Diego Martin Duro - cp's OEIS frontend

A363701 Numbers k with the property that the character table of S_k contains a zero in every nontrivial column.

0, 1, 5, 6, 8, 9, 10, 12, 14, 17, 21, 28, 30, 32, 34, 36, 37

Offset: 1

Diego Martin Duro, Jun 16 2023

This sequence contains A363675 and A363676.

Cf. A000217, A010054, A175595, A260682, A363675, A363676.

def zeros(n):
    G = SymmetricGroup(n)
    N = G.character_table()
    for j in range(1,N.ncols()):
        if 0 not in N[:,j]:
            return False
        j+=1
    return True
n=1
while n>0:
    if zeros(n) == True:
        print(n)
    n+=1

A363676 Numbers k such that the least common multiple of the degrees of the irreducible characters of A_k equals |A_k| = k!/2.

Original entry on oeis.org

0, 1, 2, 5, 6, 8, 10, 12, 17, 21, 30, 36, 57, 66, 80, 105, 120, 122, 136, 190, 192, 210, 212, 233, 276, 302, 325, 380, 408, 465, 496, 530, 561, 597, 630, 632, 666, 705, 741, 780, 782, 822, 905, 990, 992, 1081, 1130, 1176, 1225, 1433, 1540, 1542, 1596, 1772

Offset: 1

Diego Martin Duro, Jun 14 2023

nonn

Intersection of the sequences of numbers k such that there exists a 2-core partition of k or k-2 and a 3-core partition of k. This sequence contains A363675.

			The degrees of the irreducible characters of A_5 are 1,3,3,4,5 so their least common multiple is 5!/2 = 60, so 5 is a term of the sequence.

Diego Martin Duro, Arithmetic Properties of Character Degrees and the Generalised Knutson Index, arXiv:2306.12408 [math.RT], 2023.

Cf. A000217, A010054, A175595, A260682, A267137, A363675, A363701.

More terms from Alois P. Heinz, Jun 16 2023

A363675 Numbers k such that the least common multiple of the degrees of the irreducible characters of S_k equals |S_k| = k!.

Original entry on oeis.org

0, 1, 6, 10, 21, 36, 66, 105, 120, 136, 190, 210, 276, 325, 465, 496, 561, 630, 666, 741, 780, 990, 1081, 1176, 1225, 1540, 1596, 1830, 2080, 2145, 2346, 2556, 2926, 3081, 3160, 3240, 3486, 3570, 3916, 4005, 4186, 4560, 4656, 4950, 5050, 5356, 5460, 5886, 6105

Offset: 1

Diego Martin Duro, Jun 14 2023

nonn

Intersection of the sequences of numbers k such that there exists a 2-core partition of k (A267137) and a 3-core partition of k (A000217).

Alois P. Heinz, Table of n, a(n) for n = 1..100
Diego Martin Duro, Arithmetic Properties of Character Degrees and the Generalised Knutson Index, arXiv:2306.12408 [math.RT], 2023.

Cf. A000142, A000217, A010054, A033687, A175595, A260682, A267137, A363676, A363701.

From Alois P. Heinz, Jun 16 2023: (Start)
{ k : A175595(k,2) > 0 and A175595(k,3) > 0 }.
{ k : A010054(k) > 0 and A033687(k) > 0 }. (End)

More terms from Alois P. Heinz, Jun 16 2023

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User: Diego Martin Duro

A363701 Numbers k with the property that the character table of S_k contains a zero in every nontrivial column.

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A363676 Numbers k such that the least common multiple of the degrees of the irreducible characters of A_k equals |A_k| = k!/2.

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A363675 Numbers k such that the least common multiple of the degrees of the irreducible characters of S_k equals |S_k| = k!.

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