A117506 Irregular triangle read by rows: dimensions of the irreducible representations of the symmetric group S_n.

1, 1, 1, 1, 1, 2, 1, 1, 3, 2, 3, 1, 1, 4, 5, 6, 5, 4, 1, 1, 5, 9, 5, 10, 16, 5, 10, 9, 5, 1, 1, 6, 14, 14, 15, 35, 21, 21, 20, 35, 14, 15, 14, 6, 1, 1, 7, 20, 28, 14, 21, 64, 70, 56, 42, 35, 90, 56, 70, 14, 35, 64, 28, 21, 20, 7, 1

Offset: 0

Author

Wolfdieter Lang, Apr 13 2006

Keywords

• nonn
• easy
• tabf

Comments

The n-th row has partition(n) = A000041(n) entries.

Also the numbers of standard Young tableaux for Young diagrams (or partitions).

Also "generalized" Catalan numbers. For a partition of n, n=(n_1+...+n_d), this is the number of integral lattice paths from (0,...,0) to (n_1,...,n_d) such that for any point p=(p_1,...p_d) on such a path p_i is never less than p_j whenever iGraham H. Hawkes, Jul 05 2013

The irreducible representations of S_n correspond to Young diagrams or partitions.

Partitions of n are ordered according to Abramowitz-Stegun (A-St) (see the reference, pp. 831-2). In contrast to A-St, a partition has nondecreasing parts (reverse notation of A-St).

The dimension of a representation of S_n corresponding to a Young diagram or partition is a(n,k) for the k-th partition of n in this A-St order.

One could call these numbers a(n,k) M_4 (similar to M_0, M_1, M_2, M_3 given in A111786, A036038, A036039, A036040, respectively).

From Wolfdieter Lang, Oct 09 2015: (Start)

The first formula given below appears in A. Young, Q.S.A. III, PLMS 28 (1928) 255-292 (third paper on "On Quantitative Substitutional Analysis"), Theorem II on p. 260, and he calls it f; see the collected papers (CP) reference, p. 357. Note the shorthand notation for the products; see Q.S.A. II, PLMS 34 (1902) 361-397, p. 366, CP, p. 97, for the explicit one.

This formula also can be found in the Glass-Ng link, Theorem 1, p. 702, using the Vandermonde determinant in the numerator and re-indexing the denominator.

The product of the hook length numbers, called H(n, k) in this formula below, is found in A263003(n, k).

The squared row entries sum to n!. See A. Young, Q.S.A. II (see above), pp. 367-368, CP pp. 98-99. Also Q.S.A. III, p. 265, CP p. 362.

(End)

Examples

			[1];
[1];
[1, 1];
[1, 2, 1];
[1, 3, 2, 3, 1];
[1, 4, 5, 6, 5, 4, 1];
[1, 5, 9, 5, 10, 16, 5, 10, 9, 5, 1];...
a(4,4)=3 because the 4th partition of n=4 in A-St order is [2,1,1],
and H(4,4)=(4!*2!*1!)/Vandermonde([4,2,1]) = (4!*2)/6 =4*2, hence
4!/H(4,4) = 3.
a(4,4)=3 because the hook lengths of the Young diagram of [2,1,1] are [4, 1; 2; 1], hence 4!/(4*1*2*1) = 3.
The sum of the squared entries of each row gives n!: n = 5: 2*(1^1 + 4^2 + 5^2) + 6^2 = 120 = 5!. - _Wolfdieter Lang_, Oct 09 2015
		

References

• G. de B. Robinson (ed.), The Collected Papers of Alfred Young 1873-1940, University of Toronto Press, 1977.
• G. B. Wybourne, Symmetry principles and atomic spectroscopy, Wiley, New York, 1970, p. 9.

Links

• Alois P. Heinz, Rows n = 0..30, flattened
• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
• A. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, pp. 831-2.
• Kenneth Glass and Chi-Keung Ng, A Simple Proof of the Hook Length Formula, Am. Math. Monthly 111 (2004) 700 - 704.
• Graham H. Hawkes, An Elementary Proof of a Formula for SYT, arXiv preprint arXiv:1310.5919 [math.CO], 2013-2014.
• Wolfdieter Lang, First 15 rows.
• Eric Weisstein's World of Mathematics, Hook length formula.
• Doron Zeilberger, Andre's Reflection Proof Generalized to the Many-Candidate Ballot Problem, Discrete Mathematics 44 (1983) 325-326.
• Index entries for sequences related to groups

Crossrefs

Cf. A000041, A000085 (row sums), A060240 (rows sorted), A263003.

Cf. A067924.

Programs

Maple

h:= l-> (n-> mul(mul(1+l[i]-j+add(`if`(l[k]>=j, 1, 0),
             k=i+1..n), j=1..l[i]), i=1..n))(nops(l)):
g:= (n, i, l)-> `if`(n=0 or i=1, [h([l[], 1$n])],
    [g(n, i-1, l)[], g(n-i, min(n-i, i), [l[], i])[]]):
T:= n-> map(x-> n!/x, g(n$2, []))[]:
seq(T(n), n=0..10);  # Alois P. Heinz, Nov 05 2015

Mathematica

h[l_List] := Function[n, Product[Product[1 + l[[i]] - j + Sum[If[l[[k]] >= j, 1, 0], {k, i+1, n}], {j, 1, l[[i]]}], {i, 1, n}]][Length[l]]; g[n_, i_, l_List] := If[n==0 || i==1, Join[{h[Join[l, Array[1&, n]]]}], If[i<1, {}, Join[{g[n, i-1, l]}, If[i>n, {}, g[n-i, i, Join[l, {i}]]]]]] // Flatten; T[n_] := n!/ g[n, n, {}]; Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Dec 19 2015, after Alois P. Heinz *)

Formula

a(n,k) = n!/H(n,k) with H(n,k):= Product_{i=1..m(n,k)} (x_i)!/Det(x_i^(m(n,k)-j)) with the Vandermonde determinant for the variables x_i:=lambda(n,k)_i + m(n,k)-i, i,j=1..m(n,k) if m(n,k) is the number of parts of the k-th partition of n, called lambda(n,k), in the A-St order (see above). Lambda(n,k)_i denotes the i-th part of the partition lambda(n,k), sorted in decreasing order (this is the reverse of the A-St notation).

a(n,k) = n!/Product_{j=1..n}(h(n,k,j) with the hook numbers h(n,k,j) of the Young diagram of the partition lambda(n,k) in the A-St order. See the link for 'hook length formula'.

Extensions

Row n=0 prepended by Alois P. Heinz, Nov 05 2015