A176211 -id:A176211 - cp's OEIS frontend

A176212 Terms of A176211, duplicates removed.

6, 9, 13, 20, 31, 36, 49, 54, 78, 81, 117, 120, 125, 169, 180, 186, 201, 216, 260, 279, 294, 324, 400, 403, 441, 468, 486, 523, 620, 637, 702, 720, 729, 750, 845, 961, 980, 1014, 1053, 1080, 1116, 1125, 1206, 1296, 1366, 1519, 1521, 1560, 1620, 1625, 1674, 1764, 1809, 1944, 2197

Offset: 1

Vladimir Shevelev, Apr 12 2010

nonn

The terms are permanents of a set of certain symmetric (0,1)-matrices as detailed in A176211. Thus the sequence solves a symmetric version of Gristein problem: to find all the values of permanent of all square (0,1) matrices, which contain exactly three 1's in each row and column (see the list of unsolved problems in chapter 8 of Minc's book).

H. Minc, Permanents, Addison-Wesley, 1978.

V. S. Shevelev, Some problems of the theory of enumerating the permutations with restricted position, Journal of Soviet Mathematics, 61 (4) (1992) 2272-2317

Cf. A176211, A176210, A000211.

f(n) = fibonacci(n+1) + fibonacci(n-1) + 2; \\ A000211
lista(nn) = {my(v = vector(nn, k, f(k+2))); my(vmax = vecmax(v)); my(w =  vector(nn, k, [0, logint(vmax, v[k])])); my(list=List()); forvec(x = w, if (vecmax(x), my(y = prod(k=1, #v, v[k]^x[k])); if (y <= vmax, listput(list, y)););); Vec(vecsort(list,,8));}
lista(14) \\ Michel Marcus, Jan 06 2021

A185178 Number of distinct values of the permanent of an n X n (0,1)-matrix with exactly three 1's in each row and each column.

Original entry on oeis.org

1, 1, 2, 4, 8, 18, 32, 61, 109

Offset: 3

Vladimir Shevelev, Feb 19 2011

The values a(n) are unknown for n>=9.

a(12) >= 170, a(13) >= 276, a(14) >= 438, a(15) >= 547. - Robert P. P. McKone, Jul 14 2025

V. I. Bolshakov, On spectrum of permanent on Lambda_n^k, Proc. of Seminar on Discrete Math. and Appl., Moscow State Univ. (1986), 65-73 (in Russian).

Robert P. P. McKone, One example matrix for each known permanent for a(3)..a(15), with a(11)..a(15) being incomplete
Robert P. P. McKone, Complete frequence counts for each permanent for a(3)..a(11)
Robert P. P. McKone, Robert P. P. McKone, C++ code for enumerating and counting each permanent's frequency for a(n)
V. S. Shevelev, Some problems of the theory of enumerating the permutations with restricted position, Journal of Soviet Mathematics, 61 (4) (1992) 2272-2317 (see pp. 2307-2309).

Cf. A185179 (values of the permanent), A176211, A176212.

Cf. A001501 (number of n X n (0,1)-matrix with exactly three 1's in each row and each column).

a(9)-a(11) from Robert P. P. McKone, Jul 22 2025

A185179 Irregular triangle, read by rows, of the A185178(n) values of the permanent of an n X n (0,1)-matrix (n>=3) with exactly three 1's in each row and each column.

Original entry on oeis.org

6, 9, 12, 13, 17, 18, 20, 36, 24, 25, 26, 27, 30, 31, 32, 54, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 44, 45, 48, 49, 52, 72, 78, 81, 42, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 62, 63, 64, 66, 68, 70, 72, 78, 84, 102, 108, 117, 120, 216

Offset: 3

Vladimir Shevelev, Feb 19 2011

			Triangle begins:
  6
  9
  12, 13
  17, 18, 20, 36
  24, 25, 26, 27, 30, 31, 32, 54
  33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 44, 45, 48, 49, 52, 72, 78, 81
  42, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 62, 63, 64, 66, 68, 70, 72, 78, 84, 102, 108, 117, 120, 216

V. I. Bolshakov, On spectrum of permanent on Lambda_n^k, Proc. of Seminar on Discrete Math. and Appl., Moscow State Univ. (1986), 65-73 (in Russian).

Robert P. P. McKone, Complete and Known Values of the Permanents A185178(9-15)
V. S. Shevelev, Some problems of the theory of enumerating the permutations with restricted position, Journal of Soviet Mathematics, 61 (4) (1992) 2272-2317 (see pp. 2307-2309).

Cf. A185178, A185177, A176211, A176212.

More terms from Robert P. P. McKone, Jul 26 2025

A185177 Different values of the permanent in (0,1) quadratic matrices with exactly 3 ones in each row and each column, in the increasing order.

Original entry on oeis.org

6, 9, 12, 13, 17, 18, 20, 24, 25, 26, 27, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42

Offset: 1

Vladimir Shevelev, Feb 19 2011

The sequence contains all different terms of A185179 in increasing order.

V. I. Bolshakov, On spectrum of permanent on Lambda_n^k, Proc. of Seminar on Discrete Math. and Appl., Moscow State Univ. (1986), 65-73 (in Russian).

V. S. Shevelev, Some problems of the theory of enumerating the permutations with restricted position, Journal of Soviet Mathematics, 61 (4) (1992) 2272-2317 (see pp.2307-2309).

Cf. A176211, A176212, A185178, A185179.

