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Identifier

St000175: Integer partitions ⟶ ℤ

Values

[1] => 0

[2] => 0

[1,1] => 0

[3] => 0

[2,1] => 1

[1,1,1] => 0

[4] => 0

[3,1] => 1

[2,2] => 0

[2,1,1] => 2

[1,1,1,1] => 0

[5] => 0

[4,1] => 1

[3,2] => 1

[3,1,1] => 2

[2,2,1] => 2

[2,1,1,1] => 3

[1,1,1,1,1] => 0

[6] => 0

[5,1] => 1

[4,2] => 1

[4,1,1] => 2

[3,3] => 0

[3,2,1] => 3

[3,1,1,1] => 3

[2,2,2] => 0

[2,2,1,1] => 4

[2,1,1,1,1] => 4

[1,1,1,1,1,1] => 0

[7] => 0

[6,1] => 1

[5,2] => 1

[5,1,1] => 2

[4,3] => 1

[4,2,1] => 3

[4,1,1,1] => 3

[3,3,1] => 2

[3,2,2] => 2

[3,2,1,1] => 5

[3,1,1,1,1] => 4

[2,2,2,1] => 3

[2,2,1,1,1] => 6

[2,1,1,1,1,1] => 5

[1,1,1,1,1,1,1] => 0

[8] => 0

[7,1] => 1

[6,2] => 1

[6,1,1] => 2

[5,3] => 1

[5,2,1] => 3

[5,1,1,1] => 3

[4,4] => 0

[4,3,1] => 3

[4,2,2] => 2

[4,2,1,1] => 5

[4,1,1,1,1] => 4

[3,3,2] => 2

[3,3,1,1] => 4

[3,2,2,1] => 5

[3,2,1,1,1] => 7

[3,1,1,1,1,1] => 5

[2,2,2,2] => 0

[2,2,2,1,1] => 6

[2,2,1,1,1,1] => 8

[2,1,1,1,1,1,1] => 6

[1,1,1,1,1,1,1,1] => 0

[9] => 0

[8,1] => 1

[7,2] => 1

[7,1,1] => 2

[6,3] => 1

[6,2,1] => 3

[6,1,1,1] => 3

[5,4] => 1

[5,3,1] => 3

[5,2,2] => 2

[5,2,1,1] => 5

[5,1,1,1,1] => 4

[4,4,1] => 2

[4,3,2] => 3

[4,3,1,1] => 5

[4,2,2,1] => 5

[4,2,1,1,1] => 7

[4,1,1,1,1,1] => 5

[3,3,3] => 0

[3,3,2,1] => 5

[3,3,1,1,1] => 6

[3,2,2,2] => 3

[3,2,2,1,1] => 8

[3,2,1,1,1,1] => 9

[3,1,1,1,1,1,1] => 6

[2,2,2,2,1] => 4

[2,2,2,1,1,1] => 9

[2,2,1,1,1,1,1] => 10

[2,1,1,1,1,1,1,1] => 7

[1,1,1,1,1,1,1,1,1] => 0

[10] => 0

[9,1] => 1

[8,2] => 1

[8,1,1] => 2

[7,3] => 1

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Description

Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape.
Given a partition $\lambda$ with $r$ parts, the number of semi-standard Young-tableaux of shape $k\lambda$ and boxes with values in $[r]$ grows as a polynomial in $k$. This follows by setting $q=1$ in (7.105) on page 375 of [1], which yields the polynomial
$$p(k) = \prod_{i < j}\frac{k(\lambda_j-\lambda_i)+j-i}{j-i}.$$
The statistic of the degree of this polynomial.
For example, the partition $(3, 2, 1, 1, 1)$ gives
$$p(k) = \frac{-1}{36} (k - 3) (2k - 3) (k - 2)^2 (k - 1)^3$$
which has degree 7 in $k$. Thus, $[3, 2, 1, 1, 1] \mapsto 7$.
This is the same as the number of unordered pairs of different parts, which follows from:
$$\deg p(k)=\sum_{i < j}\begin{cases}1& \lambda_j \neq \lambda_i\\0&\lambda_i=\lambda_j\end{cases}=\sum_{\stackrel{i < j}{\lambda_j \neq \lambda_i}} 1$$

References

[1] Stanley, R. P. Enumerative combinatorics. Vol. 2 MathSciNet:1676282

Code

def statistic(p):
    R. = PolynomialRing(QQ)
    poly =  R(prod((k*p[j]-k*p[i]+j-i)/(j-i) for j in range(len(p)) for i in range(j)))
    return poly.degree()

#CodeLanguage: Mathematica

ips = Join @@ (IntegerPartitions /@ Range[18]);

degree[p_] := Block[{k}, Exponent[Product[(k*p[[j]] - k*p[[i]] + j - i)/(j-i), {j, 1, Length@p}, {i, 1, j - 1}], k]];

(* Generate table with data. *)
#->degree[#] & /@ ips;

Created

Feb 16, 2014 at 19:05 by Per Alexandersson

Updated

Dec 11, 2017 at 15:14 by Martin Rubey