A216118: Triangle read by rows: T(n,k) is the number of stretching pairs in all permutations in S_{n,k} (=set of permutations in S_n with k cycles) (n >= 3; 1 <= k <= n-2).

Matches ?

Row sums ? A216119

Submission text ?

Triangle begins:
  0:         0;
  1:         1,         1;
  2:        10,        15,         5;
  3:        90,       165,        90,        15;
  4:       840,      1750,      1225,       350,        35;
  5:      8400,     19180,     15750,      5950,      1050,       70;
  6:     90720,    222264,    204624,     92610,     22050,     2646,     126;
  7:   1058400,   2744280,   2757720,   1421490,    411600,    67620,    5880,    210;
  8:  13305600,  36162720,  38980920,  22203720,   7408170,  1496880,  180180,  11880,   330;
  9: 179625600, 508155120, 580486500, 358221600, 133315875, 31320135, 4677750, 430650, 22275, 495;

The row polynomials satisfy (n-1)*P(n,x) = (n^2+3*n+(n+3)*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1 + x.

P[0, x_] := 0;
P[1, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 + 3*n + (n + 3)*x)*P[n - 1, x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = ((n^2 + 3*n + (n + 3)*x)*v[i - 1])/(n - 1)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Properties

• Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
  by Iterated real-rootedness preservers
• Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
  derived from Iterated real-rootedness preservers
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
  derived from Iterated real-rootedness preservers
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m ≥ 2
  by Iterated real-rootedness preservers

Proof chains ? JSON

• Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

Related ?

Same fingerprint ?

• A225470 → basis A013610
• A132393 → reverse
• A180013: Triangular array read by rows: T(n,k) = number of fixed points in the permutations of {1,2,...,n} that have ex…
• A028338 → basis A013609
• A094638: Triangle read by rows: T(n,k) = |s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind A…
• A225471 → basis A013611
• A130534: Triangle T(n,k), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded…
• A132393: Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.

References

1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.