# Mistake in OEIS A103904?

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:

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:

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!

• @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 Feb 28, 2021 at 13:40
• 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? Feb 28, 2021 at 15:23

@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.

• sorry for the delay in replying. I think you are right, thank you so much for the answer :) Apr 9, 2021 at 8:21
• 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". Jun 5, 2022 at 12:39
• 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! Jun 5, 2022 at 12:40