# Proofs and Confirmations

Note that a reference to line 7b means the seventh line from the bottom of the page.

• page 10,
pp[75] = 37,745,732,428,153; note that the 9th and 10th digits were transposed
• page 19,
the reference to Figure 1.7 should come in the third line from the bottom, at the end of the sentence that concludes "the bottom level of each shell."
• page 22, line 9,
last number should be 7436 instead of 7435
• page 24, conjecture 9
In the formula, (r-k-1)! should be (2r-k-1)!
• page 27, conjecture 1
$A_{n.k}$ should be $A_{n,k}$
• page 30, exercise 1.3.11
This exercise is nonsense. Given a descending plane partition with largest part r, we can always insert above it a row of r (r+1)s. Every descending plane partition fits the definition of being stripped.
• page 34, line 14
y^2 z ++ y z^2 should be y^2 z + y z^2

• page 41, exercise 2.1.12
left side of displayed equation should be: $(1-q)(1-q^2)(1-q^3) \cdots$

• page 41, exercise 2.1.13
left side of displayed equation should be: $\frac{1}{(1-tq)(1-tq^2)(1-tq^3) \cdots}$
• Page 42, exercise 2.1.18

$n = (2j+1)(3j\pm 1)$ should be $n=(2j+1)(3j+1)$ or $(2j+1)(3j+2)$

• page 54, exercise 2.2.15, line 2,
delete "is"
• page 55, line 1
the binomial involving y's should have exponenets that are functions of j rather than i: $(y^{3j} - y^{1-3j})$
• page 61, exercise 2.3.11
to clarify, change last two lines to: "equation (2.24) implies that the $a_j$ must be unique."
• page 72, exercise 2.4.9
first product to right of = should be over $1 \leq i < j \leq n$
• page 82, exercise 3.1.10,
"negative for $k \equiv \pm 1 \pmod{8}$" should read "negative for $k \equiv \pm 3 \pmod{8}$".
• page 95, last line,
the upper limit on the product should be r rather than l.
• page 99, second sentence after Conclusion
This should say, "For each $i$, we factor  $(q;q)_{s+t}/(q;q)_{s-i+r}(q;q)_{t+i-1}$ out of the $i$th row of our matrix."
• page 105, Theorem 3.10
change "exactly k parts of size r," to "exactly k parts of size r in the associated shifted plane partition,"

in last displayed equation, numerator of last Gaussian polynomial should be "2r - 2 - k" rather than "r - 2 - k".
• page 109, exercise 3.4.3 should be changed to read:
Show that the conjectured generating function for cyclically symmetric plane partitions that fit inside $\cal{B}(r,r,r)$ can be written as

$\prod_{i=1}^r \frac{1-q^{3i-1}}{1-q^{3i-2}} \prod_{1 \leq i \leq j \leq r} \frac{1-q^{3(r+i+j-1)}}{1-q^{3(2i+j-1)}}.$

Use {\it Mathematica\/} to show that this conjectured generating function agrees with $\det(I_r+G_r)$ for $1 \leq r \leq 5$.
• page 117, exercise 3.5.5
sentence should end "when $\lambda = -1$.
• page 125, Proposition 4.2,
all $n$s should be $k$s.
• page 127, Exercise 4.1.19,
the last term in the displayed summation should be $(-1)^b h_{a+b+1}$.
• page 144, line 4,
the limits on the second product should be $1 \leq i < j \leq r$
• page 148, Exercise 4.3.9,
in the last line of this exercise, the exponent in the numerator of the last fraction should be $a_j+1$
• page 155, last diplayed equation
$L_m$ should be $L_r$
• page 166, beginning line 8
The phrase "For the series given in equation (5.10) with real parameters," should read:

"For the series $\sum_{k=0}^{\infty} x^k (\alpha)_k (\beta)_k / k! (\gamma)_k$ with $|x| = 1$ and real parameters, "
• page 166, third displayed mathematics
there should be a factor of $k+1$ in the denominator
• page 167, line 2
there should be a factor of 2 in the numerator
• page 173, Exercise 5.2.5:
$-c-1-k$ should be $-c-1+k$
• page 180, bottom line of Equation 5.33:
misplaced comma, should come after the vector
• page 180, line 5b
$I_r + T_j$ should be $I_r + T_r$
• page 184, line 10
$-RA_r^*R{-1}$ should be $-RT_r^*R{-1}$
• page 187, line 11
a factor of $1-q^{r-2/3}$ is missing from the denominator
• page 187, Exercise 5.3.1:
$\frac{1-q^{|\eta|+ht(\eta)}}{1-q^{ht(\eta)}}$ should be $\frac{1-q^{|\eta|(1+ht(\eta))}}{1-q^{|\eta|ht(\eta)}}$.
(There are two instances of this error.)

• page 193, expansion of f_7(x)
linear term should be $29400x$ rather than $24900x$
• page 196, line 3b
change "reflection across the $y=x$ plane" to "reflection through the center of the box."
• page 196, equation (6.4)
change $(r-i-1,s-j-1,t-k-1)$ to $(r-i+1,s-j+1,t-k+1)$
• page 198, equation (6.7) is incorrect. It should read
$N_3(r,r,r) = \left( \prod_{i=1}^r \frac{3i-1}{3i-2} \right) \left( \prod_{1 \leq i \leq j \leq r} \frac{r+i+j-1}{2i+j-1} \right).$
• page 198, equation 6.8 is incorrect. It should read
$N_4(r,r,r) = \prod_{1 \leq i \leq j \leq r} \frac{i+j+r-1}{i+2j-2}.$
• page 198, the left side of equation 6.11 should read
$N_5(2r+1,2s+1,2t)$
• page 199, equation 6.15 is incorrect. It should read
$N_8(2r,2r,2r) = \prod_{i=0} ^{r-1}\frac{(3i+1) (6i)! (2i)!}{(4i+1)! (4i)!}.$
• page 203, exercise 6.1.10
$n+j)!$ in last denominator should be $(n+j)!$
• page 216, line 7b
insert "non-negative" before "integer entries"
• page 229, line 2
delete "twice"
• page 236, line 9
"perpendicular bisector" should be "angle bisector"
• page 243, Exercise 7.2.5:
$y_j^{n-1}$ should be $y_j^{n-2}$.
• page 250, equation (7.28)
numerator of right-most fraction should be $f(x) - f(xq)$
• pages 253-254, in each of the five equations for S(P_{n-1}(x)) beginning at the bottom of page 253
If we replace $x$ by $xq$ in $D_q^m f(x)$, we get $q^{-m} D_q^m f(xq)$, so right-hand side of each of these equalities also needs a factor of $q^{-3(n-1)(n-2)/2}$
• page 254, second equation (line 9):
$(q^{3j+3}:q^3)$ should be $(q^{3j+3};q^3)$. Also in that term, a factor of $(t^6 q^{3-3n})^j$ is
missing.

Thanks to the following people who have found errors in Proofs and Confirmations: Robin Chapman, Emeric Deutsch, Neil J. A. Sloane, Paul Terwilliger and his class at UW-Madison, Ronald P. Infante, Eric Kuo, Robert Mills, Graham Hawkes