,1,1,1] generating graphics... => 232
[3,2,1,1,1,1,1,1] generating graphics... => 100
[3,1,1,1,1,1,1,1,1] generating graphics... => 20
[2,2,2,2,2,1] generating graphics... => 57
[2,2,2,2,1,1,1] generating graphics... => 72
[2,2,2,1,1,1,1,1] generating graphics... => 49
[2,2,1,1,1,1,1,1,1] generating graphics... => 20
[2,1,1,1,1,1,1,1,1,1] generating graphics... => 5
[1,1,1,1,1,1,1,1,1,1,1] generating graphics... => 0
[12] generating graphics... => 0
[11,1] generating graphics... => 1
[10,2] generating graphics... => 9
[10,1,1] generating graphics... => 9
[9,3] generating graphics... => 35
[9,2,1] generating graphics... => 72
[9,1,1,1] generating graphics... => 37
[8,4] generating graphics... => 75
[8,3,1] generating graphics... => 243
[8,2,2] generating graphics... => 168
[8,2,1,1] generating graphics... => 259
[8,1,1,1,1] generating graphics... => 91
[7,5] generating graphics... => 90
[7,4,1] generating graphics... => 432
[7,3,2] generating graphics... => 595
[7,3,1,1] generating graphics... => 740
[7,2,2,1] generating graphics... => 651
[7,2,1,1,1] generating graphics... => 546
[7,1,1,1,1,1] generating graphics... => 148
[6,6] generating graphics... => 42
[6,5,1] generating graphics... => 378
[6,4,2] generating graphics... => 891
[6,4,1,1] generating graphics... => 1036
[6,3,3] generating graphics... => 555
[6,3,2,1] generating graphics... => 1920
[6,3,1,1,1] generating graphics... => 1274
[6,2,2,2] generating graphics... => 665
[6,2,2,1,1] generating graphics... => 1239
[6,2,1,1,1,1] generating graphics... => 740
[6,1,1,1,1,1,1] generating graphics... => 166
[5,5,2] generating graphics... => 456
[5,5,1,1] generating graphics... => 518
[5,4,3] generating graphics... => 744
[5,4,2,1] generating graphics... => 2065
[5,4,1,1,1] generating graphics... => 1274
[5,3,3,1] generating graphics... => 1505
[5,3,2,2] generating graphics... => 1629
[5,3,2,1,1] generating graphics... => 2835
[5,3,1,1,1,1] generating graphics... => 1332
[5,2,2,2,1] generating graphics... => 1316
[5,2,2,1,1,1] generating graphics... => 1396
[5,2,1,1,1,1,1] generating graphics... => 664
[5,1,1,1,1,1,1,1] generating graphics... => 130
[4,4,4] generating graphics... => 168
[4,4,3,1] generating graphics... => 1104
[4,4,2,2] generating graphics... => 992
[4,4,2,1,1] generating graphics... => 1686
[4,4,1,1,1,1] generating graphics... => 740
[4,3,3,2] generating graphics... => 1132
[4,3,3,1,1] generating graphics... => 1596
[4,3,2,2,1] generating graphics... => 2240
[4,3,2,1,1,1] generating graphics... => 2208
[4,3,1,1,1,1,1] generating graphics... => 830
[4,2,2,2,2] generating graphics... => 588
[4,2,2,2,1,1] generating graphics... => 1228
[4,2,2,1,1,1,1] generating graphics... => 960
[4,2,1,1,1,1,1,1] generating graphics... => 390
[4,1,1,1,1,1,1,1,1] generating graphics... => 70
[3,3,3,3] generating graphics... => 182
[3,3,3,2,1] generating graphics... => 842
[3,3,3,1,1,1] generating graphics... => 665
[3,3,2,2,2] generating graphics... => 534
[3,3,2,2,1,1] generating graphics... => 1089
[3,3,2,1,1,1,1] generating graphics... => 795
[3,3,1,1,1,1,1,1] generating graphics... => 260
[3,2,2,2,2,1] generating graphics... => 481
[3,2,2,2,1,1,1] generating graphics... => 592
[3,2,2,1,1,1,1,1] generating graphics... => 381
[3,2,1,1,1,1,1,1,1] generating graphics... => 140
[3,1,1,1,1,1,1,1,1,1] generating graphics... => 25
[2,2,2,2,2,2] generating graphics... => 57
[2,2,2,2,2,1,1] generating graphics... => 129
[2,2,2,2,1,1,1,1] generating graphics... => 121
[2,2,2,1,1,1,1,1,1] generating graphics... => 69
[2,2,1,1,1,1,1,1,1,1] generating graphics... => 25
[2,1,1,1,1,1,1,1,1,1,1] generating graphics... => 5
[1,1,1,1,1,1,1,1,1,1,1,1] generating graphics... => 1
click to show generating function       
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