A050447 Table T(n,m) giving total degree of n-th-order elementary symmetric polynomials in m variables, -1 <= n, 1 <= m, transposed and read by upward antidiagonals.
1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 5, 1, 1, 5, 10, 14, 8, 1, 1, 6, 15, 30, 31, 13, 1, 1, 7, 21, 55, 85, 70, 21, 1, 1, 8, 28, 91, 190, 246, 157, 34, 1, 1, 9, 36, 140, 371, 671, 707, 353, 55, 1, 1, 10, 45, 204, 658, 1547, 2353, 2037, 793, 89, 1, 1, 11, 55, 285, 1086, 3164, 6405, 8272, 5864, 1782, 144, 1
Offset: 0
Examples
Table begins . 1 1 1 1 1 1 1 1 1 1 . 1 2 3 5 8 13 21 34 55 89 . 1 3 6 14 31 70 157 353 793 1782 . 1 4 10 30 85 246 707 2037 5864 16886 . 1 5 15 55 190 671 2353 8272 29056 102091 . 1 6 21 91 371 1547 6405 26585 110254 457379 . 1 7 28 140 658 3164 15106 72302 345775 1654092 . 1 8 36 204 1086 5916 31998 173502 940005 5094220 . 1 9 45 285 1695 10317 62349 377739 2286648 13846117 . 1 10 55 385 2530 17017 113641 760804 5089282 34053437
References
- J. Berman and P. Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124.
- S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 120).
- Manfred Goebel, Rewriting Techniques and Degree Bounds for Higher Order Symmetric Polynomials, Applicable Algebra in Engineering, Communication and Computing (AAECC), Volume 9, Issue 6 (1999), 559-573.
Links
- T. D. Noe, Table of 100 antidiagonals
- J. Berman and P. Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124. [Annotated scanned copy]
- G. Kreweras, Les préordres totaux compatibles avec un ordre partiel, Math. Sci. Humaines No. 53 (1976), 5-30.
- R. P. Stanley, Examples of Magic Labelings, Unpublished Notes, 1973. [Cached copy, with permission] See p. 31.
Crossrefs
Programs
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Mathematica
nmax = 12; t[n_, m_?Positive] := t[n, m] = t[n, m-1] + Sum[t[2k, m-1]*t[n-1-2k, m], {k, 0, (n-1)/2}]; t[n_, 0]=1; Flatten[ Table[ t[k-1, n-k], {n, 1, nmax}, {k, 1, n}]] (* Jean-François Alcover, Nov 14 2011 *) nmax = 10; f[0, x_] := 1; f[1, x_] := 1/(1 - x); f[n_, x_] := (x + f[n - 2, x])/(1 - x^2 - x*f[n - 2, x]); t[n_, m_] := Coefficient[Series[f[n, x], {x, 0, m}], x, m]; Grid[Table[t[n, m], {n, nmax}, {m, 0, nmax - 1}]] (* L. Edson Jeffery, Oct 19 2017 *) -
PARI
M(n)=matrix(n,n,i,j,if(sign(i+j-n)-1,0,1)); V(n)=vector(n,i,1); P(r,n)=vecmax(V(r)*M(r)^n) \\ P(r,n) is T(n,k); Benoit Cloitre, Jan 27 2003
Formula
See PARI code. See A050446 for recurrence.
G.f. for row n >= 0: f(n, x) = (x + f(n-2, x))/(1 - x^2 - x*f(n-2, x)), where f(0, x) = 1 and f(1, x) = 1/(1 - x) [R. P. Stanley]. - L. Edson Jeffery, Oct 19 2017
Extensions
More terms from Naohiro Nomoto, Jul 03 2001