A037224 -id:A037224 - cp's OEIS frontend

A373691 Triangle read by rows T(n, k) is the number of permutations on n elements whose square has k descents, for n >= 1 and 0 <= k <= n-1.

1, 2, 0, 4, 2, 0, 10, 6, 6, 2, 26, 22, 48, 22, 2, 76, 68, 276, 260, 40, 0, 232, 214, 1384, 2204, 944, 62, 0, 764, 672, 6240, 16172, 13212, 3048, 200, 12, 2620, 2204, 27096, 103588, 145160, 70740, 10936, 524, 12, 9496, 7354, 113722, 612178, 1338370, 1145614, 364366, 36838, 862, 0

Offset: 1

Michel Marcus, Jun 14 2024

			Triangle begins:
    1;
    2,   0;
    4,   2,    0;
   10,   6,    6,    2;
   26,  22,   48,   22,   2;
   76,  68,  276,  260,  40,  0;
  232, 214, 1384, 2204, 944, 62, 0;
  ...

Kassie Archer and Aaron Geary, Descents in powers of permutations, arXiv:2406.09369 [math.CO], 2024. See Table 5 p. 13.

Cf. A000085 (1st column), A000142 (row sums), A037224 (right diagonal).

Cf. A003483 (square permutations), A008292.

sq(p) = vector(#p, k, p[p[k]]);
nbd(p) = sum(i=1, #p-1, p[i+1] < p[i]);
row(n) = my(v=vector(n)); for (i=1, n!, v[nbd(sq(numtoperm(n, i)))+1]++;); v;

A383835 Number of permutations of [n] whose compositional square is the identity permutation or its reverse.

Original entry on oeis.org

1, 1, 2, 4, 12, 28, 76, 232, 776, 2632, 9496, 35696, 140272, 568624, 2390480, 10349536, 46208416, 211800992, 997313824, 4809701440, 23758694336, 119952723136, 618884638912, 3257843882624, 17492191242880, 95680444425856, 532985208200576, 3020676745975552

Offset: 0

Darío Clavijo, May 11 2025

			a(4) = A000085(4) + A037224(4) = 10 + 2 = 12: 1234, 1243, 1324, 1432, 2134, 2143, 3214, 3412, 4231, 4321, 2413, 3142.

Cf. A000085, A000142, A037224.

a(n) = A000085(n) + A037224(n) for n > 1.

cp's OEIS Frontend

A373691 Triangle read by rows T(n, k) is the number of permutations on n elements whose square has k descents, for n >= 1 and 0 <= k <= n-1.

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