A185177 -id:A185177 - cp's OEIS frontend

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.

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

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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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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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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