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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.

%I A373691 #35 Jun 14 2024 12:15:46
%S A373691 1,2,0,4,2,0,10,6,6,2,26,22,48,22,2,76,68,276,260,40,0,232,214,1384,
%T A373691 2204,944,62,0,764,672,6240,16172,13212,3048,200,12,2620,2204,27096,
%U A373691 103588,145160,70740,10936,524,12,9496,7354,113722,612178,1338370,1145614,364366,36838,862,0
%N 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.
%H A373691 Kassie Archer and Aaron Geary, <a href="https://arxiv.org/abs/2406.09369">Descents in powers of permutations</a>, arXiv:2406.09369 [math.CO], 2024. See Table 5 p. 13.
%e A373691 Triangle begins:
%e A373691     1;
%e A373691     2,   0;
%e A373691     4,   2,    0;
%e A373691    10,   6,    6,    2;
%e A373691    26,  22,   48,   22,   2;
%e A373691    76,  68,  276,  260,  40,  0;
%e A373691   232, 214, 1384, 2204, 944, 62, 0;
%e A373691   ...
%o A373691 (PARI) sq(p) = vector(#p, k, p[p[k]]);
%o A373691 nbd(p) = sum(i=1, #p-1, p[i+1] < p[i]);
%o A373691 row(n) = my(v=vector(n)); for (i=1, n!, v[nbd(sq(numtoperm(n, i)))+1]++;); v;
%Y A373691 Cf. A000085 (1st column), A000142 (row sums), A037224 (right diagonal).
%Y A373691 Cf. A003483 (square permutations), A008292.
%K A373691 nonn,tabl
%O A373691 1,2
%A A373691 _Michel Marcus_, Jun 14 2024