The Circulant Matrix of a Number and its Permanent

In a post on December 15th 2022 titled Numbers and Matrices, I wrote about the circulant matrix of a number and looked at those special numbers where the determinant of the circulant matrix was equal to the number itself. An example of such a number is 494376160 which has the circulant matrix shown in Figure 1:

Figure 1

These sorts of numbers form OEIS A219324 and the initial members are:

1, 2, 3, 4, 5, 6, 7, 8, 9, 247, 370, 378, 407, 481, 518, 592, 629, 1360, 3075, 26027, 26933, 45018, 69781, 80487, 154791, 1920261, 2137616, 2716713, 3100883, 3480140, 3934896, 4179451, 4830936, 5218958, 11955168, 80651025, 95738203, 257059332, 278945612, 456790123, 469135802, 493827160, 494376160

Instead of the determinant, we can use the permanent instead. The latter is not as well known as the former so Figure 2 provides a short explanation:

Figure 2

It's basically the same as the calculation for the determinant except that there are no alternating positive and negative signs. Figure 3 shows the calculations to determine the permanent of a rectangular matrix.

Figure 3

So now we can investigate what numbers have a circulant matrix whose permanent is equal to the number itself. It turns out that there are not that many. Here is the list up to ten million: 1, 2, 3, 4, 5, 6, 7, 8, 9, 261, 370, 407, 52036, 724212. The circulant matrix for 724212 is shown in Figure 4.

Figure 4

We can search for other numbers that the permanent might be equal to. The number 666 came to mind but if we collect the permanents of the first one million numbers, then there are only 269 numbers between 1 and 1000 (permalink). These are:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 13, 16, 17, 18, 20, 24, 25, 26, 27, 28, 29, 32, 33, 34, 35, 36, 37, 38, 40, 41, 44, 45, 48, 49, 50, 52, 53, 54, 58, 61, 64, 65, 68, 72, 73, 74, 78, 79, 80, 81, 82, 84, 85, 89, 90, 91, 97, 98, 100, 106, 113, 116, 117, 120, 121, 125, 126, 128, 130, 132, 133, 136, 142, 144, 145, 152, 154, 155, 162, 164, 165, 168, 169, 171, 177, 178, 185, 189, 194, 198, 200, 201, 216, 217, 224, 226, 232, 233, 236, 243, 244, 250, 256, 257, 261, 265, 272, 273, 275, 280, 281, 289, 298, 299, 304, 305, 306, 314, 317, 324, 326, 337, 341, 343, 344, 349, 351, 353, 359, 361, 364, 366, 369, 370, 373, 384, 388, 389, 393, 394, 395, 396, 397, 400, 404, 407, 408, 409, 413, 416, 418, 420, 425, 432, 433, 434, 440, 443, 466, 468, 472, 481, 484, 486, 492, 493, 502, 504, 512, 513, 514, 520, 523, 526, 529, 530, 538, 539, 541, 544, 548, 559, 569, 574, 576, 577, 583, 586, 606, 612, 614, 624, 625, 626, 632, 633, 637, 638, 641, 656, 665, 673, 674, 676, 686, 691, 697, 706, 720, 728, 729, 730, 737, 745, 750, 753, 756, 757, 758, 765, 769, 772, 776, 782, 783, 784, 792, 793, 801, 807, 809, 810, 819, 824, 825, 833, 834, 838, 841, 850, 853, 854, 855, 872, 873, 877, 881, 882, 884, 885, 891, 896, 902, 907, 916, 926, 927, 928, 937, 945, 952, 953, 962, 964, 978, 980, 988, 990, 1000

As can be seen, 665 is among them but not 666. If we took the permanents of numbers in excess of one million, perhaps more numbers between 1 and 1000 would appear but it's doubtful. If we do the same for the determinants, we find that there are 417 of them between 1 and 1000. Oddly enough, 666 is not among them even though the numbers on either side (665 and 667) are (permalink).