Aniruddha Biswas - cp's OEIS frontend

A378302 Number of nondegenerate balanced Boolean functions of n variables.

0, 2, 2, 58, 12618, 601016690, 1832624137336299922, 23951146041928082853307218802404658090, 5768658823449206338089748357862286887548602533639737369730665340966207267034

Offset: 0

Aniruddha Biswas, Nov 22 2024

A Boolean function is degenerate on some variable if its output does not depend on the variable, and it is said to be non-degenerate if it is not degenerate on any variable.

Aniruddha Biswas and Palash Sarkar, Counting unate and balanced monotone Boolean functions, arXiv:2304.14069 [math.CO], 2023.
Aniruddha Biswas and Palash Sarkar, Counting Unate and Monotone Boolean Functions Under Restrictions of Balancedness and Non-Degeneracy, J. Int. Seq. (2025) Vol. 28, Art. No. 25.3.4. See pp. 4, 6.

Cf. A037293, A000618.

a[n_]:=Sum[(-1)^(n-i)*Binomial[n,i]*Binomial[2^i,2^(i-1)],{i,n}]; Array[a,9,0] (* Stefano Spezia, Nov 24 2024 *)

from math import comb
def A378302(n): return sum(-comb(n,i)*comb(1<Chai Wah Wu, Dec 11 2024

a(n) = Sum_{i=1..n} (-1)^(n-i) * binomial(n,i) * binomial(2^i,2^(i-1)).

A378300 Number of NPN-equivalence classes of balanced monotone Boolean functions of n or fewer variables.

Original entry on oeis.org

1, 1, 2, 4, 15, 581

Offset: 1

Aniruddha Biswas, Nov 22 2024

a(n) is also the number of NPN-equivalence classes of balanced unate Boolean functions of n or fewer variables.

Aniruddha Biswas and Palash Sarkar, Counting unate and balanced monotone Boolean functions, arXiv:2304.14069 [math.CO], 2023.
Aniruddha Biswas and Palash Sarkar, Counting Unate and Monotone Boolean Functions Under Restrictions of Balancedness and Non-Degeneracy, J. Int. Seq. (2025) Vol. 28, Art. No. 25.3.4. See pp. 4, 17.

Cf. A000372, A003183, A037293, A341633.

A374401 Number of balanced inequivalent nondegenerate unate functions of n or fewer variables.

Original entry on oeis.org

0, 2, 0, 4, 18, 230, 49918

Offset: 0

Aniruddha Biswas, Jul 07 2024

Aniruddha Biswas and Palash Sarkar, Counting unate and balanced monotone Boolean functions, arXiv:2304.14069 [math.CO], 2023.
Aniruddha Biswas and Palash Sarkar, Counting Unate and Monotone Boolean Functions Under Restrictions of Balancedness and Non-Degeneracy, J. Int. Seq. (2025) Vol. 28, Art. No. 25.3.4. See pp. 4, 16.
Index entries for sequences related to Boolean functions

Cf. A341633, A245079, A003182, A000372, A372495, A373697, A000721.

A374400 Number of balanced inequivalent unate functions of n or fewer variables.

Original entry on oeis.org

0, 2, 2, 6, 24, 254, 50172

Offset: 0

Aniruddha Biswas, Jul 07 2024

A Boolean function is said to be balanced if it takes the values 0 and 1 an equal number of times.

Aniruddha Biswas and Palash Sarkar, Counting unate and balanced monotone Boolean functions, arXiv:2304.14069 [math.CO], 2023.
Aniruddha Biswas and Palash Sarkar, Counting Unate and Monotone Boolean Functions Under Restrictions of Balancedness and Non-Degeneracy, J. Int. Seq. (2025) Vol. 28, Art. No. 25.3.4. See pp. 4, 16.
Index entries for sequences related to Boolean functions

Cf. A341633, A245079, A003182, A000372, A372495, A000721.

A374399 Number of inequivalent nondegenerate unate functions of n or fewer variables.

Original entry on oeis.org

2, 2, 6, 24, 166, 3266, 826308

Offset: 0

Aniruddha Biswas, Jul 07 2024

A Boolean function is degenerate on some variable if its output does not depend on the variable, and it is said to be non-degenerate if it is not degenerate on any variable.

Appears to be (essentially) the partial sums of A304997. - N. J. A. Sloane, Jul 26 2024

Aniruddha Biswas and Palash Sarkar, Counting unate and balanced monotone Boolean functions, arXiv:2304.14069 [math.CO], 2023.
Aniruddha Biswas and Palash Sarkar, Counting Unate and Monotone Boolean Functions Under Restrictions of Balancedness and Non-Degeneracy, J. Int. Seq. (2025) Vol. 28, Art. No. 25.3.4. See pp. 4, 16.
Index entries for sequences related to Boolean functions

Cf. A373697, A372495, A000372, A003182, A245079.

A373697 Number of nondegenerate balanced unate functions of n or fewer variables.

