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A049401 Number of Young tableaux of height <= 5. 14

%I

%S 1,1,2,4,10,26,75,225,715,2347,7990,27908,99991,365587,1362310,

%T 5159208,19831101,77233517,304423574,1212962072,4881181036,

%U 19821471956,81165639197,334925706659,1391935877463,5823186349671,24511802558326,103772782048252,441696903185704

%N Number of Young tableaux of height <= 5.

%C 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

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

%H Alois P. Heinz, <a href="/A049401/b049401.txt">Table of n, a(n) for n = 0..1000</a>

%H F. Bergeron, L. Favreau and D. Krob, <a href="http://dx.doi.org/10.1016/0012-365X(94)00148-C">Conjectures on the enumeration of tableaux of bounded height</a>, Discrete Math, vol. 139, no. 1-3 (1995), 463-468.

%H F. Bergeron and F. Gascon, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL3/CYT/cyt.html">Counting Young tableaux of bounded height</a>, J. Integer Sequences, Vol. 3 (2000), #00.1.7.

%H Juan B. Gil, Peter R. W. McNamara, Jordan O. Tirrell, Michael D. Weiner, <a href="https://arxiv.org/abs/1708.00513">From Dyck paths to standard Young tableaux</a>, arXiv:1708.00513 [math.CO], 2017.

%H Alon Regev, Amitai Regev, Doron Zeilberger, <a href="http://arxiv.org/abs/1507.03499">Identities in character tables of S_n</a>, arXiv preprint arXiv:1507.03499 [math.CO], 2015.

%H <a href="/index/Y#Young">Index entries for sequences related to Young tableaux.</a>

%F 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).

%F a(n) ~ 3*5^(n+5)/(8*Pi*n^5). - _Vaclav Kotesovec_, Aug 18 2013

%p a:= proc(n) option remember;

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

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

%p ((n+4)*(n+6)))

%p end:

%p seq(a(n), n=0..30); # _Alois P. Heinz_, Oct 12 2012

%t 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_ *)

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

%Y Column k=5 of A182172. - _Alois P. Heinz_, May 30 2012

%K nonn,easy

%O 0,3

%A _Alford Arnold_, _N. J. A. Sloane_.

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

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Last modified February 23 00:03 EST 2018. Contains 299472 sequences. (Running on oeis4.)