# 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 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 at 12:40