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A049401 Number of Young tableaux of height <= 5. 12
1, 1, 2, 4, 10, 26, 75, 225, 715, 2347, 7990, 27908, 99991, 365587, 1362310, 5159208, 19831101, 77233517, 304423574, 1212962072, 4881181036, 19821471956, 81165639197, 334925706659, 1391935877463, 5823186349671, 24511802558326, 103772782048252, 441696903185704 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Also the number of n-length words w over alphabet {a,b,c,d,e} such that for every prefix z of w we have #(z,a) >= #(z,b) >= #(z,c) >= #(z,d) >= #(z,e), where #(z,x) counts the letters x in word z.  The a(5) = 26 words are: aaaaa, aaaab, aaaba, aabaa, abaaa, aaabb, aabab, abaab, aabba, ababa, aaabc, aabac, abaac, aabca, abaca, abcaa, aabbc, ababc, aabcb, abacb, abcab, aabcd, abacd, abcad, abcda, abcde. - Alois P. Heinz, May 30 2012

REFERENCES

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 7.16(b), y_5(n), p. 452.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

F. Bergeron, L. Favreau and D. Krob, Conjectures on the enumeration of tableaux of bounded height, Discrete Math, vol. 139, no. 1-3 (1995), 463-468.

F. Bergeron and F. Gascon, Counting Young tableaux of bounded height, J. Integer Sequences, Vol. 3 (2000), #00.1.7.

Alon Regev, Amitai Regev, Doron Zeilberger, Identities in character tables of S_n, arXiv preprint arXiv:1507.03499, 2015

Index entries for sequences related to Young tableaux.

FORMULA

E.g.f.: e^x*(BesselI(0, 2*x)^2 - BesselI(0, 2*x)*BesselI(2, 2*x) - BesselI(0, 2*x)*BesselI(4, 2*x) - BesselI(1, 2*x)^2 + 2*BesselI(1, 2*x)*BesselI(3, 2*x) + BesselI(2, 2*x)*BesselI(4, 2*x) - BesselI(3, 2*x)^2) (BesselI = modified Bessel function of first kind).

a(n) ~ 3*5^(n+5)/(8*Pi*n^5). - Vaclav Kotesovec, Aug 18 2013

MAPLE

a:= proc(n) option remember;

      `if`(n<3, [1, 1, 2][n+1], ((3*n^2+17*n+15)*a(n-1)

       +(n-1)*(13*n+9)*a(n-2) -15*(n-1)*(n-2)*a(n-3)) /

       ((n+4)*(n+6)))

    end:

seq(a(n), n=0..30);  # Alois P. Heinz, Oct 12 2012

MATHEMATICA

a[n_] := a[n] = If[n<3, {1, 1, 2}[[n+1]], ((3*n^2+17*n+15)*a[n-1] + (n-1)*(13*n+9)*a[n-2] - 15*(n-1)*(n-2)*a[n-3]) / ((n+4)*(n+6))]; Table[a[n], {n, 0, 30}] (* Jean-Fran├žois Alcover, Mar 10 2014, after Alois P. Heinz *)

CROSSREFS

Sum of first five diagonals of A047884. Cf. A007579.

Column k=5 of A182172. - Alois P. Heinz, May 30 2012

Sequence in context: A089404 A006123 A006251 * A239077 A148099 A007579

Adjacent sequences:  A049398 A049399 A049400 * A049402 A049403 A049404

KEYWORD

nonn,easy

AUTHOR

Alford Arnold, N. J. A. Sloane.

EXTENSIONS

More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jun 17 2001

STATUS

approved

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Last modified August 21 05:56 EDT 2017. Contains 290862 sequences.