A232553 Maximal values of permanent on (0,1) square matrices of order n with row and column sums 3.

Original entry on oeis.org

6, 9, 13, 36, 54, 81, 216, 324, 486, 1296, 1944, 2916, 7776, 11664, 17496, 46656, 69984, 104976, 279936, 419904, 629856, 1679616, 2519424, 3779136, 10077696, 15116544, 22674816, 60466176, 90699264, 136048896, 362797056, 544195584, 816293376, 2176782336, 3265173504, 4897760256

Offset: 3

Vladimir Shevelev, Nov 26 2013

a(n) is attained on the subset of symmetric matrices with the main diagonal all 1's.

Colin Barker, Table of n, a(n) for n = 3..1000
D. Merriell, The maximum permanents in Lambda_n,k, Linear and Multilinear Algebra, 1980, no.9, 81-91.
V. S. Shevelev, Some problems of the theory of enumerating the permutations with restricted position, Journal of Soviet Mathematics, 61 (4) (1992) 2272-2317
Index entries for linear recurrences with constant coefficients, signature (0,0,6).

Cf. A176211, A176212, A185177, A185178, A185179.

LinearRecurrence[{0, 0, 6}, {6, 9, 13, 36, 54, 81}, 50] (* Paolo Xausa, Aug 08 2025 *)

a(n) = h = n%3; floor(6^((n-h)/3)*(3/2)^h); \\ Michel Marcus, Nov 26 2013

Vec(x^3*(6+9*x+13*x^2+3*x^5)/(1-6*x^3) + O(x^50)) \\ Colin Barker, May 27 2016

a(n) = floor(6^((n-h)/3)*(3/2)^h), where h=0,1 or 2, such that n == h (mod 3).
From Colin Barker, May 27 2016: (Start)
a(n) = 6*a(n-3) for n>8. [Corrected by Paolo Xausa, Aug 08 2025]
G.f.: x^3*(6+9*x+13*x^2+3*x^5) / (1-6*x^3). (End)

A232636 The second largest value of permanent of (0,1) square matrices of order n with row and column sums equal to 3.

Original entry on oeis.org

34012224, 53747712, 131010048, 204073344, 322486272, 786060288, 1224440064, 1934917632, 4716361728, 7346640384, 11609505792, 28298170368, 44079842304, 69657034752, 169789022208, 264479053824, 417942208512

Offset: 30

Vladimir Shevelev, Nov 27 2013

nonn

The interval (a(n), A232553(n)) contains no permanents of the matrices under consideration. For n=3 and n=4, the permanent takes only one value: 6 and 9 respectively. Our method (see Shevelev link) allows us to find a simple regularity of the sequence only beginning with n=30 (see formula). However, several values for 5<=n<30 are known: a(5)=13, a(6)=20, a(7)=32, a(8)=78, a(9)=120, a(12)=729, a(15)=4374, a(18)=26244, a(21)=157464, a(24)=944784, a(27)=5668704 (Bolshakov) and a(28)=8957952.
We conjecture that formula below also holds for other values.
In cases n==0 (nod 3), n>=6, and n==2 (mod 3), at least, for n=8 and n>=32, the second largest value of permanent attains on subset of symmetric matrices with the main diagonal of 1's, and does not attain, if n==1 (mod 3), at least, for n=7 and n>=28.
All terms for n>30 not divisible by 3 are conjectural since our method is based on the following very plausible but yet not proved conjecture with confirms in many cases and without counterexample since 1992 (when it was posed, see ref. [11] in Shevelev link). Let L_n be the set of considered n X n (0,1)-matrices with all row and column sums equal to 3. Let S_n be the subset of completely indecomposable such matrices, i.e., not containing submatrices from L_m (m

V. I. Bolshakov, On spectrum of permanent on Lambda_n^k, Proc. of Seminar on Discrete Math. and Appl., Moscow State Univ. (1986), 65-73 (in Russian).

V. Shevelev, Spectrum of permanent's values and its extremal magnitudes in Lambda_n,3 and Lambda_n(alpha,beta,gamma), Journal of Optimization, Vol. 2013(2013), Article 289829.

Cf. A232553, A176211, A176212, A185177, A185178, A185179.

a(n) = 9/16 * 6^(n/3), if n>=12, n==0 (mod 3) (Bolshakov); a(n) = 8/9 * 6^((n-1)/3), if n>=28, n== 1 (mod 3); a(n) = 13/6 * 6^((n-2)/3), if n>=32, n==2 (mod 3).
Conjectures from Colin Barker, May 27 2016: (Start)
a(n) = 6*a(n-3) for n>32.
G.f.: 419904*x^30*(81+128*x+312*x^2) / (1-6*x^3).
(End)

Edited by M. F. Hasler, Nov 30 2013

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A176212 Terms of A176211, duplicates removed.

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A185178 Number of distinct values of the permanent of an n X n (0,1)-matrix with exactly three 1's in each row and each column.

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A185179 Irregular triangle, read by rows, of the A185178(n) values of the permanent of an n X n (0,1)-matrix (n>=3) with exactly three 1's in each row and each column.

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A185177 Different values of the permanent in (0,1) quadratic matrices with exactly 3 ones in each row and each column, in the increasing order.

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A232553 Maximal values of permanent on (0,1) square matrices of order n with row and column sums 3.

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A232636 The second largest value of permanent of (0,1) square matrices of order n with row and column sums equal to 3.

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