A180013: Triangular array read by rows: T(n,k) = number of fixed points in the permutations of {1,2,...,n} that have exactly k cycles; n>=1, 1<=k<=n.

A028338 → x ↦ 2x−1

A180013 ↗ nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, 1/2]interlacing (weak)

Recurrence from A180013

\(P_{n}(x) = \left(-1 + 2\,x + n\right)\,P_{n-1}(x) + \left(-2 + 3\,x - x^{2} + n - n x\right)\,P_{n-2}(x)\)

\(P_{0}(x) = 1\)

\(P_{1}(x) = 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	0	2							2
2	0	3	3						6
3	0	8	12	4					24
4	0	30	55	30	5				120
5	0	144	300	210	60	6			720
6	0	840	1918	1575	595	105	7		5040
7	0	5760	14112	12992	5880	1400	168	8	40320

All 15 rows b-files: A180013 ?

Technical details

Node id: n-46edc3c082e19c85 · revision 0 · JSON
Definition source: oeis_catalog:A180013
Fingerprint: f6-34eaf45657600f7a

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
Palindromic rows: no
γ-positive (gamma_positive): no
Nonnegative coefficients (nonn): yes
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): yes

Matches ?

A180013 triangle 55 terms, 10 rows · tabl · nonn · catalog b-file?

Row sums ? A370360, A354074, A159333, A104150, A124355, A338112, A000142, A358185

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0: 1;
  1: 0,      2;
  2: 0,      3,       3;
  3: 0,      8,      12,       4;
  4: 0,     30,      55,      30,      5;
  5: 0,    144,     300,     210,     60,      6;
  6: 0,    840,    1918,    1575,    595,    105,     7;
  7: 0,   5760,   14112,   12992,   5880,   1400,   168,    8;
  8: 0,  45360,  117612,  118188,  60921,  17640,  2898,  252,   9;
  9: 0, 403200, 1095840, 1181240, 672840, 224490, 45360, 5460, 360, 10;

%F Formula

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

%t Mathematica

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

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2*x; for(n = 2, N, my(i = n + 1); v[i] = (n - 1 + 2*x)*v[i - 1] + (n - 2 - (n - 3)*x - x^2)*v[i - 2]); 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)

polytool c48629374597

A180013 triangle, offset 1, from P₀

%e Example

Triangle begins:
   1: 1;
   2: 0,      2;
   3: 0,      3,       3;
   4: 0,      8,      12,       4;
   5: 0,     30,      55,      30,      5;
   6: 0,    144,     300,     210,     60,      6;
   7: 0,    840,    1918,    1575,    595,    105,     7;
   8: 0,   5760,   14112,   12992,   5880,   1400,   168,    8;
   9: 0,  45360,  117612,  118188,  60921,  17640,  2898,  252,   9;
  10: 0, 403200, 1095840, 1181240, 672840, 224490, 45360, 5460, 360, 10;

%F Formula

The row polynomials satisfy P(n,x) = (n-2+2*x)*P(n-1,x) + (n-3-(n-4)*x-x^2)*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2*x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 2*x;
P[n_, x_] := P[n, x] = Expand[(n - 2 + 2*x)*P[n - 1, x] + (n - 3 - (n - 4)*x - x^2)*P[n - 2, x]];
Table[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2*x; for(n = 3, N, my(i = n); v[i] = (n - 2 + 2*x)*v[i - 1] + (n - 3 - (n - 4)*x - x^2)*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 1/2]
by Affine, reversal and Möbius substitutions (elementary)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
  - Parent-row provers of polytool polytool ? step › direct check

Related ?

Same fingerprint ?

References

[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
[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.