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The sequence OEIS A103904 is described as

Number of perfect matchings of an $n \times (n+1)$ Aztec rectangle with the third vertex in the topmost row removed.

Definition of $M \times N $ Aztec rectangle (one can see such a definition by C. Krattenthaler):

Consider a $(2M+1) \times (2N+1)$ rectangular chessboard and suppose that the corners are black. Then an $M \times N$ Aztec rectangle is the graph whose vertices are the white squares and whose edges connect precisely those pairs of white squares that are diagonally adjacent. So an $n \times (n+1)$ Aztec rectangle is when $M=n$ and $N=n+1$.

see the following figures for $n \times (n+1)$ Aztec rectangles as examples: n X (n+1) Aztec rectangle

When $n=2$, I try to calculate the number of perfect matchings of $2 \times (2+1)$ Aztec rectangle with the third vertex in the topmost row removed:

example

The number of perfect matching is actually $8$, which is not in A103904. Is there any mistake in this sequence or the definition is wrong? Or maybe I misunderstand it. I am very happy to hear from anyone and thank you very much in advance!

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    $\begingroup$ @PeterLuschny, I am not sure about this, but when n=1, there is no perfert matching (the number is zero) while there is no zero in the A103904; when n=2, the number of perfect matchings is 8 (which is also not shown in the sequence). So I am confused with the meaning of A103904 $\endgroup$
    – Xuemei
    Feb 28, 2021 at 13:40
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    $\begingroup$ I can confirm, from my understanding of the OEIS entry, the graph you obtain is correct, and the number of perfect matchings for n=2 is 8. That is in contrast to what OEIS writes. Could somebody else please try to confirm too? $\endgroup$ Feb 28, 2021 at 15:23

1 Answer 1

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@Marie, I agree with you that the sequence A103904 is mistaken.

The first term is plainly wrong (incidentally the formula section on the OEIS page has every entry marked with $n>1$, as none of them produces $1$ when $n=1$), but here are two ways to fix the rest:

(1) Keep the definition. The correct sequence is then given by $$ \frac{n(n-1)}{2}\times2^{n(n+1)/2}, $$ or $$ 0, 8, 192, 6144, 327680, \dots $$ i.e., the OEIS entry has the exponent wrong: it should be $n+1\choose2$ rather than $n\choose 2$, so the sequence is off by a factor of $2^n$. Note $n=1$ does not need to be singled out any more.

(2) Change the definition to the following:

Number of perfect matchings of an $n\times(n+1)$ Aztec rectangle, where all the vertices in the top-most row and the second top-most row have been removed, except the third and the last vertex of the second top-most row.

Then the sequence can remain as it is, except the first term needs to be changed to $0$: $$ 0, 2, 24, 384, 10240, \dots $$

These results follow directly from Lemma 1 and 2 in Krattenthaler that you have quoted above. Let me know if you have any questions.

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  • $\begingroup$ sorry for the delay in replying. I think you are right, thank you so much for the answer :) $\endgroup$
    – Xuemei
    Apr 9, 2021 at 8:21
  • $\begingroup$ Krattenthaler's Eq. (2.2) correctly gives $1$ when $m=1$, so $1, 2, 24, 382\ldots$ is "Number of perfect matchings of an $n\times(n+1)$ Aztec rectangle, where all the vertices in the top-most row and the second top-most row have been removed, except the third and the last vertex of the second top-most row, unless this is the same vertex, in which case any two vertices of that row are preserved". $\endgroup$ Jun 5 at 12:39
  • $\begingroup$ Anyway, I have edited the OEIS entry in accordance with your analysis, making the simple formula the primary definition. So the mistake is now corrected. Thanks to everyone involved! $\endgroup$ Jun 5 at 12:40

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