Original entry on oeis.org

0, 2, 0, 8, 256, 16832, 31287424, 10393552784640

Offset: 0

Aniruddha Biswas, Jun 13 2024

Aniruddha Biswas and P. Sarkar, Counting unate and balanced monotone Boolean functions, arXiv:2304.14069 [math.CO], 2023.
Aniruddha Biswas and Palash Sarkar, Counting Unate and Monotone Boolean Functions Under Restrictions of Balancedness and Non-Degeneracy, J. Int. Seq. (2025) Vol. 28, Art. No. 25.3.4. See pp. 4, 13.
Index entries for sequences related to Boolean functions

Cf. A006126, A006602, A341633, A245079.

A373690 Number of balanced unate functions of n or fewer variables.

Original entry on oeis.org

0, 2, 4, 14, 296, 18202, 31392428, 10393772159334

Offset: 0

Aniruddha Biswas, Jun 13 2024

Aniruddha Biswas and P. Sarkar, Counting unate and balanced monotone Boolean functions, arXiv:2304.14069 [math.CO], 2023.
Aniruddha Biswas and Palash Sarkar, Counting Unate and Monotone Boolean Functions Under Restrictions of Balancedness and Non-Degeneracy, J. Int. Seq. (2025) Vol. 28, Art. No. 25.3.4. See pp. 4, 13.
Index entries for sequences related to Boolean functions

Cf. A341633, A245079.

Typo in a(3) corrected by Andrei Zabolotskii, Jul 29 2025

A372495 Number of inequivalent unate functions of n or fewer variables.

Original entry on oeis.org

2, 4, 10, 34, 200, 3466, 829744

Offset: 0

Aniruddha Biswas, May 03 2024

A Boolean function is unate in a variable if it is either nondecreasing or nonincreasing with respect to that variable. Therefore in the circuit representation of unate functions, each variable appears either in its original form or in complemented form. Thus x⊕y = (x∧¬y)∨(¬x∧y) is not a unate function.

Moreover, two Boolean functions are said to be equivalent if they are equivalent under the permutation of variables. For example, f(x,y)=x is equivalent to f(x,y)=y under the permutation of input variables.

			The list of all 2-variable inequivalent unate functions f(x,y) is 0,1,x,¬x,x∧y,¬x∧y,¬x∧¬y,x∨y,¬x∨y,¬x∨¬y. So a(2)=10.

Aniruddha Biswas and Palash Sarkar, Counting unate and balanced monotone Boolean functions, arXiv:2304.14069 [math.CO], 2023.
Aniruddha Biswas and Palash Sarkar, Counting Unate and Monotone Boolean Functions Under Restrictions of Balancedness and Non-Degeneracy, J. Int. Seq. (2025) Vol. 28, Art. No. 25.3.4. See pp. 4, 16.
Index entries for sequences related to Boolean functions

Cf. A000372, A003182, A245079.

A371722 Number of balanced nondegenerate monotone Boolean functions of n or fewer variables.

Original entry on oeis.org

0, 1, 0, 1, 16, 526, 488866, 81199631130

Offset: 0

Aniruddha Biswas, Apr 04 2024

Aniruddha Biswas and Palash Sarkar, Counting unate and balanced monotone Boolean functions, arXiv:2304.14069 [math.CO], 2023.
Aniruddha Biswas and Palash Sarkar, Counting Unate and Monotone Boolean Functions Under Restrictions of Balancedness and Non-Degeneracy, J. Int. Seq. (2025) Vol. 28, Art. No. 25.3.4. See pp. 4, 12.

Cf. A000372, A006126, A341633.

A371718 Number of balanced inequivalent nondegenerate monotone Boolean functions of n or fewer variables.

Original entry on oeis.org

0, 1, 0, 1, 2, 12, 935, 16439515

Offset: 0

Aniruddha Biswas, Apr 05 2024

Aniruddha Biswas and Palash Sarkar, Counting unate and balanced monotone Boolean functions, arXiv:2304.14069 [math.CO], 2023.
Aniruddha Biswas and Palash Sarkar, Counting Unate and Monotone Boolean Functions Under Restrictions of Balancedness and Non-Degeneracy, J. Int. Seq. (2025) Vol. 28, Art. No. 25.3.4. See pp. 4, 14.

Cf. A000372, A003182, A006126, A341633.

a(7) added by Aniruddha Biswas, Oct 18 2024

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User: Aniruddha Biswas

A378302 Number of nondegenerate balanced Boolean functions of n variables.

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A378300 Number of NPN-equivalence classes of balanced monotone Boolean functions of n or fewer variables.

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A374401 Number of balanced inequivalent nondegenerate unate functions of n or fewer variables.

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A374400 Number of balanced inequivalent unate functions of n or fewer variables.

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A374399 Number of inequivalent nondegenerate unate functions of n or fewer variables.

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A373697 Number of nondegenerate balanced unate functions of n or fewer variables.

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A373690 Number of balanced unate functions of n or fewer variables.

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A372495 Number of inequivalent unate functions of n or fewer variables.

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A371722 Number of balanced nondegenerate monotone Boolean functions of n or fewer variables.

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A371718 Number of balanced inequivalent nondegenerate monotone Boolean functions of n or fewer variables.

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